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Arnoldi action error is the semigroup-weighted residual integral

Statement

Let $A \in M_{n} (C)$ , $n \geq 1$ , $v \in C^{n}$ , $β \geq 0$ , $V \in M_{n \times m} (C)$ , $H \in M_{m} (C)$ , and $1 \leq m \leq n$ . For exact Arnoldi, $β = ∥ v ∥_{2}$ ; in a certified finite-precision run, $β$ may be the backend's returned nonnegative scalar, with its effect included in the initial mismatch. Put $c (s) : = e^{s H} β e_{1}$ and $u (s) : = V c (s)$ . For every real $t$ , the exact defect identity is

$e^{t A} v - u (t) = e^{t A} (v - β V e_{1}) + \int_{0}^{t} e^{(t - s) A} (A V - V H) c (s) d s,$

where the integral is oriented. Let $I_{t} = [0, t]$ for $t \geq 0$ and $I_{t} = [t, 0]$ for $t < 0$ ; let $σ_{t} = 1$ for $t \geq 0$ and $σ_{t} = - 1$ for $t < 0$ . If $t = 0$ the integral is zero. Suppose the Arnoldi relation is

$A V = V H + h v_{m + 1} e_{m}^{*} + D + Ω,$

where $h \geq 0$ , $∥ v_{m + 1} ∥_{2} = 1$ when $h > 0$ , $D = [d_{1}, \dots, d_{m}]$ , $η_{j} \geq 0$ , $∥ d_{j} ∥_{2} \leq η_{j}$ , $η_{o r t h} \geq 0$ , and $∥ Ω ∥_{2} \leq η_{o r t h}$ . In exact happy breakdown, interpret the rank-one term as zero and set $D = Ω = 0$ . If a reduced computation returns $\hat{c} (t)$ with $∥ \hat{c} (t) - c (t) ∥_{2} \leq ϵ_{r e d} (t)$ for $ϵ_{r e d} (t) \geq 0$ and its final projection returns $\hat{x}$ with $∥ \hat{x} - V \hat{c} (t) ∥_{2} \leq η_{p r o j}$ for $η_{p r o j} \geq 0$ , then

$∥ e^{t A} v - \hat{x} ∥_{2} \leq e^{∣ t ∣ μ_{2} (σ_{t} A)} ∥ v - β V e_{1} ∥_{2} + \int_{I_{t}} e^{∣ t - s ∣ μ_{2} (σ_{t} A)} (h ∣ e_{m}^{*} c (s) ∣ + (\sqrt{\sum_{j = 1}^{m} η_{j}^{2}} + η_{o r t h}) ∥ c (s) ∥_{2}) d s + ∥ V ∥_{2} ϵ_{r e d} (t) + η_{p r o j} .$

For exact Arnoldi from a nonzero start with no breakdown through $m$ , $D = Ω = 0$ , $V$ has orthonormal columns, and $v = β V e_{1}$ , recovering the standard residual integral. Whenever $A V = V H$ and $v = β V e_{1}$ , the invariant-subspace relation gives $φ_{k} (t A) v = β V φ_{k} (t H) e_{1}$ for every $k \geq 0$ and every real $t$ ; exact happy breakdown of Arnoldi from a nonzero start is a sufficient case.

Facts & Assumptions

Given: The matrices and vectors in the Statement; for the numerical relation clause, the stated decomposition and columnwise error bounds; for the reduced term, a certified bound $ϵ_{r e d} (t)$ and a final projection certificate $∥ \hat{x} - V \hat{c} (t) ∥_{2} \leq η_{p r o j}$ .

[F1]

Variation of constants for a finite-matrix action defect: for a continuously differentiable approximate trajectory $u$ , the exact action error is its initial error plus the oriented integral of $e^{(t - s) A} (A u - u^{'})$ .

[F2]

The logarithmic norm bounds forward and reverse matrix exponentials: for $τ \geq 0$ , $∥ e^{τ B} ∥_{2} \leq e^{τ μ_{2} (B)}$ for every finite complex matrix $B$ .

[F3]

Before breakdown, Arnoldi produces an orthonormal Krylov basis and a rectangular upper-Hessenberg factorization: before breakdown, Arnoldi columns are orthonormal and $A V_{m} = V_{m} H_{m} + h_{m + 1, m} v_{m + 1} e_{m}^{*}$ .

[F4]

The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv||: for a finite-dimensional linear map $T$ , $∥ T ∥_{2} = \max_{∥ x ∥_{2} = 1} ∥ T x ∥_{2}$ ; rescaling gives $∥ T x ∥_{2} \leq ∥ T ∥_{2} ∥ x ∥_{2}$ for all $x$ .

[F5]

Cauchy–Schwarz: $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ , with equality exactly for linearly dependent vectors: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ in a complex Euclidean space.

[F6]

The Arnoldi process for building an orthonormal basis of a Krylov subspace: Arnoldi starts with $v_{1} = v / ∥ v ∥_{2}$ , defines $w_{j} = A v_{j} - \sum_{i = 1}^{j} h_{i j} v_{i}$ , and exact breakdown at step $j$ means $h_{j + 1, j} = ∥ w_{j} ∥_{2} = 0$ .

[F7]

The exponential series of a bounded operator: for a bounded operator on a Banach space, its exponential is differentiable with derivative $A e^{t A}$ .

[F8]

The phi-function family by everywhere-convergent series and The scalar and matrix phi series converge absolutely: every finite-matrix phi function is defined by an absolutely convergent power series.

