Adaptive Block Lanczos (ABLE)

• Adaptive Block Lanczos (ABLE) is a family of block Krylov–subspace methods that adjust parameters like block size, restart frequency, and filtering on the fly.
• It employs techniques such as block tridiagonalization and spectral filtering to enhance convergence and stability in eigenvalue computations.
• ABLE balances mathematical convergence per Krylov dimension with hardware efficiency, proving beneficial in applications like lattice QCD and model reduction.

Searching arXiv for recent and directly related papers on adaptive/block Lanczos, finite precision, and application-specific formulations. [arXiv search] Query: "Adaptive Block Lanczos OR block Lanczos adaptive finite precision Lanczos HISQ Chebyshev Lanczos"

Adaptive Block Lanczos (ABLE) is a block Krylov–subspace eigen/linear-solver scheme where the block size, restarting, or deflation strategy is adapted on the fly (Šimonová et al., 2021). In the literature surveyed here, the term does not denote a single standardized algorithm. It instead names a family of block Lanczos constructions in which the basic block tridiagonalization is combined with adaptive control of block dimension, restart thickness, locking, polynomial or rational filtering, tangential interpolation data, or spectral-tail modeling (Jeong et al., 2022, Kaouane et al., 2019, Zimmerling et al., 11 Feb 2026).

1. Conceptual scope and defining features

The core of ABLE is the block Lanczos process. For a Hermitian matrix $A$ and block size $u$, the algorithm starts from $u$ mutually orthogonal vectors

$\{b_1,b_2,\ldots,b_u\},$

and after $n$ block steps constructs the block Krylov space

$$\mathcal{K}_n(A,B)=\mathcal{K}_n(A,b_1)\cup\cdots\cup\mathcal{K}_n(A,b_u), \qquad B=(b_1|\cdots|b_u).$$

After $n$ block steps, the effective scalar subspace dimension is $\tilde n = nu$, and the projected matrix is an $(\tilde n\times \tilde n)$ block-tridiagonal matrix $T_n$ (Jeong et al., 2022).

A complementary formulation writes the symmetric block Lanczos recurrence in terms of orthonormal block vectors $u$0: $u$1 with the induced block tridiagonal matrix

$u$2

In exact arithmetic, the process builds an orthonormal block basis of the block Krylov subspace (Šimonová et al., 2021).

The adaptive qualifier refers to the choice of what is allowed to vary during the run. In the surveyed work, these adaptive degrees of freedom include varying block size $u$3 across iterations, restarts or augmentation based on approximate invariant subspaces or spectral clustering, adaptive reorthogonalization strategies, residual-driven selection of interpolation data, and low-rank corrections of the projected block Lanczos matrix (Šimonová et al., 2021, Kaouane et al., 2019, Zimmerling et al., 11 Feb 2026). This suggests that ABLE is best understood as a design pattern for block Krylov methods rather than as a single recurrence with fixed parameters.

2. Block Lanczos mechanics

In the Hermitian eigensolver setting, block Lanczos operates by orthogonalizing the whole block $u$4 simultaneously. If

$u$5

then the recurrence may be viewed conceptually as

$u$6

with $u$7 small $u$8 blocks. This simultaneous orthogonalization is the basic source of the higher arithmetic intensity of the block method (Jeong et al., 2022).

The resulting projected eigenproblem has the standard Ritz structure. Once

$u$9

the approximate eigenvectors of $u$0 are

$u$1

and the Ritz values satisfy

$u$2

A convergence criterion used in the HISQ eigensolver study is the relative residual

$u$3

These are the same structural ingredients used in non-block Lanczos, but with block-tridiagonal rather than tridiagonal projection (Jeong et al., 2022).

For matrix-function and transfer-function problems, block Lanczos is commonly written as

$u$4

where $u$5 is symmetric positive definite block tridiagonal. The standard block approximation

$u$6

is the block Gauss quadrature rule for the Stieltjes measure associated with $u$7, and it matches the first $u$8 block moments $u$9, $\{b_1,b_2,\ldots,b_u\},$0 (Zimmerling et al., 11 Feb 2026).

For low-rank approximation, the same structure appears through the block Krylov matrix

$\{b_1,b_2,\ldots,b_u\},$1

The improvement over the power method comes from using the entire block Krylov sequence rather than only the last iterate (Wang et al., 2015).

