Arnoldi projection for exponential and phi-function actions

Inputs: $A\in M_n(\mathbb{C})$ with $n\ge 1$, $v\in {\mathbb{C}}^n$, time $t\in \mathbb{R}$, integer $k\ge 0$, Krylov budget $1\le m\le n$, nonnegative threshold $\tau$, an exact-arithmetic Arnoldi run or a finite-precision Arnoldi backend, and a reduced-action configuration. In finite precision, the backend returns a length $j$ with $1\le j\le m$, $\beta >0$ for $v\mathrm{\ne }0$, $V_j\in {\mathbb{C}}^{n\times j}$, $H_j\in M_j(\mathbb{C})$, and a computed subdiagonal; it may supply the relation certificate in Step 2 and an exact happy-breakdown certificate verifying the full Arnoldi recurrence from $v$ through $j$ steps with zero terminal residual. The reduced-action configuration supplies, for each Taylor call, an exactly represented call time, a rational upper bound on its absolute value, rational operator-norm and input-vector norm upper bounds, a positive rational tolerance, finite scaling/order budgets, and a provider satisfying Normwise certified matrix-vector and vector-step arithmetic for Taylor calls, scalar scaling, and final projection. Missing or invalid bounds produce an inconclusive status.

Outputs: a vector candidate when data are available, reduced-coefficient error bound ${ϵ}_{\mathrm{r}\mathrm{e}\mathrm{d}}\in {\mathbb{R}}_{\ge 0}\cup \{+\mathrm{\infty }\}$, final-projection error bound ${\eta }_{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}}\in {\mathbb{Q}}_{\ge 0}\cup \{+\mathrm{\infty }\}$, Arnoldi data, any relation-error certificates, arnoldi_status (zero-start, happy-breakdown, projected, numerical-breakdown, inconclusive-arnoldi, or invalid-input), relation_status (exact, certified, inconclusive, or not-run), reduced_action_status (success, inconclusive, or not-run), and projection_status (exact, certified, inconclusive, or not-run). An invalid input may return no candidate.

Procedure:

