Namespaces
namespace	gmres_detail

Classes
struct	ArnoldiFactorization
	Arnoldi factorization for a nonsymmetric matrix-free operator. More...

struct	ArnoldiRitzExtraction
	Ritz data extracted from a real Arnoldi Hessenberg projection. More...

class	BlockTensorMatrixFreeOps
	Combine fixed BlockTensor vector algebra with an owned matrix-free apply operation. More...

class	BlockTensorVectorOps
	Krylov vector algebra for one exact owning BlockTensor structure. More...

struct	ComplexHermitianEigensystem
	Eigenvalues and optional eigenvectors of a dense complex Hermitian matrix. More...

struct	ComplexNonsymmetricEigensystem
	Generalized real Schur decomposition of a dense real matrix pencil. More...

struct	ComplexSchurCompressedArnoldiFactorization
	Compressed complex Arnoldi relation after a Schur restart. More...

struct	ComplexSchurDecomposition
	Complex Schur decomposition stored in column-major DenseMatrix values. More...

struct	DenseHostVector
	Prototype host vector used by matrix-free Krylov tests and adapters. More...

class	DenseHostVectorAlgebra
	Dense host vector algebra without an assembled operator. More...

class	DenseHostVectorOps
	Dense reference operator combined with host vector algebra for prototypes. More...

struct	DenseHostVectorTransfer
	Explicit precision conversion between dense prototype vector spaces. More...

struct	EigenResult
	Matrix-free eigenvalue result with solver-owned vector objects. More...

struct	GmresErrorEstimates
	Error estimates evaluated from the residual and supplied operator data. More...

struct	GmresInfo
	Recoverable solve outcome; counters include attempted operator calls. More...

struct	GmresOperatorEstimate
	An application-supplied estimate or known upper bound. More...

struct	GmresOperatorEstimates
	Optional properties of the original operator in the vector space's 2-norm. More...

struct	GmresOptions
	Residual stopping policy and storage/iteration limits. More...

struct	GmresRefinementContext
	Bind precision roles and application operations for GMRES refinement. More...

struct	GmresRefinementInfo
	Refinement diagnostics for the last finite published working iterate. More...

struct	GmresRefinementOptions
	Outer residual targets and independently configured inner solves. More...

struct	GmresResidual
	Recomputed Euclidean norms for the published iterate. More...

struct	IdentityPreconditioner
	Identity preconditioner using the supplied vector copy operation. More...

struct	KrylovExponentialDiagnostics
	Optional diagnostic state for Krylov exponential actions. More...

struct	KrylovExponentialParams
	Parameters for Krylov exponential actions. More...

struct	KrylovExponentialResult
	Matrix-free approximation to `exp(t A) v`. More...

struct	KrylovVectorSpace
	An opaque vector space with one explicit arithmetic precision. More...

struct	NonsymmetricArnoldiDiagnostics
	Optional diagnostic state for nonsymmetric Arnoldi solves. More...

struct	NonsymmetricEigenParams
	Parameters for a native nonsymmetric eigenvalue solve. More...

struct	NonsymmetricEigenResult
	Matrix-free nonsymmetric eigenvalue result. More...

struct	RealBidiagonalReduction
	Compact Householder bidiagonal reduction of a dense real matrix. More...

struct	RealBidiagonalSvd
	Singular value decomposition of a real bidiagonal matrix. More...

struct	RealCompactLqFactorization
	Compact Householder LQ factorization of a dense real matrix. More...

struct	RealCompactQlFactorization
	Compact Householder QL factorization of a dense real matrix. More...

struct	RealCompactQrFactorization
	Compact Householder QR factorization of a dense real matrix. More...

struct	RealCompactRqFactorization
	Compact Householder RQ factorization of a dense real matrix. More...

struct	RealEquilibration
	Dense real general matrix equilibration factors. More...

struct	RealExpertLinearSolve
	Expert dense real linear solve result. More...

struct	RealGeneralBandExpertLinearSolve
	Expert real general-band linear solve result. More...

struct	RealGeneralBandFactorization
	LU factorization of a real square general band matrix. More...

struct	RealGeneralBandMatrix
	LAPACK general band storage for a real square matrix. More...

struct	RealGeneralizedHessenbergReduction
	Generalized Hessenberg reduction of a dense real matrix pencil. More...

struct	RealGeneralizedNonsymmetricBalance
	Balanced dense real nonsymmetric matrix pencil and LAPACK scaling metadata. More...

struct	RealGeneralizedNonsymmetricEigensystem
	Eigenvalues and optional right eigenvectors of a real nonsymmetric matrix pencil. More...

struct	RealGeneralizedNonsymmetricExpertEigensystem
	Expert eigensystem diagnostics for a dense real nonsymmetric matrix pencil. More...

struct	RealGeneralizedSchurConditionEstimates
	Reciprocal condition estimates for eigenpairs of a generalized real Schur form. More...

struct	RealGeneralizedSchurDecomposition
	One diagonal block of a real Schur form. More...

struct	RealGeneralizedSchurRightEigenvectors
	Right eigenvectors of a generalized real Schur form unpacked into complex columns. More...

struct	RealGeneralizedSchurSelectedSubspace
	Selected generalized real Schur subspace with LAPACK `tgsen` diagnostics. More...

struct	RealGeneralTridiagonalExpertLinearSolve
	Expert real general-tridiagonal linear solve result. More...

struct	RealGeneralTridiagonalFactorization
	LU factorization of a real square general tridiagonal matrix. More...

struct	RealGeneralTridiagonalMatrix
	LAPACK general tridiagonal storage for a real square matrix. More...

struct	RealHessenbergReduction
	Compact real Hessenberg reduction of a dense square matrix. More...

struct	RealLqFactorization
	Reduced LQ factorization of a dense real matrix. More...

struct	RealLuFactorization
	LU factorization of a dense real square matrix. More...

struct	RealNonsymmetricBalance
	Balanced dense real nonsymmetric matrix and LAPACK scaling metadata. More...

struct	RealNonsymmetricEigensystem
	Eigenvalues and optional eigenvectors of a dense real symmetric matrix. More...

struct	RealNonsymmetricExpertEigensystem
	Eigenvalues and optional right eigenvectors of a real nonsymmetric matrix. More...

struct	RealPivotedCholeskyFactorization
	Pivoted Cholesky factorization of a dense real positive semidefinite matrix. More...

struct	RealPivotedQrFactorization
	Reduced QR factorization with column pivoting of a dense real matrix. More...

struct	RealQlFactorization
	Reduced QL factorization of a dense real matrix. More...

struct	RealQrFactorization
	Reduced QR factorization of a dense real matrix. More...

struct	RealRankRevealingLeastSquares
	Rank-revealing dense real least-squares result. More...

struct	RealRefinedLinearSolve
	Dense real refined linear solve result. More...

struct	RealRqFactorization
	Reduced RQ factorization of a dense real matrix. More...

struct	RealSchurBlock
	Expert eigensystem diagnostics for a dense real nonsymmetric matrix. More...

struct	RealSchurCompressedArnoldiFactorization
	Compressed real Arnoldi relation after a Schur restart. More...

struct	RealSchurConditionEstimates
	Reciprocal condition estimates for eigenpairs of a real Schur form. More...

struct	RealSchurDecomposition

struct	RealSchurRestartSelection
	Block-aware selection result for real Schur restart. More...

struct	RealSchurRightEigenvectors
	Right eigenvectors of a real Schur form unpacked into complex columns. More...

struct	RealSchurSelectedSubspace
	Eigenvalues and optional right eigenvectors of a complex nonsymmetric matrix. More...

struct	RealSingularValueDecomposition
	Singular values and optional singular vectors of a dense real matrix. More...

struct	RealSvdLeastSquares
	SVD-based dense real least-squares result. More...

struct	RealSylvesterSolution
	Dense real Sylvester equation result. More...

struct	RealSymmetricEigensystem

struct	RealSymmetricIndefiniteExpertLinearSolve
	Expert dense real symmetric-indefinite linear solve result. More...

struct	RealSymmetricIndefiniteFactorization
	Bunch-Kaufman factorization of a dense real symmetric-indefinite matrix. More...

struct	RealSymmetricPositiveDefiniteBandExpertLinearSolve
	Expert real SPD band linear solve result. More...

struct	RealSymmetricPositiveDefiniteBandFactorization
	Cholesky factorization of a real SPD band matrix. More...

struct	RealSymmetricPositiveDefiniteBandMatrix
	LAPACK symmetric positive-definite band storage for a real square matrix. More...

struct	RealSymmetricPositiveDefiniteEquilibration
	Dense real SPD equilibration factors. More...

struct	RealSymmetricPositiveDefiniteExpertLinearSolve
	Expert dense real SPD linear solve result. More...

struct	RealSymmetricPositiveDefiniteFactorization
	Cholesky factorization of a dense real SPD matrix. More...

struct	RealSymmetricPositiveDefiniteTridiagonalExpertLinearSolve
	Expert real SPD tridiagonal linear solve result. More...

struct	RealSymmetricPositiveDefiniteTridiagonalFactorization
	Cholesky factorization of a real SPD tridiagonal matrix. More...

struct	RealSymmetricPositiveDefiniteTridiagonalMatrix
	Real symmetric positive-definite tridiagonal matrix storage. More...

struct	RealSymmetricTridiagonalReduction
	Compact real symmetric tridiagonal reduction of a dense symmetric matrix. More...

struct	RightComplexSchurDecomposition
	Complex Schur decomposition stored in prototype layout-right matrices. More...

struct	RightRealSchurDecomposition
	Real Schur decomposition stored in prototype layout-right matrices. More...

class	symmetric_tridiagonal_matrix
	Fixed-size owning real symmetric tridiagonal matrix. More...

struct	symmetric_tridiagonal_view
	Non-owning read-only view of a real symmetric tridiagonal matrix. More...

struct	SymmetricCompressedLanczosFactorization
	Compressed symmetric Lanczos factorization after implicit restart. More...

struct	SymmetricEigenParams
	Common parameters for a symmetric or Hermitian eigenvalue solve. More...

struct	SymmetricLanczosDiagnostics
	Optional diagnostic state for symmetric Lanczos solves. More...

struct	SymmetricQrShiftResult
	Result of applying implicit QR shifts to a symmetric tridiagonal projection. More...

struct	SymmetricRestartCycleDiagnostics
	Diagnostic state for one symmetric Lanczos restart cycle. More...

struct	SymmetricRestartPlan
	Restart dimensions after implicit-restart edge-case adjustment. More...

struct	SymmetricRestartSelection
	Ritz-value selection state used by symmetric implicit restart logic. More...

struct	SymmetricTransformOptions
	Native symmetric transform options for mapping `OP` Ritz values. More...

struct	TaylorExponentialDiagnostics
	Diagnostic state for Taylor exponential actions. More...

struct	TaylorExponentialParams
	Parameters for conservative Taylor scaling exponential actions. More...

struct	TaylorExponentialResult
	Result for validation-oriented Taylor exponential actions. More...

struct	TridiagonalEigensystem
	Eigenvalues and optional eigenvectors of a symmetric tridiagonal matrix. More...

Concepts
concept	KrylovPreconditioner
	Fixed linear inverse-preconditioner application, `output <- M^{-1} input`.

concept	KrylovVectorOps
	Concept for opaque vector-space operations required by Krylov solvers.

concept	KrylovOperator
	Concept for an output-first Krylov operator application `y <- OP*x`.

concept	KrylovMatrixFreeOperator
	Composite concept for current adapters that own vector ops and `OP*x`.

concept	MatrixFreeVectorOps
	Backward-compatible name for the combined vector/operator concept.

concept	KrylovMetricInnerProduct
	Optional customization point for backend-owned metric inner products.

concept	MetricInnerProductOps
	Backward-compatible name for direct metric inner-product customization.

concept	KrylovNorm
	Optional customization point for direct vector norms.

Typedefs
using	MatrixFill = uni20::linalg::MatrixRegion
	Matrix region selected by LAPACK-style copy and set operations.

template<typename Scalar >
using	RightMatrix = uni20::DenseMatrix< Scalar, uni20::RowMajor >
	Owning row-major dense matrix retained for layout-specific Krylov helpers.

using	SymmetricSpectralTransform = SymmetricTransformKind
	Backward-compatible name for symmetric transform kinds.

template<uni20::Real Scalar>
using	SymmetricSpectralTransformOptions = SymmetricTransformOptions< Scalar >
	Backward-compatible name for symmetric transform options.

Enumerations
enum class	GmresEstimateKind { estimate , upper_bound }
	Provenance of a supplied operator quantity, not of a computed error. More...

enum class	GmresRefinementStatus { converged , iteration_limit , stagnated , inner_failed , nonfinite_input , nonfinite_result }
	Outer termination, distinct from convergence of an inner correction. More...

enum class	GmresStatus { converged , iteration_limit , stagnated , breakdown , nonfinite_input , nonfinite_result }
	Numerical termination of a matrix-free linear solve. More...

enum class	KrylovDiagnosticsLevel { None , Summary , Full }
	Amount of solver-internal diagnostic state to retain. More...

enum class	MatrixNorm { MaxAbs , One , Infinity , Frobenius }
	LAPACK-compatible dense matrix norm selector. More...

enum class	MatrixSide { Left , Right }
	Matrix side selected by LAPACK-style multiply and solve operations. More...

enum class	MatrixTranspose { None , Transpose , ConjugateTranspose }
	Matrix transpose mode for local dense Krylov operations. More...

enum class	NonsymmetricStatus { Converged , NotConverged , ComplexPairEncountered , ComplexPromotionRecommended , RealSchurPairRequired , AmbiguousReality , Breakdown , InvalidInput }
	Native nonsymmetric solver status. More...

enum class	RealNonsymmetricBalanceJob { None , Permute , Scale , Both }
	LAPACK balancing operations for a dense real nonsymmetric matrix. More...

enum class	RealNonsymmetricPolicy { RequireRealEigenpairs , AllowRealSchurPairs , PromoteToComplexSuggested }
	Policy for real nonsymmetric solves when wanted Ritz values may be complex. More...

enum class	RitzReality { Real , Ambiguous , Complex }
	Numerical reality classification for a complex Ritz value from a real Arnoldi solve. More...

enum class	SpectrumPart { LargestMagnitude , SmallestMagnitude , LargestAlgebraic , SmallestAlgebraic , BothEnds , LargestReal , SmallestReal , LargestImaginary , SmallestImaginary }
	Spectrum selector for matrix-free eigenvalue solvers. More...

enum class	SymmetricTransformKind { Regular , GeneralizedInverse , ShiftInvert , Buckling , Cayley }
	Transform represented by the caller-supplied symmetric Krylov operator. More...

Functions
template<uni20::LapackReal Scalar>
detail::ColumnMajorLapackMatrix< Scalar >	apply_real_bidiagonal_left_factor (RealBidiagonalReduction< Scalar > const &reduction, detail::ColumnMajorLapackMatrix< Scalar > target, MatrixSide side=MatrixSide::Left, MatrixTranspose transpose=MatrixTranspose::None)
	Apply the left orthogonal factor `Q` from a real bidiagonal reduction.

template<uni20::LapackReal Scalar>
detail::ColumnMajorLapackMatrix< Scalar >	apply_real_bidiagonal_right_factor (RealBidiagonalReduction< Scalar > const &reduction, detail::ColumnMajorLapackMatrix< Scalar > target, MatrixSide side=MatrixSide::Right, MatrixTranspose transpose=MatrixTranspose::None)
	Apply the right orthogonal factor `P` from a real bidiagonal reduction.

template<uni20::LapackReal Scalar>
detail::ColumnMajorLapackMatrix< Scalar >	apply_real_hessenberg_orthogonal_factor (RealHessenbergReduction< Scalar > const &reduction, detail::ColumnMajorLapackMatrix< Scalar > target, MatrixSide side=MatrixSide::Left, MatrixTranspose transpose=MatrixTranspose::None)
	Apply the orthogonal factor from a compact real Hessenberg reduction.

template<uni20::LapackReal Scalar>
detail::ColumnMajorLapackMatrix< Scalar >	apply_real_lq_factor (RealCompactLqFactorization< Scalar > const &factorization, detail::ColumnMajorLapackMatrix< Scalar > target, MatrixSide side=MatrixSide::Right, MatrixTranspose transpose=MatrixTranspose::None)
	Apply the orthogonal factor from a compact real LQ factorization.

template<uni20::LapackReal Scalar>
detail::ColumnMajorLapackMatrix< Scalar >	apply_real_ql_factor (RealCompactQlFactorization< Scalar > const &factorization, detail::ColumnMajorLapackMatrix< Scalar > target, MatrixSide side=MatrixSide::Left, MatrixTranspose transpose=MatrixTranspose::None)
	Apply the orthogonal factor from a compact real QL factorization.

template<uni20::LapackReal Scalar>
detail::ColumnMajorLapackMatrix< Scalar >	apply_real_qr_factor (RealCompactQrFactorization< Scalar > const &factorization, detail::ColumnMajorLapackMatrix< Scalar > target, MatrixSide side=MatrixSide::Left, MatrixTranspose transpose=MatrixTranspose::None)
	Apply the orthogonal factor from a compact real QR factorization.

template<uni20::LapackReal Scalar>
detail::ColumnMajorLapackMatrix< Scalar >	apply_real_rq_factor (RealCompactRqFactorization< Scalar > const &factorization, detail::ColumnMajorLapackMatrix< Scalar > target, MatrixSide side=MatrixSide::Right, MatrixTranspose transpose=MatrixTranspose::None)
	Apply the orthogonal factor from a compact real RQ factorization.

template<uni20::LapackReal Scalar>
detail::ColumnMajorLapackMatrix< Scalar >	apply_real_symmetric_tridiagonal_orthogonal_factor (RealSymmetricTridiagonalReduction< Scalar > const &reduction, detail::ColumnMajorLapackMatrix< Scalar > target, MatrixSide side=MatrixSide::Left, MatrixTranspose transpose=MatrixTranspose::None)
	Apply the orthogonal factor from a compact real symmetric tridiagonal reduction.

template<uni20::Real Scalar>
SymmetricQrShiftResult< Scalar >	apply_symmetric_qr_shifts (std::vector< Scalar > diagonal, std::vector< Scalar > subdiagonal, std::vector< Scalar > const &shifts)
	Apply implicit shifted QR sweeps to a symmetric tridiagonal matrix.

template<uni20::RealOrComplex Scalar, typename Vector , KrylovMatrixFreeOperator< Vector, Scalar > Ops>
ArnoldiFactorization< Scalar, Vector >	arnoldi_factorize (Ops &ops, Vector const &initial, int max_steps, uni20::make_real_t< Scalar > breakdown_tolerance=uni20::make_real_t< Scalar >{})
	Build a full-reorthogonalized Arnoldi factorization.

template<uni20::RealOrComplex Scalar, typename Vector >
uni20::DenseMatrix< Scalar >	arnoldi_projected_hessenberg (ArnoldiFactorization< Scalar, Vector > const &factorization)
	Copy the square projected Hessenberg block from an Arnoldi factorization.

template<typename Scalar >
void	axpy (std::span< Scalar > y, Scalar const &alpha, std::span< Scalar const > x)
	Add a scaled local dense vector to another vector.

template<class Tensor , class Operator >
	BlockTensorMatrixFreeOps (Tensor const &, Operator) -> BlockTensorMatrixFreeOps< std::remove_cvref_t< Tensor >, Operator >

template<uni20::Complex C>
RitzReality	classify_ritz_reality (C theta, uni20::make_real_t< C > tolerance=uni20::make_real_t< C >{}, uni20::make_real_t< C > scale=uni20::make_real_t< C >{1}, uni20::make_real_t< C > ambiguity_factor=uni20::make_real_t< C >{10})
	Classify whether a complex Ritz value is numerically real.

template<uni20::LapackComplexReal Real>
ComplexHermitianEigensystem< Real >	complex_generalized_hermitian_eigensystem (detail::ColumnMajorLapackMatrix< uni20::detail::standard_complex< Real > > matrix, detail::ColumnMajorLapackMatrix< uni20::detail::standard_complex< Real > > metric, bool compute_vectors, MatrixFill triangle=MatrixFill::Upper)
	Solve a dense type-1 generalized complex Hermitian eigensystem through LAPACK `hegv`.

template<uni20::LapackComplexReal Real>
ComplexHermitianEigensystem< Real >	complex_generalized_hermitian_eigensystem_divide_and_conquer (detail::ColumnMajorLapackMatrix< uni20::detail::standard_complex< Real > > matrix, detail::ColumnMajorLapackMatrix< uni20::detail::standard_complex< Real > > metric, bool compute_vectors, MatrixFill triangle=MatrixFill::Upper)
	Solve a dense type-1 generalized complex Hermitian eigensystem through LAPACK `hegvd`.

template<uni20::LapackComplexReal Real>
ComplexHermitianEigensystem< Real >	complex_generalized_hermitian_eigensystem_index_range (detail::ColumnMajorLapackMatrix< uni20::detail::standard_complex< Real > > matrix, detail::ColumnMajorLapackMatrix< uni20::detail::standard_complex< Real > > metric, std::size_t first_index, std::size_t last_index, bool compute_vectors, MatrixFill triangle=MatrixFill::Upper)
	Solve a selected dense type-1 generalized complex Hermitian eigensystem through LAPACK `hegvx`.

template<uni20::LapackComplexReal Real>
ComplexHermitianEigensystem< Real >	complex_hermitian_eigensystem (detail::ColumnMajorLapackMatrix< uni20::detail::standard_complex< Real > > matrix, bool compute_vectors, MatrixFill triangle=MatrixFill::Upper)
	Solve a dense complex Hermitian eigensystem through LAPACK `heev`.

template<uni20::LapackComplexReal Real>
ComplexHermitianEigensystem< Real >	complex_hermitian_eigensystem_divide_and_conquer (detail::ColumnMajorLapackMatrix< uni20::detail::standard_complex< Real > > matrix, bool compute_vectors, MatrixFill triangle=MatrixFill::Upper)
	Solve a dense complex Hermitian eigensystem through LAPACK `heevd`.

template<uni20::LapackComplexReal Real>
ComplexHermitianEigensystem< Real >	complex_hermitian_eigensystem_index_range (detail::ColumnMajorLapackMatrix< uni20::detail::standard_complex< Real > > matrix, std::size_t first_index, std::size_t last_index, bool compute_vectors, MatrixFill triangle=MatrixFill::Upper)
	Solve a selected dense complex Hermitian eigensystem through LAPACK `heevr`.

template<uni20::LapackComplexReal Real, typename Vector , KrylovMatrixFreeOperator< Vector, uni20::complex< Real > > Ops>
NonsymmetricEigenResult< Real, Vector >	complex_nonsymmetric_arnoldi_standard (Ops &ops, Vector const &initial, NonsymmetricEigenParams< Real > const &params)
	Run a restarted complex nonsymmetric Arnoldi solver.

template<uni20::LapackReal Real>
ComplexNonsymmetricEigensystem< Real >	complex_nonsymmetric_eigensystem (uni20::DenseMatrix< uni20::detail::standard_complex< Real > > matrix, bool compute_right_vectors)
	Solve a dense real nonsymmetric eigensystem through LAPACK `geevx`.

template<uni20::LapackReal Real>
ComplexSchurDecomposition< Real >	complex_schur (uni20::DenseMatrix< uni20::detail::standard_complex< Real > > matrix, bool compute_vectors)
	Compute the complex Schur decomposition of a column-major DenseMatrix.

template<uni20::LapackReal Real>
RightComplexSchurDecomposition< Real >	complex_schur_layout_right (RightMatrix< uni20::detail::standard_complex< Real > > matrix, bool compute_vectors)
	Compute the complex Schur decomposition of a layout-right matrix through LAPACK `gees`.

template<uni20::LapackComplexReal Real, typename Vector , KrylovMatrixFreeOperator< Vector, uni20::complex< Real > > Ops>
ComplexSchurCompressedArnoldiFactorization< Real, Vector >	compress_complex_schur_arnoldi_restart (Ops &ops, ArnoldiFactorization< uni20::complex< Real >, Vector > const &factorization, NonsymmetricEigenParams< Real > const &params)
	Compress a complex Arnoldi factorization through complex Schur reordering.

template<uni20::LapackReal Scalar, typename Vector , KrylovMatrixFreeOperator< Vector, Scalar > Ops>
RealSchurCompressedArnoldiFactorization< Scalar, Vector >	compress_real_schur_arnoldi_restart (Ops &ops, ArnoldiFactorization< Scalar, Vector > const &factorization, NonsymmetricEigenParams< Scalar > const &params)
	Compress an Arnoldi factorization through real Schur block reordering.

template<uni20::LapackRealOrComplex Scalar, typename Vector , KrylovMatrixFreeOperator< Vector, Scalar > Ops>
SymmetricCompressedLanczosFactorization< uni20::make_real_t< Scalar >, Vector >	compress_symmetric_lanczos_restart (Ops &ops, std::vector< Vector > const &basis, Vector const &residual, SymmetricQrShiftResult< uni20::make_real_t< Scalar > > const &shifted, std::size_t retained_count)
	Compress a Lanczos factorization with the accumulated restart transform.

template<typename Scalar >
void	copy (std::span< Scalar > destination, std::span< Scalar const > source)
	Copy a local dense vector.

template<typename Scalar >
RightMatrix< Scalar >	copy_left_to_right (uni20::DenseMatrix< Scalar > const &matrix)
	Copy a column-major DenseMatrix into a row-major matrix.

template<typename Scalar >
uni20::DenseMatrix< Scalar >	copy_right_to_left (RightMatrix< Scalar > const &matrix)
	Copy a row-major matrix into a column-major DenseMatrix.

int	default_adaptive_exponential_krylov_dimension (std::size_t problem_dimension)
	Return the default adaptive Krylov projection cap for exponential actions.

template<uni20::Real Scalar> requires requires { uni20::numeric_limits<Scalar>::epsilon(); }
Scalar	default_complex_pair_tolerance ()
	Default imaginary-part tolerance for deciding whether a real Arnoldi Ritz value is real.

template<uni20::Real Scalar>
Scalar	default_complex_pair_tolerance (Scalar const &exemplar)
	Default reality tolerance using the exemplar's working precision.

int	default_exponential_krylov_dimension (std::size_t problem_dimension)
	Return the default Krylov projection dimension for exponential actions.

int	default_nonsymmetric_krylov_dimension (int eigenvalue_count, std::size_t problem_dimension)
	Return the default nonsymmetric Arnoldi subspace dimension.

int	default_nonsymmetric_retained_ritz_count (int eigenvalue_count, int krylov_dimension)
	Return the default nonsymmetric retained Schur dimension.

int	default_symmetric_krylov_dimension (int eigenvalue_count, std::size_t problem_dimension)
	Return the default symmetric/Hermitian Krylov subspace dimension.

int	default_symmetric_retained_ritz_count (int eigenvalue_count, int krylov_dimension)
	Return the default symmetric/Hermitian retained Ritz count.

template<uni20::Real Scalar>
Scalar	dot (std::span< Scalar const > x, std::span< Scalar const > y)
	Compute a BLAS-style real dot product.

template<typename Scalar >
Scalar	dotc (std::span< Scalar const > x, std::span< Scalar const > y)
	Compute a conjugated dot product.

template<typename Scalar >
Scalar	dotu (std::span< Scalar const > x, std::span< Scalar const > y)
	Compute an unconjugated dot product.

template<uni20::Real Scalar>
int	effective_exponential_krylov_dimension (KrylovExponentialParams< Scalar > const &params, std::size_t problem_dimension)
	Return the effective Krylov projection dimension for exponential actions.

template<uni20::Real Scalar>
int	effective_nonsymmetric_krylov_dimension (NonsymmetricEigenParams< Scalar > const &params, std::size_t problem_dimension)
	Return the effective nonsymmetric Arnoldi subspace dimension.

template<uni20::Real Scalar>
int	effective_nonsymmetric_retained_ritz_count (NonsymmetricEigenParams< Scalar > const &params, int krylov_dimension)
	Return the effective nonsymmetric retained Ritz or Schur count.

template<uni20::Real Scalar>
int	effective_symmetric_krylov_dimension (SymmetricEigenParams< Scalar > const &params, std::size_t problem_dimension)
	Return the effective symmetric/Hermitian Krylov subspace dimension.

template<uni20::Real Scalar>
int	effective_symmetric_retained_ritz_count (SymmetricEigenParams< Scalar > const &params, int krylov_dimension)
	Return the effective symmetric/Hermitian retained Ritz count.

template<uni20::LapackComplexReal Real, typename Vector , KrylovMatrixFreeOperator< Vector, uni20::complex< Real > > Ops>
ArnoldiFactorization< uni20::complex< Real >, Vector >	expand_complex_schur_arnoldi_restart (Ops &ops, ComplexSchurCompressedArnoldiFactorization< Real, Vector > const &compressed, std::size_t target_step_count, Real breakdown_tolerance=Real{})
	Expand a compressed complex Schur Arnoldi relation back to a requested dimension.

template<uni20::LapackReal Scalar, typename Vector , KrylovMatrixFreeOperator< Vector, Scalar > Ops>
ArnoldiFactorization< Scalar, Vector >	expand_real_schur_arnoldi_restart (Ops &ops, RealSchurCompressedArnoldiFactorization< Scalar, Vector > const &compressed, std::size_t target_step_count, Scalar breakdown_tolerance=Scalar{})
	Expand a compressed real Schur Arnoldi relation back to a requested dimension.

template<uni20::Real Scalar>
bool	exponential_adaptive_acceptance_enabled (KrylovExponentialParams< Scalar > const &params)

template<uni20::LapackReal Scalar, typename Vector >
ArnoldiRitzExtraction< Scalar >	extract_arnoldi_ritz (ArnoldiFactorization< Scalar, Vector > const &factorization, Scalar complex_pair_tolerance=Scalar{}, Scalar scale=Scalar{1})
	Extract projected Ritz values and residual estimates from a real Arnoldi factorization.

template<uni20::LapackComplex Complex, typename Vector >
ArnoldiRitzExtraction< uni20::make_real_t< Complex > >	extract_complex_arnoldi_ritz (ArnoldiFactorization< Complex, Vector > const &factorization, uni20::make_real_t< Complex > complex_pair_tolerance=uni20::make_real_t< Complex >{}, uni20::make_real_t< Complex > scale=uni20::make_real_t< Complex >{1})
	Extract projected Ritz values from a complex Arnoldi factorization.

template<typename Scalar >
void	gemv (std::span< Scalar > y, Scalar const &alpha, uni20::DenseMatrix< Scalar > const &matrix, std::span< Scalar const > x, Scalar const &beta, MatrixTranspose transpose=MatrixTranspose::None)
	Compute a dense matrix-vector product.

template<typename Scalar >
void	gerc (uni20::DenseMatrix< Scalar > &matrix, Scalar const &alpha, std::span< Scalar const > x, std::span< Scalar const > y)
	Apply a conjugated dense rank-one update.

template<typename Scalar >
void	geru (uni20::DenseMatrix< Scalar > &matrix, Scalar const &alpha, std::span< Scalar const > x, std::span< Scalar const > y)
	Apply a dense rank-one update.

template<class WSpace , class ISpace , class RSpace , class IOp , class ROp , class P , class Transfer >
auto	gmres_refine_with_info (GmresRefinementContext< WSpace, ISpace, RSpace, IOp, ROp, P, Transfer > &context, typename WSpace::vector_type &x, typename RSpace::vector_type const &b, GmresRefinementOptions< typename ISpace::real_type, typename RSpace::real_type > const &options={})
	Refine a supplied iterate using matrix-free GMRES correction solves.

template<RealOrComplex S, class Ops , class Op , class P , class V > requires KrylovVectorOps<Ops, V, S> && KrylovOperator<Op, V> && KrylovPreconditioner<P, V>
GmresInfo< make_real_t< S > >	gmres_with_info (Ops &ops, Op &op, P &preconditioner, V &x, V const &b, GmresOptions< make_real_t< S > > const &options, Precision precision)
	Solve `A*x=b` with restarted GMRES and a fixed right preconditioner.

template<RealOrComplex S, class Ops , class Op , class P , class V > requires (!has_runtime_precision_v<S>) && KrylovVectorOps<Ops, V, S> && KrylovOperator<Op, V> && KrylovPreconditioner<P, V>
auto	gmres_with_info (Ops &ops, Op &op, P &preconditioner, V &x, V const &b, GmresOptions< make_real_t< S > > const &options={})
	Fixed-scalar GMRES using the scalar's native precision.

template<RealOrComplex S, class Ops , class Op , class V > requires KrylovVectorOps<Ops, V, S> && KrylovOperator<Op, V>
auto	gmres_with_info (Ops &ops, Op &op, V &x, V const &b, GmresOptions< make_real_t< S > > const &options, Precision p)
	GMRES without preconditioning, at explicit scalar precision.

template<RealOrComplex S, class Ops , class Op , class V > requires (!has_runtime_precision_v<S>) && KrylovVectorOps<Ops, V, S> && KrylovOperator<Op, V>
auto	gmres_with_info (Ops &ops, Op &op, V &x, V const &b, GmresOptions< make_real_t< S > > const &options={})
	Fixed-scalar GMRES without preconditioning.

