Randomized Krylov-Schur Eigsolver

• Randomized Krylov-Schur (rKS) is a randomized eigensolver that extracts a few eigenpairs from large matrices using low-dimensional sketching.
• It replaces costly full-dimensional orthogonalization with sketch-based methods, ensuring stable eigenvalue extraction via Schur reordering and restart.
• The method incorporates deflation and locking of converged eigenpairs, accelerating convergence and maintaining backward stability in large-scale eigenproblems.

Searching arXiv for the rKS paper and related randomized orthogonalization references mentioned in the source material. First, locating the main paper by arXiv id. Randomized Krylov–Schur (rKS) is a randomized eigensolver for large-scale eigenvalue problems that seeks a small subset of eigenpairs of a matrix $A\in\mathbb R^{n\times n}$ with $n\gg 1$. It is introduced as a variant of the Krylov–Schur framework in which full-dimensional orthogonalization is replaced by low-dimensional sketching, while Schur truncation, restart, and a practical deflation technique for converged eigenpairs are retained. The method is designed for settings in which only a few largest-magnitude or interior eigenvalues are needed, and a full Schur or QR factorization is impractical. Its defining features are sketch-orthogonalization in a reduced dimension, stable reordering of small Schur factorizations, and locking of converged Ritz vectors so that the eigenspace associated with a prescribed spectral region can be computed efficiently (Damas et al., 7 Aug 2025).

1. Problem formulation and relation to Krylov–Schur

The method addresses the eigenvalue problem

$$A x = \lambda x,\qquad A\in\mathbb R^{n\times n},\; n\gg 1,$$

under the assumption that only a small subset $k\ll n$ of eigenpairs is sought. The motivating examples stated for this regime include structural mechanics, quantum chemistry, and data science, where the dimension is too large for direct dense factorizations and only part of the spectrum is relevant (Damas et al., 7 Aug 2025).

Like classical Krylov methods, rKS operates through low-dimensional projection onto a Krylov subspace,

$$\mathcal K_m(A,v_1)=\operatorname{span}\{v_1,Av_1,\dots,A^{m-1}v_1\},\quad m\ll n,$$

from which approximate eigenpairs are extracted by Rayleigh–Ritz projection. The description explicitly places rKS in the lineage of the classical Krylov–Schur method of Stewart (2002), which alternates between constructing an Arnoldi-like decomposition of length $m$, computing a small Schur decomposition and truncating unwanted directions, and restarting by re-expanding back to length $m$ (Damas et al., 7 Aug 2025).

The principal distinction is computational. In the classical setting, repeated orthonormalization of $m$ vectors of length $n$ incurs a cost of approximately $O(nm^2)$, while each restart also requires an $n\gg 1$0 Schur factorization with cost $n\gg 1$1. rKS modifies the orthogonalization stage by maintaining a sketch of the basis in a much lower-dimensional space, so that orthogonality is checked and enforced in the sketch rather than directly in $n\gg 1$2. This suggests that rKS should be viewed not as a new spectral projection principle, but as a randomized implementation of the existing Krylov–Schur cycle with a different orthogonalization mechanism (Damas et al., 7 Aug 2025).

2. Randomized sketching and the Arnoldi factorization

The randomization mechanism is formulated through an $n\gg 1$3-subspace embedding $n\gg 1$4, with $n\gg 1$5, such that for every $n\gg 1$6 in the relevant $n\gg 1$7-dimensional subspace,

$n\gg 1$8

By Johnson–Lindenstrauss arguments, the description states that a Gaussian or sparse-sign $n\gg 1$9 of size $$A x = \lambda x,\qquad A\in\mathbb R^{n\times n},\; n\gg 1,$$0 has this property with high probability, so orthogonality in the sketch reflects true orthogonality up to $$A x = \lambda x,\qquad A\in\mathbb R^{n\times n},\; n\gg 1,$$1 distortion (Damas et al., 7 Aug 2025).

The algorithm begins by drawing an oblivious $$A x = \lambda x,\qquad A\in\mathbb R^{n\times n},\; n\gg 1,$$2-embedding $$A x = \lambda x,\qquad A\in\mathbb R^{n\times n},\; n\gg 1,$$3 with $$A x = \lambda x,\qquad A\in\mathbb R^{n\times n},\; n\gg 1,$$4, for example using sparse $$A x = \lambda x,\qquad A\in\mathbb R^{n\times n},\; n\gg 1,$$5 signs, and by selecting a random unit vector $$A x = \lambda x,\qquad A\in\mathbb R^{n\times n},\; n\gg 1,$$6 whose sketch is normalized so that $$A x = \lambda x,\qquad A\in\mathbb R^{n\times n},\; n\gg 1,$$7. It then constructs a length-$$A x = \lambda x,\qquad A\in\mathbb R^{n\times n},\; n\gg 1,$$8 randomized Arnoldi factorization

$$A x = \lambda x,\qquad A\in\mathbb R^{n\times n},\; n\gg 1,$$9

where $k\ll n$0 is sketch-orthonormal and $k\ll n$1 is upper-Hessenberg (Damas et al., 7 Aug 2025).