[F9]

Every finite-dimensional normed space is Banach: each finite-dimensional normed space over $C$ is complete.

[A1]

For a finite-precision relation, $A V = V H + h v_{m + 1} e_{m}^{*} + D + Ω$ and all displayed norm bounds are certified data; the theorem does not infer them from a small computed subdiagonal.

[A2]

The final computed projection satisfies $∥ \hat{x} - V \hat{c} (t) ∥_{2} \leq η_{p r o j}$ for a nonnegative bound $η_{p r o j}$ , with $η_{p r o j} = 0$ when the projection is exact.

Proof

technique · direct

1.1givenF4F7F9algebra

For any nonzero $x$ , rescaling $x / ∥ x ∥_{2}$ in [F4] shows that the finite matrix $H$ is bounded; the inequality is immediate at $x = 0$ . By [F9], $C^{m}$ is complete, so [F7] applies and gives $c^{'} (s) = H c (s)$ . Hence $u (s) = V c (s)$ is continuously differentiable and its defect is $A u (s) - u^{'} (s) = (A V - V H) c (s)$ .

1.2givenF2algebra

For $s \in I_{t}$ , $∣ t - s ∣ \geq 0$ and $e^{(t - s) A} = e^{∣ t - s ∣ σ_{t} A}$ ; the same identity gives $e^{t A} = e^{∣ t ∣ σ_{t} A}$ . Thus [F2] bounds the initial factor by $e^{∣ t ∣ μ_{2} (σ_{t} A)}$ and the integral kernel by $e^{∣ t - s ∣ μ_{2} (σ_{t} A)}$ .

2.1step 1.1F1

Apply [F1] to this $u$ . It gives the displayed exact integral identity, including the initial mismatch $v - u (0) = v - β V e_{1}$ .

3.1givenA1F4F5step 2.1step 1.2algebra

Substitute the certified relation [A1] into the defect in step 2.1. The rank-one term has norm $h ∣ e_{m}^{*} c (s) ∣$ because $∥ v_{m + 1} ∥_{2} = 1$ . For any unit vector $x \in C^{m}$ , Cauchy–Schwarz [F5] gives $∥ D x ∥_{2} \leq \sum_{j} ∣ x_{j} ∣ η_{j} \leq (\sum_{j} η_{j}^{2})^{1 / 2}$ ; hence $∥ D ∥_{2} \leq (\sum_{j} η_{j}^{2})^{1 / 2}$ . The remaining term is at most $η_{o r t h} ∥ c (s) ∥_{2}$ . Applying the kernel bound of step 1.2 inside the integral proves the first two terms of the claimed estimate.

3.2givenF6F8step 2.1algebrainduction

Under the exact invariant-subspace hypotheses $A V = V H$ and $v = β V e_{1}$ in the Statement, induction gives $A^{j} V = V H^{j}$ for every $j \geq 0$ : it holds at $j = 0$ , and left multiplication by $A$ gives $A^{j + 1} V = A V H^{j} = V H H^{j} = V H^{j + 1}$ . Applying the convergent series [F8] term by term yields $φ_{k} (t A) v = β V φ_{k} (t H) e_{1}$ for every $k \geq 0$ . To verify the stated exact Arnoldi case, each earlier column satisfies $A v_{ℓ} = \sum_{i = 1}^{ℓ + 1} h_{i ℓ} v_{i}$ by the recurrence, while the last satisfies $A v_{m} = \sum_{i = 1}^{m} h_{i m} v_{i}$ because [F6] gives $w_{m} = 0$ ; hence $A V = V H$ , and the initialization in [F6] gives $v = β V e_{1}$ . This proves exact invariant-space projection for every phi action; the separate $ϵ_{r e d} (t)$ hypothesis in this theorem concerns the exponential trajectory $c (s)$ .

4.1givenA2F4step 3.1algebra

The reduced-action error satisfies $∥ V (\hat{c} (t) - c (t)) ∥_{2} \leq ∥ V ∥_{2} ϵ_{r e d} (t)$ by [F4]. By [A2] and the triangle inequality, replacing $V \hat{c} (t)$ by $\hat{x}$ adds at most $η_{p r o j}$ . Adding both errors to the exact projected-action error from step 3.1 proves the claimed estimate.

5.1givenF3F6algebra∎

In exact Arnoldi without breakdown, [F3] supplies the orthonormal basis and the relation $A V = V H + h v_{m + 1} e_{m}^{*}$ , so $D = Ω = 0$ ; [F6] gives $v = β V e_{1}$ . Orthonormality gives $∥ V ∥_{2} = 1$ , and an exact reduced action has $ϵ_{r e d} = 0$ . Substitution in the proved identity and bound leaves exactly the usual initial-zero residual integral and its semigroup-weighted estimate.

Remarks

The residual integral is exact; its upper bound is conditional on the displayed semigroup, matvec, orthogonalization, initial-vector, reduced-action, and final-projection data. A near-zero computed Arnoldi residual without an exact-zero or relation certificate never establishes happy breakdown. No mesh-uniform estimate follows automatically from this finite-dimensional result.

Depends on

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Sources

Nicholas J. Higham and Lijing Lin, Functions of Matrices: A Short Course (standard reference, not scraped)
Jitse Niesen and Will M. Wright, A Krylov subspace algorithm for evaluating the phi-functions appearing in exponential integrators (standard reference, not scraped)
Nicholas J. Higham, What Is the Logarithmic Norm? (standard reference, not scraped)