template<uni20::LapackScalar Scalar, typename TimeScalar , typename Vector , KrylovMatrixFreeOperator< Vector, Scalar > Ops> requires (std::constructible_from<uni20::make_real_t<Scalar>, TimeScalar> \|\| (uni20::Complex<Scalar> && std::constructible_from<Scalar, TimeScalar>))
KrylovExponentialResult< Scalar, Vector >	hermitian_krylov_exponential_action (Ops &ops, Vector const &initial, TimeScalar time, KrylovExponentialParams< uni20::make_real_t< Scalar > > const &params={})
	Approximate `exp(t A) v` for a Hermitian matrix-free operator.

template<RealOrComplex S, class Ops , class V > requires (!has_runtime_precision_v<S>) && requires(Ops& ops, V&& source) { ops.prototype_view(std::forward<V>(source)); }
auto	krylov_space (Ops &ops, V &&source)
	Bind fixed-scalar vector operations at native precision.

template<RealOrComplex S, class Ops , class V > requires requires(Ops& ops, V&& source) { ops.prototype_view(std::forward<V>(source)); }
auto	krylov_space (Ops &ops, V &&source, Precision p)
	Bind vector operations and an allocation prototype to finite precision.

template<typename Scalar >
void	lacpy (uni20::DenseMatrix< Scalar > &destination, uni20::DenseMatrix< Scalar > const &source, MatrixFill fill)
	Copy a selected region of a local dense matrix.

template<typename Scalar >
void	laset (uni20::DenseMatrix< Scalar > &matrix, Scalar const &diagonal, Scalar const &off_diagonal, MatrixFill fill)
	Set a selected region of a local dense matrix.

template<uni20::Real Scalar>
Scalar	map_symmetric_transformed_eigenvalue (Scalar transformed_value, SymmetricSpectralTransformOptions< Scalar > const &options)
	Map a native symmetric transformed Ritz value back to the original eigenvalue scale.

template<typename Scalar , typename Vector , typename MetricOps , typename BOps , typename InnerOps >
Scalar	metric_inner_product_or_apply (MetricOps &metric_ops, BOps &b_ops, InnerOps &inner_ops, Vector const &x, Vector const &y, Vector &scratch)
	Evaluate a metric inner product, using a backend override when available.

template<uni20::LapackScalar Scalar, typename TimeScalar , typename Vector , KrylovMatrixFreeOperator< Vector, Scalar > Ops> requires (std::constructible_from<uni20::make_real_t<Scalar>, TimeScalar> \|\| (uni20::Complex<Scalar> && std::constructible_from<Scalar, TimeScalar>))
KrylovExponentialResult< Scalar, Vector >	nonsymmetric_krylov_exponential_action (Ops &ops, Vector const &initial, TimeScalar time, KrylovExponentialParams< uni20::make_real_t< Scalar > > const &params={})
	Approximate `exp(t A) v` for a nonsymmetric matrix-free operator.

template<typename Scalar , typename Vector , typename Ops >
uni20::make_real_t< Scalar >	norm_or_inner_product (Ops &ops, Vector const &x)
	Evaluate a vector norm, using a backend override when available.

template<uni20::Real Scalar>
SymmetricRestartPlan< Scalar >	plan_symmetric_restart (SymmetricEigenParams< Scalar > const &params, std::vector< Scalar > const &values, std::vector< Scalar > const &residual_bounds, int converged_count)
	Adjust restart dimensions for implicit-restart edge cases.

template<uni20::RealOrComplex Scalar>
uni20::make_real_t< Scalar >	projected_departure_from_normality (uni20::DenseMatrix< Scalar > const &matrix)
	Measure departure from normality of a projected matrix.

template<uni20::LapackReal Scalar>
detail::ColumnMajorLapackMatrix< Scalar >	real_bidiagonal_left_orthogonal_factor (RealBidiagonalReduction< Scalar > const &reduction)
	Materialize the left orthogonal factor `Q` from a real bidiagonal reduction.

template<uni20::LapackReal Scalar>
RealBidiagonalReduction< Scalar >	real_bidiagonal_reduction (detail::ColumnMajorLapackMatrix< Scalar > matrix)
	Reduce a dense real matrix to bidiagonal form through LAPACK `gebrd`.

template<uni20::LapackReal Scalar>
detail::ColumnMajorLapackMatrix< Scalar >	real_bidiagonal_right_orthogonal_factor_transpose (RealBidiagonalReduction< Scalar > const &reduction)
	Materialize the right transposed factor `P^T` from a real bidiagonal reduction.

template<uni20::LapackReal Scalar>
std::vector< Scalar >	real_bidiagonal_singular_values (RealBidiagonalReduction< Scalar > const &reduction)
	Compute singular values from a compact real bidiagonal reduction.

template<uni20::LapackReal Scalar>
std::vector< Scalar >	real_bidiagonal_singular_values (std::vector< Scalar > diagonal, std::vector< Scalar > offdiagonal, bool upper=true)
	Compute singular values of a real bidiagonal matrix through LAPACK `bdsqr`.

template<uni20::LapackReal Scalar>
RealBidiagonalSvd< Scalar >	real_bidiagonal_svd (RealBidiagonalReduction< Scalar > const &reduction, bool compute_vectors)
	Compute the SVD from a compact real bidiagonal reduction.

template<uni20::LapackReal Scalar>
RealBidiagonalSvd< Scalar >	real_bidiagonal_svd (std::vector< Scalar > diagonal, std::vector< Scalar > offdiagonal, bool compute_vectors, bool upper=true)
	Compute the SVD of a real bidiagonal matrix through LAPACK `bdsdc`.

template<uni20::LapackReal Scalar>
RealBidiagonalSvd< Scalar >	real_bidiagonal_svd_index_range (RealBidiagonalReduction< Scalar > const &reduction, std::size_t first_index, std::size_t last_index, bool compute_vectors)
	Compute a selected-index SVD from a compact real bidiagonal reduction.

template<uni20::LapackReal Scalar>
RealBidiagonalSvd< Scalar >	real_bidiagonal_svd_index_range (std::vector< Scalar > diagonal, std::vector< Scalar > offdiagonal, std::size_t first_index, std::size_t last_index, bool compute_vectors, bool upper=true)
	Compute a selected-index SVD of a real bidiagonal matrix through LAPACK `bdsvdx`.

template<uni20::LapackReal Scalar>
RealCompactLqFactorization< Scalar >	real_compact_lq_factorization (detail::ColumnMajorLapackMatrix< Scalar > matrix)
	Compute a compact dense real LQ factorization through LAPACK `gelqf`.

template<uni20::LapackReal Scalar>
RealCompactQlFactorization< Scalar >	real_compact_ql_factorization (detail::ColumnMajorLapackMatrix< Scalar > matrix)
	Compute a compact dense real QL factorization through LAPACK `geqlf`.

template<uni20::LapackReal Scalar>
RealCompactQrFactorization< Scalar >	real_compact_qr_factorization (detail::ColumnMajorLapackMatrix< Scalar > matrix)
	Compute a compact dense real QR factorization through LAPACK `geqrf`.

template<uni20::LapackReal Scalar>
RealCompactRqFactorization< Scalar >	real_compact_rq_factorization (detail::ColumnMajorLapackMatrix< Scalar > matrix)
	Compute a compact dense real RQ factorization through LAPACK `gerqf`.

template<uni20::LapackReal Scalar>
detail::ColumnMajorLapackMatrix< Scalar >	real_dense_inverse (detail::ColumnMajorLapackMatrix< Scalar > matrix)
	Invert a dense real square matrix through LAPACK `getrf` and `getri`.

template<uni20::LapackReal Scalar>
detail::ColumnMajorLapackMatrix< Scalar >	real_dense_solve_linear_system (detail::ColumnMajorLapackMatrix< Scalar > coefficients, detail::ColumnMajorLapackMatrix< Scalar > right_hand_sides)
	Solve a dense real linear system through LAPACK `gesv`.

template<uni20::LapackReal Scalar>
RealSvdLeastSquares< Scalar >	real_divide_and_conquer_svd_least_squares (detail::ColumnMajorLapackMatrix< Scalar > coefficients, detail::ColumnMajorLapackMatrix< Scalar > right_hand_sides, Scalar rcond=Scalar{100} *uni20::numeric_limits< Scalar >::epsilon())
	Solve a dense real divide-and-conquer SVD least-squares problem through LAPACK `gelsd`.

template<uni20::LapackReal Scalar>
RealEquilibration< Scalar >	real_equilibration (detail::ColumnMajorLapackMatrix< Scalar > matrix)
	Compute dense real row/column equilibration factors through LAPACK `geequ`.

template<uni20::LapackReal Scalar>
RealExpertLinearSolve< Scalar >	real_expert_linear_solve (detail::ColumnMajorLapackMatrix< Scalar > coefficients, detail::ColumnMajorLapackMatrix< Scalar > right_hand_sides)
	Solve a dense real linear system with LAPACK `gesvx` diagnostics.

template<uni20::LapackReal Scalar>
RealEquilibration< Scalar >	real_general_band_equilibration (RealGeneralBandMatrix< Scalar > band_matrix)
	Compute real general-band row/column equilibration factors through LAPACK `gbequ`.

template<uni20::LapackReal Scalar>
RealGeneralBandExpertLinearSolve< Scalar >	real_general_band_expert_linear_solve (RealGeneralBandMatrix< Scalar > coefficients, detail::ColumnMajorLapackMatrix< Scalar > right_hand_sides, MatrixTranspose transpose=MatrixTranspose::None)
	Solve a real general-band linear system with LAPACK `gbsvx` diagnostics.

template<uni20::LapackReal Scalar>
RealGeneralBandFactorization< Scalar >	real_general_band_factorization (RealGeneralBandMatrix< Scalar > matrix)
	Compute a real general-band LU factorization through LAPACK `gbtrf`.

template<uni20::LapackReal Scalar>
RealGeneralBandMatrix< Scalar >	real_general_band_from_dense (detail::ColumnMajorLapackMatrix< Scalar > const &matrix, std::size_t lower_bandwidth, std::size_t upper_bandwidth)
	Pack a dense real square matrix into LAPACK general-band factor storage.

template<uni20::LapackReal Scalar>
Scalar	real_general_band_matrix_norm (RealGeneralBandMatrix< Scalar > band_matrix, MatrixNorm norm)
	Compute a real general-band matrix norm through LAPACK `langb`.

template<uni20::LapackReal Scalar>
Scalar	real_general_band_one_norm_reciprocal_condition_number (RealGeneralBandFactorization< Scalar > const &factorization, Scalar original_one_norm)
	Estimate a real general-band one-norm reciprocal condition number through LAPACK `gbcon`.

template<uni20::LapackReal Scalar>
Scalar	real_general_band_one_norm_reciprocal_condition_number (RealGeneralBandMatrix< Scalar > matrix)
	Estimate a real general-band one-norm reciprocal condition number.

template<uni20::LapackReal Scalar>
RealEquilibration< Scalar >	real_general_band_power_of_two_equilibration (RealGeneralBandMatrix< Scalar > band_matrix)
	Compute power-of-two real general-band equilibration factors through LAPACK `gbequb`.

template<uni20::LapackReal Scalar>
RealRefinedLinearSolve< Scalar >	real_general_band_refined_solve (RealGeneralBandMatrix< Scalar > coefficients, detail::ColumnMajorLapackMatrix< Scalar > right_hand_sides, MatrixTranspose transpose=MatrixTranspose::None)
	Solve and refine a real general-band system through LAPACK `gbrfs`.

template<uni20::LapackReal Scalar>
detail::ColumnMajorLapackMatrix< Scalar >	real_general_band_solve (RealGeneralBandFactorization< Scalar > const &factorization, detail::ColumnMajorLapackMatrix< Scalar > right_hand_sides, MatrixTranspose transpose=MatrixTranspose::None)
	Solve using an existing real general-band LU factorization through LAPACK `gbtrs`.

template<uni20::LapackReal Scalar>
detail::ColumnMajorLapackMatrix< Scalar >	real_general_band_solve (RealGeneralBandMatrix< Scalar > coefficients, detail::ColumnMajorLapackMatrix< Scalar > right_hand_sides)
	Solve a real general-band linear system through LAPACK `gbsv`.

template<uni20::LapackReal Scalar>
RealGeneralTridiagonalExpertLinearSolve< Scalar >	real_general_tridiagonal_expert_linear_solve (RealGeneralTridiagonalMatrix< Scalar > coefficients, detail::ColumnMajorLapackMatrix< Scalar > right_hand_sides, MatrixTranspose transpose=MatrixTranspose::None)
	Solve a real general-tridiagonal linear system with LAPACK `gtsvx` diagnostics.

template<uni20::LapackReal Scalar>
RealGeneralTridiagonalFactorization< Scalar >	real_general_tridiagonal_factorization (RealGeneralTridiagonalMatrix< Scalar > matrix)
	Compute a real general tridiagonal LU factorization through LAPACK `gttrf`.

template<uni20::LapackReal Scalar>
RealGeneralTridiagonalMatrix< Scalar >	real_general_tridiagonal_from_dense (detail::ColumnMajorLapackMatrix< Scalar > const &matrix)
	Extract real general tridiagonal storage from a dense matrix.

template<uni20::LapackReal Scalar>
Scalar	real_general_tridiagonal_one_norm (RealGeneralTridiagonalMatrix< Scalar > const &matrix)
	Compute the one-norm of a real general tridiagonal matrix.

template<uni20::LapackReal Scalar>
Scalar	real_general_tridiagonal_one_norm_reciprocal_condition_number (RealGeneralTridiagonalFactorization< Scalar > const &factorization, Scalar original_one_norm)
	Estimate a real general-tridiagonal one-norm reciprocal condition number through LAPACK `gtcon`.

template<uni20::LapackReal Scalar>
Scalar	real_general_tridiagonal_one_norm_reciprocal_condition_number (RealGeneralTridiagonalMatrix< Scalar > matrix)
	Estimate a real general-tridiagonal one-norm reciprocal condition number from a matrix.

template<uni20::LapackReal Scalar>
RealRefinedLinearSolve< Scalar >	real_general_tridiagonal_refined_solve (RealGeneralTridiagonalMatrix< Scalar > coefficients, detail::ColumnMajorLapackMatrix< Scalar > right_hand_sides, MatrixTranspose transpose=MatrixTranspose::None)
	Solve and refine a real general-tridiagonal linear system through LAPACK `gtrfs`.

template<uni20::LapackReal Scalar>
detail::ColumnMajorLapackMatrix< Scalar >	real_general_tridiagonal_solve (RealGeneralTridiagonalFactorization< Scalar > const &factorization, detail::ColumnMajorLapackMatrix< Scalar > right_hand_sides, MatrixTranspose transpose=MatrixTranspose::None)
	Solve using an existing real general tridiagonal LU factorization through LAPACK `gttrs`.

template<uni20::LapackReal Scalar>
detail::ColumnMajorLapackMatrix< Scalar >	real_general_tridiagonal_solve (RealGeneralTridiagonalMatrix< Scalar > coefficients, detail::ColumnMajorLapackMatrix< Scalar > right_hand_sides)
	Solve a real general tridiagonal linear system through LAPACK `gtsv`.

template<uni20::LapackReal Real>
RealGeneralizedHessenbergReduction< Real >	real_generalized_hessenberg_reduction (detail::ColumnMajorLapackMatrix< Real > matrix, detail::ColumnMajorLapackMatrix< Real > upper_triangular_metric, bool compute_vectors)
	Reduce a real matrix pencil to generalized upper Hessenberg form through LAPACK `gghrd`.

template<uni20::LapackReal Real>
RealGeneralizedSchurDecomposition< Real >	real_generalized_hessenberg_schur (detail::ColumnMajorLapackMatrix< Real > hessenberg, detail::ColumnMajorLapackMatrix< Real > upper_triangular_metric, bool compute_vectors)
	Compute generalized real Schur form from a generalized Hessenberg pencil through LAPACK `hgeqz`.

template<uni20::LapackReal Real>
RealGeneralizedNonsymmetricBalance< Real >	real_generalized_nonsymmetric_balance (detail::ColumnMajorLapackMatrix< Real > matrix, detail::ColumnMajorLapackMatrix< Real > metric, RealNonsymmetricBalanceJob job=RealNonsymmetricBalanceJob::Both)
	Balance a dense real nonsymmetric matrix pencil through LAPACK `ggbal`.

template<uni20::LapackReal Real>
detail::ColumnMajorLapackMatrix< Real >	real_generalized_nonsymmetric_balance_backtransform_right_vectors (detail::ColumnMajorLapackMatrix< Real > vectors, RealGeneralizedNonsymmetricBalance< Real > const &balance, RealNonsymmetricBalanceJob job=RealNonsymmetricBalanceJob::Both)
	Backtransform real right eigenvectors after generalized pencil balancing.

template<uni20::LapackReal Real>
RealGeneralizedNonsymmetricEigensystem< Real >	real_generalized_nonsymmetric_eigensystem (detail::ColumnMajorLapackMatrix< Real > matrix, detail::ColumnMajorLapackMatrix< Real > metric, bool compute_right_vectors)
	Solve a dense real generalized nonsymmetric eigensystem through LAPACK `ggev`.

template<uni20::LapackReal Real>
RealGeneralizedNonsymmetricExpertEigensystem< Real >	real_generalized_nonsymmetric_expert_eigensystem (detail::ColumnMajorLapackMatrix< Real > matrix, detail::ColumnMajorLapackMatrix< Real > metric, bool compute_right_vectors)
	Solve a dense real generalized nonsymmetric eigensystem through LAPACK `ggevx`.

template<uni20::LapackReal Real>
RealGeneralizedSchurDecomposition< Real >	real_generalized_schur (detail::ColumnMajorLapackMatrix< Real > matrix, detail::ColumnMajorLapackMatrix< Real > metric, bool compute_vectors)
	Compute the generalized real Schur decomposition of a matrix pencil through LAPACK `gges`.

template<uni20::LapackReal Real>
RealGeneralizedSchurConditionEstimates< Real >	real_generalized_schur_condition_estimates (RealGeneralizedSchurDecomposition< Real > decomposition)
	Estimate generalized Schur eigenvalue and eigenvector conditioning through LAPACK `tgsna`.

template<uni20::LapackReal Real>
RealGeneralizedSchurRightEigenvectors< Real >	real_generalized_schur_right_eigenvectors (RealGeneralizedSchurDecomposition< Real > decomposition)
	Compute right eigenvectors of a generalized real Schur form through LAPACK `tgevc`.

template<uni20::LapackReal Real>
RealGeneralizedSchurSelectedSubspace< Real >	real_generalized_schur_selected_subspace (RealGeneralizedSchurDecomposition< Real > decomposition, std::vector< std::size_t > const &selected_blocks)
	Move selected generalized real Schur blocks to the leading deflating subspace and estimate conditioning.

template<uni20::LapackReal Scalar>
RealSymmetricEigensystem< Scalar >	real_generalized_symmetric_eigensystem (detail::ColumnMajorLapackMatrix< Scalar > matrix, detail::ColumnMajorLapackMatrix< Scalar > metric, bool compute_vectors, MatrixFill triangle=MatrixFill::Upper)
	Solve a dense type-1 generalized real symmetric eigensystem through LAPACK `sygv`.

template<uni20::LapackReal Scalar>
RealSymmetricEigensystem< Scalar >	real_generalized_symmetric_eigensystem_divide_and_conquer (detail::ColumnMajorLapackMatrix< Scalar > matrix, detail::ColumnMajorLapackMatrix< Scalar > metric, bool compute_vectors, MatrixFill triangle=MatrixFill::Upper)
	Solve a dense type-1 generalized real symmetric eigensystem through LAPACK `sygvd`.

template<uni20::LapackReal Scalar>
RealSymmetricEigensystem< Scalar >	real_generalized_symmetric_eigensystem_index_range (detail::ColumnMajorLapackMatrix< Scalar > matrix, detail::ColumnMajorLapackMatrix< Scalar > metric, std::size_t first_index, std::size_t last_index, bool compute_vectors, MatrixFill triangle=MatrixFill::Upper)
	Solve a selected dense type-1 generalized real symmetric eigensystem through LAPACK `sygvx`.

template<uni20::LapackReal Scalar>
detail::ColumnMajorLapackMatrix< Scalar >	real_hessenberg_orthogonal_factor (RealHessenbergReduction< Scalar > const &reduction)
	Generate the orthogonal factor from a compact real Hessenberg reduction.

template<uni20::LapackReal Scalar>
RealHessenbergReduction< Scalar >	real_hessenberg_reduction (detail::ColumnMajorLapackMatrix< Scalar > matrix)
	Reduce a dense real square matrix to upper Hessenberg form through LAPACK `gehrd`.

template<uni20::LapackReal Scalar>
RealSchurDecomposition< Scalar >	real_hessenberg_schur (uni20::DenseMatrix< Scalar > hessenberg, bool compute_vectors)
	Compute the real Schur decomposition of an upper Hessenberg matrix through LAPACK `hseqr`.

template<uni20::LapackReal Scalar>
detail::ColumnMajorLapackMatrix< Scalar >	real_least_squares (detail::ColumnMajorLapackMatrix< Scalar > coefficients, detail::ColumnMajorLapackMatrix< Scalar > right_hand_sides)
	Solve a dense real least-squares or minimum-norm problem through LAPACK `gels`.

template<uni20::LapackReal Scalar>
RealLqFactorization< Scalar >	real_lq_factorization (detail::ColumnMajorLapackMatrix< Scalar > matrix)
	Compute a reduced dense real LQ factorization through LAPACK `gelqf` and `orglq`.

template<uni20::LapackReal Scalar>
RealLuFactorization< Scalar >	real_lu_factorization (detail::ColumnMajorLapackMatrix< Scalar > matrix)
	Compute a dense real LU factorization through LAPACK `getrf`.

template<uni20::LapackReal Scalar>
Scalar	real_lu_one_norm_reciprocal_condition_number (RealLuFactorization< Scalar > const &factorization, Scalar original_one_norm)
	Estimate the one-norm reciprocal condition number from an LU factorization.

template<uni20::LapackReal Scalar>
detail::ColumnMajorLapackMatrix< Scalar >	real_lu_solve (RealLuFactorization< Scalar > const &factorization, detail::ColumnMajorLapackMatrix< Scalar > right_hand_sides, MatrixTranspose transpose=MatrixTranspose::None)
	Solve using an existing dense real LU factorization through LAPACK `getrs`.

template<uni20::LapackReal Scalar>
Scalar	real_matrix_norm (detail::ColumnMajorLapackMatrix< Scalar > matrix, MatrixNorm norm)

template<uni20::LapackReal Scalar, typename Vector , KrylovMatrixFreeOperator< Vector, Scalar > Ops>
NonsymmetricEigenResult< Scalar, Vector >	real_nonsymmetric_arnoldi_restarted_standard (Ops &ops, Vector const &initial, NonsymmetricEigenParams< Scalar > const &params)
	Run a restarted native real nonsymmetric Arnoldi solver.

template<uni20::LapackReal Scalar, typename Vector , KrylovMatrixFreeOperator< Vector, Scalar > Ops>
NonsymmetricEigenResult< Scalar, Vector >	real_nonsymmetric_arnoldi_standard (Ops &ops, Vector const &initial, NonsymmetricEigenParams< Scalar > const &params)
	Run the first native non-restarted real nonsymmetric Arnoldi solver.

template<uni20::LapackReal Real>
RealNonsymmetricBalance< Real >	real_nonsymmetric_balance (detail::ColumnMajorLapackMatrix< Real > matrix, RealNonsymmetricBalanceJob job=RealNonsymmetricBalanceJob::Both)
	Reorder a layout-right complex Schur decomposition by moving selected entries to the front.

template<uni20::LapackReal Real>
detail::ColumnMajorLapackMatrix< Real >	real_nonsymmetric_balance_backtransform_right_vectors (detail::ColumnMajorLapackMatrix< Real > vectors, RealNonsymmetricBalance< Real > const &balance, RealNonsymmetricBalanceJob job=RealNonsymmetricBalanceJob::Both)
	Backtransform real right eigenvectors after `real_nonsymmetric_balance`.

template<uni20::LapackReal Real>
RealNonsymmetricEigensystem< Real >	real_nonsymmetric_eigensystem (uni20::DenseMatrix< Real > matrix, bool compute_right_vectors)

template<uni20::LapackReal Real>
RealNonsymmetricExpertEigensystem< Real >	real_nonsymmetric_expert_eigensystem (detail::ColumnMajorLapackMatrix< Real > matrix, bool compute_right_vectors)
	Solve a dense real nonsymmetric eigensystem through LAPACK `geev`.

template<uni20::LapackReal Scalar>
Scalar	real_one_norm_reciprocal_condition_number (detail::ColumnMajorLapackMatrix< Scalar > matrix)
	Estimate the one-norm reciprocal condition number of a dense real matrix.

template<uni20::LapackReal Scalar>
RealPivotedCholeskyFactorization< Scalar >	real_pivoted_cholesky_factorization (detail::ColumnMajorLapackMatrix< Scalar > matrix, Scalar tolerance=Scalar{-1}, MatrixFill triangle=MatrixFill::Upper)
	Compute a dense real pivoted Cholesky factorization through LAPACK `pstrf`.

template<uni20::LapackReal Scalar>
RealPivotedQrFactorization< Scalar >	real_pivoted_qr_factorization (detail::ColumnMajorLapackMatrix< Scalar > matrix)
	Compute a reduced dense real pivoted QR factorization through LAPACK `geqp3`.

template<uni20::LapackReal Scalar>
RealQlFactorization< Scalar >	real_ql_factorization (detail::ColumnMajorLapackMatrix< Scalar > matrix)
	Compute a reduced dense real QL factorization through LAPACK `geqlf` and `orgql`.

template<uni20::LapackReal Scalar>
RealQrFactorization< Scalar >	real_qr_factorization (detail::ColumnMajorLapackMatrix< Scalar > matrix)
	Compute a reduced dense real QR factorization through LAPACK `geqrf` and `orgqr`.

template<uni20::LapackReal Scalar>
RealRankRevealingLeastSquares< Scalar >	real_rank_revealing_least_squares (detail::ColumnMajorLapackMatrix< Scalar > coefficients, detail::ColumnMajorLapackMatrix< Scalar > right_hand_sides, Scalar rcond=Scalar{100} *uni20::numeric_limits< Scalar >::epsilon())
	Solve a dense real rank-revealing least-squares problem through LAPACK `gelsy`.

template<uni20::LapackReal Scalar>
RealRefinedLinearSolve< Scalar >	real_refined_linear_solve (detail::ColumnMajorLapackMatrix< Scalar > coefficients, detail::ColumnMajorLapackMatrix< Scalar > right_hand_sides, MatrixTranspose transpose=MatrixTranspose::None)
	Solve and refine a dense real linear system through LAPACK `gerfs`.

template<uni20::LapackReal Scalar>
RealRqFactorization< Scalar >	real_rq_factorization (detail::ColumnMajorLapackMatrix< Scalar > matrix)
	Compute a reduced dense real RQ factorization through LAPACK `gerqf` and `orgrq`.

template<uni20::LapackReal Scalar>
RealSchurDecomposition< Scalar >	real_schur (uni20::DenseMatrix< Scalar > matrix, bool compute_vectors)
	Compute the real Schur decomposition of a column-major DenseMatrix.

template<uni20::LapackReal Scalar>
RealSchurConditionEstimates< Scalar >	real_schur_condition_estimates (RealSchurDecomposition< Scalar > decomposition)
	Estimate Schur-form eigenvalue and eigenvector conditioning through LAPACK `trsna`.

template<uni20::LapackReal Scalar>
RightRealSchurDecomposition< Scalar >	real_schur_layout_right (RightMatrix< Scalar > matrix, bool compute_vectors)
	Solve a real symmetric tridiagonal eigensystem through LAPACK `stevd`.

template<uni20::LapackReal Scalar>
RealSchurRightEigenvectors< Scalar >	real_schur_right_eigenvectors (RealSchurDecomposition< Scalar > decomposition)
	Compute right eigenvectors of a real Schur form through LAPACK `trevc`.

template<uni20::LapackReal Scalar>
RealSchurSelectedSubspace< Scalar >	real_schur_selected_subspace (RealSchurDecomposition< Scalar > decomposition, std::vector< std::size_t > const &selected_blocks)
	Compute the real Schur decomposition of a layout-right matrix through LAPACK `gees`.

template<uni20::LapackReal Scalar>
RealSingularValueDecomposition< Scalar >	real_singular_value_decomposition (detail::ColumnMajorLapackMatrix< Scalar > matrix, bool compute_vectors)
	Compute a dense real singular value decomposition through LAPACK `gesvd`.

template<uni20::LapackReal Scalar>
RealSingularValueDecomposition< Scalar >	real_singular_value_decomposition_divide_and_conquer (detail::ColumnMajorLapackMatrix< Scalar > matrix, bool compute_vectors)
	Compute a dense real singular value decomposition through LAPACK `gesdd`.

template<uni20::LapackReal Scalar>
RealSingularValueDecomposition< Scalar >	real_singular_value_decomposition_index_range (detail::ColumnMajorLapackMatrix< Scalar > matrix, std::size_t first_index, std::size_t last_index, bool compute_vectors)
	Compute a selected-index dense real SVD through LAPACK `gesvdx`.

template<uni20::LapackReal Scalar>
RealSvdLeastSquares< Scalar >	real_svd_least_squares (detail::ColumnMajorLapackMatrix< Scalar > coefficients, detail::ColumnMajorLapackMatrix< Scalar > right_hand_sides, Scalar rcond=Scalar{100} *uni20::numeric_limits< Scalar >::epsilon())
	Solve a dense real SVD-based least-squares problem through LAPACK `gelss`.

template<uni20::LapackReal Scalar>
RealSylvesterSolution< Scalar >	real_sylvester_solve (detail::ColumnMajorLapackMatrix< Scalar > left, detail::ColumnMajorLapackMatrix< Scalar > right, detail::ColumnMajorLapackMatrix< Scalar > rhs, int sign=1, MatrixTranspose transpose_left=MatrixTranspose::None, MatrixTranspose transpose_right=MatrixTranspose::None)
	Solve a dense real Sylvester equation through LAPACK `trsyl`.

template<uni20::LapackReal Scalar>
RealSymmetricEigensystem< Scalar >	real_symmetric_eigensystem (detail::ColumnMajorLapackMatrix< Scalar > matrix, bool compute_vectors, MatrixFill triangle=MatrixFill::Upper)
	Solve a dense real symmetric eigensystem through LAPACK `syev`.

template<uni20::LapackReal Scalar>
RealSymmetricEigensystem< Scalar >	real_symmetric_eigensystem_divide_and_conquer (detail::ColumnMajorLapackMatrix< Scalar > matrix, bool compute_vectors, MatrixFill triangle=MatrixFill::Upper)
	Solve a dense real symmetric eigensystem through LAPACK `syevd`.

template<uni20::LapackReal Scalar>
RealSymmetricEigensystem< Scalar >	real_symmetric_eigensystem_index_range (detail::ColumnMajorLapackMatrix< Scalar > matrix, std::size_t first_index, std::size_t last_index, bool compute_vectors, MatrixFill triangle=MatrixFill::Upper)
	Solve a selected dense real symmetric eigensystem through LAPACK `syevr`.

template<uni20::LapackReal Scalar>
RealSymmetricIndefiniteExpertLinearSolve< Scalar >	real_symmetric_indefinite_expert_solve (detail::ColumnMajorLapackMatrix< Scalar > coefficients, detail::ColumnMajorLapackMatrix< Scalar > right_hand_sides, MatrixFill triangle=MatrixFill::Upper)
	Solve a dense real symmetric-indefinite system with LAPACK `sysvx` diagnostics.

template<uni20::LapackReal Scalar>
RealSymmetricIndefiniteFactorization< Scalar >	real_symmetric_indefinite_factorization (detail::ColumnMajorLapackMatrix< Scalar > matrix, MatrixFill triangle=MatrixFill::Upper)
	Compute a dense real symmetric-indefinite Bunch-Kaufman factorization through LAPACK `sytrf`.

template<uni20::LapackReal Scalar>
detail::ColumnMajorLapackMatrix< Scalar >	real_symmetric_indefinite_inverse (detail::ColumnMajorLapackMatrix< Scalar > matrix, MatrixFill triangle=MatrixFill::Upper)
	Invert a dense real symmetric-indefinite matrix through LAPACK `sytrf` and `sytri`.

template<uni20::LapackReal Scalar>
Scalar	real_symmetric_indefinite_one_norm_reciprocal_condition_number (detail::ColumnMajorLapackMatrix< Scalar > matrix, MatrixFill triangle=MatrixFill::Upper)
	Estimate a dense real symmetric-indefinite one-norm reciprocal condition number through LAPACK `sycon`.

template<uni20::LapackReal Scalar>
Scalar	real_symmetric_indefinite_one_norm_reciprocal_condition_number (RealSymmetricIndefiniteFactorization< Scalar > const &factorization, Scalar original_one_norm)
	Estimate a symmetric-indefinite one-norm reciprocal condition number from an existing factorization.