The parallel sketch variables are

$k\ll n$2

At each step $k\ll n$3, the method computes $k\ll n$4, forms its sketch $k\ll n$5, projects in the sketch space via

$k\ll n$6

uses the resulting coefficients in the high-dimensional update

$k\ll n$7

and normalizes by the sketched norm,

$k\ll n$8

After augmentation of $k\ll n$9 and $$\mathcal K_m(A,v_1)=\operatorname{span}\{v_1,Av_1,\dots,A^{m-1}v_1\},\quad m\ll n,$$0, the resulting factorization satisfies

$$\mathcal K_m(A,v_1)=\operatorname{span}\{v_1,Av_1,\dots,A^{m-1}v_1\},\quad m\ll n,$$1

with

$$\mathcal K_m(A,v_1)=\operatorname{span}\{v_1,Av_1,\dots,A^{m-1}v_1\},\quad m\ll n,$$2

in the stated formulation (Damas et al., 7 Aug 2025).

The reduction in orthogonalization cost is central. Replacing full Gram–Schmidt on $$\mathcal K_m(A,v_1)=\operatorname{span}\{v_1,Av_1,\dots,A^{m-1}v_1\},\quad m\ll n,$$3-vectors by Gram–Schmidt on $$\mathcal K_m(A,v_1)=\operatorname{span}\{v_1,Av_1,\dots,A^{m-1}v_1\},\quad m\ll n,$$4-dimensional sketches changes the orthogonalization cost from $$\mathcal K_m(A,v_1)=\operatorname{span}\{v_1,Av_1,\dots,A^{m-1}v_1\},\quad m\ll n,$$5 to $$\mathcal K_m(A,v_1)=\operatorname{span}\{v_1,Av_1,\dots,A^{m-1}v_1\},\quad m\ll n,$$6, with only an $$\mathcal K_m(A,v_1)=\operatorname{span}\{v_1,Av_1,\dots,A^{m-1}v_1\},\quad m\ll n,$$7 degradation in conditioning. In the paper’s formulation, the randomized Arnoldi basis is therefore controlled through the geometry of the sketch rather than through explicit Euclidean orthogonalization in the ambient space (Damas et al., 7 Aug 2025).

3. Schur contraction, Ritz extraction, and restart

Once the length-$$\mathcal K_m(A,v_1)=\operatorname{span}\{v_1,Av_1,\dots,A^{m-1}v_1\},\quad m\ll n,$$8 factorization is available, rKS returns to a standard Krylov–Schur pattern in reduced dimension. The projected Hessenberg matrix $$\mathcal K_m(A,v_1)=\operatorname{span}\{v_1,Av_1,\dots,A^{m-1}v_1\},\quad m\ll n,$$9 is factorized in real Schur form,

$m$0

where $m$1 is orthonormal and $m$2 is block-upper triangular. The Schur form is then reordered so that the $m$3 wanted Ritz values occupy the leading $m$4 block. Writing $m$5 with $m$6, the method contracts the decomposition to

$m$7

so that

$m$8

The small eigenproblem

$m$9

is then solved, and each eigenpair $m$0 lifts to the Ritz pair

$m$1

(Damas et al., 7 Aug 2025).

The paper’s pseudocode presents the rKS cycle as five stages: expansion to length $m$2 via randomized Arnoldi; Schur factorization and reordering; Ritz extraction and residual checking; deflation or locking of converged Ritz pairs; and contraction to the unlocked invariant component before the next restart. In that pseudocode, the residual proxy is written as

$m$3

with reference to equation (2.33) in the paper, and Ritz vectors with $m$4 are locked (Damas et al., 7 Aug 2025).

This design preserves the Arnoldi $m$5 Schur $m$6 truncate $m$7 restart logic of classical Krylov–Schur while moving the most expensive orthogonalization work into the sketch space. A plausible implication is that the method’s behavior remains structurally familiar to practitioners of implicitly restarted Arnoldi and Krylov–Schur, even though the basis management strategy is randomized.

4. Deflation and locking of converged eigenpairs

A distinguishing feature of the method is its practical deflation technique for converged eigenpairs. In the single-vector case, after contraction the decomposition is assumed to have the form

$m$8

with $m$9. Under this condition, $m$0 approximately satisfies $m$1, and the vector is locked by replacing $m$2 with the deflated operator

$m$3

The description states that $m$4 is then an exact Schur vector of $m$5 with eigenvalue $m$6, while the remaining factorization becomes

$m$7

Subsequent Arnoldi steps enforce orthogonality to the locked vector through

$m$8

(Damas et al., 7 Aug 2025).

