This note records the exponential-action convergence diagnostics measured by examples/krylov/krylov_exponential_probe_example.cpp. For a smaller toy illustration of an orthogonality-driven tolerance floor, run examples/krylov/krylov_exponential_orthogonality_example.cpp. For a real Matrix Market stress case copied from the Cytnx TDVP/Lanczos harmonic-oscillator investigation, run examples/krylov/krylov_exponential_matrix_market_probe_example.cpp; its default complex<double> configuration uses tests/krylov/matrix_market/tdvp_lanczos/ and shows the old raw projected-tail indicator below 1e-8 while the Taylor-reference action error is still above 1e-8.

The notation is

A V_m = V_m H_m + h_{m+1,m} v_{m+1} e_m^*

exp(t A) v ~= ||v|| V_m exp(t H_m) e_1

For Hermitian Lanczos, H_m is tridiagonal. For the diagnostic diagonal examples, the probe can compute exact errors, so the estimates are reported against the true action error.

Measured Quantities

probe column	formula	purpose
`tail coeff`	`abs(e_m^* exp(t H_m) e_1)`	Raw projected-exponential last coefficient. This mirrors the historical Cytnx-style `abs(B_mat(i,0))` stopping indicator.

| defect/||v|| | h_{m+1,m} abs(e_m^* exp(t H_m) e_1) | Final-time defect norm per input norm. The native exponential result reports this quantity multiplied by ||v|| as endpoint_defect_estimate; for Hermitian actions, error_estimate is instead the direct defect integral. See the residual/defect framing in [BotchevGrimmHochbruck2013] and the xi_2-style discussion in [JiaLv2015]. | | Hquad/||v|| | h_{m+1,m} |t| abs(e_m^* exp(t H_m) e_1) / m | Hermite-quadrature defect-integral estimate from the [HochbruckLubich1997] / [JaweckiAuzingerKoch2020] line of analysis. | | Saad phi1/||v|| | h_{m+1,m} |t| abs(e_m^* phi_1(t H_m) e_1) | First-term error expansion estimate from [Saad1992], also used as an a posteriori estimate in [JiaLv2015]. Here phi_1(z) = (exp(z)-1)/z; the probe computes it by a small augmented matrix exponential. | | HL bound/||v|| | h_{m+1,m} gamma_m |t|^m / m! | Nonexpansive-case leading upper bound/asymptotic term from [HochbruckLubich1997], with computable forms in [JaweckiAuzingerKoch2020]; gamma_m = prod_{j=1}^{m-1} h_{j+1,j}. The probe evaluates this in log space to avoid artificial overflow/underflow in float. | | orth offdiag | max_{i != j} abs(q_i^* q_j) | Final-basis orthogonality defect. This is not an exponential error estimate; it indicates when the Krylov basis itself has reached a precision floor. | | reorth ratio | max abs(q_i^* r) / ||r|| during residual reorthogonalization | Measures how much the raw Lanczos residual had to be corrected. A large value means the three-term recurrence is no longer producing a nearly orthogonal residual before explicit reorthogonalization. | | passes | maximum number of reorthogonalization passes used for any residual in that run | Distinguishes ordinary one-pass cleanup from the second-pass Daniel-Gragg-Kaufman-Stewart style refinement used by the implementation. |

The Hquad, Saad phi1, and HL bound estimates are most meaningful once the Krylov approximation has entered its convergence regime. They can all become misleading after the true error has reached the scalar precision floor; the probe intentionally samples beyond that point to expose underflow and over-solving behavior. The orthogonality columns help identify whether that floor is dominated by loss of basis orthogonality rather than by the projected exponential estimator itself. Simple diagonal examples are intentionally limited: they are good for showing false stopping signals and basis loss, but real TDVP operators are much more likely to exhibit severe post-threshold residual rebound.