template<uni20::LapackReal Scalar>
RealRefinedLinearSolve< Scalar >	real_symmetric_indefinite_refined_solve (detail::ColumnMajorLapackMatrix< Scalar > coefficients, detail::ColumnMajorLapackMatrix< Scalar > right_hand_sides, MatrixFill triangle=MatrixFill::Upper)
	Solve and refine a dense real symmetric-indefinite system through LAPACK `syrfs`.

template<uni20::LapackReal Scalar>
detail::ColumnMajorLapackMatrix< Scalar >	real_symmetric_indefinite_solve (detail::ColumnMajorLapackMatrix< Scalar > coefficients, detail::ColumnMajorLapackMatrix< Scalar > right_hand_sides, MatrixFill triangle=MatrixFill::Upper)
	Solve a dense real symmetric indefinite system through LAPACK `sytrf` and `sytrs`.

template<uni20::LapackReal Scalar>
detail::ColumnMajorLapackMatrix< Scalar >	real_symmetric_indefinite_solve (RealSymmetricIndefiniteFactorization< Scalar > const &factorization, detail::ColumnMajorLapackMatrix< Scalar > right_hand_sides)
	Solve a dense real symmetric-indefinite system from an existing Bunch-Kaufman factorization.

template<uni20::LapackReal Scalar>
Scalar	real_symmetric_matrix_norm (detail::ColumnMajorLapackMatrix< Scalar > matrix, MatrixNorm norm, MatrixFill triangle=MatrixFill::Upper)
	Compute a dense real symmetric matrix norm through LAPACK `lansy`.

template<uni20::LapackReal Scalar>
RealSymmetricPositiveDefiniteBandExpertLinearSolve< Scalar >	real_symmetric_positive_definite_band_expert_linear_solve (RealSymmetricPositiveDefiniteBandMatrix< Scalar > coefficients, detail::ColumnMajorLapackMatrix< Scalar > right_hand_sides)
	Solve a real SPD band linear system with LAPACK `pbsvx` diagnostics.

template<uni20::LapackReal Scalar>
RealSymmetricPositiveDefiniteBandFactorization< Scalar >	real_symmetric_positive_definite_band_factorization (RealSymmetricPositiveDefiniteBandMatrix< Scalar > matrix)
	Compute a real SPD band Cholesky factorization through LAPACK `pbtrf`.

template<uni20::LapackReal Scalar>
RealSymmetricPositiveDefiniteBandMatrix< Scalar >	real_symmetric_positive_definite_band_from_dense (detail::ColumnMajorLapackMatrix< Scalar > const &matrix, std::size_t bandwidth, MatrixFill triangle=MatrixFill::Upper)
	Pack a dense real SPD matrix into LAPACK symmetric-band storage.

template<uni20::LapackReal Scalar>
Scalar	real_symmetric_positive_definite_band_one_norm (RealSymmetricPositiveDefiniteBandMatrix< Scalar > const &matrix)
	Compute the one-norm of a real SPD band matrix.

template<uni20::LapackReal Scalar>
Scalar	real_symmetric_positive_definite_band_one_norm_reciprocal_condition_number (RealSymmetricPositiveDefiniteBandFactorization< Scalar > const &factorization, Scalar original_one_norm)
	Estimate a real SPD band one-norm reciprocal condition number through LAPACK `pbcon`.

template<uni20::LapackReal Scalar>
Scalar	real_symmetric_positive_definite_band_one_norm_reciprocal_condition_number (RealSymmetricPositiveDefiniteBandMatrix< Scalar > matrix)
	Estimate a real SPD band one-norm reciprocal condition number from a matrix.

template<uni20::LapackReal Scalar>
RealRefinedLinearSolve< Scalar >	real_symmetric_positive_definite_band_refined_solve (RealSymmetricPositiveDefiniteBandMatrix< Scalar > coefficients, detail::ColumnMajorLapackMatrix< Scalar > right_hand_sides)
	Solve and refine a real SPD band linear system through LAPACK `pbrfs`.

template<uni20::LapackReal Scalar>
detail::ColumnMajorLapackMatrix< Scalar >	real_symmetric_positive_definite_band_solve (RealSymmetricPositiveDefiniteBandFactorization< Scalar > const &factorization, detail::ColumnMajorLapackMatrix< Scalar > right_hand_sides)
	Solve using an existing real SPD band Cholesky factorization through LAPACK `pbtrs`.

template<uni20::LapackReal Scalar>
detail::ColumnMajorLapackMatrix< Scalar >	real_symmetric_positive_definite_band_solve (RealSymmetricPositiveDefiniteBandMatrix< Scalar > coefficients, detail::ColumnMajorLapackMatrix< Scalar > right_hand_sides)
	Solve a real SPD band system through LAPACK `pbsv`.

template<uni20::LapackReal Scalar>
RealSymmetricPositiveDefiniteEquilibration< Scalar >	real_symmetric_positive_definite_equilibration (detail::ColumnMajorLapackMatrix< Scalar > matrix)
	Compute dense real SPD equilibration factors through LAPACK `poequ`.

template<uni20::LapackReal Scalar>
RealSymmetricPositiveDefiniteExpertLinearSolve< Scalar >	real_symmetric_positive_definite_expert_solve (detail::ColumnMajorLapackMatrix< Scalar > coefficients, detail::ColumnMajorLapackMatrix< Scalar > right_hand_sides, MatrixFill triangle=MatrixFill::Upper)
	Solve a dense real SPD linear system with LAPACK `posvx` diagnostics.

template<uni20::LapackReal Scalar>
RealSymmetricPositiveDefiniteFactorization< Scalar >	real_symmetric_positive_definite_factorization (detail::ColumnMajorLapackMatrix< Scalar > matrix, MatrixFill triangle=MatrixFill::Upper)
	Compute a dense real SPD Cholesky factorization through LAPACK `potrf`.

template<uni20::LapackReal Scalar>
detail::ColumnMajorLapackMatrix< Scalar >	real_symmetric_positive_definite_inverse (detail::ColumnMajorLapackMatrix< Scalar > matrix, MatrixFill triangle=MatrixFill::Upper)
	Invert a dense real SPD matrix through LAPACK `potrf` and `potri`.

template<uni20::LapackReal Scalar>
Scalar	real_symmetric_positive_definite_one_norm_reciprocal_condition_number (detail::ColumnMajorLapackMatrix< Scalar > matrix, MatrixFill triangle=MatrixFill::Upper)
	Estimate a dense real SPD one-norm reciprocal condition number through LAPACK `pocon`.

template<uni20::LapackReal Scalar>
Scalar	real_symmetric_positive_definite_one_norm_reciprocal_condition_number (RealSymmetricPositiveDefiniteFactorization< Scalar > const &factorization, Scalar original_one_norm)
	Estimate an SPD one-norm reciprocal condition number from an existing Cholesky factorization.

template<uni20::LapackReal Scalar>
RealRefinedLinearSolve< Scalar >	real_symmetric_positive_definite_refined_solve (detail::ColumnMajorLapackMatrix< Scalar > coefficients, detail::ColumnMajorLapackMatrix< Scalar > right_hand_sides, MatrixFill triangle=MatrixFill::Upper)
	Solve and refine a dense real SPD system through LAPACK `porfs`.

template<uni20::LapackReal Scalar>
detail::ColumnMajorLapackMatrix< Scalar >	real_symmetric_positive_definite_solve (detail::ColumnMajorLapackMatrix< Scalar > coefficients, detail::ColumnMajorLapackMatrix< Scalar > right_hand_sides, MatrixFill triangle=MatrixFill::Upper)
	Solve a dense real symmetric positive definite system through LAPACK `potrf` and `potrs`.

template<uni20::LapackReal Scalar>
detail::ColumnMajorLapackMatrix< Scalar >	real_symmetric_positive_definite_solve (RealSymmetricPositiveDefiniteFactorization< Scalar > const &factorization, detail::ColumnMajorLapackMatrix< Scalar > right_hand_sides)
	Solve a dense real SPD system from an existing Cholesky factorization.

template<uni20::LapackReal Scalar>
RealSymmetricPositiveDefiniteTridiagonalExpertLinearSolve< Scalar >	real_symmetric_positive_definite_tridiagonal_expert_linear_solve (RealSymmetricPositiveDefiniteTridiagonalMatrix< Scalar > coefficients, detail::ColumnMajorLapackMatrix< Scalar > right_hand_sides)
	Solve a real SPD tridiagonal linear system with LAPACK `ptsvx` diagnostics.

template<uni20::LapackReal Scalar>
RealSymmetricPositiveDefiniteTridiagonalFactorization< Scalar >	real_symmetric_positive_definite_tridiagonal_factorization (RealSymmetricPositiveDefiniteTridiagonalMatrix< Scalar > matrix)
	Compute a real SPD tridiagonal factorization through LAPACK `pttrf`.

template<uni20::LapackReal Scalar>
RealSymmetricPositiveDefiniteTridiagonalMatrix< Scalar >	real_symmetric_positive_definite_tridiagonal_from_dense (detail::ColumnMajorLapackMatrix< Scalar > const &matrix)
	Extract real SPD tridiagonal storage from a dense matrix.

template<uni20::LapackReal Scalar>
Scalar	real_symmetric_positive_definite_tridiagonal_one_norm (RealSymmetricPositiveDefiniteTridiagonalMatrix< Scalar > const &matrix)
	Compute the one-norm of a real SPD tridiagonal matrix.

template<uni20::LapackReal Scalar>
Scalar	real_symmetric_positive_definite_tridiagonal_one_norm_reciprocal_condition_number (RealSymmetricPositiveDefiniteTridiagonalFactorization< Scalar > const &factorization, Scalar original_one_norm)
	Estimate a real SPD tridiagonal one-norm reciprocal condition number through LAPACK `ptcon`.

template<uni20::LapackReal Scalar>
Scalar	real_symmetric_positive_definite_tridiagonal_one_norm_reciprocal_condition_number (RealSymmetricPositiveDefiniteTridiagonalMatrix< Scalar > matrix)
	Estimate a real SPD tridiagonal one-norm reciprocal condition number from a matrix.

template<uni20::LapackReal Scalar>
RealRefinedLinearSolve< Scalar >	real_symmetric_positive_definite_tridiagonal_refined_solve (RealSymmetricPositiveDefiniteTridiagonalMatrix< Scalar > coefficients, detail::ColumnMajorLapackMatrix< Scalar > right_hand_sides)
	Solve and refine a real SPD tridiagonal linear system through LAPACK `ptrfs`.

template<uni20::LapackReal Scalar>
detail::ColumnMajorLapackMatrix< Scalar >	real_symmetric_positive_definite_tridiagonal_solve (RealSymmetricPositiveDefiniteTridiagonalFactorization< Scalar > const &factorization, detail::ColumnMajorLapackMatrix< Scalar > right_hand_sides)
	Solve using an existing real SPD tridiagonal factorization through LAPACK `pttrs`.

template<uni20::LapackReal Scalar>
detail::ColumnMajorLapackMatrix< Scalar >	real_symmetric_positive_definite_tridiagonal_solve (RealSymmetricPositiveDefiniteTridiagonalMatrix< Scalar > coefficients, detail::ColumnMajorLapackMatrix< Scalar > right_hand_sides)
	Solve a real SPD tridiagonal system through LAPACK `ptsv`.

template<uni20::LapackReal Scalar>
detail::ColumnMajorLapackMatrix< Scalar >	real_symmetric_tridiagonal_orthogonal_factor (RealSymmetricTridiagonalReduction< Scalar > const &reduction)
	Generate the orthogonal factor from a compact real symmetric tridiagonal reduction.

template<uni20::LapackReal Scalar>
RealSymmetricTridiagonalReduction< Scalar >	real_symmetric_tridiagonal_reduction (detail::ColumnMajorLapackMatrix< Scalar > matrix, MatrixFill triangle=MatrixFill::Upper)
	Reduce a dense real symmetric matrix to tridiagonal form through LAPACK `sytrd`.

template<uni20::LapackReal Scalar>
detail::ColumnMajorLapackMatrix< Scalar >	real_triangular_inverse (detail::ColumnMajorLapackMatrix< Scalar > matrix, MatrixFill triangle=MatrixFill::Upper, bool unit_diagonal=false)
	Invert a dense real triangular matrix through LAPACK `trtri`.

template<uni20::LapackReal Scalar>
Scalar	real_triangular_matrix_norm (detail::ColumnMajorLapackMatrix< Scalar > matrix, MatrixNorm norm, MatrixFill triangle=MatrixFill::Upper, bool unit_diagonal=false)
	Compute a dense real triangular or trapezoidal matrix norm through LAPACK `lantr`.

template<uni20::LapackReal Scalar>
Scalar	real_triangular_one_norm_reciprocal_condition_number (detail::ColumnMajorLapackMatrix< Scalar > matrix, MatrixFill triangle=MatrixFill::Upper, bool unit_diagonal=false)
	Estimate a dense real triangular one-norm reciprocal condition number through LAPACK `trcon`.

template<uni20::LapackReal Scalar>
RealRefinedLinearSolve< Scalar >	real_triangular_refined_solve (detail::ColumnMajorLapackMatrix< Scalar > coefficients, detail::ColumnMajorLapackMatrix< Scalar > right_hand_sides, MatrixFill triangle=MatrixFill::Upper, MatrixTranspose transpose=MatrixTranspose::None, bool unit_diagonal=false)
	Solve and refine a dense real triangular system through LAPACK `trrfs`.

template<uni20::LapackReal Scalar>
detail::ColumnMajorLapackMatrix< Scalar >	real_triangular_solve (detail::ColumnMajorLapackMatrix< Scalar > coefficients, detail::ColumnMajorLapackMatrix< Scalar > right_hand_sides, MatrixFill triangle=MatrixFill::Upper, MatrixTranspose transpose=MatrixTranspose::None, bool unit_diagonal=false)
	Solve a dense real triangular system through LAPACK `trtrs`.

template<uni20::LapackReal Real>
ComplexSchurDecomposition< Real >	reorder_complex_schur (ComplexSchurDecomposition< Real > decomposition, std::vector< std::size_t > const &leading_order)
	Reorder a column-major complex Schur decomposition by moving selected entries to the front.

template<uni20::LapackReal Real>
RightComplexSchurDecomposition< Real >	reorder_complex_schur_layout_right (RightComplexSchurDecomposition< Real > decomposition, std::vector< std::size_t > const &leading_order)
	Move selected real Schur blocks to the leading invariant subspace and estimate conditioning.

template<uni20::LapackReal Real>
RealGeneralizedSchurDecomposition< Real >	reorder_real_generalized_schur (RealGeneralizedSchurDecomposition< Real > decomposition, std::vector< std::size_t > const &leading_block_order)
	Reorder a generalized real Schur decomposition by moving selected blocks to the front.

template<uni20::LapackReal Scalar>
RealSchurDecomposition< Scalar >	reorder_real_schur (RealSchurDecomposition< Scalar > decomposition, std::vector< std::size_t > const &leading_block_order)
	Reorder a column-major real Schur decomposition by moving selected blocks to the front.

template<uni20::LapackReal Scalar>
RightRealSchurDecomposition< Scalar >	reorder_real_schur_layout_right (RightRealSchurDecomposition< Scalar > decomposition, std::vector< std::size_t > const &leading_block_order)
	Reorder a layout-right real Schur decomposition by moving selected blocks to the front.

template<typename Scalar >
auto	right_mdspan (RightMatrix< Scalar > &matrix)
	Return a mutable layout-right mdspan view of a prototype right-layout matrix.

template<typename Scalar >
auto	right_mdspan (RightMatrix< Scalar > const &matrix)
	Return an immutable layout-right mdspan view of a prototype right-layout matrix.

template<typename Scalar >
void	scal (Scalar const &alpha, std::span< Scalar > vector)
	Scale a local dense vector in place.

template<uni20::Real Scalar>
std::vector< std::size_t >	select_nonsymmetric_restart_indices (std::vector< uni20::complex< Scalar > > const &values, NonsymmetricEigenParams< Scalar > const &params)
	Select retained Ritz or Schur indices for a nonsymmetric thick restart.

template<uni20::Real Scalar>
std::vector< std::size_t >	select_nonsymmetric_ritz_indices (std::vector< uni20::complex< Scalar > > const &values, NonsymmetricEigenParams< Scalar > const &params)
	Select wanted Ritz indices for a nonsymmetric projected spectrum.

template<uni20::LapackReal Scalar>
RealSchurRestartSelection< Scalar >	select_real_schur_restart_blocks (std::vector< RealSchurBlock< Scalar > > const &blocks, NonsymmetricEigenParams< Scalar > const &params)
	Select real Schur blocks for a nonsymmetric thick restart.

template<uni20::Real Scalar>
SymmetricRestartSelection< Scalar >	select_symmetric_restart (std::vector< Scalar > const &values, std::vector< Scalar > const &residual_bounds, SymmetricEigenParams< Scalar > const &params, std::size_t shift_count, std::span< std::size_t const > protected_shift_indices={})
	Select wanted Ritz values and exact-shift candidates for symmetric restart.

template<uni20::LapackRealOrComplex Scalar, typename Vector , KrylovMatrixFreeOperator< Vector, Scalar > OpOps, KrylovMatrixFreeOperator< Vector, Scalar > BOps, typename MetricOps >
EigenResult< uni20::make_real_t< Scalar >, Vector >	symmetric_lanczos_restarted_generalized_transformed (OpOps &op_ops, BOps &b_ops, MetricOps &metric_ops, Vector const &initial, SymmetricEigenParams< uni20::make_real_t< Scalar > > const &params, SymmetricSpectralTransformOptions< uni20::make_real_t< Scalar > > const &options)
	Run restarted symmetric Lanczos on a transformed generalized operator with a `B` metric.

template<uni20::LapackRealOrComplex Scalar, typename Vector , KrylovMatrixFreeOperator< Vector, Scalar > OpOps, KrylovMatrixFreeOperator< Vector, Scalar > BOps>
EigenResult< uni20::make_real_t< Scalar >, Vector >	symmetric_lanczos_restarted_generalized_transformed (OpOps &op_ops, BOps &b_ops, Vector const &initial, SymmetricEigenParams< uni20::make_real_t< Scalar > > const &params, SymmetricSpectralTransformOptions< uni20::make_real_t< Scalar > > const &options)
	Run restarted symmetric Lanczos on a transformed generalized operator with default metric handling.

template<uni20::LapackRealOrComplex Scalar, typename Vector , KrylovMatrixFreeOperator< Vector, Scalar > Ops>
EigenResult< uni20::make_real_t< Scalar >, Vector >	symmetric_lanczos_restarted_standard (Ops &ops, Vector const &initial, SymmetricEigenParams< uni20::make_real_t< Scalar > > const &params)
	Run a native symmetric/Hermitian Lanczos solve with implicit restarts.

template<uni20::LapackRealOrComplex Scalar, typename Vector , KrylovMatrixFreeOperator< Vector, Scalar > Ops>
EigenResult< uni20::make_real_t< Scalar >, Vector >	symmetric_lanczos_restarted_transformed (Ops &op_ops, Vector const &initial, SymmetricEigenParams< uni20::make_real_t< Scalar > > const &params, SymmetricSpectralTransformOptions< uni20::make_real_t< Scalar > > const &options)
	Run restarted symmetric Lanczos on a caller-supplied transformed operator.

template<uni20::LapackRealOrComplex Scalar, typename Vector , KrylovMatrixFreeOperator< Vector, Scalar > Ops>
EigenResult< uni20::make_real_t< Scalar >, Vector >	symmetric_lanczos_standard (Ops &ops, Vector const &initial, SymmetricEigenParams< uni20::make_real_t< Scalar > > const &params)
	Run the native symmetric/Hermitian Lanczos full-projection solver.

template<uni20::Real Real>
symmetric_tridiagonal_view< Real >	symmetric_tridiagonal (std::span< Real const > diagonal, std::span< Real const > offdiagonal)

template<uni20::LapackReal Scalar>
TridiagonalEigensystem< Scalar >	symmetric_tridiagonal_eigensystem (std::vector< Scalar > diagonal, std::vector< Scalar > subdiagonal, bool compute_vectors)
	Solve a real symmetric tridiagonal eigensystem through LAPACK `sterf` or `steqr`.

template<uni20::LapackReal Scalar>
TridiagonalEigensystem< Scalar >	symmetric_tridiagonal_eigensystem_divide_and_conquer (std::vector< Scalar > diagonal, std::vector< Scalar > subdiagonal, bool compute_vectors)
	Solve a real symmetric tridiagonal eigenvalue problem through LAPACK `sterf`.

template<uni20::LapackReal Scalar>
TridiagonalEigensystem< Scalar >	symmetric_tridiagonal_eigensystem_index_range (std::vector< Scalar > diagonal, std::vector< Scalar > subdiagonal, std::size_t first_index, std::size_t last_index, bool compute_vectors)
	Solve a selected real symmetric tridiagonal eigensystem through LAPACK `stevr`.

template<uni20::LapackReal Scalar>
std::vector< Scalar >	symmetric_tridiagonal_eigenvalues (std::vector< Scalar > diagonal, std::vector< Scalar > subdiagonal)

template<uni20::RealOrComplex Scalar, typename TimeScalar , typename Vector , KrylovMatrixFreeOperator< Vector, Scalar > Ops> requires (std::constructible_from<uni20::make_real_t<Scalar>, TimeScalar> \|\| (uni20::Complex<Scalar> && std::constructible_from<Scalar, TimeScalar>))
TaylorExponentialResult< Scalar, Vector >	taylor_exponential_action (Ops &ops, Vector const &initial, TimeScalar time, uni20::make_real_t< Scalar > operator_norm_bound, TaylorExponentialParams< uni20::make_real_t< Scalar > > const &params={})
	Compute `exp(t A) v` using conservative scaled Taylor series.

template<typename Ops , typename Vector >
std::size_t	validate_matrix_free_dimensions (Ops const &ops, Vector const &vector, char const *context)
	Validate that an initial/prototype vector matches the matrix-free problem dimension.

Typedef Documentation

◆ MatrixFill

using uni20::krylov::MatrixFill = typedef uni20::linalg::MatrixRegion

Matrix region selected by LAPACK-style copy and set operations.

◆ RightMatrix

template<typename Scalar >

using uni20::krylov::RightMatrix = typedef uni20::DenseMatrix<Scalar, uni20::RowMajor>

Owning row-major dense matrix retained for layout-specific Krylov helpers.

◆ SymmetricSpectralTransform

using uni20::krylov::SymmetricSpectralTransform = typedef SymmetricTransformKind

Backward-compatible name for symmetric transform kinds.

◆ SymmetricSpectralTransformOptions

template<uni20::Real Scalar>

using uni20::krylov::SymmetricSpectralTransformOptions = typedef SymmetricTransformOptions<Scalar>

Backward-compatible name for symmetric transform options.

Enumeration Type Documentation

◆ GmresEstimateKind

enum class uni20::krylov::GmresEstimateKind

strong

Provenance of a supplied operator quantity, not of a computed error.

Enumerator
estimate
upper_bound

◆ GmresRefinementStatus

enum class uni20::krylov::GmresRefinementStatus

strong

Outer termination, distinct from convergence of an inner correction.

Enumerator
converged
iteration_limit
stagnated
inner_failed
nonfinite_input
nonfinite_result

◆ GmresStatus

enum class uni20::krylov::GmresStatus

strong

Numerical termination of a matrix-free linear solve.

Enumerator
converged
iteration_limit
stagnated
breakdown
nonfinite_input
nonfinite_result

◆ KrylovDiagnosticsLevel

enum class uni20::krylov::KrylovDiagnosticsLevel

strong

Amount of solver-internal diagnostic state to retain.

Enumerator
None
Summary
Full

◆ MatrixNorm

enum class uni20::krylov::MatrixNorm

strong

LAPACK-compatible dense matrix norm selector.

Enumerator
MaxAbs
One
Infinity
Frobenius

◆ MatrixSide

enum class uni20::krylov::MatrixSide

strong

Matrix side selected by LAPACK-style multiply and solve operations.

Enumerator
Left
Right

◆ MatrixTranspose

enum class uni20::krylov::MatrixTranspose

strong

Matrix transpose mode for local dense Krylov operations.

Enumerator
None
Transpose
ConjugateTranspose

◆ NonsymmetricStatus

enum class uni20::krylov::NonsymmetricStatus

strong

Native nonsymmetric solver status.

Enumerator
Converged
NotConverged
ComplexPairEncountered
ComplexPromotionRecommended
RealSchurPairRequired
AmbiguousReality
Breakdown
InvalidInput

◆ RealNonsymmetricBalanceJob

enum class uni20::krylov::RealNonsymmetricBalanceJob

strong

LAPACK balancing operations for a dense real nonsymmetric matrix.

Enumerator
None
Permute
Scale
Both

◆ RealNonsymmetricPolicy

enum class uni20::krylov::RealNonsymmetricPolicy

strong

Policy for real nonsymmetric solves when wanted Ritz values may be complex.

Enumerator
RequireRealEigenpairs	Accept only Ritz values that are numerically real; report genuinely complex wanted pairs.
AllowRealSchurPairs	Allow real Schur two-plane output for complex conjugate pairs once that output is implemented.
PromoteToComplexSuggested	Stop with a promotion recommendation when wanted Ritz values are genuinely complex.

◆ RitzReality

enum class uni20::krylov::RitzReality

strong

Numerical reality classification for a complex Ritz value from a real Arnoldi solve.

Enumerator
Real
Ambiguous
Complex

◆ SpectrumPart

enum class uni20::krylov::SpectrumPart

strong

Spectrum selector for matrix-free eigenvalue solvers.

Enumerator
LargestMagnitude
SmallestMagnitude
LargestAlgebraic
SmallestAlgebraic
BothEnds
LargestReal
SmallestReal
LargestImaginary
SmallestImaginary

◆ SymmetricTransformKind

enum class uni20::krylov::SymmetricTransformKind

strong

Transform represented by the caller-supplied symmetric Krylov operator.

Enumerator
Regular
GeneralizedInverse
ShiftInvert
Buckling
Cayley

Function Documentation

◆ apply_real_bidiagonal_left_factor()

template<uni20::LapackReal Scalar>

detail::ColumnMajorLapackMatrix< Scalar > uni20::krylov::apply_real_bidiagonal_left_factor	(	RealBidiagonalReduction< Scalar > const &	reduction,
		detail::ColumnMajorLapackMatrix< Scalar >	target,
		MatrixSide	side = `MatrixSide::Left`,
		MatrixTranspose	transpose = `MatrixTranspose::None`
	)

Apply the left orthogonal factor Q from a real bidiagonal reduction.

Applies Q or Q^T using LAPACK ormbr.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

reduction	Compact bidiagonal reduction returned by `real_bidiagonal_reduction`.
target	Matrix overwritten by `op(Q) * target` or `target * op(Q)`.
side	Whether `op(Q)` multiplies `target` from the left or right.
transpose	Matrix operation applied to `Q`.

Returns: Matrix after applying the encoded orthogonal factor.

◆ apply_real_bidiagonal_right_factor()

template<uni20::LapackReal Scalar>

detail::ColumnMajorLapackMatrix< Scalar > uni20::krylov::apply_real_bidiagonal_right_factor	(	RealBidiagonalReduction< Scalar > const &	reduction,
		detail::ColumnMajorLapackMatrix< Scalar >	target,
		MatrixSide	side = `MatrixSide::Right`,
		MatrixTranspose	transpose = `MatrixTranspose::None`
	)

Apply the right orthogonal factor P from a real bidiagonal reduction.

Applies P or P^T using LAPACK ormbr.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

reduction	Compact bidiagonal reduction returned by `real_bidiagonal_reduction`.
target	Matrix overwritten by `op(P) * target` or `target * op(P)`.
side	Whether `op(P)` multiplies `target` from the left or right.
transpose	Matrix operation applied to `P`.

Returns: Matrix after applying the encoded orthogonal factor.

◆ apply_real_hessenberg_orthogonal_factor()

template<uni20::LapackReal Scalar>

detail::ColumnMajorLapackMatrix< Scalar > uni20::krylov::apply_real_hessenberg_orthogonal_factor	(	RealHessenbergReduction< Scalar > const &	reduction,
		detail::ColumnMajorLapackMatrix< Scalar >	target,
		MatrixSide	side = `MatrixSide::Left`,
		MatrixTranspose	transpose = `MatrixTranspose::None`
	)

Apply the orthogonal factor from a compact real Hessenberg reduction.

Applies Q, or Q^T, from the Householder data returned by real_hessenberg_reduction to a dense target matrix using LAPACK ormhr.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

reduction	Compact Hessenberg reduction returned by `real_hessenberg_reduction`.
target	Matrix overwritten by `op(Q) * target` or `target * op(Q)`.
side	Whether `op(Q)` multiplies `target` from the left or right.
transpose	Matrix operation applied to `Q`.

Returns: Matrix after applying the encoded orthogonal factor.

◆ apply_real_lq_factor()

template<uni20::LapackReal Scalar>

detail::ColumnMajorLapackMatrix< Scalar > uni20::krylov::apply_real_lq_factor	(	RealCompactLqFactorization< Scalar > const &	factorization,
		detail::ColumnMajorLapackMatrix< Scalar >	target,
		MatrixSide	side = `MatrixSide::Right`,
		MatrixTranspose	transpose = `MatrixTranspose::None`
	)

Apply the orthogonal factor from a compact real LQ factorization.

Applies Q, or Q^T, from a Householder LQ factorization to a dense target matrix using LAPACK ormlq.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

factorization	Compact LQ factorization returned by `real_compact_lq_factorization`.
target	Matrix overwritten by `op(Q) * target` or `target * op(Q)`.
side	Whether `op(Q)` multiplies `target` from the left or right.
transpose	Matrix operation applied to `Q`.

Returns: Matrix after applying the encoded orthogonal factor.

◆ apply_real_ql_factor()

template<uni20::LapackReal Scalar>

detail::ColumnMajorLapackMatrix< Scalar > uni20::krylov::apply_real_ql_factor	(	RealCompactQlFactorization< Scalar > const &	factorization,
		detail::ColumnMajorLapackMatrix< Scalar >	target,
		MatrixSide	side = `MatrixSide::Left`,
		MatrixTranspose	transpose = `MatrixTranspose::None`
	)

Apply the orthogonal factor from a compact real QL factorization.

Applies Q, or Q^T, from a Householder QL factorization to a dense target matrix using LAPACK ormql.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

factorization	Compact QL factorization returned by `real_compact_ql_factorization`.
target	Matrix overwritten by `op(Q) * target` or `target * op(Q)`.
side	Whether `op(Q)` multiplies `target` from the left or right.
transpose	Matrix operation applied to `Q`.

Returns: Matrix after applying the encoded orthogonal factor.

◆ apply_real_qr_factor()

template<uni20::LapackReal Scalar>

detail::ColumnMajorLapackMatrix< Scalar > uni20::krylov::apply_real_qr_factor	(	RealCompactQrFactorization< Scalar > const &	factorization,
		detail::ColumnMajorLapackMatrix< Scalar >	target,
		MatrixSide	side = `MatrixSide::Left`,
		MatrixTranspose	transpose = `MatrixTranspose::None`
	)

Apply the orthogonal factor from a compact real QR factorization.

Applies Q, or Q^T, from a Householder QR factorization to a dense target matrix using LAPACK ormqr.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

factorization	Compact QR factorization returned by `real_compact_qr_factorization`.
target	Matrix overwritten by `op(Q) * target` or `target * op(Q)`.
side	Whether `op(Q)` multiplies `target` from the left or right.
transpose	Matrix operation applied to `Q`.

Returns: Matrix after applying the encoded orthogonal factor.

◆ apply_real_rq_factor()

template<uni20::LapackReal Scalar>

detail::ColumnMajorLapackMatrix< Scalar > uni20::krylov::apply_real_rq_factor	(	RealCompactRqFactorization< Scalar > const &	factorization,
		detail::ColumnMajorLapackMatrix< Scalar >	target,
		MatrixSide	side = `MatrixSide::Right`,
		MatrixTranspose	transpose = `MatrixTranspose::None`
	)

Apply the orthogonal factor from a compact real RQ factorization.

Applies Q, or Q^T, from a Householder RQ factorization to a dense target matrix using LAPACK ormrq.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

factorization	Compact RQ factorization returned by `real_compact_rq_factorization`.
target	Matrix overwritten by `op(Q) * target` or `target * op(Q)`.
side	Whether `op(Q)` multiplies `target` from the left or right.
transpose	Matrix operation applied to `Q`.

Returns: Matrix after applying the encoded orthogonal factor.

◆ apply_real_symmetric_tridiagonal_orthogonal_factor()

template<uni20::LapackReal Scalar>

detail::ColumnMajorLapackMatrix< Scalar > uni20::krylov::apply_real_symmetric_tridiagonal_orthogonal_factor	(	RealSymmetricTridiagonalReduction< Scalar > const &	reduction,
		detail::ColumnMajorLapackMatrix< Scalar >	target,
		MatrixSide	side = `MatrixSide::Left`,
		MatrixTranspose	transpose = `MatrixTranspose::None`
	)

Apply the orthogonal factor from a compact real symmetric tridiagonal reduction.