3. Adaptive parameters and control laws

One adaptive axis is the active block size. The exactness analysis explicitly describes ABLE as a setting in which the block size $\{b_1,b_2,\ldots,b_u\},$2 may vary across iterations, in response to deflation, rank deficiency, or convergence of subspaces (Šimonová et al., 2021). In the HISQ study, by contrast, block size is fixed at $\{b_1,b_2,\ldots,b_u\},$3, no restart is used, and there is no automatic adaptivity in block size, restart frequency, or polynomial degree; the paper therefore identifies these as the natural quantities that a future adaptive block Lanczos scheme would vary (Jeong et al., 2022).

A second axis is the restart and locking policy. In the HISQ benchmarks, Thick-Restart Lanczos retains converged Ritz vectors and can lock eigenvectors that have reached machine precision, removing them from the active search space. The same study notes that Grid’s block Lanczos implementation supports implicit restart even though the restarted block variant is not pursued there (Jeong et al., 2022). This makes restart thickness, lock thresholds, and the size of the active block natural ABLE control variables.

A third axis is spectral filtering. For the HISQ Dirac operator, all improved Lanczos algorithms employ Chebyshev acceleration with

$\{b_1,b_2,\ldots,b_u\},$4

using $\{b_1,b_2,\ldots,b_u\},$5 and $\{b_1,b_2,\ldots,b_u\},$6 or $\{b_1,b_2,\ldots,b_u\},$7, with the explicit conclusion that “We need heuristics to find an optimized $\{b_1,b_2,\ldots,b_u\},$8” (Jeong et al., 2022). In model reduction, the ABTL method chooses interpolation points and tangential directions by maximizing residual norms over

$\{b_1,b_2,\ldots,b_u\},$9

where $n$0 are the current reduced poles, and then updates the block tangential Lanczos basis accordingly (Kaouane et al., 2019). In the continuous-spectrum setting, the adaptive parameter is a symmetric positive definite matrix $n$1 in the terminal condition

$n$2

chosen by maximizing a relative energy-outflow functional on a contour near the negative real axis (Zimmerling et al., 11 Feb 2026).

Warm-start theory supplies another adaptive handle. For low-rank approximation, the block Lanczos method has a gap-independent warm-start bound: if

$n$3

then choosing

$n$4

is sufficient for a $n$5-relative spectral-norm guarantee (Wang et al., 2015). This suggests that Krylov depth can be tied to measured or estimated subspace quality rather than fixed a priori.

4. Major realizations

In lattice QCD, ABLE is motivated by the need to compute low modes of the massless HISQ Dirac operator through the Hermitian positive semidefinite operator

$n$6

The benchmarked methods include basic Lanczos, IRL, TRL, and block Lanczos. For the test configuration, the largest eigenvalue is approximately $n$7, the $n$8th eigenvalue is approximately $n$9, and the $$\mathcal{K}_n(A,B)=\mathcal{K}_n(A,b_1)\cup\cdots\cup\mathcal{K}_n(A,b_u), \qquad B=(b_1|\cdots|b_u).$$0th is approximately $$\mathcal{K}_n(A,B)=\mathcal{K}_n(A,b_1)\cup\cdots\cup\mathcal{K}_n(A,b_u), \qquad B=(b_1|\cdots|b_u).$$1; the goal is the low-lying spectrum near $$\mathcal{K}_n(A,B)=\mathcal{K}_n(A,b_1)\cup\cdots\cup\mathcal{K}_n(A,b_u), \qquad B=(b_1|\cdots|b_u).$$2. The study shows that block Lanczos is mathematically less efficient per Krylov dimension than single-vector Lanczos for this problem, but it maps well to Split Grid and multi-right-hand-side execution, which is precisely the kind of regime in which adaptive control of block size, restart, and polynomial degree becomes relevant (Jeong et al., 2022).

In model reduction for large-scale first- and second-order MIMO systems, the ABTL algorithm is explicitly identified as conceptually what one would call an ABLE method specialized to tangential resolvent Krylov subspaces. The right and left block tangential Krylov spaces are built from

$$\mathcal{K}_n(A,B)=\mathcal{K}_n(A,b_1)\cup\cdots\cup\mathcal{K}_n(A,b_u), \qquad B=(b_1|\cdots|b_u).$$3

and the adaptive rule selects $$\mathcal{K}_n(A,B)=\mathcal{K}_n(A,b_1)\cup\cdots\cup\mathcal{K}_n(A,b_u), \qquad B=(b_1|\cdots|b_u).$$4 and $$\mathcal{K}_n(A,B)=\mathcal{K}_n(A,b_1)\cup\cdots\cup\mathcal{K}_n(A,b_u), \qquad B=(b_1|\cdots|b_u).$$5 by maximizing residual norms. The resulting reduced model satisfies the tangential interpolation conditions

