mode=:error_estimate gives wrong results for long times

[axsk]

Describe the bug 🐞

expv(t, A, b, mode=:error_estimate) gives vastly different results for longer time-steps.

Expected behavior

I would expect error_estimate and happy_breakdown results to at least have the same magnitudes

Minimal Reproducible Example 👇

A = I - sprand(100,100,.1) * .1
b = rand(100)
@assert expv(10, A,b) ≈ expv(10, A,b ,mode=:error_estimate)

Error & Stacktrace ⚠️

julia>  expv(10, A,b)
100-element Vector{Float64}:
  19372.458459100264
 -57030.13477800453
   3852.973450336655
  -7684.450057227177

julia>  expv(10, A,b ,mode=:error_estimate)
100-element Vector{Float64}:
 -3.1939130045290945e15
  4.314219301303509e15
 -1.6738861310296682e15
 -4.276422508317541e15

Environment (please complete the following information):

• Output of using Pkg; Pkg.status()

[d4d017d3] ExponentialUtilities v1.25.0

• Output of versioninfo()

julia> versioninfo()
Julia Version 1.10.0
Commit 3120989f39b (2023-12-25 18:01 UTC)
Build Info:
  Official https://julialang.org/ release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 64 × AMD EPYC 7502P 32-Core Processor
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-15.0.7 (ORCJIT, znver2)
  Threads: 11 on 64 virtual cores
Environment:
  JULIA_HISTORY = ./.history.jl_

[daviehh]

It's been a while since I last read through the code, but iirc the error estimation routine only implements the Lanczos algorithm, i.e. mostly for Hermitian (or, in your real-valued case, symmetric) matrices.

ExponentialUtilities.jl/src/krylov_phiv_error_estimate.jl

Line 3 in 18e5161

# Currently only expv for Lanczos is implemented.

ExponentialUtilities.jl/src/krylov_phiv_error_estimate.jl

Lines 86 to 101 in 18e5161

	for j in 1:m
	lanczos_step!(j, A, V, α, β)
	expT!(@view(α[1:j]), @view(β[1:j]), t, cache)
	# This is practical error estimate Er₂ from
	#
	# Saad, Y. (1992). Analysis of some Krylov subspace
	# approximations. SIAM Journal on Numerical Analysis.
	σ = β[j] * Ks.beta * abs(cache.v[j])
	verbose && @printf("iter %d, α[%d] %e, β[%d] %e, σ %e\n", j, j, α[j], j, β[j], σ)
	if σ < ε
	Ks.m = j
	break
	end
	end
	verbose && println("Krylov subspace size: ", Ks.m)

Note with your example, with a symmetric A A += A' the results do agree w/

expv(10, A, b) - expv(10, A, b, mode=:error_estimate)

--
more details:
the non error-estimation expv call uses the arnoldi method which only switches to the Lanczos step based on the isherimtian argument

ExponentialUtilities.jl/src/arnoldi.jl

Lines 246 to 247 in 18e5161

	ishermitian &&
	return lanczos!(Ks, A, b; tol = tol, m = m, init = init, t = t, mu = mu, l = l)

however, the implementation of the error-estimation just checks the KrylovSubspace type

ExponentialUtilities.jl/src/krylov_phiv_error_estimate.jl

Lines 37 to 39 in 18e5161

	function get_subspace_cache(Ks::KrylovSubspace{T, U}) where {T, U <: Complex}
	error("Subspace exponential caches not yet available for non-Hermitian matrices.")
	end

[ChrisRackauckas]

So if you do :error_estimate and are non Hermition, we should throw an error saying that it's an inappropriate choice of error estimator?

[daviehh]

@ChrisRackauckas Yes, would make sense since it assumes Lanczos/tridagonal KrylovSubspace.H to use with the stegr call

ExponentialUtilities.jl/src/krylov_phiv_error_estimate.jl

Line 29 in 18e5161

LAPACK.stegr!(α, β, cache.sw)

correction in PR. Thanks!