Applies Q, or Q^T, from the Householder data returned by real_symmetric_tridiagonal_reduction to a dense target matrix using LAPACK ormtr.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

reduction	Compact tridiagonal reduction returned by `real_symmetric_tridiagonal_reduction`.
target	Matrix overwritten by `op(Q) * target` or `target * op(Q)`.
side	Whether `op(Q)` multiplies `target` from the left or right.
transpose	Matrix operation applied to `Q`.

Returns: Matrix after applying the encoded orthogonal factor.

◆ apply_symmetric_qr_shifts()

template<uni20::Real Scalar>

SymmetricQrShiftResult< Scalar > uni20::krylov::apply_symmetric_qr_shifts	(	std::vector< Scalar >	diagonal,
		std::vector< Scalar >	subdiagonal,
		std::vector< Scalar > const &	shifts
	)

Apply implicit shifted QR sweeps to a symmetric tridiagonal matrix.

Each shift is applied by Givens rotations that chase the QR bulge through the tridiagonal, while the accumulated orthogonal transform is returned for basis compression. Updating the opaque Lanczos basis and residual is handled by a later restart slice.

Template Parameters

Scalar Real scalar type.

Parameters

diagonal	Main diagonal of the tridiagonal matrix.
subdiagonal	Subdiagonal entries, length `diagonal.size() - 1`.
shifts	Exact shifts to apply.

Returns: Updated tridiagonal coefficients and accumulated orthogonal transform.

◆ arnoldi_factorize()

template<uni20::RealOrComplex Scalar, typename Vector , KrylovMatrixFreeOperator< Vector, Scalar > Ops>

ArnoldiFactorization< Scalar, Vector > uni20::krylov::arnoldi_factorize	(	Ops &	ops,
		Vector const &	initial,
		int	max_steps,
		uni20::make_real_t< Scalar >	breakdown_tolerance = `uni20::make_real_t<Scalar>{}`
	)

Build a full-reorthogonalized Arnoldi factorization.

This constructs A V_m = V_{m+1} H_m when no happy breakdown is detected. On happy breakdown, the final subdiagonal is zero and the basis spans an invariant subspace. Vector storage and operations are entirely delegated to ops; only the small Hessenberg projection is stored locally.

Template Parameters

Scalar	Vector-space scalar type.
Vector	Opaque vector type.
Ops	Matrix-free operator/vector operation type.

Parameters

ops	Matrix-free vector operations and operator application.
initial	Initial vector used to seed the Arnoldi basis.
max_steps	Maximum number of Arnoldi steps.
breakdown_tolerance	Absolute residual norm threshold for happy breakdown.

Returns: Arnoldi factorization.

◆ arnoldi_projected_hessenberg()

template<uni20::RealOrComplex Scalar, typename Vector >

uni20::DenseMatrix< Scalar > uni20::krylov::arnoldi_projected_hessenberg ( ArnoldiFactorization< Scalar, Vector > const & factorization )

Copy the square projected Hessenberg block from an Arnoldi factorization.

Template Parameters

Scalar	Vector-space scalar type.
Vector	Opaque vector type used by the Arnoldi basis.

Parameters

factorization Arnoldi factorization to extract from.

Returns: The leading m-by-m projected Hessenberg matrix.

◆ axpy()

template<typename Scalar >

void uni20::krylov::axpy	(	std::span< Scalar >	y,
		Scalar const &	alpha,
		std::span< Scalar const >	x
	)

Add a scaled local dense vector to another vector.

Template Parameters

Scalar Element type.

Parameters

y	Destination vector updated as `y += alpha * x`.
alpha	Scale factor.
x	Source vector.

◆ BlockTensorMatrixFreeOps()

template<class Tensor , class Operator >

uni20::krylov::BlockTensorMatrixFreeOps	(	Tensor const &	,
		Operator
	)		-> BlockTensorMatrixFreeOps< std::remove_cvref_t< Tensor >, Operator >

◆ classify_ritz_reality()

template<uni20::Complex C>

RitzReality uni20::krylov::classify_ritz_reality	(	C	theta,
		uni20::make_real_t< C >	tolerance = `uni20::make_real_t<C>{}`,
		uni20::make_real_t< C >	scale = `uni20::make_real_t<C>{1}`,
		uni20::make_real_t< C >	ambiguity_factor = `uni20::make_real_t<C>{10}`
	)

Classify whether a complex Ritz value is numerically real.

This is intended for real nonsymmetric Arnoldi. A real operator can have complex-conjugate Ritz pairs; transfer-matrix workflows often want to remain in a real path when the wanted fixed point is real, but detect genuine or borderline time-reversal-symmetry breaking. Values with |imag(theta)| <= tolerance * scale are classified as real. Values within ambiguity_factor of that boundary are classified as ambiguous.

Template Parameters

C	Complex scalar type deduced from the Ritz value.

Parameters

theta	Ritz value to classify.
tolerance	Relative imaginary-part tolerance. If zero or negative, a machine-precision default is used, derived from theta for runtime scalars. An exact theta requires an explicit finite-precision tolerance.
scale	Problem or projected-matrix scale. Values below one are raised to one.
ambiguity_factor	Width of the ambiguous band above the real threshold.

Returns: Reality classification for theta.

◆ complex_generalized_hermitian_eigensystem()

template<uni20::LapackComplexReal Real>

ComplexHermitianEigensystem< Real > uni20::krylov::complex_generalized_hermitian_eigensystem	(	detail::ColumnMajorLapackMatrix< uni20::detail::standard_complex< Real > >	matrix,
		detail::ColumnMajorLapackMatrix< uni20::detail::standard_complex< Real > >	metric,
		bool	compute_vectors,
		MatrixFill	triangle = `MatrixFill::Upper`
	)

Solve a dense type-1 generalized complex Hermitian eigensystem through LAPACK hegv.

Solves A * x = lambda * B * x, where A is complex Hermitian and B is complex Hermitian positive definite. Eigenvalues are returned in ascending order.

Template Parameters

Real	Underlying real precision.

Parameters

matrix	Complex Hermitian square matrix `A`. Only the selected triangle is read.
metric	Complex Hermitian positive definite square matrix `B`.
compute_vectors	Whether to compute generalized eigenvectors.
triangle	Triangle of `matrix` and `metric` supplied to LAPACK.

Returns: Eigenvalues and, optionally, generalized eigenvectors.

◆ complex_generalized_hermitian_eigensystem_divide_and_conquer()

template<uni20::LapackComplexReal Real>

ComplexHermitianEigensystem< Real > uni20::krylov::complex_generalized_hermitian_eigensystem_divide_and_conquer	(	detail::ColumnMajorLapackMatrix< uni20::detail::standard_complex< Real > >	matrix,
		detail::ColumnMajorLapackMatrix< uni20::detail::standard_complex< Real > >	metric,
		bool	compute_vectors,
		MatrixFill	triangle = `MatrixFill::Upper`
	)

Solve a dense type-1 generalized complex Hermitian eigensystem through LAPACK hegvd.

Uses the divide-and-conquer generalized Hermitian eigensystem driver for A * x = lambda * B * x.

Template Parameters

Real	Underlying real precision.

Parameters

matrix	Complex Hermitian square matrix `A`. Only the selected triangle is read.
metric	Complex Hermitian positive definite square matrix `B`.
compute_vectors	Whether to compute generalized eigenvectors.
triangle	Triangle of `matrix` and `metric` supplied to LAPACK.

Returns: Eigenvalues and, optionally, generalized eigenvectors.

◆ complex_generalized_hermitian_eigensystem_index_range()

template<uni20::LapackComplexReal Real>

ComplexHermitianEigensystem< Real > uni20::krylov::complex_generalized_hermitian_eigensystem_index_range	(	detail::ColumnMajorLapackMatrix< uni20::detail::standard_complex< Real > >	matrix,
		detail::ColumnMajorLapackMatrix< uni20::detail::standard_complex< Real > >	metric,
		std::size_t	first_index,
		std::size_t	last_index,
		bool	compute_vectors,
		MatrixFill	triangle = `MatrixFill::Upper`
	)

Solve a selected dense type-1 generalized complex Hermitian eigensystem through LAPACK hegvx.

Selects eigenpairs by 0-based inclusive index in ascending eigenvalue order for A * x = lambda * B * x.

Template Parameters

Real	Underlying real precision.

Parameters

matrix	Complex Hermitian square matrix `A`. Only the selected triangle is read.
metric	Complex Hermitian positive definite square matrix `B`.
first_index	First 0-based eigenvalue index to compute.
last_index	Last 0-based eigenvalue index to compute.
compute_vectors	Whether to compute selected generalized eigenvectors.
triangle	Triangle of `matrix` and `metric` supplied to LAPACK.

Returns: Selected eigenvalues and, optionally, selected generalized eigenvectors.

◆ complex_hermitian_eigensystem()

template<uni20::LapackComplexReal Real>

ComplexHermitianEigensystem< Real > uni20::krylov::complex_hermitian_eigensystem	(	detail::ColumnMajorLapackMatrix< uni20::detail::standard_complex< Real > >	matrix,
		bool	compute_vectors,
		MatrixFill	triangle = `MatrixFill::Upper`
	)

Solve a dense complex Hermitian eigensystem through LAPACK heev.

Eigenvalues are returned in ascending order. If compute_vectors is true, eigenvectors is an n-by-n column-major matrix whose columns are normalized eigenvectors.

Template Parameters

Real	Underlying real precision.

Parameters

matrix	Complex Hermitian square matrix. Only the selected triangle is read.
compute_vectors	Whether to compute eigenvectors.
triangle	Triangle of `matrix` supplied to LAPACK.

Returns: Eigenvalues and, optionally, eigenvectors.

◆ complex_hermitian_eigensystem_divide_and_conquer()

template<uni20::LapackComplexReal Real>

ComplexHermitianEigensystem< Real > uni20::krylov::complex_hermitian_eigensystem_divide_and_conquer	(	detail::ColumnMajorLapackMatrix< uni20::detail::standard_complex< Real > >	matrix,
		bool	compute_vectors,
		MatrixFill	triangle = `MatrixFill::Upper`
	)

Solve a dense complex Hermitian eigensystem through LAPACK heevd.

Uses the divide-and-conquer Hermitian eigensystem driver. Eigenvalues are returned in ascending order.

Template Parameters

Real	Underlying real precision.

Parameters

matrix	Complex Hermitian square matrix. Only the selected triangle is read.
compute_vectors	Whether to compute eigenvectors.
triangle	Triangle of `matrix` supplied to LAPACK.

Returns: Eigenvalues and, optionally, eigenvectors.

◆ complex_hermitian_eigensystem_index_range()

template<uni20::LapackComplexReal Real>

ComplexHermitianEigensystem< Real > uni20::krylov::complex_hermitian_eigensystem_index_range	(	detail::ColumnMajorLapackMatrix< uni20::detail::standard_complex< Real > >	matrix,
		std::size_t	first_index,
		std::size_t	last_index,
		bool	compute_vectors,
		MatrixFill	triangle = `MatrixFill::Upper`
	)

Solve a selected dense complex Hermitian eigensystem through LAPACK heevr.

Selects eigenpairs by 0-based inclusive index in ascending eigenvalue order.

Template Parameters

Real	Underlying real precision.

Parameters

matrix	Complex Hermitian square matrix. Only the selected triangle is read.
first_index	First 0-based eigenvalue index to compute.
last_index	Last 0-based eigenvalue index to compute.
compute_vectors	Whether to compute selected eigenvectors.
triangle	Triangle of `matrix` supplied to LAPACK.

Returns: Selected eigenvalues and, optionally, selected eigenvectors.

◆ complex_nonsymmetric_arnoldi_standard()

template<uni20::LapackComplexReal Real, typename Vector , KrylovMatrixFreeOperator< Vector, uni20::complex< Real > > Ops>

NonsymmetricEigenResult< Real, Vector > uni20::krylov::complex_nonsymmetric_arnoldi_standard	(	Ops &	ops,
		Vector const &	initial,
		NonsymmetricEigenParams< Real > const &	params
	)

Run a restarted complex nonsymmetric Arnoldi solver.

This Phase 5 entry point keeps the Arnoldi basis and matrix-free operations complex, extracts complex projected Ritz values, and uses complex Schur thick restarts when the initial projected subspace does not satisfy the convergence criterion.

Template Parameters

Real	Underlying real precision.
Vector	Opaque vector type.
Ops	Matrix-free operator/vector operation type.

Parameters

ops	Matrix-free vector operations and operator application.
initial	Initial residual/start vector.
params	Solver parameters. Tolerances and residuals use `Real`.

Returns: Selected Ritz values, residual bounds, optional complex Ritz vectors, and diagnostics.

◆ complex_nonsymmetric_eigensystem()

template<uni20::LapackReal Real>

ComplexNonsymmetricEigensystem< Real > uni20::krylov::complex_nonsymmetric_eigensystem	(	uni20::DenseMatrix< uni20::detail::standard_complex< Real > >	matrix,
		bool	compute_right_vectors
	)

Solve a dense real nonsymmetric eigensystem through LAPACK geevx.

The expert driver balances the matrix (‘BALANC='B’) and computes reciprocal condition estimates for both eigenvalues and right eigenvectors (SENSE='B'). It computes left and right eigenvectors internally because LAPACK needs them for these estimates. The input matrix is copied because LAPACK overwrites it. If right eigenvectors are requested, real eigenvectors are returned as complex vectors with zero imaginary part, and complex conjugate pairs are unpacked from LAPACK's adjacent real-vector representation. \tparam Real Real scalar type satisfyinguni20::LapackReal`.

Parameters

matrix	Real square matrix in column-major local storage.
compute_right_vectors	Whether to compute right eigenvectors.

Returns: Complex eigenvalues, optional right eigenvectors, balancing data, and reciprocal condition estimates.

◆ complex_schur()

template<uni20::LapackReal Real>

ComplexSchurDecomposition< Real > uni20::krylov::complex_schur	(	uni20::DenseMatrix< uni20::detail::standard_complex< Real > >	matrix,
		bool	compute_vectors
	)

Compute the complex Schur decomposition of a column-major DenseMatrix.

Template Parameters

Real	`float` or `double`.

Parameters

matrix	Complex square matrix in current Krylov local storage.
compute_vectors	Whether to compute Schur vectors.

Returns: Complex Schur form, optional Schur vectors, and eigenvalues.

◆ complex_schur_layout_right()

template<uni20::LapackReal Real>

RightComplexSchurDecomposition< Real > uni20::krylov::complex_schur_layout_right	(	RightMatrix< uni20::detail::standard_complex< Real > >	matrix,
		bool	compute_vectors
	)

Compute the complex Schur decomposition of a layout-right matrix through LAPACK gees.

The public wrapper accepts the prototype row-major RightMatrix and copies to a column-major DenseMatrix before calling Fortran LAPACK. Complex Schur blocks are all scalar 1x1 entries, so eigenvalues are the diagonal of the Schur form.

Template Parameters

Real	`float` or `double`.

Parameters

matrix	Complex square matrix in logical layout-right storage.
compute_vectors	Whether to compute Schur vectors.

Returns: Complex Schur form, optional Schur vectors, and eigenvalues.

◆ compress_complex_schur_arnoldi_restart()

template<uni20::LapackComplexReal Real, typename Vector , KrylovMatrixFreeOperator< Vector, uni20::complex< Real > > Ops>

ComplexSchurCompressedArnoldiFactorization< Real, Vector > uni20::krylov::compress_complex_schur_arnoldi_restart	(	Ops &	ops,
		ArnoldiFactorization< uni20::complex< Real >, Vector > const &	factorization,
		NonsymmetricEigenParams< Real > const &	params
	)

Compress a complex Arnoldi factorization through complex Schur reordering.

Computes the complex Schur form of the projected Hessenberg matrix, reorders selected Schur entries to the leading part, transforms the opaque Arnoldi basis by the Schur vectors, and returns the compressed Krylov-Schur relation A V_k = V_k T_k + f b^T.

Template Parameters

Real	Real precision whose `uni20::complex<Real>` has dense LAPACK coverage.
Vector	Opaque vector type.
Ops	Matrix-free vector operation provider.

Parameters

ops	Matrix-free vector operations.
factorization	Input Arnoldi factorization.
params	Nonsymmetric solve parameters used for Schur entry selection.

Returns: Compressed restart relation ready for Krylov-Schur expansion.

◆ compress_real_schur_arnoldi_restart()

template<uni20::LapackReal Scalar, typename Vector , KrylovMatrixFreeOperator< Vector, Scalar > Ops>

RealSchurCompressedArnoldiFactorization< Scalar, Vector > uni20::krylov::compress_real_schur_arnoldi_restart	(	Ops &	ops,
		ArnoldiFactorization< Scalar, Vector > const &	factorization,
		NonsymmetricEigenParams< Scalar > const &	params
	)

Compress an Arnoldi factorization through real Schur block reordering.

Computes the real Schur form of the projected Hessenberg matrix, reorders selected Schur blocks to the leading part, transforms the opaque Arnoldi basis by the Schur vectors, and returns the compressed Krylov-Schur relation A V_k = V_k T_k + f b^T.

Template Parameters

Scalar	Real scalar type satisfying `uni20::LapackReal`.
Vector	Opaque vector type.
Ops	Matrix-free vector operation provider.

Parameters

ops	Matrix-free vector operations.
factorization	Input Arnoldi factorization.
params	Nonsymmetric solve parameters used for block selection.

Returns: Compressed restart relation ready for Krylov-Schur expansion.

◆ compress_symmetric_lanczos_restart()

template<uni20::LapackRealOrComplex Scalar, typename Vector , KrylovMatrixFreeOperator< Vector, Scalar > Ops>

SymmetricCompressedLanczosFactorization< uni20::make_real_t< Scalar >, Vector > uni20::krylov::compress_symmetric_lanczos_restart	(	Ops &	ops,
		std::vector< Vector > const &	basis,
		Vector const &	residual,
		SymmetricQrShiftResult< uni20::make_real_t< Scalar > > const &	shifted,
		std::size_t	retained_count
	)

Compress a Lanczos factorization with the accumulated restart transform.

Given a factorization of order kplusp, it keeps the first retained_count columns of V Q, truncates the transformed tridiagonal projection, and updates the residual as r <- sigma_k r + beta_k (V Q)[:, k + 1], where sigma_k = Q(kplusp, k) and beta_k = T(k + 1, k). All application-space vector work goes through the matrix-free operation object.

Template Parameters

Scalar	Real scalar type.
Vector	Opaque vector type.
Ops	Matrix-free vector operation provider.

Parameters

ops	Matrix-free operation object.
basis	Original Lanczos basis vectors.
residual	Original residual vector.
shifted	Shifted projected factorization and accumulated transform.
retained_count	Number of transformed basis vectors to keep.

Returns: Compressed factorization ready for renewed basis expansion.

◆ copy()

template<typename Scalar >

void uni20::krylov::copy	(	std::span< Scalar >	destination,
		std::span< Scalar const >	source
	)

Copy a local dense vector.

Template Parameters

Scalar Element type.

Parameters

destination	Destination vector overwritten with `source`.
source	Source vector.

◆ copy_left_to_right()

template<typename Scalar >

RightMatrix< Scalar > uni20::krylov::copy_left_to_right ( uni20::DenseMatrix< Scalar > const & matrix )

Copy a column-major DenseMatrix into a row-major matrix.

Template Parameters

Scalar Element type.

Parameters

matrix Source column-major matrix.

Returns: Row-major matrix with the same logical entries.

◆ copy_right_to_left()

template<typename Scalar >

uni20::DenseMatrix< Scalar > uni20::krylov::copy_right_to_left ( RightMatrix< Scalar > const & matrix )

Copy a row-major matrix into a column-major DenseMatrix.

Template Parameters

Scalar Element type.

Parameters

matrix Source row-major matrix.

Returns: Column-major matrix with the same logical entries.

◆ default_adaptive_exponential_krylov_dimension()

int uni20::krylov::default_adaptive_exponential_krylov_dimension ( std::size_t problem_dimension )

inline

Return the default adaptive Krylov projection cap for exponential actions.

◆ default_complex_pair_tolerance() [1/2]

template<uni20::Real Scalar>
requires requires { uni20::numeric_limits<Scalar>::epsilon(); }

Scalar uni20::krylov::default_complex_pair_tolerance ( )

Default imaginary-part tolerance for deciding whether a real Arnoldi Ritz value is real.

Template Parameters

Scalar Real scalar type.

Returns: A conservative scale multiplier derived from machine precision.

◆ default_complex_pair_tolerance() [2/2]

template<uni20::Real Scalar>

Scalar uni20::krylov::default_complex_pair_tolerance ( Scalar const & exemplar )

Default reality tolerance using the exemplar's working precision.

Exceptions

std::logic_error If a runtime-precision exemplar is exact or unset.

◆ default_exponential_krylov_dimension()

int uni20::krylov::default_exponential_krylov_dimension ( std::size_t problem_dimension )

inline

Return the default Krylov projection dimension for exponential actions.

◆ default_nonsymmetric_krylov_dimension()

int uni20::krylov::default_nonsymmetric_krylov_dimension	(	int	eigenvalue_count,
		std::size_t	problem_dimension
	)

inline

Return the default nonsymmetric Arnoldi subspace dimension.

Nonsymmetric restarts are more sensitive to nonnormality and complex conjugate Schur blocks than Hermitian Lanczos restarts, so the native default is deliberately roomier than the minimal symmetric/Hermitian default: ncv = min(n, max(20, 6*nev + 8)).

◆ default_nonsymmetric_retained_ritz_count()

int uni20::krylov::default_nonsymmetric_retained_ritz_count	(	int	eigenvalue_count,
		int	krylov_dimension
	)

inline

Return the default nonsymmetric retained Schur dimension.

The native nonsymmetric default keeps guard vectors by retaining max(2*nev, nev+4) Schur dimensions, clamped below ncv. This improves robustness on nonnormal and clustered cases while still leaving room for restart filtering.

◆ default_symmetric_krylov_dimension()

int uni20::krylov::default_symmetric_krylov_dimension	(	int	eigenvalue_count,
		std::size_t	problem_dimension
	)

inline

Return the default symmetric/Hermitian Krylov subspace dimension.

The default follows the common implicitly restarted Krylov guideline ncv = min(n, max(20, 2*nev + 1)). Callers may still set krylov_dimension explicitly when memory is tight or a hard clustered problem needs a larger search space.

◆ default_symmetric_retained_ritz_count()

int uni20::krylov::default_symmetric_retained_ritz_count	(	int	eigenvalue_count,
		int	krylov_dimension
	)

inline

Return the default symmetric/Hermitian retained Ritz count.

Hermitian problems default to a minimal implicit restart: keep exactly the requested invariant subspace and use the remaining Krylov directions as shifts. Clustered problems may set retained_ritz_count larger to get thick-restart guard vectors.

◆ dot()

template<uni20::Real Scalar>

Scalar uni20::krylov::dot	(	std::span< Scalar const >	x,
		std::span< Scalar const >	y
	)

Compute a BLAS-style real dot product.

Template Parameters

Scalar Real scalar type.

Parameters

x	Left vector.
y	Right vector.

Returns: Sum of x[i] * y[i].

◆ dotc()

template<typename Scalar >

Scalar uni20::krylov::dotc	(	std::span< Scalar const >	x,
		std::span< Scalar const >	y
	)

Compute a conjugated dot product.

Template Parameters

Scalar Element type.

Parameters

x	Left vector, conjugated elementwise for complex scalar types.
y	Right vector.

Returns: Sum of conj(x[i]) * y[i].

◆ dotu()

template<typename Scalar >

Scalar uni20::krylov::dotu	(	std::span< Scalar const >	x,
		std::span< Scalar const >	y
	)

Compute an unconjugated dot product.

Template Parameters

Scalar Element type.

Parameters

x	Left vector.
y	Right vector.

Returns: Sum of x[i] * y[i].

◆ effective_exponential_krylov_dimension()

template<uni20::Real Scalar>

int uni20::krylov::effective_exponential_krylov_dimension	(	KrylovExponentialParams< Scalar > const &	params,
		std::size_t	problem_dimension
	)

Return the effective Krylov projection dimension for exponential actions.

◆ effective_nonsymmetric_krylov_dimension()

template<uni20::Real Scalar>

int uni20::krylov::effective_nonsymmetric_krylov_dimension	(	NonsymmetricEigenParams< Scalar > const &	params,
		std::size_t	problem_dimension
	)

Return the effective nonsymmetric Arnoldi subspace dimension.

◆ effective_nonsymmetric_retained_ritz_count()

template<uni20::Real Scalar>

int uni20::krylov::effective_nonsymmetric_retained_ritz_count	(	NonsymmetricEigenParams< Scalar > const &	params,
		int	krylov_dimension
	)

Return the effective nonsymmetric retained Ritz or Schur count.

◆ effective_symmetric_krylov_dimension()

template<uni20::Real Scalar>

int uni20::krylov::effective_symmetric_krylov_dimension	(	SymmetricEigenParams< Scalar > const &	params,
		std::size_t	problem_dimension
	)

Return the effective symmetric/Hermitian Krylov subspace dimension.

◆ effective_symmetric_retained_ritz_count()

template<uni20::Real Scalar>

int uni20::krylov::effective_symmetric_retained_ritz_count	(	SymmetricEigenParams< Scalar > const &	params,
		int	krylov_dimension
	)

Return the effective symmetric/Hermitian retained Ritz count.

◆ expand_complex_schur_arnoldi_restart()

template<uni20::LapackComplexReal Real, typename Vector , KrylovMatrixFreeOperator< Vector, uni20::complex< Real > > Ops>

ArnoldiFactorization< uni20::complex< Real >, Vector > uni20::krylov::expand_complex_schur_arnoldi_restart	(	Ops &	ops,
		ComplexSchurCompressedArnoldiFactorization< Real, Vector > const &	compressed,
		std::size_t	target_step_count,
		Real	breakdown_tolerance = `Real{}`
	)

Expand a compressed complex Schur Arnoldi relation back to a requested dimension.

The compressed relation has the form A V_k = V_k T_k + f b^T. If f is nonzero, this routine normalizes it as the next Arnoldi vector, writes ||f|| b^T into the next projected row, and resumes full reorthogonalized Arnoldi expansion until target_step_count is reached or happy breakdown occurs.

Template Parameters

Real	Real precision whose `uni20::complex<Real>` has dense LAPACK coverage.
Vector	Opaque vector type.
Ops	Matrix-free vector operation provider.

Parameters

ops	Matrix-free vector operations and operator application.
compressed	Compressed Schur restart relation.
target_step_count	Desired projected dimension after expansion.
breakdown_tolerance	Absolute residual norm threshold for happy breakdown.

Returns: Expanded Arnoldi relation.

◆ expand_real_schur_arnoldi_restart()

template<uni20::LapackReal Scalar, typename Vector , KrylovMatrixFreeOperator< Vector, Scalar > Ops>

ArnoldiFactorization< Scalar, Vector > uni20::krylov::expand_real_schur_arnoldi_restart	(	Ops &	ops,
		RealSchurCompressedArnoldiFactorization< Scalar, Vector > const &	compressed,
		std::size_t	target_step_count,
		Scalar	breakdown_tolerance = `Scalar{}`
	)

Expand a compressed real Schur Arnoldi relation back to a requested dimension.

The compressed relation has the form A V_k = V_k T_k + f b^T. If f is nonzero, this routine normalizes it as the next Arnoldi vector, writes ||f|| b^T into the next projected row, and resumes full reorthogonalized Arnoldi expansion until target_step_count is reached or happy breakdown occurs. The returned factorization uses the same storage convention as arnoldi_factorize.

Template Parameters

Scalar	Real scalar type satisfying `uni20::LapackReal`.
Vector	Opaque vector type.
Ops	Matrix-free vector operation provider.

Parameters

ops	Matrix-free vector operations and operator application.
compressed	Compressed Schur restart relation.
target_step_count	Desired projected dimension after expansion.
breakdown_tolerance	Absolute residual norm threshold for happy breakdown.

Returns: Expanded Arnoldi relation.

◆ exponential_adaptive_acceptance_enabled()

template<uni20::Real Scalar>

bool uni20::krylov::exponential_adaptive_acceptance_enabled ( KrylovExponentialParams< Scalar > const & params )

◆ extract_arnoldi_ritz()

template<uni20::LapackReal Scalar, typename Vector >

ArnoldiRitzExtraction< Scalar > uni20::krylov::extract_arnoldi_ritz	(	ArnoldiFactorization< Scalar, Vector > const &	factorization,
		Scalar	complex_pair_tolerance = `Scalar{}`,
		Scalar	scale = `Scalar{1}`
	)

Extract projected Ritz values and residual estimates from a real Arnoldi factorization.

The Arnoldi relation gives A V_m y - theta V_m y = h_{m+1,m} (e_m^T y) v_{m+1} before happy breakdown. This routine solves the small projected Hessenberg problem and reports that residual bound without touching the caller-owned vector backend.

Template Parameters

Scalar	Real scalar type satisfying `uni20::LapackReal`.
Vector	Opaque vector type used by the Arnoldi basis.

Parameters

factorization	Arnoldi factorization to extract from.
complex_pair_tolerance	Relative imaginary-part tolerance for reality classification.
scale	Optional projected-problem scale for reality classification.

Returns: Complex Ritz values, projected right eigenvectors, residual bounds, and reality classifications.

◆ extract_complex_arnoldi_ritz()

template<uni20::LapackComplex Complex, typename Vector >

ArnoldiRitzExtraction< uni20::make_real_t< Complex > > uni20::krylov::extract_complex_arnoldi_ritz	(	ArnoldiFactorization< Complex, Vector > const &	factorization,
		uni20::make_real_t< Complex >	complex_pair_tolerance = `uni20::make_real_t<Complex>{}`,
		uni20::make_real_t< Complex >	scale = `uni20::make_real_t<Complex>{1}`
	)

Extract projected Ritz values from a complex Arnoldi factorization.

The Arnoldi relation gives A V_m y - theta V_m y = h_{m+1,m} (e_m^T y) v_{m+1} before happy breakdown. This routine solves the small complex projected Hessenberg problem and reports that residual bound.

Template Parameters

Complex	Complex scalar type with dense LAPACK coverage, deduced from the factorization.
Vector	Opaque vector type used by the Arnoldi basis.

Parameters

factorization	Complex Arnoldi factorization to extract from.
complex_pair_tolerance	Relative imaginary-part tolerance for reality classification.
scale	Optional projected-problem scale for reality classification.

Returns: Complex Ritz values, projected right eigenvectors, residual bounds, and reality classifications.

◆ gemv()

template<typename Scalar >

void uni20::krylov::gemv	(	std::span< Scalar >	y,
		Scalar const &	alpha,
		uni20::DenseMatrix< Scalar > const &	matrix,
		std::span< Scalar const >	x,
		Scalar const &	beta,
		MatrixTranspose	transpose = `MatrixTranspose::None`
	)

Compute a dense matrix-vector product.

Template Parameters

Scalar Element type.

Parameters

y	Output vector updated as `y = alpha * op(matrix) * x + beta * y`.
alpha	Scale factor for the matrix-vector product.
matrix	Input matrix.
x	Input vector.
beta	Scale factor for the original `y` value.
transpose	Matrix operation applied before multiplication.

◆ gerc()

template<typename Scalar >

void uni20::krylov::gerc	(	uni20::DenseMatrix< Scalar > &	matrix,
		Scalar const &	alpha,
		std::span< Scalar const >	x,
		std::span< Scalar const >	y
	)

Apply a conjugated dense rank-one update.

Template Parameters

Scalar Element type.

Parameters

matrix	Matrix updated as `matrix += alpha * x * conj(y)^T`.
alpha	Scale factor.
x	Left vector.
y	Right vector, conjugated elementwise for complex scalar types.

◆ geru()

template<typename Scalar >

void uni20::krylov::geru	(	uni20::DenseMatrix< Scalar > &	matrix,
		Scalar const &	alpha,
		std::span< Scalar const >	x,
		std::span< Scalar const >	y
	)

Apply a dense rank-one update.

Template Parameters

Scalar Element type.

Parameters

matrix	Matrix updated as `matrix += alpha * x * y^T`.
alpha	Scale factor.
x	Left vector.
y	Right vector.

◆ gmres_refine_with_info()

template<class WSpace , class ISpace , class RSpace , class IOp , class ROp , class P , class Transfer >

auto uni20::krylov::gmres_refine_with_info	(	GmresRefinementContext< WSpace, ISpace, RSpace, IOp, ROp, P, Transfer > &	context,
		typename WSpace::vector_type &	x,
		typename RSpace::vector_type const &	b,
		GmresRefinementOptions< typename ISpace::real_type, typename RSpace::real_type > const &	options = `{}`
	)

Refine a supplied iterate using matrix-free GMRES correction solves.

Evaluates r=b-A*x in the residual space, normalizes r before converting to the inner space, solves for a normalized correction, rescales it in residual precision and updates x in working precision. Inner convergence alone never implies outer convergence. An inner failure stops refinement without applying its partial correction. Candidate iterates are published only after successful evaluation of their true residual. No automatic precision escalation occurs.

Precondition: Spaces represent the same mathematical vector space. Transfer operations preserve its symmetry and layout semantics; there is no dense fallback.