Defect-Integral Stopping Rule

The Hermitian exponential action reports both the final-time defect norm,

h_{m+1,m} |delta_m(t)| ||v||,

delta_m(s) = e_m^* exp(sigma s H_m) e_1.

as endpoint_defect_estimate, and the direct defect integral as defect_integral_estimate. The top-level Hermitian error_estimate uses the direct integral, since endpoint sampling is often far too conservative and the raw tail coefficient is not reliable.

For a nonexpansive exponential, JAK starts from

L_m(t)v = int_0^t exp(sigma (t-s) A) D_m(s)v ds,

D_m(s)v = sigma h_{m+1,m} delta_m(s) v_{m+1}.

This gives the computable bound

||L_m(t)v|| <= h_{m+1,m} int_0^t |delta_m(s)| ds.

The direct integral is therefore the natural quantity to target for adaptive stopping. The cheap HL/JAK bound

h_{m+1,m} gamma_m |t|^m / m!

is rigorous and asymptotically correct under the nonexpansive assumptions, but it can be far too conservative in hot TDVP loops.

For the TDVP harmonic-oscillator fixture with tau = -0.1968473663975394 i and ||v|| = 1, direct quadrature of the defect integral is much tighter:

m	true error	final defect	direct defect integral	HL bound
33	`1.398e-8`	`1.136e-6`	`3.225e-8`	`2.364e2`
34	`9.686e-9`	`7.735e-7`	`2.289e-8`	`1.753e2`
40	`3.254e-9`	`1.726e-7`	`8.369e-9`	`1.828e1`
64	`4.554e-12`	`1.105e-9`	`5.819e-12`	`8.160e-7`

For a 1e-8 target, the direct integral accepts m=40 on this fixture. The endpoint-defect estimator accepts much later, around m=64, and the HL bound later still, around m=69.

The native Hermitian action implements this direct-integral target for the nonexpansive path:

Diagonalize the projected Hermitian matrix H_m = Q Lambda Q^*.
Evaluate delta_m(s) = e_m^* Q exp(sigma s Lambda) Q^* e_1.
Integrate abs(delta_m(s)) on [0, |t|] with a deterministic 1024-panel Simpson rule.
Accept the first projected dimension whose direct integral is below relative_tolerance * ||v||, after applying the configured safety factor.

Pure-imaginary Hermitian time is recognized automatically as the unitary exp(-i t H) case. Other Hermitian nonexpansive actions require the caller to set assume_nonexpansive, since the Krylov layer cannot infer positivity or contractivity from the matrix-free operator alone. The raw tail, endpoint defect, and direct integral remain available as diagnostics; the probe also continues to report Hermite/trapezoid-style, effective-order, and HL-family quantities for comparison.

Non-Hermitian Defect Integrals

The defect identity itself is not Hermitian-specific. For an Arnoldi projection,

A V_m = V_m H_m + h_{m+1,m} v_{m+1} e_m^*,

the same residual/defect construction gives

D_m(s)v = h_{m+1,m} e_m^* exp(s H_m) e_1 v_{m+1} ||v||.

The difference is the propagation factor in the error integral. The safe bound is

||L_m(t)v|| <= h_{m+1,m} ||v||

int_0^t ||exp((t-s) A)|| |e_m^* exp(s H_m) e_1| ds.

For Hermitian unitary evolution, ||exp((t-s) A)|| = 1, so this reduces to the projected scalar integral used above. For a general non-Hermitian or non-normal operator, that simplification is not available: transient growth in ||exp((t-s) A)|| can be large even when the spectrum looks stable. Therefore a future Arnoldi adaptive rule can use the projected defect integral only as a diagnostic unless the caller supplies, or the algorithm proves, a usable nonexpansive or semigroup-growth bound. [WangYe2017] is the current reference starting point for that non-Hermitian error-control work.

The trapezoid estimate,

h_{m+1,m} |t| |delta_m(t)| / 2,

is a useful simple fallback. On the same fixture it would accept around m=48. The Hermite estimate h |t| |delta_m(t)| / m is often too optimistic to use as a sole stopping criterion, while the effective-order estimate can be very tight but needs robust handling near roundoff and cancellation.

