Size the BlockLanczos thick restart by R_size instead of the initial block size

[leburgel]

When looking into why my attempted warm-start of BlockLanczos in #170 still required more applications than a cold start, I stumbled into some inconsistencies in the shrinking procedure compared to a regular Lanczos run.

The BlockLanczos restart used the initial block size bs to set how many vectors to keep and which rows of U couple to the residual. It should use the current residual block size fact.R_size, which becomes smaller than bs whenever block_qr! drops vectors. This PR switches the shrink branch to R_size.

The restarted factorization was already correct. The extra rows of U don't couple to the residual and don't change the kept subspace. But keep was rounded down to a multiple of bs. For example, with bs = 6 and krylovdim = 20, it stayed at 12, whereas Lanczos keeps 12 to 14 as vectors converge. With this change, a block that has shrunk to R_size = 1 restarts the same way as Lanczos. When R_size == bs, the first commit changes nothing; see the second commit below.

The reproducer below starts from five exact eigenvectors plus a random vector, so R_size drops to 1 after the first step:

using KrylovKit, LinearAlgebra, Random
Random.seed!(1)
n = 4000
A = SymTridiagonal(2 .+ 0.5 .* rand(n), -ones(n - 1))
opts = (; tol = 1e-10, ishermitian = true, verbosity = 0, maxiter = 10_000)

E = eigen(A, 1:5).vectors
v = rand(n)
x₀ = Block([[E[:, i] for i in 1:5]; [v]])

for kd in (20, 30, 60)
    lanczos = eigsolve(A, v, 6, :SR; opts..., krylovdim = kd)[3]
    block = eigsolve(A, copy(x₀), 6, :SR; opts..., krylovdim = kd)[3]
    println("krylovdim = $kd: Lanczos(v) numops = $(lanczos.numops), BlockLanczos(x₀) numops = $(block.numops), converged = $(block.converged)")
end

krylovdim	Lanczos(v)	BlockLanczos (master)	BlockLanczos (this PR)
20	905	1229	993
30	633	535	541
60	555	445	435

This matters most together with #170. Take a warm start from the output vectors of an earlier unconverged run at krylovdim = 20: it needs 1173 matvecs with #170 alone and 749 with both PRs, compared with 886 for a cold start.

Second commit: never keep fewer vectors than have converged. Lanczos keeps div(3 * krylovdim + 2 * converged, 5) vectors, which always satisfies converged <= keep < krylovdim. Rounding that down to a multiple of R_size can take it below converged, and the restart then throws away Ritz vectors that had already converged. This needs a residual block that stays large while many vectors converge, which typically means degenerate eigenvalues, and krylovdim - converged < 5(R_size - 1)/3. With the default krylovdim = 100 and R_size = 16, that only happens once more than 75 vectors have converged. In that case the restart now falls back to the unrounded Lanczos value. Otherwise nothing changes, and the reproducer above gives identical numbers.

using KrylovKit, LinearAlgebra, Random
BLAS.set_num_threads(1)  # reproducible numbers
Random.seed!(2)
n, b, mult = 240, 16, 16
λ = repeat(vcat(1.0:5.0, 200 .+ 10 .* rand(10)), inner = mult)  # 15 distinct eigenvalues, each 16-fold
Q = Matrix(qr(randn(n, n)).Q)
A = Hermitian(Q * Diagonal(λ) * Q')
x₀ = Block([randn(n) for _ in 1:b])
alg = BlockLanczos(; krylovdim = 40, maxiter = 300, tol = 1e-10)
_, _, info = eigsolve(A, x₀, 32, :SR, alg)
@show info.converged info.numiter info.numops

	converged	numiter	numops
first commit only	32	218	4271
both commits	32	138	3625

Without the second commit, restart 111 has converged = 28 with R_size = 12, which gives keep = 24, and the next restart counts only 24 converged.

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 86.65%. Comparing base (531b074) to head (17acdcb).

Additional details and impacted files

@@           Coverage Diff           @@
##           master     #172   +/-   ##
=======================================
  Coverage   86.65%   86.65%           
=======================================
  Files          36       36           
  Lines        3918     3919    +1     
=======================================
+ Hits         3395     3396    +1     
  Misses        523      523           

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