1. Reject invalid dimensions, $m$, $k$, or threshold by returning no candidate, both error bounds $+\mathrm{\infty }$, no Arnoldi data or relation certificates, arnoldi_status=invalid-input, and relation_status=reduced_action_status=projection_status=not-run. If $v=0$, return the zero vector with both error bounds zero, arnoldi_status=zero-start, and relation_status=reduced_action_status=projection_status=not-run, without division.
2. In exact-arithmetic mode, run The Arnoldi process for building an orthonormal basis of a Krylov subspace for at most $m$ steps and retain $\beta :=\mathrm{\parallel }v{\mathrm{\parallel }}_2$, $V_j=[v_1,\dots ,v_j]$, and the leading square Hessenberg matrix $H_j$; set relation_status=exact. Stop when the Arnoldi residual norm is exactly zero and set arnoldi_status=happy-breakdown; otherwise continue until $m$ steps and leave the Arnoldi status pending. In finite-precision mode, if the backend data have invalid dimensions or fail $\beta >0$ for $v\mathrm{\ne }0$, return the zero candidate, both bounds $+\mathrm{\infty }$, arnoldi_status=inconclusive-arnoldi, relation_status=inconclusive, and reduced_action_status=projection_status=not-run. Otherwise use the backend's $V_j,H_j,\beta$ and computed subdiagonal. When available, it supplies a relation certificate $A{V}_j=V_j{H}_j+h_{j+1,j}{v}_{j+1}{e}_j^{\ast }+D+\mathrm{\Omega }$, with $\mathrm{\parallel }{v}_{j+1}{\mathrm{\parallel }}_2=1$, column bounds $\mathrm{\parallel }{d}_i{\mathrm{\parallel }}_2\le {\eta }_i$ for $D=[d_1,\dots ,d_j]$, and a norm bound for $\mathrm{\Omega }$. Set relation_status=certified if this relation is supplied and verified, and relation_status=inconclusive otherwise. A verified exact happy-breakdown certificate sets relation_status=certified and arnoldi_status=happy-breakdown; it verifies the exact Arnoldi recurrence, including $A{V}_j=V_j{H}_j$ and $v=\beta V_j{e}_1$. Without that full certificate, a computed subdiagonal at most $\tau$ sets arnoldi_status=numerical-breakdown; continue to the reduced evaluation but never label it happy breakdown. Otherwise leave the Arnoldi status pending until Step 5.
3. For $k=0$, request a reduced configuration for $(H_j,\beta e_1,t)$. If it is missing or invalid, including if it cannot supply the time $t$ exactly and a rational upper bound on $\mathrm{\mid }t\mathrm{\mid }$, treat the Taylor request as returning no candidate without calling it. Otherwise call Finite scaled Taylor action for the matrix exponential to approximate $e^{t{H}_j}\beta e_1$. If no reduced candidate is available, set ${ϵ}_{\mathrm{r}\mathrm{e}\mathrm{d}}={\eta }_{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}}=+\mathrm{\infty }$, set reduced_action_status=inconclusive and projection_status=not-run, set a pending arnoldi_status to inconclusive-arnoldi when relation_status=inconclusive and to projected otherwise, and return no candidate. Otherwise set $\widehat{c}$ and ${ϵ}_{\mathrm{r}\mathrm{e}\mathrm{d}}$ to its candidate and bound. Set reduced_action_status=success only when the Taylor status is success; otherwise set it to inconclusive. Continue to Step 5 with the returned candidate in either case.
4. For $k\ge 1$, form $B_k(t{H}_j,e_1)$ from Phi recurrence and augmented-matrix action identities, request a reduced configuration for $(B_k(t{H}_j,e_1),e_{j+k},1)$. If it is missing or invalid, treat the Taylor request as returning no candidate without calling it. Otherwise call Finite scaled Taylor action for the matrix exponential with the exactly represented time $1$ and rational time bound $1$. If no reduced candidate is available, set ${ϵ}_{\mathrm{r}\mathrm{e}\mathrm{d}}={\eta }_{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}}=+\mathrm{\infty }$, set reduced_action_status=inconclusive and projection_status=not-run, set a pending arnoldi_status to inconclusive-arnoldi when relation_status=inconclusive and to projected otherwise, and return no candidate. Otherwise let $q$ be the first $j$ entries of its candidate and use the certified vector-scaling provider with scalar $\beta$ to obtain $\widehat{c}$ and local bound ${\xi }_{\beta }$. Set ${ϵ}_{\mathrm{r}\mathrm{e}\mathrm{d}}:=\beta {ϵ}_{\mathrm{a}\mathrm{u}\mathrm{g}}+{\xi }_{\beta }$, where ${ϵ}_{\mathrm{a}\mathrm{u}\mathrm{g}}$ is the augmented Taylor bound; if either certificate is missing set ${ϵ}_{\mathrm{r}\mathrm{e}\mathrm{d}}=+\mathrm{\infty }$ and reduced_action_status=inconclusive. Set reduced_action_status=success when the Taylor status is success and the scaling certificate is sound; otherwise set it to inconclusive. Continue to Step 5 in either case.
5. In exact-arithmetic mode set $x:=V_j\widehat{c}$, ${\eta }_{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}}=0$, and projection_status=exact. In finite precision, use the certified matrix-vector provider with matrix $V_j$, scalar $1$, and vector $\widehat{c}$ to return $x$ and bound ${\eta }_{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}}$; a sound certificate sets projection_status=certified, and missing certification sets ${\eta }_{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}}=+\mathrm{\infty }$ and projection_status=inconclusive. Return $x$. If arnoldi_status is still pending, set it to inconclusive-arnoldi when relation_status=inconclusive and projected otherwise.

In exact arithmetic the projected formula is $\beta V_j{\phi }_k(t{H}_j)e_1$. Exact happy breakdown means the generated Krylov subspace is invariant and this projected formula equals ${\phi }_k(tA)v$; a finite Taylor evaluation still has its separately reported reduced-action error. A numerical-breakdown status asserts no exact invariance.

Input validation precedes the norm division. The zero-start branch returns immediately, and in the exact The Arnoldi process for building an orthonormal basis of a Krylov subspace every basis normalization occurs only after a nonzero residual norm has been established. The Arnoldi loop stops after at most $m$ steps. The finite-precision backend must supply a finite basis and reduced matrix; absent a verified relation, relation_status=inconclusive preserves that gap. Each reduced action delegates to the finite-budget Finite scaled Taylor action for the matrix exponential, whose missing-bound and exhausted-budget outcomes are returned as inconclusive.

happy-breakdown is assigned only in exact arithmetic after a zero Arnoldi residual, or from a verified exact Arnoldi-recurrence certificate with zero terminal residual. A small computed subdiagonal without that full certificate is numerical-breakdown; it makes no claim about whether the true subspace is invariant. The four statuses separately report Arnoldi breakdown evidence, relation certification, reduced-action certification, and final-projection certification. For $k=0$, the local action-error theorem adds the reduced-coefficient and final-projection error bounds under its hypotheses. For $k\ge 1$, the local Taylor theorem certifies the augmented reduced action, and the vector-scaling provider certifies multiplication by $\beta$.