The subspace version generalizes this mechanism. If $m$9 vectors $n$0 have converged, the projector is formed as

$n$1

and the operator is replaced by

$n$2

The stated consequence is that the locked subspace $n$3 remains invariant under $n$4, with zero eigenvalues in $n$5, while the rKS cycle continues on the complement $n$6 (Damas et al., 7 Aug 2025).

In operational terms, the deflation mechanism reduces the active subspace dimension after convergence of part of the spectrum. The numerical summary explicitly reports that locking converged vectors reduced subsequent subspace dimension and accelerated convergence, with backward error consistent with theory (Damas et al., 7 Aug 2025). This suggests that deflation is not merely an implementation convenience, but an integral component of the method’s scalability when a sequence of targeted eigenpairs is required.

5. Complexity, conditioning, and stability

The per-restart computational cost is decomposed into matrix–vector products, sketching, low-dimensional orthogonalization, and a small Schur factorization. The paper states the following costs for one cycle: $n$7 products $n$8 with complexity $n$9; sketch formation $O(nm^2)$0 at $O(nm^2)$1 per step, or $O(nm^2)$2 if $O(nm^2)$3 is very sparse; computation of $O(nm^2)$4 and related Gram–Schmidt operations in sketch space at $O(nm^2)$5; and Schur factorization of $O(nm^2)$6 at $O(nm^2)$7. The resulting total is summarized as

$O(nm^2)$8

in contrast to the classical cost

$O(nm^2)$9

(Damas et al., 7 Aug 2025).

The stability discussion is likewise phrased in terms of the sketch. Because $n\gg 1$00 is an $n\gg 1$01-embedding for the evolving Krylov subspace, the sketched basis satisfies

$n\gg 1$02

Using Corollary 2.2 in Balabanov–Grigori ’22 as cited in the paper, the condition number of the full basis $n\gg 1$03 remains $n\gg 1$04, which is used to justify that the projected small problem is well-conditioned (Damas et al., 7 Aug 2025).

For a Ritz pair $n\gg 1$05 extracted from

$n\gg 1$06

the residual norm is bounded according to Theorem 2.14 of the paper by

$n\gg 1$07

The deflation stage introduces an additional tolerance parameter $n\gg 1$08, and the paper states that the backward error on each locked eigenpair is $n\gg 1$09, so the method is backward stable up to the sketch distortion $n\gg 1$10 (Damas et al., 7 Aug 2025).

The stated interpretation is that randomization changes the orthogonalization model but does not forfeit the stability guarantees typically required of restarted Krylov eigensolvers. A plausible implication is that parameter selection for $n\gg 1$11, $n\gg 1$12, and $n\gg 1$13 governs an explicit trade-off between orthogonalization cost and spectral accuracy.

6. Parameterization, implementation, and numerical behavior

The implementation guidance in the paper is specific. The sketch dimension is taken as $n\gg 1$14 with $n\gg 1$15, and $n\gg 1$16 is reported to give $n\gg 1$17 in practice. The Krylov dimension is often recommended as $n\gg 1$18; if $n\gg 1$19 is too small, convergence slows, whereas excessively large $n\gg 1$20 increases per-cycle cost. For the embedding $n\gg 1$21, the two listed options are dense Gaussian and sparse $n\gg 1$22 signs with a few nonzeros per column. For sketch-orthonormalization, the recommended choices are Randomized Gram–Schmidt (RGS) of Balabanov–Grigori ’22 and Randomized Householder QR of Grigori–Timsit ’24, which the description states cost half the flops of classical Gram–Schmidt and maintain sketch-orthonormality to high accuracy. For Schur reordering, LAPACK’s dtrsen or Kressner’s block-reordering algorithms are suggested. Storage requires both $n\gg 1$23 and its sketch $n\gg 1$24, and contraction applies the same orthonormal transformations to both (Damas et al., 7 Aug 2025).

The numerical experiments summarized in the source description cover both synthetic and real sparse problems. For synthetic tri-diagonal tests with $n\gg 1$25 up to $n\gg 1$26, $n\gg 1$27, and $n\gg 1$28, using exponential, logarithmic, harmonic, and geometric diagonal spectra with Gaussian sub-diagonal noise, rKS was reported as $n\gg 1$29–$n\gg 1$30 faster than classical IRA/KS, often converging in fewer restarts, while producing identical Ritz values and residuals below $n\gg 1$31. For real sparse matrices from the UF collection, including Atmosmodl and Vas_stokes_4M, with $n\gg 1$32 and $n\gg 1$33, the method ran $n\gg 1$34–$n\gg 1$35 faster than both MATLAB’s eigs (IRA) and the authors’ deterministic Krylov–Schur implementation, and was described as robust to the choice of $n\gg 1$36 (Damas et al., 7 Aug 2025).

These results are presented as evidence that the method scales on large nonsymmetric problems while preserving the overall Krylov–Schur architecture. Within that framing, rKS is characterized by three coupled design choices: low-dimensional sketching in place of full-dimensional orthogonalization, Schur-based restart and truncation on the projected problem, and locking of converged eigenpairs through deflation.