Exceptions

std::invalid_argument For incompatible dimensions or invalid options. Callback exceptions propagate.

◆ gmres_with_info() [1/4]

template<RealOrComplex S, class Ops , class Op , class P , class V >
requires KrylovVectorOps<Ops, V, S> && KrylovOperator<Op, V> && KrylovPreconditioner<P, V>

GmresInfo< make_real_t< S > > uni20::krylov::gmres_with_info	(	Ops &	ops,
		Op &	op,
		P &	preconditioner,
		V &	x,
		V const &	b,
		GmresOptions< make_real_t< S > > const &	options,
		Precision	precision
	)

Solve A*x=b with restarted GMRES and a fixed right preconditioner.

Preserves the supplied initial guess. Acceptance uses a recomputed residual, ||b-A*x|| <= max(atol, rtol*||b||). Modified Gram-Schmidt uses selective second-pass orthogonalization. Only the Arnoldi basis is stored; a combined basis correction is preconditioned once per cycle. Callbacks own backend selection, memory placement and any higher-precision operator/preconditioner arithmetic. Scalar returns and Hessenberg arithmetic use precision; vector operations must implement that working precision without discarding vector metadata.

Precondition: Vector copies allocate independent values. Norms/inner products must expose nonfinite data as nonfinite results. The operator and preconditioner are fixed linear maps throughout the call.

Exceptions

std::invalid_argument For incompatible dimensions, invalid tolerances or a zero restart length. Callback exceptions propagate.

Returns: Numerical status and diagnostics of the last published iterate.

◆ gmres_with_info() [2/4]

template<RealOrComplex S, class Ops , class Op , class P , class V >
requires (!has_runtime_precision_v<S>) && KrylovVectorOps<Ops, V, S> && KrylovOperator<Op, V> && KrylovPreconditioner<P, V>

auto uni20::krylov::gmres_with_info	(	Ops &	ops,
		Op &	op,
		P &	preconditioner,
		V &	x,
		V const &	b,
		GmresOptions< make_real_t< S > > const &	options = `{}`
	)

Fixed-scalar GMRES using the scalar's native precision.

◆ gmres_with_info() [3/4]

template<RealOrComplex S, class Ops , class Op , class V >
requires KrylovVectorOps<Ops, V, S> && KrylovOperator<Op, V>

auto uni20::krylov::gmres_with_info	(	Ops &	ops,
		Op &	op,
		V &	x,
		V const &	b,
		GmresOptions< make_real_t< S > > const &	options,
		Precision	p
	)

GMRES without preconditioning, at explicit scalar precision.

◆ gmres_with_info() [4/4]

template<RealOrComplex S, class Ops , class Op , class V >
requires (!has_runtime_precision_v<S>) && KrylovVectorOps<Ops, V, S> && KrylovOperator<Op, V>

auto uni20::krylov::gmres_with_info	(	Ops &	ops,
		Op &	op,
		V &	x,
		V const &	b,
		GmresOptions< make_real_t< S > > const &	options = `{}`
	)

Fixed-scalar GMRES without preconditioning.

◆ hermitian_krylov_exponential_action()

template<uni20::LapackScalar Scalar, typename TimeScalar , typename Vector , KrylovMatrixFreeOperator< Vector, Scalar > Ops>
requires (std::constructible_from<uni20::make_real_t<Scalar>, TimeScalar> || (uni20::Complex<Scalar> && std::constructible_from<Scalar, TimeScalar>))

KrylovExponentialResult< Scalar, Vector > uni20::krylov::hermitian_krylov_exponential_action	(	Ops &	ops,
		Vector const &	initial,
		TimeScalar	time,
		KrylovExponentialParams< uni20::make_real_t< Scalar > > const &	params = `{}`
	)

Approximate exp(t A) v for a Hermitian matrix-free operator.

Builds a Lanczos projection T_m, evaluates exp(t T_m) with the current CPU dense Padé backend, and returns ||v|| V_m exp(t T_m) e_1. When relative_tolerance > 0, pure imaginary time is treated as the usual unitary exp(-i t H) case and the routine stops at the first dimension whose direct defect-integral estimate satisfies the relative target. Other Hermitian nonexpansive actions may opt in with assume_nonexpansive. error_estimate is the time-integrated Hermitian defect estimate, while endpoint_defect_estimate keeps the final-time sampled defect. Real vector spaces require real time. Complex vector spaces also accept complex time, so exp(-i t H) v can use the Hermitian Lanczos projection without hiding the phase in the operator.

Template Parameters

Scalar	Real or complex vector-field scalar type with projected dense LAPACK coverage.
Vector	Opaque vector type.
Ops	Matrix-free operation provider.

◆ krylov_space() [1/2]

template<RealOrComplex S, class Ops , class V >
requires (!has_runtime_precision_v<S>) && requires(Ops& ops, V&& source) { ops.prototype_view(std::forward<V>(source)); }

auto uni20::krylov::krylov_space	(	Ops &	ops,
		V &&	source
	)

Bind fixed-scalar vector operations at native precision.

◆ krylov_space() [2/2]

template<RealOrComplex S, class Ops , class V >
requires requires(Ops& ops, V&& source) { ops.prototype_view(std::forward<V>(source)); }

auto uni20::krylov::krylov_space	(	Ops &	ops,
		V &&	source,
		Precision	p
	)

Bind vector operations and an allocation prototype to finite precision.

ops.prototype_view(source) produces a descriptor retained by value. The adapter accepts temporary borrowed views, rejects temporary owners, and allocates owning work vectors from the descriptor.

◆ lacpy()

template<typename Scalar >

void uni20::krylov::lacpy	(	uni20::DenseMatrix< Scalar > &	destination,
		uni20::DenseMatrix< Scalar > const &	source,
		MatrixFill	fill
	)

Copy a selected region of a local dense matrix.

Template Parameters

Scalar Element type.

Parameters

destination	Destination matrix with the same shape as `source`.
source	Source matrix.
fill	Region to copy.

◆ laset()

template<typename Scalar >

void uni20::krylov::laset	(	uni20::DenseMatrix< Scalar > &	matrix,
		Scalar const &	diagonal,
		Scalar const &	off_diagonal,
		MatrixFill	fill
	)

Set a selected region of a local dense matrix.

Template Parameters

Scalar Element type.

Parameters

matrix	Matrix to update.
diagonal	Value written on the diagonal.
off_diagonal	Value written away from the diagonal.
fill	Region to set.

◆ map_symmetric_transformed_eigenvalue()

template<uni20::Real Scalar>

Scalar uni20::krylov::map_symmetric_transformed_eigenvalue	(	Scalar	transformed_value,
		SymmetricSpectralTransformOptions< Scalar > const &	options
	)

Map a native symmetric transformed Ritz value back to the original eigenvalue scale.

Template Parameters

Scalar Real or complex vector-field scalar type.

Parameters

transformed_value	Ritz value of the caller-supplied `OP`.
options	Spectral transform represented by `OP`.

Returns: Eigenvalue in the original symmetric problem.

◆ metric_inner_product_or_apply()

template<typename Scalar , typename Vector , typename MetricOps , typename BOps , typename InnerOps >

Scalar uni20::krylov::metric_inner_product_or_apply	(	MetricOps &	metric_ops,
		BOps &	b_ops,
		InnerOps &	inner_ops,
		Vector const &	x,
		Vector const &	y,
		Vector &	scratch
	)

Evaluate a metric inner product, using a backend override when available.

If metric_ops has metric_inner_product(x, y), that operation is used directly. Otherwise the default computes B*y into scratch with b_ops and returns inner_ops.inner_product(x, scratch).

◆ nonsymmetric_krylov_exponential_action()

template<uni20::LapackScalar Scalar, typename TimeScalar , typename Vector , KrylovMatrixFreeOperator< Vector, Scalar > Ops>
requires (std::constructible_from<uni20::make_real_t<Scalar>, TimeScalar> || (uni20::Complex<Scalar> && std::constructible_from<Scalar, TimeScalar>))

KrylovExponentialResult< Scalar, Vector > uni20::krylov::nonsymmetric_krylov_exponential_action	(	Ops &	ops,
		Vector const &	initial,
		TimeScalar	time,
		KrylovExponentialParams< uni20::make_real_t< Scalar > > const &	params = `{}`
	)

Approximate exp(t A) v for a nonsymmetric matrix-free operator.

Builds a full-reorthogonalized Arnoldi projection H_m, evaluates exp(t H_m) with the current CPU dense Padé backend, and returns ||v|| V_m exp(t H_m) e_1. This covers real and complex nonsymmetric operators without inspecting vector storage. Complex vector spaces also accept complex time.

Template Parameters

Scalar	Real or complex vector-field scalar type with projected dense LAPACK coverage.
Vector	Opaque vector type.
Ops	Matrix-free operation provider.

◆ norm_or_inner_product()

template<typename Scalar , typename Vector , typename Ops >

uni20::make_real_t< Scalar > uni20::krylov::norm_or_inner_product	(	Ops &	ops,
		Vector const &	x
	)

Evaluate a vector norm, using a backend override when available.

The fallback computes sqrt(max(0, real(inner_product(x, x)))). The result type is the real counterpart of Scalar, so complex vector spaces return a real norm without duplicating scalar-type metadata in the matrix-free interface.

◆ plan_symmetric_restart()

template<uni20::Real Scalar>

SymmetricRestartPlan< Scalar > uni20::krylov::plan_symmetric_restart	(	SymmetricEigenParams< Scalar > const &	params,
		std::vector< Scalar > const &	values,
		std::vector< Scalar > const &	residual_bounds,
		int	converged_count
	)

Adjust restart dimensions for implicit-restart edge cases.

The restart protects unwanted Ritz values whose residual estimates are exactly zero, because they indicate a split-off leading block that should not be removed by shifting. It also increases the retained block when some, but not all, requested Ritz values have converged, which prevents stagnation from repeatedly applying too many shifts.

Template Parameters

Scalar Real scalar type.

Parameters

params	Solver parameters defining `nev`, optional `n_keep`, and selector.
values	Projected Ritz values.
residual_bounds	Ritz residual estimates with the same length as `values`.
converged_count	Number of currently converged requested Ritz values.

Returns: Adjusted restart dimensions.

◆ projected_departure_from_normality()

template<uni20::RealOrComplex Scalar>

uni20::make_real_t< Scalar > uni20::krylov::projected_departure_from_normality ( uni20::DenseMatrix< Scalar > const & matrix )

Measure departure from normality of a projected matrix.

Returns ||H^* H - H H^*||_F / max(1, ||H||_F^2). The value is zero for normal projected matrices and grows as the projected Hessenberg matrix becomes non-normal. This is a diagnostic of the local projected problem only; it does not inspect the original matrix-free operator.

Template Parameters

Scalar Real or complex scalar type.

Parameters

matrix Square projected matrix.

Returns: Scale-normalized Frobenius departure from normality.

◆ real_bidiagonal_left_orthogonal_factor()

template<uni20::LapackReal Scalar>

detail::ColumnMajorLapackMatrix< Scalar > uni20::krylov::real_bidiagonal_left_orthogonal_factor ( RealBidiagonalReduction< Scalar > const & reduction )

Materialize the left orthogonal factor Q from a real bidiagonal reduction.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

reduction Compact bidiagonal reduction returned by real_bidiagonal_reduction.

Returns: Full Q factor satisfying A = Q * B * P^T.

◆ real_bidiagonal_reduction()

template<uni20::LapackReal Scalar>

RealBidiagonalReduction< Scalar > uni20::krylov::real_bidiagonal_reduction ( detail::ColumnMajorLapackMatrix< Scalar > matrix )

Reduce a dense real matrix to bidiagonal form through LAPACK gebrd.

Returns the clean bidiagonal matrix plus LAPACK's compact Householder storage. For an m-by-n input, the decomposition is A = Q * B * P^T, where B is upper bidiagonal if m >= n and lower bidiagonal otherwise.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix Real matrix to reduce.

Returns: Bidiagonal form plus compact Householder reflectors.

◆ real_bidiagonal_right_orthogonal_factor_transpose()

template<uni20::LapackReal Scalar>

detail::ColumnMajorLapackMatrix< Scalar > uni20::krylov::real_bidiagonal_right_orthogonal_factor_transpose ( RealBidiagonalReduction< Scalar > const & reduction )

Materialize the right transposed factor P^T from a real bidiagonal reduction.

This follows LAPACK ‘orgbr('P’), which generatesP^T, notP. \tparam Scalar Real scalar type satisfyinguni20::LapackReal. \param reduction Compact bidiagonal reduction returned by real_bidiagonal_reduction. \return FullP^Tfactor satisfyingA = Q * B * P^T`.

◆ real_bidiagonal_singular_values() [1/2]

template<uni20::LapackReal Scalar>

std::vector< Scalar > uni20::krylov::real_bidiagonal_singular_values ( RealBidiagonalReduction< Scalar > const & reduction )

Compute singular values from a compact real bidiagonal reduction.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

reduction Compact bidiagonal reduction returned by real_bidiagonal_reduction.

Returns: Singular values in decreasing order.

◆ real_bidiagonal_singular_values() [2/2]

template<uni20::LapackReal Scalar>

std::vector< Scalar > uni20::krylov::real_bidiagonal_singular_values	(	std::vector< Scalar >	diagonal,
		std::vector< Scalar >	offdiagonal,
		bool	upper = `true`
	)

Compute singular values of a real bidiagonal matrix through LAPACK bdsqr.

The singular values are returned in decreasing order. The offdiagonal has length diagonal.size() - 1 unless the diagonal is empty.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

diagonal	Main diagonal of the bidiagonal matrix.
offdiagonal	Superdiagonal entries if `upper` is true, otherwise subdiagonal entries.
upper	Whether the bidiagonal matrix is upper or lower bidiagonal.

Returns: Singular values in decreasing order.

◆ real_bidiagonal_svd() [1/2]

template<uni20::LapackReal Scalar>

RealBidiagonalSvd< Scalar > uni20::krylov::real_bidiagonal_svd	(	RealBidiagonalReduction< Scalar > const &	reduction,
		bool	compute_vectors
	)

Compute the SVD from a compact real bidiagonal reduction.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

reduction	Compact bidiagonal reduction returned by `real_bidiagonal_reduction`.
compute_vectors	Whether to compute explicit singular vectors.

Returns: Bidiagonal singular values and, optionally, explicit singular vectors.

◆ real_bidiagonal_svd() [2/2]

template<uni20::LapackReal Scalar>

RealBidiagonalSvd< Scalar > uni20::krylov::real_bidiagonal_svd	(	std::vector< Scalar >	diagonal,
		std::vector< Scalar >	offdiagonal,
		bool	compute_vectors,
		bool	upper = `true`
	)

Compute the SVD of a real bidiagonal matrix through LAPACK bdsdc.

The singular values are returned in decreasing order. If compute_vectors is true, u and vt are n-by-n column-major matrices satisfying B = u * diag(s) * vt, where B is the input upper or lower bidiagonal matrix.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

diagonal	Main diagonal of the bidiagonal matrix.
offdiagonal	Superdiagonal entries if `upper` is true, otherwise subdiagonal entries.
compute_vectors	Whether to compute explicit singular vectors.
upper	Whether the bidiagonal matrix is upper or lower bidiagonal.

Returns: Singular values and, optionally, explicit singular vectors.

◆ real_bidiagonal_svd_index_range() [1/2]

template<uni20::LapackReal Scalar>

RealBidiagonalSvd< Scalar > uni20::krylov::real_bidiagonal_svd_index_range	(	RealBidiagonalReduction< Scalar > const &	reduction,
		std::size_t	first_index,
		std::size_t	last_index,
		bool	compute_vectors
	)

Compute a selected-index SVD from a compact real bidiagonal reduction.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

reduction	Compact bidiagonal reduction returned by `real_bidiagonal_reduction`.
first_index	First 0-based decreasing singular-value index to compute.
last_index	Last 0-based decreasing singular-value index to compute.
compute_vectors	Whether to compute explicit singular vectors.

Returns: Selected singular values and, optionally, explicit singular vectors.

◆ real_bidiagonal_svd_index_range() [2/2]

template<uni20::LapackReal Scalar>

RealBidiagonalSvd< Scalar > uni20::krylov::real_bidiagonal_svd_index_range	(	std::vector< Scalar >	diagonal,
		std::vector< Scalar >	offdiagonal,
		std::size_t	first_index,
		std::size_t	last_index,
		bool	compute_vectors,
		bool	upper = `true`
	)

Compute a selected-index SVD of a real bidiagonal matrix through LAPACK bdsvdx.

Selects singular triplets by 0-based inclusive index in decreasing singular-value order. If compute_vectors is true, u is n-by-m and vt is m-by-n, where m is the number of selected singular values. The selected contribution is u * diag(s) * vt.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

diagonal	Main diagonal of the bidiagonal matrix.
offdiagonal	Superdiagonal entries if `upper` is true, otherwise subdiagonal entries.
first_index	First 0-based decreasing singular-value index to compute.
last_index	Last 0-based decreasing singular-value index to compute.
compute_vectors	Whether to compute explicit singular vectors.
upper	Whether the bidiagonal matrix is upper or lower bidiagonal.

Returns: Selected singular values and, optionally, explicit singular vectors.

◆ real_compact_lq_factorization()

template<uni20::LapackReal Scalar>

RealCompactLqFactorization< Scalar > uni20::krylov::real_compact_lq_factorization ( detail::ColumnMajorLapackMatrix< Scalar > matrix )

Compute a compact dense real LQ factorization through LAPACK gelqf.

Stores Householder reflectors and scalar factors in LAPACK's compact LQ representation. Use apply_real_lq_factor to apply the encoded orthogonal factor without explicitly materializing Q.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix Real matrix to factor.

Returns: Compact LQ reflector data and scalar factors.

◆ real_compact_ql_factorization()

template<uni20::LapackReal Scalar>

RealCompactQlFactorization< Scalar > uni20::krylov::real_compact_ql_factorization ( detail::ColumnMajorLapackMatrix< Scalar > matrix )

Compute a compact dense real QL factorization through LAPACK geqlf.

Stores Householder reflectors and scalar factors in LAPACK's compact QL representation. Use apply_real_ql_factor to apply the encoded orthogonal factor without explicitly materializing Q.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix Real matrix to factor.

Returns: Compact QL reflector data and scalar factors.

◆ real_compact_qr_factorization()

template<uni20::LapackReal Scalar>

RealCompactQrFactorization< Scalar > uni20::krylov::real_compact_qr_factorization ( detail::ColumnMajorLapackMatrix< Scalar > matrix )

Compute a compact dense real QR factorization through LAPACK geqrf.

Stores Householder reflectors and scalar factors in LAPACK's compact QR representation. Use apply_real_qr_factor to apply the encoded orthogonal factor without explicitly materializing Q.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix Real matrix to factor.

Returns: Compact QR reflector data and scalar factors.

◆ real_compact_rq_factorization()

template<uni20::LapackReal Scalar>

RealCompactRqFactorization< Scalar > uni20::krylov::real_compact_rq_factorization ( detail::ColumnMajorLapackMatrix< Scalar > matrix )

Compute a compact dense real RQ factorization through LAPACK gerqf.

Stores Householder reflectors and scalar factors in LAPACK's compact RQ representation. Use apply_real_rq_factor to apply the encoded orthogonal factor without explicitly materializing Q.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix Real matrix to factor.

Returns: Compact RQ reflector data and scalar factors.

◆ real_dense_inverse()

template<uni20::LapackReal Scalar>

detail::ColumnMajorLapackMatrix< Scalar > uni20::krylov::real_dense_inverse ( detail::ColumnMajorLapackMatrix< Scalar > matrix )

Invert a dense real square matrix through LAPACK getrf and getri.

The input matrix is copied by value and overwritten by LAPACK. The returned matrix stores the inverse.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix Real square matrix to invert.

Returns: Dense inverse matrix.

◆ real_dense_solve_linear_system()

template<uni20::LapackReal Scalar>

detail::ColumnMajorLapackMatrix< Scalar > uni20::krylov::real_dense_solve_linear_system	(	detail::ColumnMajorLapackMatrix< Scalar >	coefficients,
		detail::ColumnMajorLapackMatrix< Scalar >	right_hand_sides
	)

Solve a dense real linear system through LAPACK gesv.

Solves A * X = B for one or more dense right-hand sides. The coefficient and right-hand-side matrices are copied by value and may be overwritten by LAPACK. The returned matrix stores X.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

coefficients	Real square coefficient matrix.
right_hand_sides	Dense right-hand-side matrix with matching row count.

Returns: Dense solution matrix.

◆ real_divide_and_conquer_svd_least_squares()

template<uni20::LapackReal Scalar>

RealSvdLeastSquares< Scalar > uni20::krylov::real_divide_and_conquer_svd_least_squares	(	detail::ColumnMajorLapackMatrix< Scalar >	coefficients,
		detail::ColumnMajorLapackMatrix< Scalar >	right_hand_sides,
		Scalar	rcond = `Scalar{100} * uni20::numeric_limits<Scalar>::epsilon()`
	)

Solve a dense real divide-and-conquer SVD least-squares problem through LAPACK gelsd.

Solves A * X = B using a divide-and-conquer singular value decomposition. The returned singular values are sorted in descending order, and the numerical rank is determined by rcond.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

coefficients	Real rectangular coefficient matrix `A`.
right_hand_sides	Dense right-hand-side matrix `B` with matching row count.
rcond	Relative singular-value threshold used by LAPACK to determine rank.

Returns: Dense solution, numerical rank, and singular values of A.

◆ real_equilibration()

template<uni20::LapackReal Scalar>

RealEquilibration< Scalar > uni20::krylov::real_equilibration ( detail::ColumnMajorLapackMatrix< Scalar > matrix )

Compute dense real row/column equilibration factors through LAPACK geequ.

Returns row scales R and column scales C suitable for forming diag(R) * A * diag(C), along with LAPACK's scale-condition diagnostics and maximum absolute matrix entry.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix Real dense matrix.

Returns: Row and column scaling factors plus equilibration diagnostics.

◆ real_expert_linear_solve()

template<uni20::LapackReal Scalar>

RealExpertLinearSolve< Scalar > uni20::krylov::real_expert_linear_solve	(	detail::ColumnMajorLapackMatrix< Scalar >	coefficients,
		detail::ColumnMajorLapackMatrix< Scalar >	right_hand_sides
	)

Solve a dense real linear system with LAPACK gesvx diagnostics.

Solves A * X = B and returns the solution, LU factors, pivot metadata, reciprocal condition estimate, componentwise backward errors, forward error bounds, and equilibration/condition diagnostics reported by LAPACK. The current wrapper requests a fresh factorization (‘FACT='N’) and does not expose LAPACK's transpose or caller-supplied equilibration modes. \tparam Scalar Real scalar type satisfyinguni20::LapackReal`.

Parameters

coefficients	Real square coefficient matrix.
right_hand_sides	Dense right-hand-side matrix with matching row count.

Returns: Dense solution and LAPACK expert-driver diagnostics.

◆ real_general_band_equilibration()

template<uni20::LapackReal Scalar>

RealEquilibration< Scalar > uni20::krylov::real_general_band_equilibration ( RealGeneralBandMatrix< Scalar > band_matrix )

Compute real general-band row/column equilibration factors through LAPACK gbequ.

Returns row scales R and column scales C suitable for forming diag(R) * A * diag(C), along with LAPACK's scale-condition diagnostics and maximum absolute matrix entry. The band matrix must use the factor-storage layout returned by real_general_band_from_dense.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

band_matrix Real square general-band matrix in LAPACK factor-storage layout.

Returns: Row and column scaling factors plus equilibration diagnostics.

◆ real_general_band_expert_linear_solve()

template<uni20::LapackReal Scalar>

RealGeneralBandExpertLinearSolve< Scalar > uni20::krylov::real_general_band_expert_linear_solve	(	RealGeneralBandMatrix< Scalar >	coefficients,
		detail::ColumnMajorLapackMatrix< Scalar >	right_hand_sides,
		MatrixTranspose	transpose = `MatrixTranspose::None`
	)

Solve a real general-band linear system with LAPACK gbsvx diagnostics.

Solves op(A) * X = B and returns the solution, banded LU factors, pivot metadata, reciprocal condition estimate, componentwise backward errors, forward error bounds, and condition diagnostics reported by LAPACK. The current wrapper requests a fresh factorization (‘FACT='N’). \tparam Scalar Real scalar type satisfyinguni20::LapackReal`.

Parameters

coefficients	Real square general-band coefficient matrix.
right_hand_sides	Dense right-hand-side matrix with matching row count.
transpose	Matrix operation applied to the coefficient matrix.

Returns: Banded solution and LAPACK expert-driver diagnostics.

◆ real_general_band_factorization()

template<uni20::LapackReal Scalar>

RealGeneralBandFactorization< Scalar > uni20::krylov::real_general_band_factorization ( RealGeneralBandMatrix< Scalar > matrix )

Compute a real general-band LU factorization through LAPACK gbtrf.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix Real square general-band matrix in LAPACK factor-storage layout.

Returns: Banded LU factors and row-pivot metadata.

◆ real_general_band_from_dense()

template<uni20::LapackReal Scalar>

RealGeneralBandMatrix< Scalar > uni20::krylov::real_general_band_from_dense	(	detail::ColumnMajorLapackMatrix< Scalar > const &	matrix,
		std::size_t	lower_bandwidth,
		std::size_t	upper_bandwidth
	)

Pack a dense real square matrix into LAPACK general-band factor storage.

Entries outside the requested band are dropped. The returned storage has 2 * lower_bandwidth + upper_bandwidth + 1 rows so it can be passed directly to gbsv, gbtrf, and gbtrs.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix	Real square matrix to pack.
lower_bandwidth	Number of stored subdiagonals.
upper_bandwidth	Number of stored superdiagonals.

Returns: General-band matrix in LAPACK factor-storage layout.

◆ real_general_band_matrix_norm()

template<uni20::LapackReal Scalar>

Scalar uni20::krylov::real_general_band_matrix_norm	(	RealGeneralBandMatrix< Scalar >	band_matrix,
		MatrixNorm	norm
	)

Compute a real general-band matrix norm through LAPACK langb.

The band matrix must use the factor-storage layout returned by real_general_band_from_dense. Only the original matrix band is referenced; fill rows reserved for factorization are ignored.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

band_matrix	Real general-band matrix in LAPACK factor-storage layout.
norm	Matrix norm to compute.

Returns: Requested band-matrix norm.

◆ real_general_band_one_norm_reciprocal_condition_number() [1/2]

template<uni20::LapackReal Scalar>

Scalar uni20::krylov::real_general_band_one_norm_reciprocal_condition_number	(	RealGeneralBandFactorization< Scalar > const &	factorization,
		Scalar	original_one_norm
	)

Estimate a real general-band one-norm reciprocal condition number through LAPACK gbcon.

Uses an existing banded LU factorization. The supplied matrix norm must be the one-norm of the original unfactored band matrix.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

factorization	Factors returned by `real_general_band_factorization`.
original_one_norm	One-norm of the original coefficient matrix.

Returns: Estimate of 1 / (||A||_1 * ||inv(A)||_1).

◆ real_general_band_one_norm_reciprocal_condition_number() [2/2]

template<uni20::LapackReal Scalar>

Scalar uni20::krylov::real_general_band_one_norm_reciprocal_condition_number ( RealGeneralBandMatrix< Scalar > matrix )

Estimate a real general-band one-norm reciprocal condition number.

Computes the band one-norm, factors the matrix with LAPACK gbtrf, and estimates the reciprocal condition number with LAPACK gbcon.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix Real square general-band matrix in LAPACK factor-storage layout.

Returns: Estimate of 1 / (||A||_1 * ||inv(A)||_1).

◆ real_general_band_power_of_two_equilibration()

template<uni20::LapackReal Scalar>

RealEquilibration< Scalar > uni20::krylov::real_general_band_power_of_two_equilibration ( RealGeneralBandMatrix< Scalar > band_matrix )

Compute power-of-two real general-band equilibration factors through LAPACK gbequb.

Returns row and column scale factors constrained to radix powers by LAPACK, along with scale-condition diagnostics and the maximum absolute matrix entry.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

band_matrix Real square general-band matrix in LAPACK factor-storage layout.

Returns: Row and column scaling factors plus equilibration diagnostics.

◆ real_general_band_refined_solve()

template<uni20::LapackReal Scalar>

RealRefinedLinearSolve< Scalar > uni20::krylov::real_general_band_refined_solve	(	RealGeneralBandMatrix< Scalar >	coefficients,
		detail::ColumnMajorLapackMatrix< Scalar >	right_hand_sides,
		MatrixTranspose	transpose = `MatrixTranspose::None`
	)

Solve and refine a real general-band system through LAPACK gbrfs.

Computes a banded LU factorization, solves op(A) * X = B, then applies iterative refinement and returns LAPACK's forward and backward error estimates. The original band matrix is retained for the refinement residual calculation.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

coefficients	Real square general-band coefficient matrix.
right_hand_sides	Dense right-hand-side matrix with matching row count.
transpose	Matrix operation applied to the coefficient matrix.

Returns: Refined solution and LAPACK error estimates.

◆ real_general_band_solve() [1/2]

template<uni20::LapackReal Scalar>

detail::ColumnMajorLapackMatrix< Scalar > uni20::krylov::real_general_band_solve	(	RealGeneralBandFactorization< Scalar > const &	factorization,
		detail::ColumnMajorLapackMatrix< Scalar >	right_hand_sides,
		MatrixTranspose	transpose = `MatrixTranspose::None`
	)

Solve using an existing real general-band LU factorization through LAPACK gbtrs.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

factorization	Factors returned by `real_general_band_factorization`.
right_hand_sides	Dense right-hand-side matrix with matching row count.
transpose	Matrix operation applied to the original coefficient matrix.

Returns: Dense solution matrix.

◆ real_general_band_solve() [2/2]

template<uni20::LapackReal Scalar>

detail::ColumnMajorLapackMatrix< Scalar > uni20::krylov::real_general_band_solve	(	RealGeneralBandMatrix< Scalar >	coefficients,
		detail::ColumnMajorLapackMatrix< Scalar >	right_hand_sides
	)

Solve a real general-band linear system through LAPACK gbsv.

Solves A * X = B for one or more dense right-hand sides. The band matrix must use LAPACK factor-storage layout.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

coefficients	Real square general-band coefficient matrix.
right_hand_sides	Dense right-hand-side matrix with matching row count.

Returns: Dense solution matrix.

◆ real_general_tridiagonal_expert_linear_solve()

template<uni20::LapackReal Scalar>

RealGeneralTridiagonalExpertLinearSolve< Scalar > uni20::krylov::real_general_tridiagonal_expert_linear_solve	(	RealGeneralTridiagonalMatrix< Scalar >	coefficients,
		detail::ColumnMajorLapackMatrix< Scalar >	right_hand_sides,
		MatrixTranspose	transpose = `MatrixTranspose::None`
	)

Solve a real general-tridiagonal linear system with LAPACK gtsvx diagnostics.

Solves op(A) * X = B and returns the solution, tridiagonal LU factors, pivot metadata, reciprocal condition estimate, componentwise backward errors, forward error bounds, and condition diagnostics reported by LAPACK. The current wrapper requests a fresh factorization (‘FACT='N’). \tparam Scalar Real scalar type satisfyinguni20::LapackReal`.

Parameters

coefficients	Real square general tridiagonal coefficient matrix.
right_hand_sides	Dense right-hand-side matrix with matching row count.
transpose	Matrix operation applied to the coefficient matrix.

Returns: Solution, factorization, and diagnostics.

◆ real_general_tridiagonal_factorization()

template<uni20::LapackReal Scalar>

RealGeneralTridiagonalFactorization< Scalar > uni20::krylov::real_general_tridiagonal_factorization ( RealGeneralTridiagonalMatrix< Scalar > matrix )

Compute a real general tridiagonal LU factorization through LAPACK gttrf.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix Real square general tridiagonal matrix.

Returns: Tridiagonal LU factors and row-pivot metadata.

◆ real_general_tridiagonal_from_dense()

template<uni20::LapackReal Scalar>

RealGeneralTridiagonalMatrix< Scalar > uni20::krylov::real_general_tridiagonal_from_dense ( detail::ColumnMajorLapackMatrix< Scalar > const & matrix )

Extract real general tridiagonal storage from a dense matrix.

The diagonal, first subdiagonal, and first superdiagonal are copied. Entries outside the tridiagonal band are ignored.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix Real square tridiagonal matrix.

Returns: General tridiagonal matrix storage suitable for LAPACK gtsv and gttrf.

◆ real_general_tridiagonal_one_norm()

template<uni20::LapackReal Scalar>

Scalar uni20::krylov::real_general_tridiagonal_one_norm ( RealGeneralTridiagonalMatrix< Scalar > const & matrix )

Compute the one-norm of a real general tridiagonal matrix.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix Real square general tridiagonal matrix.

Returns: Matrix one-norm.

