A one-dimensional Arnoldi projection misses a slow mode of a stiff diagonal matrix

Given: The stated real diagonal matrix, nonzero starting vector, time, and exact-arithmetic Arnoldi run with $m=1$.

The Arnoldi process for building an orthonormal basis of a Krylov subspace and Before breakdown, Arnoldi produces an orthonormal Krylov basis and a rectangular upper-Hessenberg factorization: the first Arnoldi vector is $q_1=v\mathrm{/}\mathrm{\parallel }v{\mathrm{\parallel }}_2$, $H_1=q_1^{\ast }A{q}_1$, and $A{q}_1=H_1{q}_1+h_{2,1}{q}_2$ when the residual is nonzero.

Arnoldi projection for exponential and phi-function actions: for $k=0$, the exact projected action is $\mathrm{\parallel }v{\mathrm{\parallel }}_2{V}_1{e}^{t{H}_1}{e}_1$.

Arnoldi action error is the semigroup-weighted residual integral: with exact Arnoldi and exact reduced action, zero initial mismatch, and $D=\mathrm{\Omega }=0$, the error is bounded by the semigroup-weighted integral of $h_{2,1}\mathrm{\mid }{e}_1^{\ast }c(s)\mathrm{\mid }$, where $c(s)=e^{s{H}_1}\mathrm{\parallel }v{\mathrm{\parallel }}_2{e}_1$.

The phi-function family by everywhere-convergent series: the diagonal exponential acts entrywise by the scalar exponential series.

The Euclidean logarithmic norm of a complex matrix: ${\mu }_2(A)$ is the largest eigenvalue of the Hermitian part $(A+A^{\ast })\mathrm{/}2$.

The exponential is a continuous bijection from $\mathbb{R}$ onto $(0,\mathrm{\infty })$: the real exponential is injective on $\mathbb{R}$.

$\exp (z+w)=\exp z\exp w$, and the complex exponential extends the real exponential: the complex exponential satisfies its addition law and agrees with the real exponential on real inputs.

The exponential function is smooth and $(\exp {)}^{\mathrm{\prime }}=\exp$: $(\exp x{)}^{\mathrm{\prime }}=\exp x$ for real $x$.

The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g(c)$, then $f\circ g$ is differentiable at $c$ with $(f\circ g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(g(c))g^{\mathrm{\prime }}(c)$: differentiating a scalar exponential composed with a real affine map multiplies by that affine slope.

If $f:[a,b]\to {\mathbb{R}}^m$ is differentiable with integrable $f^{\mathrm{\prime }}$ then ${\int }_a^b{f}^{\mathrm{\prime }}=f(b)-f(a)$; and a bounded derivative makes $f$ Lipschitz: for a continuously differentiable real scalar function, its definite integral is the endpoint difference of an antiderivative.

A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion: a continuous real-valued function on a nondegenerate closed bounded interval is Riemann integrable.