References

The code comments use the same citation keys as this table.

key	reference	relevant idea
`[Saad1992]`	Y. Saad, "Analysis of some Krylov subspace approximations to the matrix exponential operator", SIAM J. Numer. Anal. 29(1):209-228, 1992. DOI: https://doi.org/10.1137/0729014.	Error expansion and the leading-term `phi_1` style estimate.
`[HochbruckLubich1997]`	M. Hochbruck and C. Lubich, "On Krylov subspace approximations to the matrix exponential operator", SIAM J. Numer. Anal. 34(5):1911-1925, 1997. DOI: https://doi.org/10.1137/S0036142995280572.	Asymptotic convergence analysis for Hermitian/skew-Hermitian exponential Krylov approximations.
`[BotchevGrimmHochbruck2013]`	M. Botchev, V. Grimm, and M. Hochbruck, "Residual, restarting and Richardson iteration for the matrix exponential", SIAM J. Sci. Comput. 35(3):A1376-A1397, 2013. DOI: https://doi.org/10.1137/110820191. arXiv: https://arxiv.org/abs/1112.5670.	Residual/defect viewpoint and restarted exponential Krylov context.
`[JiaLv2015]`	Z. Jia and H. Lv, "A posteriori error estimates of Krylov subspace approximations to matrix functions", Numer. Algorithms 69:1-28, 2015. DOI: https://doi.org/10.1007/s11075-014-9885-x. arXiv: https://arxiv.org/abs/1307.7219.	Justifies `xi_1` and `xi_2` a posteriori estimates, including the `phi_1` leading-term estimate.
`[JaweckiAuzingerKoch2020]`	T. Jawecki, W. Auzinger, and O. Koch, "Computable upper error bounds for Krylov approximations to matrix exponentials and associated phi-functions", BIT Numer. Math. 60:157-197, 2020. DOI: https://doi.org/10.1007/s10543-019-00782-z. arXiv: https://arxiv.org/abs/1809.03369.	Computable upper bound `h gamma_m t^m/m!`, Hermite quadrature estimate, effective-order quadrature, and step-size-control discussion.
`[WangYe2017]`	S. Wang and Q. Ye, "Error bounds for the Krylov subspace methods for computations of matrix exponentials", SIAM J. Matrix Anal. Appl. 38(1):155-187, 2017. arXiv: https://arxiv.org/abs/1603.07358.	Useful later for non-Hermitian Arnoldi exponential bounds; not yet measured by the current Hermitian probe.

Probe Scope

The probe reports two local recurrence variants:

implementation	role
full reorthogonalized Lanczos	Mirrors the native Hermitian exponential path closely enough to inspect estimator behavior using projected data.
legacy three-term Lanczos	Reproduces the simpler recurrence style used by older Lanczos exponential implementations, useful for detecting precision floors and loss-of-orthogonality sensitivity.

The example is diagnostic code, not a fallback policy. The native Hermitian action now uses the direct defect integral for adaptive nonexpansive acceptance, while the probe remains useful for comparing that choice against older tail, endpoint-defect, Hermite/quadrature, and HL-style indicators.

Future Estimators

Useful extensions not yet part of the native adaptive stopping rule:

| estimator | reason deferred | | — | — | | Trapezoid defect-integral estimate h |t| abs(e_m^* exp(t H_m)e_1) / 2 | Cheap middle ground between the endpoint defect and direct quadrature. It is only justified under effective-order assumptions, but is less optimistic than the Hermite 1/m factor. | | Improved Hermite quadrature from the defect derivative | Requires extra projected derivative terms and, in matrix-free form, may need one additional matvec. | | Effective-order quadrature | Useful for adaptive step-size control, but needs a robust definition of effective order near roundoff. | | Non-Hermitian defect/residual bounds | Needed before Arnoldi exponential actions can use defect-integral stopping as a bound. The projected scalar integral is only diagnostic without a nonexpansive or semigroup-growth bound; [WangYe2017] is the current starting point. |