◆ real_general_tridiagonal_one_norm_reciprocal_condition_number() [1/2]

template<uni20::LapackReal Scalar>

Scalar uni20::krylov::real_general_tridiagonal_one_norm_reciprocal_condition_number	(	RealGeneralTridiagonalFactorization< Scalar > const &	factorization,
		Scalar	original_one_norm
	)

Estimate a real general-tridiagonal one-norm reciprocal condition number through LAPACK gtcon.

Uses an existing tridiagonal LU factorization. The supplied matrix norm must be the one-norm of the original unfactored tridiagonal matrix.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

factorization	Factors returned by `real_general_tridiagonal_factorization`.
original_one_norm	One-norm of the original coefficient matrix.

Returns: Estimate of 1 / (||A||_1 * ||inv(A)||_1).

◆ real_general_tridiagonal_one_norm_reciprocal_condition_number() [2/2]

template<uni20::LapackReal Scalar>

Scalar uni20::krylov::real_general_tridiagonal_one_norm_reciprocal_condition_number ( RealGeneralTridiagonalMatrix< Scalar > matrix )

Estimate a real general-tridiagonal one-norm reciprocal condition number from a matrix.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix Real square general tridiagonal matrix.

Returns: Estimate of 1 / (||A||_1 * ||inv(A)||_1).

◆ real_general_tridiagonal_refined_solve()

template<uni20::LapackReal Scalar>

RealRefinedLinearSolve< Scalar > uni20::krylov::real_general_tridiagonal_refined_solve	(	RealGeneralTridiagonalMatrix< Scalar >	coefficients,
		detail::ColumnMajorLapackMatrix< Scalar >	right_hand_sides,
		MatrixTranspose	transpose = `MatrixTranspose::None`
	)

Solve and refine a real general-tridiagonal linear system through LAPACK gtrfs.

Computes an initial solution using gttrs, then refines it with gtrfs and returns LAPACK's forward and backward error estimates.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

coefficients	Real square general tridiagonal coefficient matrix.
right_hand_sides	Dense right-hand-side matrix with matching row count.
transpose	Matrix operation applied to the coefficient matrix.

Returns: Solution and error-bound diagnostics.

◆ real_general_tridiagonal_solve() [1/2]

template<uni20::LapackReal Scalar>

detail::ColumnMajorLapackMatrix< Scalar > uni20::krylov::real_general_tridiagonal_solve	(	RealGeneralTridiagonalFactorization< Scalar > const &	factorization,
		detail::ColumnMajorLapackMatrix< Scalar >	right_hand_sides,
		MatrixTranspose	transpose = `MatrixTranspose::None`
	)

Solve using an existing real general tridiagonal LU factorization through LAPACK gttrs.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

factorization	Factors returned by `real_general_tridiagonal_factorization`.
right_hand_sides	Dense right-hand-side matrix with matching row count.
transpose	Matrix operation applied to the original coefficient matrix.

Returns: Dense solution matrix.

◆ real_general_tridiagonal_solve() [2/2]

template<uni20::LapackReal Scalar>

detail::ColumnMajorLapackMatrix< Scalar > uni20::krylov::real_general_tridiagonal_solve	(	RealGeneralTridiagonalMatrix< Scalar >	coefficients,
		detail::ColumnMajorLapackMatrix< Scalar >	right_hand_sides
	)

Solve a real general tridiagonal linear system through LAPACK gtsv.

Solves A * X = B for one or more dense right-hand sides.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

coefficients	Real square general tridiagonal coefficient matrix.
right_hand_sides	Dense right-hand-side matrix with matching row count.

Returns: Dense solution matrix.

◆ real_generalized_hessenberg_reduction()

template<uni20::LapackReal Real>

RealGeneralizedHessenbergReduction< Real > uni20::krylov::real_generalized_hessenberg_reduction	(	detail::ColumnMajorLapackMatrix< Real >	matrix,
		detail::ColumnMajorLapackMatrix< Real >	upper_triangular_metric,
		bool	compute_vectors
	)

Reduce a real matrix pencil to generalized upper Hessenberg form through LAPACK gghrd.

LAPACK gghrd assumes the second matrix is already upper triangular. This helper preserves that boundary: callers that have a general second matrix should triangularize it explicitly before using this lower-level QZ building block. The reduction satisfies A = Q * H * Z^T and B = Q * T * Z^T when vectors are requested.

Template Parameters

Real	Real scalar type satisfying `uni20::LapackReal`.

Parameters

matrix	Real square matrix `A` in column-major local storage.
upper_triangular_metric	Real upper-triangular square matrix `B`.
compute_vectors	Whether to compute the left and right orthogonal transformations.

Returns: Generalized upper Hessenberg forms and optional orthogonal factors.

◆ real_generalized_hessenberg_schur()

template<uni20::LapackReal Real>

RealGeneralizedSchurDecomposition< Real > uni20::krylov::real_generalized_hessenberg_schur	(	detail::ColumnMajorLapackMatrix< Real >	hessenberg,
		detail::ColumnMajorLapackMatrix< Real >	upper_triangular_metric,
		bool	compute_vectors
	)

Compute generalized real Schur form from a generalized Hessenberg pencil through LAPACK hgeqz.

LAPACK hgeqz assumes the first matrix is upper Hessenberg and the second matrix is upper triangular. This helper preserves that boundary for projected QZ pipelines that have already called gghrd. When vectors are requested, the returned forms satisfy H = Q * S * Z^T and T = Q * P * Z^T.

Template Parameters

Real	Real scalar type satisfying `uni20::LapackReal`.

Parameters

hessenberg	Real upper-Hessenberg square matrix `H`.
upper_triangular_metric	Real upper-triangular square matrix `T`.
compute_vectors	Whether to compute left and right generalized Schur vectors.

Returns: Generalized Schur forms, optional Schur vectors, and projective eigenvalue data.

◆ real_generalized_nonsymmetric_balance()

template<uni20::LapackReal Real>

RealGeneralizedNonsymmetricBalance< Real > uni20::krylov::real_generalized_nonsymmetric_balance	(	detail::ColumnMajorLapackMatrix< Real >	matrix,
		detail::ColumnMajorLapackMatrix< Real >	metric,
		RealNonsymmetricBalanceJob	job = `RealNonsymmetricBalanceJob::Both`
	)

Balance a dense real nonsymmetric matrix pencil through LAPACK ggbal.

The input matrices are copied by value and overwritten with the balanced pencil. The returned balanced interval is zero-based and half-open; pass the full result to real_generalized_nonsymmetric_balance_backtransform_right_vectors to backtransform real right eigenvectors of the balanced pencil.

Template Parameters

Real	Real scalar type satisfying `uni20::LapackReal`.

Parameters

matrix	Real square matrix `A` in column-major local storage.
metric	Real square matrix `B` in column-major local storage.
job	Balancing operation to apply.

Returns: Balanced pencil, LAPACK left/right scale data, and balanced active interval.

◆ real_generalized_nonsymmetric_balance_backtransform_right_vectors()

template<uni20::LapackReal Real>

detail::ColumnMajorLapackMatrix< Real > uni20::krylov::real_generalized_nonsymmetric_balance_backtransform_right_vectors	(	detail::ColumnMajorLapackMatrix< Real >	vectors,
		RealGeneralizedNonsymmetricBalance< Real > const &	balance,
		RealNonsymmetricBalanceJob	job = `RealNonsymmetricBalanceJob::Both`
	)

Backtransform real right eigenvectors after generalized pencil balancing.

The input vectors use LAPACK's real storage convention, so complex conjugate pairs are still represented by adjacent real columns. Use this before unpacking those columns into complex vectors.

Template Parameters

Real	Real scalar type satisfying `uni20::LapackReal`.

Parameters

vectors	Real right eigenvector columns of the balanced pencil.
balance	Balancing metadata returned by `real_generalized_nonsymmetric_balance`.
job	The same balancing operation used for the pencil.

Returns: Real right eigenvector columns backtransformed to the original pencil.

◆ real_generalized_nonsymmetric_eigensystem()

template<uni20::LapackReal Real>

RealGeneralizedNonsymmetricEigensystem< Real > uni20::krylov::real_generalized_nonsymmetric_eigensystem	(	detail::ColumnMajorLapackMatrix< Real >	matrix,
		detail::ColumnMajorLapackMatrix< Real >	metric,
		bool	compute_right_vectors
	)

Solve a dense real generalized nonsymmetric eigensystem through LAPACK ggev.

Solves the matrix pencil A * x = lambda * B * x. LAPACK reports generalized eigenvalues as projective pairs alpha / beta; this wrapper preserves both alpha and beta and also stores the ratio in eigenvalues. If right eigenvectors are requested, real eigenvectors are returned as complex vectors with zero imaginary part, and complex conjugate pairs are unpacked from LAPACK's adjacent real-vector representation.

Template Parameters

Real	Real scalar type satisfying `uni20::LapackReal`.

Parameters

matrix	Real square matrix `A` in column-major local storage.
metric	Real square matrix `B` in column-major local storage.
compute_right_vectors	Whether to compute right eigenvectors.

Returns: Projective eigenvalue data and, optionally, right eigenvectors.

◆ real_generalized_nonsymmetric_expert_eigensystem()

template<uni20::LapackReal Real>

RealGeneralizedNonsymmetricExpertEigensystem< Real > uni20::krylov::real_generalized_nonsymmetric_expert_eigensystem	(	detail::ColumnMajorLapackMatrix< Real >	matrix,
		detail::ColumnMajorLapackMatrix< Real >	metric,
		bool	compute_right_vectors
	)

Solve a dense real generalized nonsymmetric eigensystem through LAPACK ggevx.

Solves the matrix pencil A * x = lambda * B * x with balancing (‘BALANC='B’) and reciprocal condition estimates for both eigenvalues and right eigenvectors (SENSE='B'). It computes left and right eigenvectors internally because LAPACK needs them for these estimates. The wrapper preserves LAPACK's projective alpha / betaeigenvalue representation and stores finite ratios ineigenvalues. \tparam Real Real scalar type satisfyinguni20::LapackReal. \param matrix Real square matrixAin column-major local storage. \param metric Real square matrixB` in column-major local storage.

Parameters

compute_right_vectors Whether to return right eigenvectors.

Returns: Projective eigenvalue data, optional right eigenvectors, balancing data, and reciprocal condition estimates.

◆ real_generalized_schur()

template<uni20::LapackReal Real>

RealGeneralizedSchurDecomposition< Real > uni20::krylov::real_generalized_schur	(	detail::ColumnMajorLapackMatrix< Real >	matrix,
		detail::ColumnMajorLapackMatrix< Real >	metric,
		bool	compute_vectors
	)

Compute the generalized real Schur decomposition of a matrix pencil through LAPACK gges.

Computes unsorted QZ form for A - lambda B. LAPACK overwrites matrix with the quasi-triangular generalized Schur form S and metric with the upper-triangular form T. Generalized eigenvalues are preserved as projective pairs alpha / beta, with finite ratios also stored in eigenvalues.

Template Parameters

Real	Real scalar type satisfying `uni20::LapackReal`.

Parameters

matrix	Real square matrix `A` in column-major local storage.
metric	Real square matrix `B` in column-major local storage.
compute_vectors	Whether to compute left and right generalized Schur vectors.

Returns: Generalized Schur forms, optional Schur vectors, and projective eigenvalue data.

◆ real_generalized_schur_condition_estimates()

template<uni20::LapackReal Real>

RealGeneralizedSchurConditionEstimates< Real > uni20::krylov::real_generalized_schur_condition_estimates ( RealGeneralizedSchurDecomposition< Real > decomposition )

Estimate generalized Schur eigenvalue and eigenvector conditioning through LAPACK tgsna.

Computes left and right generalized Schur eigenvectors internally with tgevc, then estimates reciprocal condition numbers for all eigenpairs of the generalized Schur pencil.

Template Parameters

Real	Real scalar type satisfying `uni20::LapackReal`.

Parameters

decomposition Generalized real Schur decomposition whose forms are used.

Returns: Reciprocal generalized eigenvalue and eigenvector condition estimates.

◆ real_generalized_schur_right_eigenvectors()

template<uni20::LapackReal Real>

RealGeneralizedSchurRightEigenvectors< Real > uni20::krylov::real_generalized_schur_right_eigenvectors ( RealGeneralizedSchurDecomposition< Real > decomposition )

Compute right eigenvectors of a generalized real Schur form through LAPACK tgevc.

The returned vectors solve S v = lambda T v for the generalized Schur forms S and T. They are not multiplied by the right Schur vectors, so callers that need eigenvectors of the original pencil must backtransform them separately.

Template Parameters

Real	Real scalar type satisfying `uni20::LapackReal`.

Parameters

decomposition Generalized real Schur decomposition whose forms are used.

Returns: Complex right eigenvectors of the generalized Schur form and the number computed.

◆ real_generalized_schur_selected_subspace()

template<uni20::LapackReal Real>

RealGeneralizedSchurSelectedSubspace< Real > uni20::krylov::real_generalized_schur_selected_subspace	(	RealGeneralizedSchurDecomposition< Real >	decomposition,
		std::vector< std::size_t > const &	selected_blocks
	)

Move selected generalized real Schur blocks to the leading deflating subspace and estimate conditioning.

Uses LAPACK tgsen with IJOB=4, returning lower bounds for the left/right projection quantities (PL, PR) and F-norm estimates of Difu and Difl. Selection is by real Schur block, so real-arithmetic 2x2 complex-conjugate blocks cannot be split.

Template Parameters

Real	Real scalar type satisfying `uni20::LapackReal`.

Parameters

decomposition	Generalized real Schur decomposition to reorder and diagnose.
selected_blocks	Block indices to move into the leading deflating subspace.

Returns: Reordered generalized Schur decomposition and selected-subspace diagnostics.

◆ real_generalized_symmetric_eigensystem()

template<uni20::LapackReal Scalar>

RealSymmetricEigensystem< Scalar > uni20::krylov::real_generalized_symmetric_eigensystem	(	detail::ColumnMajorLapackMatrix< Scalar >	matrix,
		detail::ColumnMajorLapackMatrix< Scalar >	metric,
		bool	compute_vectors,
		MatrixFill	triangle = `MatrixFill::Upper`
	)

Solve a dense type-1 generalized real symmetric eigensystem through LAPACK sygv.

Solves A * x = lambda * B * x, where A is real symmetric and B is real symmetric positive definite. Eigenvalues are returned in ascending order. If compute_vectors is true, eigenvectors is an n-by-n column-major matrix whose columns are normalized in the B metric.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix	Real symmetric square matrix `A`. Only the selected triangle is read.
metric	Real symmetric positive definite square matrix `B`.
compute_vectors	Whether to compute generalized eigenvectors.
triangle	Triangle of `matrix` and `metric` supplied to LAPACK.

Returns: Eigenvalues and, optionally, generalized eigenvectors.

◆ real_generalized_symmetric_eigensystem_divide_and_conquer()

template<uni20::LapackReal Scalar>

RealSymmetricEigensystem< Scalar > uni20::krylov::real_generalized_symmetric_eigensystem_divide_and_conquer	(	detail::ColumnMajorLapackMatrix< Scalar >	matrix,
		detail::ColumnMajorLapackMatrix< Scalar >	metric,
		bool	compute_vectors,
		MatrixFill	triangle = `MatrixFill::Upper`
	)

Solve a dense type-1 generalized real symmetric eigensystem through LAPACK sygvd.

Uses the divide-and-conquer generalized symmetric eigensystem driver. Solves A * x = lambda * B * x, where A is real symmetric and B is real symmetric positive definite. Eigenvalues are returned in ascending order. If compute_vectors is true, eigenvectors is an n-by-n column-major matrix whose columns are normalized in the B metric.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix	Real symmetric square matrix `A`. Only the selected triangle is read.
metric	Real symmetric positive definite square matrix `B`.
compute_vectors	Whether to compute generalized eigenvectors.
triangle	Triangle of `matrix` and `metric` supplied to LAPACK.

Returns: Eigenvalues and, optionally, generalized eigenvectors.

◆ real_generalized_symmetric_eigensystem_index_range()

template<uni20::LapackReal Scalar>

RealSymmetricEigensystem< Scalar > uni20::krylov::real_generalized_symmetric_eigensystem_index_range	(	detail::ColumnMajorLapackMatrix< Scalar >	matrix,
		detail::ColumnMajorLapackMatrix< Scalar >	metric,
		std::size_t	first_index,
		std::size_t	last_index,
		bool	compute_vectors,
		MatrixFill	triangle = `MatrixFill::Upper`
	)

Solve a selected dense type-1 generalized real symmetric eigensystem through LAPACK sygvx.

Selects eigenpairs by 0-based inclusive index in ascending eigenvalue order for A * x = lambda * B * x, where A is real symmetric and B is real symmetric positive definite. If compute_vectors is true, eigenvectors is an n-by-m column-major matrix whose columns are normalized in the B metric.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix	Real symmetric square matrix `A`. Only the selected triangle is read.
metric	Real symmetric positive definite square matrix `B`.
first_index	First 0-based eigenvalue index to compute.
last_index	Last 0-based eigenvalue index to compute.
compute_vectors	Whether to compute selected generalized eigenvectors.
triangle	Triangle of `matrix` and `metric` supplied to LAPACK.

Returns: Selected eigenvalues and, optionally, selected generalized eigenvectors.

◆ real_hessenberg_orthogonal_factor()

template<uni20::LapackReal Scalar>

detail::ColumnMajorLapackMatrix< Scalar > uni20::krylov::real_hessenberg_orthogonal_factor ( RealHessenbergReduction< Scalar > const & reduction )

Generate the orthogonal factor from a compact real Hessenberg reduction.

The returned matrix is the Q in A = Q * H * Q^T for the result returned by real_hessenberg_reduction.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

reduction Compact Hessenberg reduction returned by real_hessenberg_reduction.

Returns: Dense orthogonal matrix represented by the compact Householder data.

◆ real_hessenberg_reduction()

template<uni20::LapackReal Scalar>

RealHessenbergReduction< Scalar > uni20::krylov::real_hessenberg_reduction ( detail::ColumnMajorLapackMatrix< Scalar > matrix )

Reduce a dense real square matrix to upper Hessenberg form through LAPACK gehrd.

Returns both a clean upper Hessenberg matrix and the raw compact Householder storage needed by real_hessenberg_orthogonal_factor. The reduction satisfies A = Q * H * Q^T when Q is generated from the returned compact data.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix Real square matrix to reduce.

Returns: Upper Hessenberg form plus compact Householder reflectors.

◆ real_hessenberg_schur()

template<uni20::LapackReal Scalar>

RealSchurDecomposition< Scalar > uni20::krylov::real_hessenberg_schur	(	uni20::DenseMatrix< Scalar >	hessenberg,
		bool	compute_vectors
	)

Compute the real Schur decomposition of an upper Hessenberg matrix through LAPACK hseqr.

This helper does not reduce a dense matrix to Hessenberg form. It is intended for projected Arnoldi matrices that are already upper Hessenberg, avoiding an unnecessary gees reduction step.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

hessenberg	Upper Hessenberg matrix in column-major local storage.
compute_vectors	Whether to compute Schur vectors of the Hessenberg matrix.

Returns: Real Schur form, optional Schur vectors, eigenvalues, and block metadata.

◆ real_least_squares()

template<uni20::LapackReal Scalar>

detail::ColumnMajorLapackMatrix< Scalar > uni20::krylov::real_least_squares	(	detail::ColumnMajorLapackMatrix< Scalar >	coefficients,
		detail::ColumnMajorLapackMatrix< Scalar >	right_hand_sides
	)

Solve a dense real least-squares or minimum-norm problem through LAPACK gels.

Solves A * X = B in the least-squares sense for overdetermined systems and the minimum-norm sense for underdetermined systems. The returned matrix has A.cols() rows and B.cols() columns.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

coefficients	Real rectangular coefficient matrix `A`.
right_hand_sides	Dense right-hand-side matrix `B` with matching row count.

Returns: Dense solution matrix X.

◆ real_lq_factorization()

template<uni20::LapackReal Scalar>

RealLqFactorization< Scalar > uni20::krylov::real_lq_factorization ( detail::ColumnMajorLapackMatrix< Scalar > matrix )

Compute a reduced dense real LQ factorization through LAPACK gelqf and orglq.

Returns A = L * Q, where Q has orthonormal rows and min(rows, cols) rows. L has min(rows, cols) columns.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix Real matrix to factor.

Returns: Reduced LQ factors.

◆ real_lu_factorization()

template<uni20::LapackReal Scalar>

RealLuFactorization< Scalar > uni20::krylov::real_lu_factorization ( detail::ColumnMajorLapackMatrix< Scalar > matrix )

Compute a dense real LU factorization through LAPACK getrf.

Factorizes a square matrix as P * A = L * U. Pivot rows are returned as 0-based indices.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix Real square matrix to factor.

Returns: LU factors and row-pivot metadata.

◆ real_lu_one_norm_reciprocal_condition_number()

template<uni20::LapackReal Scalar>

Scalar uni20::krylov::real_lu_one_norm_reciprocal_condition_number	(	RealLuFactorization< Scalar > const &	factorization,
		Scalar	original_one_norm
	)

Estimate the one-norm reciprocal condition number from an LU factorization.

Uses LAPACK gecon on an existing LU factorization. The supplied matrix norm must be the one-norm of the original unfactored matrix.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

factorization	LU factors returned by `real_lu_factorization`.
original_one_norm	One-norm of the original coefficient matrix.

Returns: Estimate of 1 / (||A||_1 * ||inv(A)||_1).

◆ real_lu_solve()

template<uni20::LapackReal Scalar>

detail::ColumnMajorLapackMatrix< Scalar > uni20::krylov::real_lu_solve	(	RealLuFactorization< Scalar > const &	factorization,
		detail::ColumnMajorLapackMatrix< Scalar >	right_hand_sides,
		MatrixTranspose	transpose = `MatrixTranspose::None`
	)

Solve using an existing dense real LU factorization through LAPACK getrs.

Solves op(A) * X = B, where A is represented by the supplied LU factorization.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

factorization	LU factors returned by `real_lu_factorization`.
right_hand_sides	Dense right-hand-side matrix with matching row count.
transpose	Matrix operation applied to the original coefficient matrix.

Returns: Dense solution matrix.

◆ real_matrix_norm()

template<uni20::LapackReal Scalar>

Scalar uni20::krylov::real_matrix_norm	(	detail::ColumnMajorLapackMatrix< Scalar >	matrix,
		MatrixNorm	norm
	)

◆ real_nonsymmetric_arnoldi_restarted_standard()

template<uni20::LapackReal Scalar, typename Vector , KrylovMatrixFreeOperator< Vector, Scalar > Ops>

NonsymmetricEigenResult< Scalar, Vector > uni20::krylov::real_nonsymmetric_arnoldi_restarted_standard	(	Ops &	ops,
		Vector const &	initial,
		NonsymmetricEigenParams< Scalar > const &	params
	)

Run a restarted native real nonsymmetric Arnoldi solver.

This is a Krylov-Schur style real Arnoldi restart loop. Each cycle expands a real Arnoldi factorization to krylov_dimension, checks Ritz convergence, reorders the projected real Schur form to retain wanted Schur blocks, compresses the opaque basis through matrix-free axpy operations, and resumes expansion from the compressed relation. Complex wanted Ritz values are tolerated while unconverged, then reported through the configured real-mode policy.

Template Parameters

Vector	Opaque vector type.
Ops	Matrix-free operator/vector operation type.

Parameters

ops	Matrix-free vector operations and operator application.
initial	Initial residual/start vector.
params	Solver parameters.

Returns: Selected Ritz values, residual bounds, optional real Ritz vectors, and diagnostics.

◆ real_nonsymmetric_arnoldi_standard()

template<uni20::LapackReal Scalar, typename Vector , KrylovMatrixFreeOperator< Vector, Scalar > Ops>

NonsymmetricEigenResult< Scalar, Vector > uni20::krylov::real_nonsymmetric_arnoldi_standard	(	Ops &	ops,
		Vector const &	initial,
		NonsymmetricEigenParams< Scalar > const &	params
	)

Run the first native non-restarted real nonsymmetric Arnoldi solver.

This Phase 4 entry point keeps the Arnoldi basis and matrix-free operations real, extracts complex projected Ritz values, and reports when wanted Ritz values require complex arithmetic or a future real-Schur two-plane output path.

Template Parameters

Vector	Opaque vector type.
Ops	Matrix-free operator/vector operation type.

Parameters

ops	Matrix-free vector operations and operator application.
initial	Initial residual/start vector.
params	Solver parameters.

Returns: Selected Ritz values, residual bounds, optional real Ritz vectors, and diagnostics.

◆ real_nonsymmetric_balance()

template<uni20::LapackReal Real>

RealNonsymmetricBalance< Real > uni20::krylov::real_nonsymmetric_balance	(	detail::ColumnMajorLapackMatrix< Real >	matrix,
		RealNonsymmetricBalanceJob	job = `RealNonsymmetricBalanceJob::Both`
	)

Reorder a layout-right complex Schur decomposition by moving selected entries to the front.

Complex Schur form contains only scalar 1x1 diagonal entries. leading_order names entries in the input decomposition's current order. Entries not named are left after the requested prefix in their original relative order.

Template Parameters

Real	`float` or `double`.

Parameters

decomposition	Complex Schur decomposition to reorder.
leading_order	Input entry indices to move to the leading positions.

Returns: Reordered complex Schur decomposition.

◆ real_nonsymmetric_balance_backtransform_right_vectors()

template<uni20::LapackReal Real>

detail::ColumnMajorLapackMatrix< Real > uni20::krylov::real_nonsymmetric_balance_backtransform_right_vectors	(	detail::ColumnMajorLapackMatrix< Real >	vectors,
		RealNonsymmetricBalance< Real > const &	balance,
		RealNonsymmetricBalanceJob	job = `RealNonsymmetricBalanceJob::Both`
	)

Backtransform real right eigenvectors after real_nonsymmetric_balance.

The input vectors use LAPACK's real storage convention, so complex conjugate pairs are still represented by adjacent real columns. Use this before unpacking those columns into complex vectors.

Template Parameters

Real	Real scalar type satisfying `uni20::LapackReal`.

Parameters

vectors	Real right eigenvector columns of the balanced matrix.
balance	Balancing metadata returned by `real_nonsymmetric_balance`.
job	The same balancing operation used for the matrix.

Returns: Real right eigenvector columns backtransformed to the original matrix.

◆ real_nonsymmetric_eigensystem()

template<uni20::LapackReal Real>

RealNonsymmetricEigensystem< Real > uni20::krylov::real_nonsymmetric_eigensystem	(	uni20::DenseMatrix< Real >	matrix,
		bool	compute_right_vectors
	)

◆ real_nonsymmetric_expert_eigensystem()

template<uni20::LapackReal Real>

RealNonsymmetricExpertEigensystem< Real > uni20::krylov::real_nonsymmetric_expert_eigensystem	(	detail::ColumnMajorLapackMatrix< Real >	matrix,
		bool	compute_right_vectors
	)

Solve a dense real nonsymmetric eigensystem through LAPACK geev.

The input matrix is copied because LAPACK overwrites it. Eigenvalues are returned in LAPACK order. If right eigenvectors are requested, real eigenvectors are returned as complex vectors with zero imaginary part, and complex conjugate pairs are unpacked from LAPACK's adjacent real-vector representation.

Template Parameters

Real	Real scalar type satisfying `uni20::LapackReal`.

Parameters

matrix	Real square matrix in column-major local storage.
compute_right_vectors	Whether to compute right eigenvectors.

Returns: Complex eigenvalues and, optionally, right eigenvectors.

◆ real_one_norm_reciprocal_condition_number()

template<uni20::LapackReal Scalar>

Scalar uni20::krylov::real_one_norm_reciprocal_condition_number ( detail::ColumnMajorLapackMatrix< Scalar > matrix )

Estimate the one-norm reciprocal condition number of a dense real matrix.

Computes the one-norm, factors the matrix with LAPACK getrf, and estimates the reciprocal condition number with LAPACK gecon.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix Real square matrix.

Returns: Estimate of 1 / (||A||_1 * ||inv(A)||_1).

◆ real_pivoted_cholesky_factorization()

template<uni20::LapackReal Scalar>

RealPivotedCholeskyFactorization< Scalar > uni20::krylov::real_pivoted_cholesky_factorization	(	detail::ColumnMajorLapackMatrix< Scalar >	matrix,
		Scalar	tolerance = `Scalar{-1}`,
		MatrixFill	triangle = `MatrixFill::Upper`
	)

Compute a dense real pivoted Cholesky factorization through LAPACK pstrf.

Computes a complete-pivoting Cholesky factorization of a symmetric positive semidefinite matrix. For upper storage, LAPACK returns P^T A P = U^T U; for lower storage, it returns P^T A P = L L^T. A nonnegative tolerance is used as the stopping threshold; a negative value requests LAPACK's default n * eps * max(diag(A)) threshold.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix	Real symmetric positive semidefinite square matrix.
tolerance	Pivot stopping threshold, or negative for LAPACK's default.
triangle	Triangle of `matrix` supplied to LAPACK.

Returns: Pivoted Cholesky factors, 0-based pivot order, rank, and rank-deficiency flag.

◆ real_pivoted_qr_factorization()

template<uni20::LapackReal Scalar>

RealPivotedQrFactorization< Scalar > uni20::krylov::real_pivoted_qr_factorization ( detail::ColumnMajorLapackMatrix< Scalar > matrix )

Compute a reduced dense real pivoted QR factorization through LAPACK geqp3.

Returns A * P = Q * R, where Q has orthonormal columns and min(rows, cols) columns. pivot_columns[j] is the original 0-based column index stored in pivoted column j of A * P.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix Real matrix to factor.

Returns: Reduced pivoted QR factors and column pivot order.

◆ real_ql_factorization()

template<uni20::LapackReal Scalar>

RealQlFactorization< Scalar > uni20::krylov::real_ql_factorization ( detail::ColumnMajorLapackMatrix< Scalar > matrix )

Compute a reduced dense real QL factorization through LAPACK geqlf and orgql.

Returns A = Q * L, where Q has orthonormal columns and min(rows, cols) columns. L has min(rows, cols) rows.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix Real matrix to factor.

Returns: Reduced QL factors.

◆ real_qr_factorization()

template<uni20::LapackReal Scalar>

RealQrFactorization< Scalar > uni20::krylov::real_qr_factorization ( detail::ColumnMajorLapackMatrix< Scalar > matrix )

Compute a reduced dense real QR factorization through LAPACK geqrf and orgqr.

Returns A = Q * R, where Q has orthonormal columns and min(rows, cols) columns. R has min(rows, cols) rows.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix Real matrix to factor.

Returns: Reduced QR factors.

◆ real_rank_revealing_least_squares()

template<uni20::LapackReal Scalar>

RealRankRevealingLeastSquares< Scalar > uni20::krylov::real_rank_revealing_least_squares	(	detail::ColumnMajorLapackMatrix< Scalar >	coefficients,
		detail::ColumnMajorLapackMatrix< Scalar >	right_hand_sides,
		Scalar	rcond = `Scalar{100} * uni20::numeric_limits<Scalar>::epsilon()`
	)

Solve a dense real rank-revealing least-squares problem through LAPACK gelsy.

Solves A * X = B using complete orthogonal factorization with column pivoting. The returned pivot order is 0-based: entry j gives the original column placed at pivoted position j.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

coefficients	Real rectangular coefficient matrix `A`.
right_hand_sides	Dense right-hand-side matrix `B` with matching row count.
rcond	Relative threshold used by LAPACK to determine the effective rank.

Returns: Dense solution, numerical rank, and column pivot order.

◆ real_refined_linear_solve()

template<uni20::LapackReal Scalar>

RealRefinedLinearSolve< Scalar > uni20::krylov::real_refined_linear_solve	(	detail::ColumnMajorLapackMatrix< Scalar >	coefficients,
		detail::ColumnMajorLapackMatrix< Scalar >	right_hand_sides,
		MatrixTranspose	transpose = `MatrixTranspose::None`
	)

Solve and refine a dense real linear system through LAPACK gerfs.

Computes an LU factorization, solves op(A) * X = B, then applies iterative refinement and returns LAPACK's forward and backward error estimates. The original coefficient matrix is retained for the refinement residual calculation.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

coefficients	Real square coefficient matrix.
right_hand_sides	Dense right-hand-side matrix with matching row count.
transpose	Matrix operation applied to the coefficient matrix.

Returns: Refined solution and LAPACK error estimates.

◆ real_rq_factorization()

template<uni20::LapackReal Scalar>

RealRqFactorization< Scalar > uni20::krylov::real_rq_factorization ( detail::ColumnMajorLapackMatrix< Scalar > matrix )

Compute a reduced dense real RQ factorization through LAPACK gerqf and orgrq.

Returns A = R * Q, where Q has orthonormal rows and min(rows, cols) rows. R has min(rows, cols) columns.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix Real matrix to factor.

Returns: Reduced RQ factors.

◆ real_schur()

template<uni20::LapackReal Scalar>

RealSchurDecomposition< Scalar > uni20::krylov::real_schur	(	uni20::DenseMatrix< Scalar >	matrix,
		bool	compute_vectors
	)

Compute the real Schur decomposition of a column-major DenseMatrix.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix	Real square matrix in current Krylov local storage.
compute_vectors	Whether to compute Schur vectors.

Returns: Real Schur form, optional Schur vectors, eigenvalues, and block metadata.

◆ real_schur_condition_estimates()

template<uni20::LapackReal Scalar>

RealSchurConditionEstimates< Scalar > uni20::krylov::real_schur_condition_estimates ( RealSchurDecomposition< Scalar > decomposition )

Estimate Schur-form eigenvalue and eigenvector conditioning through LAPACK trsna.

The returned arrays are aligned with decomposition.eigenvalues. This helper asks LAPACK for all eigenpairs and computes the required real-packed left and right Schur eigenvectors internally.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

decomposition Real Schur decomposition whose Schur form is used.

Returns: Reciprocal eigenvalue and right-eigenvector condition estimates.

◆ real_schur_layout_right()

template<uni20::LapackReal Scalar>

RightRealSchurDecomposition< Scalar > uni20::krylov::real_schur_layout_right	(	RightMatrix< Scalar >	matrix,
		bool	compute_vectors
	)

Solve a real symmetric tridiagonal eigensystem through LAPACK stevd.

Uses the divide-and-conquer symmetric tridiagonal eigensystem driver. diagonal has length n; subdiagonal has length n - 1. Eigenvalues are returned in ascending order. If compute_vectors is true, eigenvectors is an n-by-n column-major matrix whose columns are the normalized eigenvectors.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

diagonal	Main diagonal of the tridiagonal matrix.
subdiagonal	Subdiagonal/superdiagonal entries.
compute_vectors	Whether to compute eigenvectors.

Returns: Eigenvalues and, optionally, eigenvectors.

◆ real_schur_right_eigenvectors()

template<uni20::LapackReal Scalar>

RealSchurRightEigenvectors< Scalar > uni20::krylov::real_schur_right_eigenvectors ( RealSchurDecomposition< Scalar > decomposition )

Compute right eigenvectors of a real Schur form through LAPACK trevc.

The returned vectors are eigenvectors of decomposition.schur_form. They are not multiplied by the Schur-vector matrix, so callers that need eigenvectors of the original matrix must backtransform them separately.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

decomposition Real Schur decomposition whose Schur form is used.

Returns: Complex right eigenvectors of the Schur form and the number computed.

◆ real_schur_selected_subspace()

template<uni20::LapackReal Scalar>

RealSchurSelectedSubspace< Scalar > uni20::krylov::real_schur_selected_subspace	(	RealSchurDecomposition< Scalar >	decomposition,
		std::vector< std::size_t > const &	selected_blocks
	)

Compute the real Schur decomposition of a layout-right matrix through LAPACK gees.

The public wrapper accepts the prototype row-major RightMatrix used by the mdspan-facing LAPACK layer. Internally it copies to a column-major DenseMatrix before calling Fortran LAPACK, so the returned Schur vectors correspond to the logical input matrix rather than its transpose.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

◆ real_singular_value_decomposition()

template<uni20::LapackReal Scalar>

RealSingularValueDecomposition< Scalar > uni20::krylov::real_singular_value_decomposition	(	detail::ColumnMajorLapackMatrix< Scalar >	matrix,
		bool	compute_vectors
	)

Compute a dense real singular value decomposition through LAPACK gesvd.

Singular values are returned in descending order. If compute_vectors is true, this computes the full decomposition A = U * Sigma * VT, with U stored in left_singular_vectors and VT stored in right_singular_vectors_transpose.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix	Real matrix to decompose.
compute_vectors	Whether to compute full left and right singular vectors.

Returns: Singular values and, optionally, full singular-vector matrices.

◆ real_singular_value_decomposition_divide_and_conquer()

template<uni20::LapackReal Scalar>

RealSingularValueDecomposition< Scalar > uni20::krylov::real_singular_value_decomposition_divide_and_conquer	(	detail::ColumnMajorLapackMatrix< Scalar >	matrix,
		bool	compute_vectors
	)

Compute a dense real singular value decomposition through LAPACK gesdd.

Uses the divide-and-conquer SVD driver. Singular values are returned in descending order. If compute_vectors is true, this computes the full decomposition A = U * Sigma * VT, with U stored in left_singular_vectors and VT stored in right_singular_vectors_transpose.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix	Real matrix to decompose.
compute_vectors	Whether to compute full left and right singular vectors.

Returns: Singular values and, optionally, full singular-vector matrices.

◆ real_singular_value_decomposition_index_range()

template<uni20::LapackReal Scalar>

RealSingularValueDecomposition< Scalar > uni20::krylov::real_singular_value_decomposition_index_range	(	detail::ColumnMajorLapackMatrix< Scalar >	matrix,
		std::size_t	first_index,
		std::size_t	last_index,
		bool	compute_vectors
	)

Compute a selected-index dense real SVD through LAPACK gesvdx.

Selects singular triplets by 0-based inclusive index in decreasing singular-value order. If compute_vectors is true, this computes the selected contribution A_selected = U * Sigma * VT, with U stored in left_singular_vectors and VT stored in right_singular_vectors_transpose.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix	Real matrix to decompose.
first_index	First 0-based decreasing singular-value index to compute.
last_index	Last 0-based decreasing singular-value index to compute.
compute_vectors	Whether to compute selected left and right singular vectors.

Returns: Selected singular values and, optionally, selected singular-vector matrices.

◆ real_svd_least_squares()

template<uni20::LapackReal Scalar>

RealSvdLeastSquares< Scalar > uni20::krylov::real_svd_least_squares	(	detail::ColumnMajorLapackMatrix< Scalar >	coefficients,
		detail::ColumnMajorLapackMatrix< Scalar >	right_hand_sides,
		Scalar	rcond = `Scalar{100} * uni20::numeric_limits<Scalar>::epsilon()`
	)

Solve a dense real SVD-based least-squares problem through LAPACK gelss.

Solves A * X = B using a singular value decomposition. The returned singular values are sorted in descending order, and the numerical rank is determined by rcond.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

coefficients	Real rectangular coefficient matrix `A`.
right_hand_sides	Dense right-hand-side matrix `B` with matching row count.
rcond	Relative singular-value threshold used by LAPACK to determine rank.

Returns: Dense solution, numerical rank, and singular values of A.

◆ real_sylvester_solve()

template<uni20::LapackReal Scalar>

RealSylvesterSolution< Scalar > uni20::krylov::real_sylvester_solve	(	detail::ColumnMajorLapackMatrix< Scalar >	left,
		detail::ColumnMajorLapackMatrix< Scalar >	right,
		detail::ColumnMajorLapackMatrix< Scalar >	rhs,
		int	sign = `1`,
		MatrixTranspose	transpose_left = `MatrixTranspose::None`,
		MatrixTranspose	transpose_right = `MatrixTranspose::None`
	)

Solve a dense real Sylvester equation through LAPACK trsyl.

Solves op(A) * X + sign * X * op(B) = scale * C. LAPACK overwrites C with X; this wrapper returns the solution, the scale factor, and whether LAPACK perturbed nearly common eigenvalues while solving.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

left	Real square matrix `A`.
right	Real square matrix `B`.
rhs	Right-hand side matrix `C` with dimensions `A.rows()` by `B.rows()`.
sign	Equation sign, either `+1` or `-1`.
transpose_left	Operation applied to `A`.
transpose_right	Operation applied to `B`.

Returns: Dense Sylvester solution, scale factor, and perturbation diagnostic.

◆ real_symmetric_eigensystem()

template<uni20::LapackReal Scalar>

RealSymmetricEigensystem< Scalar > uni20::krylov::real_symmetric_eigensystem	(	detail::ColumnMajorLapackMatrix< Scalar >	matrix,
		bool	compute_vectors,
		MatrixFill	triangle = `MatrixFill::Upper`
	)

Solve a dense real symmetric eigensystem through LAPACK syev.

Eigenvalues are returned in ascending order. If compute_vectors is true, eigenvectors is an n-by-n column-major matrix whose columns are the normalized eigenvectors.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix	Real symmetric square matrix. Only the selected triangle is read.
compute_vectors	Whether to compute eigenvectors.
triangle	Triangle of `matrix` supplied to LAPACK.

Returns: Eigenvalues and, optionally, eigenvectors.

◆ real_symmetric_eigensystem_divide_and_conquer()

template<uni20::LapackReal Scalar>

RealSymmetricEigensystem< Scalar > uni20::krylov::real_symmetric_eigensystem_divide_and_conquer	(	detail::ColumnMajorLapackMatrix< Scalar >	matrix,
		bool	compute_vectors,
		MatrixFill	triangle = `MatrixFill::Upper`
	)

Solve a dense real symmetric eigensystem through LAPACK syevd.

Uses the divide-and-conquer symmetric eigensystem driver. Eigenvalues are returned in ascending order. If compute_vectors is true, eigenvectors is an n-by-n column-major matrix whose columns are the normalized eigenvectors.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix	Real symmetric square matrix. Only the selected triangle is read.
compute_vectors	Whether to compute eigenvectors.
triangle	Triangle of `matrix` supplied to LAPACK.

Returns: Eigenvalues and, optionally, eigenvectors.

◆ real_symmetric_eigensystem_index_range()

template<uni20::LapackReal Scalar>

RealSymmetricEigensystem< Scalar > uni20::krylov::real_symmetric_eigensystem_index_range	(	detail::ColumnMajorLapackMatrix< Scalar >	matrix,
		std::size_t	first_index,
		std::size_t	last_index,
		bool	compute_vectors,
		MatrixFill	triangle = `MatrixFill::Upper`
	)

Solve a selected dense real symmetric eigensystem through LAPACK syevr.

Selects eigenpairs by 0-based inclusive index in ascending eigenvalue order. If compute_vectors is true, eigenvectors is an n-by-m column-major matrix whose columns are the selected normalized eigenvectors.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix	Real symmetric square matrix. Only the selected triangle is read.
first_index	First 0-based eigenvalue index to compute.
last_index	Last 0-based eigenvalue index to compute.
compute_vectors	Whether to compute selected eigenvectors.
triangle	Triangle of `matrix` supplied to LAPACK.

Returns: Selected eigenvalues and, optionally, selected eigenvectors.

◆ real_symmetric_indefinite_expert_solve()

template<uni20::LapackReal Scalar>

RealSymmetricIndefiniteExpertLinearSolve< Scalar > uni20::krylov::real_symmetric_indefinite_expert_solve	(	detail::ColumnMajorLapackMatrix< Scalar >	coefficients,
		detail::ColumnMajorLapackMatrix< Scalar >	right_hand_sides,
		MatrixFill	triangle = `MatrixFill::Upper`
	)

Solve a dense real symmetric-indefinite system with LAPACK sysvx diagnostics.

Solves A * X = B using a Bunch-Kaufman factorization and returns the solution, factors, signed pivot-block metadata, reciprocal condition estimate, componentwise backward errors, and forward error bounds reported by LAPACK.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

coefficients	Real symmetric square coefficient matrix.
right_hand_sides	Dense right-hand-side matrix with matching row count.
triangle	Triangle of `coefficients` supplied to LAPACK.

Returns: Dense solution and LAPACK expert-driver diagnostics.

◆ real_symmetric_indefinite_factorization()

template<uni20::LapackReal Scalar>

RealSymmetricIndefiniteFactorization< Scalar > uni20::krylov::real_symmetric_indefinite_factorization	(	detail::ColumnMajorLapackMatrix< Scalar >	matrix,
		MatrixFill	triangle = `MatrixFill::Upper`
	)

Compute a dense real symmetric-indefinite Bunch-Kaufman factorization through LAPACK sytrf.

Only the selected triangle of matrix is factorized. The signed pivot block metadata is kept in LAPACK's ipiv encoding so it can be reused by sytrs, sytri, syrfs, and sycon wrappers.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix	Real symmetric square matrix.
triangle	Triangle of `matrix` supplied to LAPACK.

Returns: Bunch-Kaufman factors, signed pivot blocks, and triangle metadata.

◆ real_symmetric_indefinite_inverse()

template<uni20::LapackReal Scalar>

detail::ColumnMajorLapackMatrix< Scalar > uni20::krylov::real_symmetric_indefinite_inverse	(	detail::ColumnMajorLapackMatrix< Scalar >	matrix,
		MatrixFill	triangle = `MatrixFill::Upper`
	)

Invert a dense real symmetric-indefinite matrix through LAPACK sytrf and sytri.

Computes a Bunch-Kaufman factorization, inverts it with sytri, and mirrors the selected triangle so the returned matrix is a full dense symmetric inverse.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix	Real symmetric square matrix.
triangle	Triangle of `matrix` supplied to LAPACK.

Returns: Dense symmetric inverse matrix.

◆ real_symmetric_indefinite_one_norm_reciprocal_condition_number() [1/2]

template<uni20::LapackReal Scalar>

Scalar uni20::krylov::real_symmetric_indefinite_one_norm_reciprocal_condition_number	(	detail::ColumnMajorLapackMatrix< Scalar >	matrix,
		MatrixFill	triangle = `MatrixFill::Upper`
	)

Estimate a dense real symmetric-indefinite one-norm reciprocal condition number through LAPACK sycon.

Computes a Bunch-Kaufman factorization with sytrf, then estimates 1 / (||A||_1 * ||inv(A)||_1) from the factorization. Only the selected triangle is read, but the one-norm is computed as if the full symmetric matrix were materialized.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix	Real symmetric indefinite square matrix.
triangle	Triangle of `matrix` supplied to LAPACK.

Returns: Estimated one-norm reciprocal condition number.

◆ real_symmetric_indefinite_one_norm_reciprocal_condition_number() [2/2]

template<uni20::LapackReal Scalar>

Scalar uni20::krylov::real_symmetric_indefinite_one_norm_reciprocal_condition_number	(	RealSymmetricIndefiniteFactorization< Scalar > const &	factorization,
		Scalar	original_one_norm
	)

Estimate a symmetric-indefinite one-norm reciprocal condition number from an existing factorization.

The supplied matrix norm must be the one-norm of the original unfactored symmetric matrix.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

factorization	Bunch-Kaufman factors returned by `real_symmetric_indefinite_factorization`.
original_one_norm	One-norm of the original coefficient matrix.

Returns: Estimate of 1 / (||A||_1 * ||inv(A)||_1).

◆ real_symmetric_indefinite_refined_solve()

template<uni20::LapackReal Scalar>

RealRefinedLinearSolve< Scalar > uni20::krylov::real_symmetric_indefinite_refined_solve	(	detail::ColumnMajorLapackMatrix< Scalar >	coefficients,
		detail::ColumnMajorLapackMatrix< Scalar >	right_hand_sides,
		MatrixFill	triangle = `MatrixFill::Upper`
	)

Solve and refine a dense real symmetric-indefinite system through LAPACK syrfs.

Computes a Bunch-Kaufman factorization, solves A * X = B, then applies iterative refinement and returns LAPACK's forward and backward error estimates. Only the selected triangle of coefficients is read by LAPACK.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

coefficients	Real symmetric square coefficient matrix.
right_hand_sides	Dense right-hand-side matrix with matching row count.
triangle	Triangle of `coefficients` supplied to LAPACK.

Returns: Refined solution and LAPACK error estimates.

◆ real_symmetric_indefinite_solve() [1/2]

template<uni20::LapackReal Scalar>

detail::ColumnMajorLapackMatrix< Scalar > uni20::krylov::real_symmetric_indefinite_solve	(	detail::ColumnMajorLapackMatrix< Scalar >	coefficients,
		detail::ColumnMajorLapackMatrix< Scalar >	right_hand_sides,
		MatrixFill	triangle = `MatrixFill::Upper`
	)

Solve a dense real symmetric indefinite system through LAPACK sytrf and sytrs.

Solves A * X = B for one or more dense right-hand sides using a Bunch-Kaufman diagonal-pivoting factorization. The coefficient and right-hand-side matrices are copied by value and may be overwritten by LAPACK. The returned matrix stores X.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

coefficients	Real symmetric square coefficient matrix.
right_hand_sides	Dense right-hand-side matrix with matching row count.
triangle	Triangle of `coefficients` supplied to LAPACK.

Returns: Dense solution matrix.

◆ real_symmetric_indefinite_solve() [2/2]

template<uni20::LapackReal Scalar>

detail::ColumnMajorLapackMatrix< Scalar > uni20::krylov::real_symmetric_indefinite_solve	(	RealSymmetricIndefiniteFactorization< Scalar > const &	factorization,
		detail::ColumnMajorLapackMatrix< Scalar >	right_hand_sides
	)

Solve a dense real symmetric-indefinite system from an existing Bunch-Kaufman factorization.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

factorization	Factors returned by `real_symmetric_indefinite_factorization`.
right_hand_sides	Dense right-hand-side matrix with matching row count.

Returns: Dense solution matrix.

◆ real_symmetric_matrix_norm()

template<uni20::LapackReal Scalar>

Scalar uni20::krylov::real_symmetric_matrix_norm	(	detail::ColumnMajorLapackMatrix< Scalar >	matrix,
		MatrixNorm	norm,
		MatrixFill	triangle = `MatrixFill::Upper`
	)

Compute a dense real symmetric matrix norm through LAPACK lansy.

Only the selected triangle of matrix is referenced.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix	Real symmetric square matrix.
norm	Matrix norm to compute.
triangle	Triangle of `matrix` supplied to LAPACK.

Returns: Requested symmetric matrix norm.

◆ real_symmetric_positive_definite_band_expert_linear_solve()

template<uni20::LapackReal Scalar>

RealSymmetricPositiveDefiniteBandExpertLinearSolve< Scalar > uni20::krylov::real_symmetric_positive_definite_band_expert_linear_solve	(	RealSymmetricPositiveDefiniteBandMatrix< Scalar >	coefficients,
		detail::ColumnMajorLapackMatrix< Scalar >	right_hand_sides
	)

Solve a real SPD band linear system with LAPACK pbsvx diagnostics.

Solves A * X = B and returns the solution, band Cholesky factors, reciprocal condition estimate, componentwise backward errors, forward error bounds, equilibration, and condition diagnostics reported by LAPACK. The current wrapper requests a fresh factorization (‘FACT='N’) and does not expose caller-supplied equilibration modes. \tparam Scalar Real scalar type satisfyinguni20::LapackReal`.

Parameters

coefficients	Real SPD band coefficient matrix.
right_hand_sides	Dense right-hand-side matrix with matching row count.

Returns: Solution, factorization, and diagnostics.

◆ real_symmetric_positive_definite_band_factorization()

template<uni20::LapackReal Scalar>

RealSymmetricPositiveDefiniteBandFactorization< Scalar > uni20::krylov::real_symmetric_positive_definite_band_factorization ( RealSymmetricPositiveDefiniteBandMatrix< Scalar > matrix )

Compute a real SPD band Cholesky factorization through LAPACK pbtrf.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix Real square SPD band matrix in LAPACK storage layout.

Returns: SPD band Cholesky factors and storage metadata.

◆ real_symmetric_positive_definite_band_from_dense()

template<uni20::LapackReal Scalar>

RealSymmetricPositiveDefiniteBandMatrix< Scalar > uni20::krylov::real_symmetric_positive_definite_band_from_dense	(	detail::ColumnMajorLapackMatrix< Scalar > const &	matrix,
		std::size_t	bandwidth,
		MatrixFill	triangle = `MatrixFill::Upper`
	)

Pack a dense real SPD matrix into LAPACK symmetric-band storage.

Entries outside the requested triangle and bandwidth are dropped. The returned storage can be passed directly to pbsv, pbtrf, and pbtrs.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix	Real square SPD matrix to pack.
bandwidth	Number of stored offdiagonals.
triangle	Triangle of `matrix` supplied to LAPACK.

Returns: SPD band matrix in LAPACK storage layout.

◆ real_symmetric_positive_definite_band_one_norm()

template<uni20::LapackReal Scalar>

Scalar uni20::krylov::real_symmetric_positive_definite_band_one_norm ( RealSymmetricPositiveDefiniteBandMatrix< Scalar > const & matrix )

Compute the one-norm of a real SPD band matrix.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix Real SPD band matrix in LAPACK storage layout.

Returns: Matrix one-norm.

◆ real_symmetric_positive_definite_band_one_norm_reciprocal_condition_number() [1/2]

template<uni20::LapackReal Scalar>

Scalar uni20::krylov::real_symmetric_positive_definite_band_one_norm_reciprocal_condition_number	(	RealSymmetricPositiveDefiniteBandFactorization< Scalar > const &	factorization,
		Scalar	original_one_norm
	)

Estimate a real SPD band one-norm reciprocal condition number through LAPACK pbcon.

Uses an existing SPD band Cholesky factorization. The supplied matrix norm must be the one-norm of the original unfactored band matrix.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

factorization	Factors returned by `real_symmetric_positive_definite_band_factorization`.
original_one_norm	One-norm of the original coefficient matrix.

Returns: Estimate of 1 / (||A||_1 * ||inv(A)||_1).

◆ real_symmetric_positive_definite_band_one_norm_reciprocal_condition_number() [2/2]

template<uni20::LapackReal Scalar>

Scalar uni20::krylov::real_symmetric_positive_definite_band_one_norm_reciprocal_condition_number ( RealSymmetricPositiveDefiniteBandMatrix< Scalar > matrix )

Estimate a real SPD band one-norm reciprocal condition number from a matrix.

Computes the band one-norm, factors the matrix with LAPACK pbtrf, and estimates the reciprocal condition number with LAPACK pbcon.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix Real SPD band matrix in LAPACK storage layout.

Returns: Estimate of 1 / (||A||_1 * ||inv(A)||_1).

◆ real_symmetric_positive_definite_band_refined_solve()

template<uni20::LapackReal Scalar>

RealRefinedLinearSolve< Scalar > uni20::krylov::real_symmetric_positive_definite_band_refined_solve	(	RealSymmetricPositiveDefiniteBandMatrix< Scalar >	coefficients,
		detail::ColumnMajorLapackMatrix< Scalar >	right_hand_sides
	)

Solve and refine a real SPD band linear system through LAPACK pbrfs.

Computes a band Cholesky factorization, solves A * X = B, then applies iterative refinement and returns LAPACK's forward and backward error estimates.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

coefficients	Real SPD band coefficient matrix.
right_hand_sides	Dense right-hand-side matrix with matching row count.

Returns: Refined solution and LAPACK error estimates.

◆ real_symmetric_positive_definite_band_solve() [1/2]

template<uni20::LapackReal Scalar>

detail::ColumnMajorLapackMatrix< Scalar > uni20::krylov::real_symmetric_positive_definite_band_solve	(	RealSymmetricPositiveDefiniteBandFactorization< Scalar > const &	factorization,
		detail::ColumnMajorLapackMatrix< Scalar >	right_hand_sides
	)

Solve using an existing real SPD band Cholesky factorization through LAPACK pbtrs.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

factorization	Factors returned by `real_symmetric_positive_definite_band_factorization`.
right_hand_sides	Dense right-hand-side matrix with matching row count.

Returns: Dense solution matrix.

◆ real_symmetric_positive_definite_band_solve() [2/2]

template<uni20::LapackReal Scalar>

detail::ColumnMajorLapackMatrix< Scalar > uni20::krylov::real_symmetric_positive_definite_band_solve	(	RealSymmetricPositiveDefiniteBandMatrix< Scalar >	coefficients,
		detail::ColumnMajorLapackMatrix< Scalar >	right_hand_sides
	)

Solve a real SPD band system through LAPACK pbsv.

Solves A * X = B for one or more dense right-hand sides. The band matrix must use LAPACK symmetric positive-definite band storage.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

coefficients	Real square SPD band coefficient matrix.
right_hand_sides	Dense right-hand-side matrix with matching row count.

Returns: Dense solution matrix.

◆ real_symmetric_positive_definite_equilibration()

template<uni20::LapackReal Scalar>

RealSymmetricPositiveDefiniteEquilibration< Scalar > uni20::krylov::real_symmetric_positive_definite_equilibration ( detail::ColumnMajorLapackMatrix< Scalar > matrix )

Compute dense real SPD equilibration factors through LAPACK poequ.

Returns diagonal scales S suitable for forming diag(S) * A * diag(S), along with LAPACK's scale-condition diagnostic and maximum absolute diagonal entry.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix Real symmetric positive definite square matrix.

Returns: Symmetric scaling factors plus equilibration diagnostics.

◆ real_symmetric_positive_definite_expert_solve()

template<uni20::LapackReal Scalar>

RealSymmetricPositiveDefiniteExpertLinearSolve< Scalar > uni20::krylov::real_symmetric_positive_definite_expert_solve	(	detail::ColumnMajorLapackMatrix< Scalar >	coefficients,
		detail::ColumnMajorLapackMatrix< Scalar >	right_hand_sides,
		MatrixFill	triangle = `MatrixFill::Upper`
	)

Solve a dense real SPD linear system with LAPACK posvx diagnostics.

Solves A * X = B and returns the solution, Cholesky factors, reciprocal condition estimate, componentwise backward errors, forward error bounds, and equilibration/condition diagnostics reported by LAPACK. The current wrapper requests a fresh factorization (‘FACT='N’) and does not expose caller-supplied equilibration modes. \tparam Scalar Real scalar type satisfyinguni20::LapackReal`.

Parameters

coefficients	Real symmetric positive definite square matrix.
right_hand_sides	Dense right-hand-side matrix with matching row count.
triangle	Triangle of `coefficients` supplied to LAPACK.

Returns: Dense solution and LAPACK expert-driver diagnostics.

◆ real_symmetric_positive_definite_factorization()

template<uni20::LapackReal Scalar>

RealSymmetricPositiveDefiniteFactorization< Scalar > uni20::krylov::real_symmetric_positive_definite_factorization	(	detail::ColumnMajorLapackMatrix< Scalar >	matrix,
		MatrixFill	triangle = `MatrixFill::Upper`
	)

Compute a dense real SPD Cholesky factorization through LAPACK potrf.

Only the selected triangle of matrix is factorized. The returned factors keep LAPACK's triangular storage.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix	Real symmetric positive definite square matrix.
triangle	Triangle of `matrix` supplied to LAPACK.

Returns: Cholesky factors and triangle metadata.

◆ real_symmetric_positive_definite_inverse()

template<uni20::LapackReal Scalar>

detail::ColumnMajorLapackMatrix< Scalar > uni20::krylov::real_symmetric_positive_definite_inverse	(	detail::ColumnMajorLapackMatrix< Scalar >	matrix,
		MatrixFill	triangle = `MatrixFill::Upper`
	)

Invert a dense real SPD matrix through LAPACK potrf and potri.

Computes a Cholesky factorization, inverts it with potri, and mirrors the selected triangle so the returned matrix is a full dense symmetric inverse.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix	Real symmetric positive definite square matrix.
triangle	Triangle of `matrix` supplied to LAPACK.

Returns: Dense symmetric inverse matrix.

◆ real_symmetric_positive_definite_one_norm_reciprocal_condition_number() [1/2]

template<uni20::LapackReal Scalar>

Scalar uni20::krylov::real_symmetric_positive_definite_one_norm_reciprocal_condition_number	(	detail::ColumnMajorLapackMatrix< Scalar >	matrix,
		MatrixFill	triangle = `MatrixFill::Upper`
	)

Estimate a dense real SPD one-norm reciprocal condition number through LAPACK pocon.

Computes a Cholesky factorization with potrf, then estimates 1 / (||A||_1 * ||inv(A)||_1) from the factorization. Only the selected triangle is read, but the one-norm is computed as if the full symmetric matrix were materialized.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix	Real symmetric positive definite square matrix.
triangle	Triangle of `matrix` supplied to LAPACK.

Returns: Estimated one-norm reciprocal condition number.

◆ real_symmetric_positive_definite_one_norm_reciprocal_condition_number() [2/2]

template<uni20::LapackReal Scalar>

Scalar uni20::krylov::real_symmetric_positive_definite_one_norm_reciprocal_condition_number	(	RealSymmetricPositiveDefiniteFactorization< Scalar > const &	factorization,
		Scalar	original_one_norm
	)

Estimate an SPD one-norm reciprocal condition number from an existing Cholesky factorization.

The supplied matrix norm must be the one-norm of the original unfactored symmetric positive definite matrix.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

factorization	Cholesky factors returned by `real_symmetric_positive_definite_factorization`.
original_one_norm	One-norm of the original coefficient matrix.

Returns: Estimate of 1 / (||A||_1 * ||inv(A)||_1).

◆ real_symmetric_positive_definite_refined_solve()

template<uni20::LapackReal Scalar>

RealRefinedLinearSolve< Scalar > uni20::krylov::real_symmetric_positive_definite_refined_solve	(	detail::ColumnMajorLapackMatrix< Scalar >	coefficients,
		detail::ColumnMajorLapackMatrix< Scalar >	right_hand_sides,
		MatrixFill	triangle = `MatrixFill::Upper`
	)

Solve and refine a dense real SPD system through LAPACK porfs.

Computes a Cholesky factorization, solves A * X = B, then applies iterative refinement and returns LAPACK's forward and backward error estimates. Only the selected triangle of coefficients is read by LAPACK.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

coefficients	Real symmetric positive definite square matrix.
right_hand_sides	Dense right-hand-side matrix with matching row count.
triangle	Triangle of `coefficients` supplied to LAPACK.

Returns: Refined solution and LAPACK error estimates.

◆ real_symmetric_positive_definite_solve() [1/2]

template<uni20::LapackReal Scalar>

detail::ColumnMajorLapackMatrix< Scalar > uni20::krylov::real_symmetric_positive_definite_solve	(	detail::ColumnMajorLapackMatrix< Scalar >	coefficients,
		detail::ColumnMajorLapackMatrix< Scalar >	right_hand_sides,
		MatrixFill	triangle = `MatrixFill::Upper`
	)

Solve a dense real symmetric positive definite system through LAPACK potrf and potrs.

Solves A * X = B for one or more dense right-hand sides. The coefficient and right-hand-side matrices are copied by value and may be overwritten by LAPACK. The returned matrix stores X.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

coefficients	Real symmetric positive definite square matrix.
right_hand_sides	Dense right-hand-side matrix with matching row count.
triangle	Triangle of `coefficients` supplied to LAPACK.

Returns: Dense solution matrix.

◆ real_symmetric_positive_definite_solve() [2/2]

template<uni20::LapackReal Scalar>

detail::ColumnMajorLapackMatrix< Scalar > uni20::krylov::real_symmetric_positive_definite_solve	(	RealSymmetricPositiveDefiniteFactorization< Scalar > const &	factorization,
		detail::ColumnMajorLapackMatrix< Scalar >	right_hand_sides
	)

Solve a dense real SPD system from an existing Cholesky factorization.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

factorization	Cholesky factors returned by `real_symmetric_positive_definite_factorization`.
right_hand_sides	Dense right-hand-side matrix with matching row count.

Returns: Dense solution matrix.

◆ real_symmetric_positive_definite_tridiagonal_expert_linear_solve()

template<uni20::LapackReal Scalar>

RealSymmetricPositiveDefiniteTridiagonalExpertLinearSolve< Scalar > uni20::krylov::real_symmetric_positive_definite_tridiagonal_expert_linear_solve	(	RealSymmetricPositiveDefiniteTridiagonalMatrix< Scalar >	coefficients,
		detail::ColumnMajorLapackMatrix< Scalar >	right_hand_sides
	)

Solve a real SPD tridiagonal linear system with LAPACK ptsvx diagnostics.

Solves A * X = B and returns the solution, SPD tridiagonal factors, reciprocal condition estimate, componentwise backward errors, forward error bounds, and condition diagnostics reported by LAPACK. The current wrapper requests a fresh factorization (‘FACT='N’). \tparam Scalar Real scalar type satisfyinguni20::LapackReal`.

Parameters

coefficients	Real SPD tridiagonal coefficient matrix.
right_hand_sides	Dense right-hand-side matrix with matching row count.

Returns: Solution, factorization, and diagnostics.

◆ real_symmetric_positive_definite_tridiagonal_factorization()

template<uni20::LapackReal Scalar>

RealSymmetricPositiveDefiniteTridiagonalFactorization< Scalar > uni20::krylov::real_symmetric_positive_definite_tridiagonal_factorization ( RealSymmetricPositiveDefiniteTridiagonalMatrix< Scalar > matrix )

Compute a real SPD tridiagonal factorization through LAPACK pttrf.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix Real SPD tridiagonal matrix.

Returns: SPD tridiagonal Cholesky factors.

◆ real_symmetric_positive_definite_tridiagonal_from_dense()

template<uni20::LapackReal Scalar>

RealSymmetricPositiveDefiniteTridiagonalMatrix< Scalar > uni20::krylov::real_symmetric_positive_definite_tridiagonal_from_dense ( detail::ColumnMajorLapackMatrix< Scalar > const & matrix )

Extract real SPD tridiagonal storage from a dense matrix.

The diagonal and first subdiagonal are copied. Entries outside the tridiagonal band are ignored.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix Real square SPD tridiagonal matrix.

Returns: Tridiagonal matrix storage suitable for LAPACK ptsv and pttrf.

◆ real_symmetric_positive_definite_tridiagonal_one_norm()

template<uni20::LapackReal Scalar>

Scalar uni20::krylov::real_symmetric_positive_definite_tridiagonal_one_norm ( RealSymmetricPositiveDefiniteTridiagonalMatrix< Scalar > const & matrix )

Compute the one-norm of a real SPD tridiagonal matrix.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix Real SPD tridiagonal matrix.

Returns: Matrix one-norm.

◆ real_symmetric_positive_definite_tridiagonal_one_norm_reciprocal_condition_number() [1/2]

template<uni20::LapackReal Scalar>

Scalar uni20::krylov::real_symmetric_positive_definite_tridiagonal_one_norm_reciprocal_condition_number	(	RealSymmetricPositiveDefiniteTridiagonalFactorization< Scalar > const &	factorization,
		Scalar	original_one_norm
	)

Estimate a real SPD tridiagonal one-norm reciprocal condition number through LAPACK ptcon.

Uses an existing SPD tridiagonal factorization. The supplied matrix norm must be the one-norm of the original unfactored tridiagonal matrix.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

factorization	Factors returned by `real_symmetric_positive_definite_tridiagonal_factorization`.
original_one_norm	One-norm of the original coefficient matrix.

Returns: Estimate of 1 / (||A||_1 * ||inv(A)||_1).

◆ real_symmetric_positive_definite_tridiagonal_one_norm_reciprocal_condition_number() [2/2]

template<uni20::LapackReal Scalar>

Scalar uni20::krylov::real_symmetric_positive_definite_tridiagonal_one_norm_reciprocal_condition_number ( RealSymmetricPositiveDefiniteTridiagonalMatrix< Scalar > matrix )

Estimate a real SPD tridiagonal one-norm reciprocal condition number from a matrix.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix Real SPD tridiagonal matrix.

Returns: Estimate of 1 / (||A||_1 * ||inv(A)||_1).

◆ real_symmetric_positive_definite_tridiagonal_refined_solve()

template<uni20::LapackReal Scalar>

RealRefinedLinearSolve< Scalar > uni20::krylov::real_symmetric_positive_definite_tridiagonal_refined_solve	(	RealSymmetricPositiveDefiniteTridiagonalMatrix< Scalar >	coefficients,
		detail::ColumnMajorLapackMatrix< Scalar >	right_hand_sides
	)

Solve and refine a real SPD tridiagonal linear system through LAPACK ptrfs.

Computes an initial solution using pttrs, then refines it with ptrfs and returns LAPACK's forward and backward error estimates.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

coefficients	Real SPD tridiagonal coefficient matrix.
right_hand_sides	Dense right-hand-side matrix with matching row count.

Returns: Solution and error-bound diagnostics.

◆ real_symmetric_positive_definite_tridiagonal_solve() [1/2]

template<uni20::LapackReal Scalar>

detail::ColumnMajorLapackMatrix< Scalar > uni20::krylov::real_symmetric_positive_definite_tridiagonal_solve	(	RealSymmetricPositiveDefiniteTridiagonalFactorization< Scalar > const &	factorization,
		detail::ColumnMajorLapackMatrix< Scalar >	right_hand_sides
	)

Solve using an existing real SPD tridiagonal factorization through LAPACK pttrs.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

factorization	Factors returned by `real_symmetric_positive_definite_tridiagonal_factorization`.
right_hand_sides	Dense right-hand-side matrix with matching row count.

Returns: Dense solution matrix.

◆ real_symmetric_positive_definite_tridiagonal_solve() [2/2]

template<uni20::LapackReal Scalar>

detail::ColumnMajorLapackMatrix< Scalar > uni20::krylov::real_symmetric_positive_definite_tridiagonal_solve	(	RealSymmetricPositiveDefiniteTridiagonalMatrix< Scalar >	coefficients,
		detail::ColumnMajorLapackMatrix< Scalar >	right_hand_sides
	)

Solve a real SPD tridiagonal system through LAPACK ptsv.

Solves A * X = B for one or more dense right-hand sides.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

coefficients	Real SPD tridiagonal coefficient matrix.
right_hand_sides	Dense right-hand-side matrix with matching row count.

Returns: Dense solution matrix.

◆ real_symmetric_tridiagonal_orthogonal_factor()

template<uni20::LapackReal Scalar>

detail::ColumnMajorLapackMatrix< Scalar > uni20::krylov::real_symmetric_tridiagonal_orthogonal_factor ( RealSymmetricTridiagonalReduction< Scalar > const & reduction )

Generate the orthogonal factor from a compact real symmetric tridiagonal reduction.

The returned matrix is the Q in A = Q * T * Q^T for the result returned by real_symmetric_tridiagonal_reduction.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

reduction Compact tridiagonal reduction returned by real_symmetric_tridiagonal_reduction.

Returns: Dense orthogonal matrix represented by the compact Householder data.

◆ real_symmetric_tridiagonal_reduction()

template<uni20::LapackReal Scalar>

RealSymmetricTridiagonalReduction< Scalar > uni20::krylov::real_symmetric_tridiagonal_reduction	(	detail::ColumnMajorLapackMatrix< Scalar >	matrix,
		MatrixFill	triangle = `MatrixFill::Upper`
	)

Reduce a dense real symmetric matrix to tridiagonal form through LAPACK sytrd.

Uses the selected triangle of matrix and returns both a clean symmetric tridiagonal matrix and the compact Householder storage needed by real_symmetric_tridiagonal_orthogonal_factor and apply_real_symmetric_tridiagonal_orthogonal_factor. The reduction satisfies A = Q * T * Q^T when Q is generated from the returned compact data.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix	Real symmetric matrix to reduce.
triangle	Which triangle of `matrix` contains the symmetric input.

Returns: Tridiagonal form plus compact Householder reflectors.

◆ real_triangular_inverse()

template<uni20::LapackReal Scalar>

detail::ColumnMajorLapackMatrix< Scalar > uni20::krylov::real_triangular_inverse	(	detail::ColumnMajorLapackMatrix< Scalar >	matrix,
		MatrixFill	triangle = `MatrixFill::Upper`,
		bool	unit_diagonal = `false`
	)

Invert a dense real triangular matrix through LAPACK trtri.

The input matrix is copied by value and overwritten by LAPACK. Entries outside the selected triangle are zeroed before returning so the result is an ordinary dense triangular inverse.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix	Real square triangular matrix to invert.
triangle	Triangle of `matrix` supplied to LAPACK.
unit_diagonal	Whether the triangular diagonal is implicitly unit.

Returns: Dense triangular inverse matrix.

◆ real_triangular_matrix_norm()

template<uni20::LapackReal Scalar>

Scalar uni20::krylov::real_triangular_matrix_norm	(	detail::ColumnMajorLapackMatrix< Scalar >	matrix,
		MatrixNorm	norm,
		MatrixFill	triangle = `MatrixFill::Upper`,
		bool	unit_diagonal = `false`
	)

Compute a dense real triangular or trapezoidal matrix norm through LAPACK lantr.

Only the selected triangle/trapezoid of matrix is referenced. When unit_diagonal is true, the diagonal is treated as unit regardless of stored values.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix	Real rectangular matrix.
norm	Matrix norm to compute.
triangle	Triangle or trapezoid of `matrix` supplied to LAPACK.
unit_diagonal	Whether to treat the diagonal as all ones.

Returns: Requested triangular/trapezoidal matrix norm.

◆ real_triangular_one_norm_reciprocal_condition_number()

template<uni20::LapackReal Scalar>

Scalar uni20::krylov::real_triangular_one_norm_reciprocal_condition_number	(	detail::ColumnMajorLapackMatrix< Scalar >	matrix,
		MatrixFill	triangle = `MatrixFill::Upper`,
		bool	unit_diagonal = `false`
	)

Estimate a dense real triangular one-norm reciprocal condition number through LAPACK trcon.

The input matrix is copied by value because LAPACK accepts a mutable pointer. Only the selected triangle is read. The estimate is for 1 / (||A||_1 * ||inv(A)||_1).

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

matrix	Real square triangular matrix.
triangle	Triangle of `matrix` supplied to LAPACK.
unit_diagonal	Whether the triangular diagonal is implicitly unit.

Returns: Estimated one-norm reciprocal condition number.

◆ real_triangular_refined_solve()

template<uni20::LapackReal Scalar>

RealRefinedLinearSolve< Scalar > uni20::krylov::real_triangular_refined_solve	(	detail::ColumnMajorLapackMatrix< Scalar >	coefficients,
		detail::ColumnMajorLapackMatrix< Scalar >	right_hand_sides,
		MatrixFill	triangle = `MatrixFill::Upper`,
		MatrixTranspose	transpose = `MatrixTranspose::None`,
		bool	unit_diagonal = `false`
	)

Solve and refine a dense real triangular system through LAPACK trrfs.

Solves op(A) * X = B with trtrs, then asks LAPACK trrfs for forward and backward error estimates. The coefficient matrix is triangular and is not factorized.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

coefficients	Real square triangular coefficient matrix.
right_hand_sides	Dense right-hand-side matrix with matching row count.
triangle	Triangle of `coefficients` supplied to LAPACK.
transpose	Matrix operation applied to `coefficients` before solving.
unit_diagonal	Whether the triangular diagonal is implicitly unit.

Returns: Refined solution and LAPACK error estimates.

◆ real_triangular_solve()

template<uni20::LapackReal Scalar>

detail::ColumnMajorLapackMatrix< Scalar > uni20::krylov::real_triangular_solve	(	detail::ColumnMajorLapackMatrix< Scalar >	coefficients,
		detail::ColumnMajorLapackMatrix< Scalar >	right_hand_sides,
		MatrixFill	triangle = `MatrixFill::Upper`,
		MatrixTranspose	transpose = `MatrixTranspose::None`,
		bool	unit_diagonal = `false`
	)

Solve a dense real triangular system through LAPACK trtrs.

Solves op(A) * X = B for one or more dense right-hand sides, where A is triangular and op(A) is selected by transpose. The coefficient and right-hand-side matrices are copied by value and may be overwritten by LAPACK. The returned matrix stores X.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

coefficients	Real square triangular coefficient matrix.
right_hand_sides	Dense right-hand-side matrix with matching row count.
triangle	Triangle of `coefficients` supplied to LAPACK.
transpose	Matrix operation applied to `coefficients` before solving.
unit_diagonal	Whether the triangular diagonal is implicitly unit.

Returns: Dense solution matrix.

◆ reorder_complex_schur()

template<uni20::LapackReal Real>

ComplexSchurDecomposition< Real > uni20::krylov::reorder_complex_schur	(	ComplexSchurDecomposition< Real >	decomposition,
		std::vector< std::size_t > const &	leading_order
	)

Reorder a column-major complex Schur decomposition by moving selected entries to the front.

Template Parameters

Real	`float` or `double`.

Parameters

decomposition	Complex Schur decomposition to reorder.
leading_order	Input entry indices to move to the leading positions.

Returns: Reordered complex Schur decomposition.

◆ reorder_complex_schur_layout_right()

template<uni20::LapackReal Real>

RightComplexSchurDecomposition< Real > uni20::krylov::reorder_complex_schur_layout_right	(	RightComplexSchurDecomposition< Real >	decomposition,
		std::vector< std::size_t > const &	leading_order
	)

Move selected real Schur blocks to the leading invariant subspace and estimate conditioning.

Uses LAPACK trsen with ‘JOB='B’, returning reciprocal condition estimates for the selected eigenvalue cluster and invariant subspace. Selection is by Schur block, so real-arithmetic 2x2 complex-conjugate blocks cannot be split accidentally. \tparam Scalar Real scalar type satisfyinguni20::LapackReal`.

Parameters

decomposition Real Schur decomposition to reorder and diagnose.

◆ reorder_real_generalized_schur()

template<uni20::LapackReal Real>

RealGeneralizedSchurDecomposition< Real > uni20::krylov::reorder_real_generalized_schur	(	RealGeneralizedSchurDecomposition< Real >	decomposition,
		std::vector< std::size_t > const &	leading_block_order
	)

Reorder a generalized real Schur decomposition by moving selected blocks to the front.

Uses LAPACK tgexc to reorder the QZ forms of A - lambda B. Real-arithmetic 2x2 complex-conjugate blocks are moved as indivisible units, and the projective alpha/beta data is kept aligned with the new block order.

Template Parameters

Real	Real scalar type satisfying `uni20::LapackReal`.

Parameters

decomposition	Generalized real Schur decomposition to reorder.
leading_block_order	Input block indices to move to the leading positions.

Returns: Reordered generalized real Schur decomposition.

◆ reorder_real_schur()

template<uni20::LapackReal Scalar>

RealSchurDecomposition< Scalar > uni20::krylov::reorder_real_schur	(	RealSchurDecomposition< Scalar >	decomposition,
		std::vector< std::size_t > const &	leading_block_order
	)

Reorder a column-major real Schur decomposition by moving selected blocks to the front.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

decomposition	Real Schur decomposition to reorder.
leading_block_order	Input block indices to move to the leading positions.

Returns: Reordered real Schur decomposition.

◆ reorder_real_schur_layout_right()

template<uni20::LapackReal Scalar>

RightRealSchurDecomposition< Scalar > uni20::krylov::reorder_real_schur_layout_right	(	RightRealSchurDecomposition< Scalar >	decomposition,
		std::vector< std::size_t > const &	leading_block_order
	)

Reorder a layout-right real Schur decomposition by moving selected blocks to the front.

leading_block_order names blocks in the input decomposition's current block order. Any blocks not named are left after the requested prefix in their original relative order. Complex conjugate 2x2 real Schur blocks are moved as indivisible blocks. The Schur form and, when present, Schur vectors are updated through LAPACK trexc.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

decomposition	Real Schur decomposition to reorder.
leading_block_order	Input block indices to move to the leading positions.

Returns: Reordered real Schur decomposition.

◆ right_mdspan() [1/2]

template<typename Scalar >

auto uni20::krylov::right_mdspan ( RightMatrix< Scalar > & matrix )

Return a mutable layout-right mdspan view of a prototype right-layout matrix.

Template Parameters

Scalar Element type.

Parameters

matrix Matrix whose storage is exposed.

Returns: Mutable rank-2 layout-right mdspan view.

◆ right_mdspan() [2/2]

template<typename Scalar >

auto uni20::krylov::right_mdspan ( RightMatrix< Scalar > const & matrix )

Return an immutable layout-right mdspan view of a prototype right-layout matrix.

Template Parameters

Scalar Element type.

Parameters

matrix Matrix whose storage is exposed.

Returns: Immutable rank-2 layout-right mdspan view.

◆ scal()

template<typename Scalar >

void uni20::krylov::scal	(	Scalar const &	alpha,
		std::span< Scalar >	vector
	)

Scale a local dense vector in place.

Template Parameters

Scalar Element type.

Parameters

alpha	Scale factor.
vector	Vector to scale.

◆ select_nonsymmetric_restart_indices()

template<uni20::Real Scalar>

std::vector< std::size_t > uni20::krylov::select_nonsymmetric_restart_indices	(	std::vector< uni20::complex< Scalar > > const &	values,
		NonsymmetricEigenParams< Scalar > const &	params
	)

Select retained Ritz or Schur indices for a nonsymmetric thick restart.

Template Parameters

Scalar Real scalar type.

Parameters

values	Projected Ritz or Schur eigenvalues.
params	Solver parameters defining selector and retained count.

Returns: Indices of retained values in wanted order.

◆ select_nonsymmetric_ritz_indices()

template<uni20::Real Scalar>

std::vector< std::size_t > uni20::krylov::select_nonsymmetric_ritz_indices	(	std::vector< uni20::complex< Scalar > > const &	values,
		NonsymmetricEigenParams< Scalar > const &	params
	)

Select wanted Ritz indices for a nonsymmetric projected spectrum.

Nonsymmetric projected eigenvalues may be complex. LargestReal and SmallestReal use the real part, imaginary selectors use the signed imaginary part, and magnitude selectors use complex magnitude. Algebraic and both-end selectors are deliberately rejected because they are symmetric/Hermitian concepts.

Template Parameters

Scalar Real scalar type.

Parameters

values	Projected Ritz values.
params	Solver parameters defining `nev` and selector.

Returns: Indices of selected wanted Ritz values.

◆ select_real_schur_restart_blocks()

template<uni20::LapackReal Scalar>

RealSchurRestartSelection< Scalar > uni20::krylov::select_real_schur_restart_blocks	(	std::vector< RealSchurBlock< Scalar > > const &	blocks,
		NonsymmetricEigenParams< Scalar > const &	params
	)

Select real Schur blocks for a nonsymmetric thick restart.

The requested retained dimension is interpreted as a number of scalar Schur dimensions. Since real arithmetic represents complex conjugate eigenpairs as 2x2 Schur blocks, the returned scalar_count may exceed the requested count by one when needed to avoid cutting a complex block.

Template Parameters

Scalar Real scalar type.

Parameters

blocks	Real Schur blocks in current projected order.
params	Nonsymmetric solve parameters defining selector and retained count.

Returns: Selected block indices in wanted order and the corresponding scalar dimension.

◆ select_symmetric_restart()

template<uni20::Real Scalar>

SymmetricRestartSelection< Scalar > uni20::krylov::select_symmetric_restart	(	std::vector< Scalar > const &	values,
		std::vector< Scalar > const &	residual_bounds,
		SymmetricEigenParams< Scalar > const &	params,
		std::size_t	shift_count,
		std::span< std::size_t const >	protected_shift_indices = `{}`
	)

Select wanted Ritz values and exact-shift candidates for symmetric restart.

Ritz values are ordered so the retained values occupy the final n_keep positions, and the unwanted shift candidates are then ordered by decreasing Ritz estimate. When retained_ritz_count is not set, n_keep defaults to nev and the behavior is the minimal implicit restart. The filtering step itself is separate.

Template Parameters

Scalar Real vector-field scalar or complex vector-field scalar.

Parameters

values	Projected Ritz values.
residual_bounds	Ritz residual estimates with the same length as `values`.
params	Solver parameters defining `nev`, optional `n_keep`, selector, and tolerance.
shift_count	Number of exact shifts to select from the unwanted values.

Returns: Restart-selection state.

◆ symmetric_lanczos_restarted_generalized_transformed() [1/2]

template<uni20::LapackRealOrComplex Scalar, typename Vector , KrylovMatrixFreeOperator< Vector, Scalar > OpOps, KrylovMatrixFreeOperator< Vector, Scalar > BOps, typename MetricOps >

EigenResult< uni20::make_real_t< Scalar >, Vector > uni20::krylov::symmetric_lanczos_restarted_generalized_transformed	(	OpOps &	op_ops,
		BOps &	b_ops,
		MetricOps &	metric_ops,
		Vector const &	initial,
		SymmetricEigenParams< uni20::make_real_t< Scalar > > const &	params,
		SymmetricSpectralTransformOptions< uni20::make_real_t< Scalar > > const &	options
	)

Run restarted symmetric Lanczos on a transformed generalized operator with a B metric.

This entry point is for generalized symmetric problems where the caller supplies OP, such as B^-1 A or (A - sigma B)^-1 B, and a separate matrix-free B operation. The Lanczos recurrence uses the B inner product, so the transformed operator is treated as self-adjoint in the generalized problem metric. The returned eigenvalues are mapped back through options, and the returned Ritz vectors are application-space vectors for the original generalized problem.

Template Parameters

Scalar	Real or complex vector-field scalar type with real LAPACK coverage.
Vector	Opaque vector type.
OpOps	Matrix-free transformed-operator provider.
BOps	Matrix-free `B` operation provider.
MetricOps	Optional backend-owned metric inner-product provider.

Parameters

op_ops	Matrix-free operation object applying `OP`.
b_ops	Matrix-free operation object applying `B`.
metric_ops	Optional metric operation object. If it provides `metric_inner_product(x, y)`, that is used directly; otherwise the solver applies `B*y` into reusable scratch and calls `op_ops.inner_product`.
initial	Initial residual/start vector.
params	Solver parameters for the transformed problem.
options	Transform metadata used for eigenvalue back-transformation.

Returns: Eigenvalues mapped to the original problem, with generalized Ritz vectors.

◆ symmetric_lanczos_restarted_generalized_transformed() [2/2]

template<uni20::LapackRealOrComplex Scalar, typename Vector , KrylovMatrixFreeOperator< Vector, Scalar > OpOps, KrylovMatrixFreeOperator< Vector, Scalar > BOps>

EigenResult< uni20::make_real_t< Scalar >, Vector > uni20::krylov::symmetric_lanczos_restarted_generalized_transformed	(	OpOps &	op_ops,
		BOps &	b_ops,
		Vector const &	initial,
		SymmetricEigenParams< uni20::make_real_t< Scalar > > const &	params,
		SymmetricSpectralTransformOptions< uni20::make_real_t< Scalar > > const &	options
	)

Run restarted symmetric Lanczos on a transformed generalized operator with default metric handling.

This convenience overload uses b_ops as the optional metric customization provider. Backends can implement b_ops.metric_inner_product(x, y) to own scratch, caching, or fused backend collectives; otherwise the generic fallback applies B*y into reusable solver scratch.

Template Parameters

Scalar	Real or complex vector-field scalar type with real LAPACK coverage.
Vector	Opaque vector type.
OpOps	Matrix-free transformed-operator provider.
BOps	Matrix-free `B` operation provider.

Parameters

op_ops	Matrix-free operation object applying `OP`.
b_ops	Matrix-free operation object applying `B`.
initial	Initial residual/start vector.
params	Solver parameters for the transformed problem.
options	Transform metadata used for eigenvalue back-transformation.

Returns: Eigenvalues mapped to the original problem, with generalized Ritz vectors.

◆ symmetric_lanczos_restarted_standard()

template<uni20::LapackRealOrComplex Scalar, typename Vector , KrylovMatrixFreeOperator< Vector, Scalar > Ops>

EigenResult< uni20::make_real_t< Scalar >, Vector > uni20::krylov::symmetric_lanczos_restarted_standard	(	Ops &	ops,
		Vector const &	initial,
		SymmetricEigenParams< uni20::make_real_t< Scalar > > const &	params
	)

Run a native symmetric/Hermitian Lanczos solve with implicit restarts.

This expands a Lanczos factorization to ncv, computes Ritz values and residual bounds, applies exact shifts to the projected tridiagonal, compresses the opaque basis through the accumulated transform, and resumes expansion from the updated residual.

Template Parameters

Scalar	Real or complex vector-field scalar type with real LAPACK coverage.
Vector	Opaque vector type.
Ops	Matrix-free vector operation provider.

Parameters

ops	Matrix-free operation object.
initial	Initial residual/start vector.
params	Solver parameters.

Returns: Ritz values and optional Ritz vectors from the final projection.

◆ symmetric_lanczos_restarted_transformed()

template<uni20::LapackRealOrComplex Scalar, typename Vector , KrylovMatrixFreeOperator< Vector, Scalar > Ops>

EigenResult< uni20::make_real_t< Scalar >, Vector > uni20::krylov::symmetric_lanczos_restarted_transformed	(	Ops &	op_ops,
		Vector const &	initial,
		SymmetricEigenParams< uni20::make_real_t< Scalar > > const &	params,
		SymmetricSpectralTransformOptions< uni20::make_real_t< Scalar > > const &	options
	)

Run restarted symmetric Lanczos on a caller-supplied transformed operator.

The matrix-free operation object must implement the transformed operator OP for the requested mode, for example (A - sigma B)^-1 B for shift-invert. The native solver remains independent of the workspace layout or linear-solver implementation used to apply OP; this wrapper only maps converged Ritz values back to the original eigenvalue scale. Reported residual bounds are those of the transformed problem.

Template Parameters

Scalar	Real scalar type.
Vector	Opaque vector type.
Ops	Matrix-free transformed-operator provider.

Parameters

op_ops	Matrix-free operation object applying `OP`.
initial	Initial residual/start vector.
params	Solver parameters for the transformed problem.
options	Transform metadata used for eigenvalue back-transformation.

Returns: Eigenvalues mapped to the original problem, with OP Ritz vectors.

◆ symmetric_lanczos_standard()

template<uni20::LapackRealOrComplex Scalar, typename Vector , KrylovMatrixFreeOperator< Vector, Scalar > Ops>

EigenResult< uni20::make_real_t< Scalar >, Vector > uni20::krylov::symmetric_lanczos_standard	(	Ops &	ops,
		Vector const &	initial,
		SymmetricEigenParams< uni20::make_real_t< Scalar > > const &	params
	)

Run the native symmetric/Hermitian Lanczos full-projection solver.

This implementation keeps the vector storage opaque and communicates with the application only through the matrix-free operation object. The current implementation uses full reorthogonalization for deterministic, robust basis construction.

Template Parameters

Scalar	Real or complex vector-field scalar type with real LAPACK coverage.
Vector	Opaque vector type.
Ops	Matrix-free vector operation provider.

Parameters

ops	Matrix-free operation object.
initial	Initial residual/start vector.
params	Solver parameters.

Returns: Ritz values and optional Ritz vectors from the Lanczos projection.

◆ symmetric_tridiagonal()

template<uni20::Real Real>

symmetric_tridiagonal_view< Real > uni20::krylov::symmetric_tridiagonal	(	std::span< Real const >	diagonal,
		std::span< Real const >	offdiagonal
	)

◆ symmetric_tridiagonal_eigensystem()

template<uni20::LapackReal Scalar>

TridiagonalEigensystem< Scalar > uni20::krylov::symmetric_tridiagonal_eigensystem	(	std::vector< Scalar >	diagonal,
		std::vector< Scalar >	subdiagonal,
		bool	compute_vectors
	)

Solve a real symmetric tridiagonal eigensystem through LAPACK sterf or steqr.

diagonal has length n; subdiagonal has length n - 1. Eigenvalues are returned in ascending order. If compute_vectors is false, LAPACK sterf is used. If compute_vectors is true, LAPACK steqr is used and eigenvectors is an n-by-n column-major matrix whose columns are the normalized eigenvectors.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

diagonal	Main diagonal of the tridiagonal matrix.
subdiagonal	Subdiagonal/superdiagonal entries.
compute_vectors	Whether to compute eigenvectors.

Returns: Eigenvalues and, optionally, eigenvectors.

◆ symmetric_tridiagonal_eigensystem_divide_and_conquer()

template<uni20::LapackReal Scalar>

TridiagonalEigensystem< Scalar > uni20::krylov::symmetric_tridiagonal_eigensystem_divide_and_conquer	(	std::vector< Scalar >	diagonal,
		std::vector< Scalar >	subdiagonal,
		bool	compute_vectors
	)

Solve a real symmetric tridiagonal eigenvalue problem through LAPACK sterf.

diagonal has length n; subdiagonal has length n - 1. Eigenvalues are returned in ascending order.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

diagonal	Main diagonal of the tridiagonal matrix.
subdiagonal	Subdiagonal/superdiagonal entries.

Returns: Eigenvalues in ascending order.

◆ symmetric_tridiagonal_eigensystem_index_range()

template<uni20::LapackReal Scalar>

TridiagonalEigensystem< Scalar > uni20::krylov::symmetric_tridiagonal_eigensystem_index_range	(	std::vector< Scalar >	diagonal,
		std::vector< Scalar >	subdiagonal,
		std::size_t	first_index,
		std::size_t	last_index,
		bool	compute_vectors
	)

Solve a selected real symmetric tridiagonal eigensystem through LAPACK stevr.

Selects eigenpairs by 0-based inclusive index in ascending eigenvalue order. diagonal has length n; subdiagonal has length n - 1. If compute_vectors is true, eigenvectors is an n-by-m column-major matrix whose columns are the selected normalized eigenvectors.

Template Parameters

Scalar Real scalar type satisfying uni20::LapackReal.

Parameters

diagonal	Main diagonal of the tridiagonal matrix.
subdiagonal	Subdiagonal/superdiagonal entries.
first_index	First 0-based eigenvalue index to compute.
last_index	Last 0-based eigenvalue index to compute.
compute_vectors	Whether to compute selected eigenvectors.

Returns: Selected eigenvalues and, optionally, selected eigenvectors.

◆ symmetric_tridiagonal_eigenvalues()

template<uni20::LapackReal Scalar>

std::vector< Scalar > uni20::krylov::symmetric_tridiagonal_eigenvalues	(	std::vector< Scalar >	diagonal,
		std::vector< Scalar >	subdiagonal
	)

◆ taylor_exponential_action()

template<uni20::RealOrComplex Scalar, typename TimeScalar , typename Vector , KrylovMatrixFreeOperator< Vector, Scalar > Ops>
requires (std::constructible_from<uni20::make_real_t<Scalar>, TimeScalar> || (uni20::Complex<Scalar> && std::constructible_from<Scalar, TimeScalar>))

TaylorExponentialResult< Scalar, Vector > uni20::krylov::taylor_exponential_action	(	Ops &	ops,
		Vector const &	initial,
		TimeScalar	time,
		uni20::make_real_t< Scalar >	operator_norm_bound,
		TaylorExponentialParams< uni20::make_real_t< Scalar > > const &	params = `{}`
	)

Compute exp(t A) v using conservative scaled Taylor series.

This validation-oriented action is algorithmically independent of the Lanczos/Arnoldi projected exponential paths. The caller supplies a finite upper bound for ||A||; the routine chooses a number of scaled steps so that each Taylor series sees ||h A|| <= step_norm_limit, then truncates using a geometric tail bound. It is intentionally slower and more conservative than the Krylov projected action, and it is not used as an automatic fallback.

Template Parameters

Scalar	Real or complex vector-field scalar type.
TimeScalar	Real time for real vector spaces; real or complex time for complex vector spaces.
Vector	Opaque vector type.
Ops	Matrix-free operation provider.

Parameters

ops	Matrix-free operation object.
initial	Initial vector.
time	Exponential coefficient.
operator_norm_bound	Finite upper bound for `\|\|A\|\|`.
params	Taylor action parameters.

Returns: Approximation to exp(t A) initial and diagnostics.

◆ validate_matrix_free_dimensions()

template<typename Ops , typename Vector >

std::size_t uni20::krylov::validate_matrix_free_dimensions	(	Ops const &	ops,
		Vector const &	vector,
		char const *	context
	)

Validate that an initial/prototype vector matches the matrix-free problem dimension.

Namespaces

Classes

Concepts

Typedefs

Enumerations

Functions

Typedef Documentation

◆ MatrixFill

◆ RightMatrix

◆ SymmetricSpectralTransform

◆ SymmetricSpectralTransformOptions

Enumeration Type Documentation

◆ GmresEstimateKind

◆ GmresRefinementStatus

◆ GmresStatus

◆ KrylovDiagnosticsLevel

◆ MatrixNorm

◆ MatrixSide

◆ MatrixTranspose

◆ NonsymmetricStatus

◆ RealNonsymmetricBalanceJob

◆ RealNonsymmetricPolicy

◆ RitzReality

◆ SpectrumPart

◆ SymmetricTransformKind

Function Documentation

◆ apply_real_bidiagonal_left_factor()

◆ apply_real_bidiagonal_right_factor()

◆ apply_real_hessenberg_orthogonal_factor()

◆ apply_real_lq_factor()

◆ apply_real_ql_factor()

◆ apply_real_qr_factor()

◆ apply_real_rq_factor()

◆ apply_real_symmetric_tridiagonal_orthogonal_factor()

◆ apply_symmetric_qr_shifts()

◆ arnoldi_factorize()

◆ arnoldi_projected_hessenberg()

◆ axpy()

◆ BlockTensorMatrixFreeOps()

◆ classify_ritz_reality()

◆ complex_generalized_hermitian_eigensystem()

◆ complex_generalized_hermitian_eigensystem_divide_and_conquer()

◆ complex_generalized_hermitian_eigensystem_index_range()

◆ complex_hermitian_eigensystem()

◆ complex_hermitian_eigensystem_divide_and_conquer()

◆ complex_hermitian_eigensystem_index_range()

◆ complex_nonsymmetric_arnoldi_standard()

◆ complex_nonsymmetric_eigensystem()

◆ complex_schur()

◆ complex_schur_layout_right()

◆ compress_complex_schur_arnoldi_restart()

◆ compress_real_schur_arnoldi_restart()

◆ compress_symmetric_lanczos_restart()

◆ copy()

◆ copy_left_to_right()

◆ copy_right_to_left()

◆ default_adaptive_exponential_krylov_dimension()

◆ default_complex_pair_tolerance() [1/2]

◆ default_complex_pair_tolerance() [2/2]

◆ default_exponential_krylov_dimension()

◆ default_nonsymmetric_krylov_dimension()

◆ default_nonsymmetric_retained_ritz_count()

◆ default_symmetric_krylov_dimension()

◆ default_symmetric_retained_ritz_count()

◆ dot()

◆ dotc()

◆ dotu()

◆ effective_exponential_krylov_dimension()

◆ effective_nonsymmetric_krylov_dimension()

◆ effective_nonsymmetric_retained_ritz_count()

◆ effective_symmetric_krylov_dimension()

◆ effective_symmetric_retained_ritz_count()

◆ expand_complex_schur_arnoldi_restart()

◆ expand_real_schur_arnoldi_restart()

◆ exponential_adaptive_acceptance_enabled()

◆ extract_arnoldi_ritz()

◆ extract_complex_arnoldi_ritz()

◆ gemv()

◆ gerc()

◆ geru()