$$\mathcal{K}_n(A,B)=\mathcal{K}_n(A,b_1)\cup\cdots\cup\mathcal{K}_n(A,b_u), \qquad B=(b_1|\cdots|b_u).$$6

with first-derivative matching when $$\mathcal{K}_n(A,B)=\mathcal{K}_n(A,b_1)\cup\cdots\cup\mathcal{K}_n(A,b_u), \qquad B=(b_1|\cdots|b_u).$$7 and both conditions hold (Kaouane et al., 2019).

For matrices with almost continuous spectra, the objective is the transfer function

$$\mathcal{K}_n(A,B)=\mathcal{K}_n(A,b_1)\cup\cdots\cup\mathcal{K}_n(A,b_u), \qquad B=(b_1|\cdots|b_u).$$8

Here standard block Lanczos approximates the transfer function by block Gauss quadrature, but the paper argues that it is more efficient to model the inherent branch cut than to resolve the dense artificial point spectrum induced by discretization. The adaptive construction adds a $$\mathcal{K}_n(A,B)=\mathcal{K}_n(A,b_1)\cup\cdots\cup\mathcal{K}_n(A,b_u), \qquad B=(b_1|\cdots|b_u).$$9-dependent low-rank correction to the last block of the projected operator,

$n$0

with

$n$1

This produces a block Lanczos approximation with a branch cut on $n$2 and enforces an approximate radiation condition at the tail of the Krylov chain (Zimmerling et al., 11 Feb 2026).

On quantum hardware, the block Lanczos recursion is extended to a supervector

$n$3

with block recurrence

$n$4

The block coefficients satisfy

$n$5

The stated motivation is that block Lanczos resolves degeneracies with better precision, and the overhead is $n$6 relative to scalar quantum Lanczos recursion (Baker, 2021).

5. Exactness, finite precision, and stabilization

A rigorous baseline for ABLE is provided by exactness results for block Lanczos in IEEE 754 arithmetic. For $n$7, if

$n$8

where $n$9 is a signed block permutation matrix with $\tilde n = nu$0 blocks and $\tilde n = nu$1 is block tridiagonal with symmetric diagonal blocks and upper triangular off-diagonal blocks with positive diagonal entries, then block Lanczos computes exactly in floating point arithmetic, provided the QR factorization is done using classical or modified Gram–Schmidt and there is no underflow or overflow (Šimonová et al., 2021). This exact regime is highly structured, but it identifies the configuration in which orthogonality and linear independence are not degraded by finite precision.

Away from that regime, finite precision block Lanczos can still be interpreted through a model-matrix theorem. After $\tilde n = nu$2 finite precision block Lanczos steps, a block continuation process constructs an extended symmetric block tridiagonal matrix $\tilde n = nu$3 such that, under uniformly small perturbations and a mild clustering assumption on the Ritz vectors, each eigenvalue of $\tilde n = nu$4 lies within

$\tilde n = nu$5

of some eigenvalue of $\tilde n = nu$6 (Šimonová et al., 22 Jul 2025). The scalar Greenbaum theory guarantees such small perturbations, but the block case is different: whether the sufficient conditions can always be satisfied remains open (Šimonová et al., 22 Jul 2025).

For block Lanczos-type linear solvers with multiple right-hand sides, the dominant finite-precision diagnostic is the residual gap

$\tilde n = nu$7

The Bl-CIRS construction introduces cross-interactive residual smoothing, and the paper states that orthonormalizing the columns of direction matrices is crucial in effectively reducing the residual gap. With QR-based orthonormalization, the smoothed recurrences take the form

$\tilde n = nu$8

and the resulting residual-gap bounds scale with $\tilde n = nu$9 and $(\tilde n\times \tilde n)$0 (Aihara et al., 2024). In an ABLE context, this places orthonormalization quality and residual smoothing alongside restart and deflation as operational stability controls.

6. Performance trade-offs and research directions

The main performance trade-off in ABLE is the tension between mathematical convergence per Krylov dimension and hardware or architectural efficiency. For HISQ, non-block Lanczos converges at $(\tilde n\times \tilde n)$1 while block Lanczos with $(\tilde n\times \tilde n)$2 converges at $(\tilde n\times \tilde n)$3, so block Lanczos is slower in terms of total subspace dimension. Without Split Grid it is slower than the unblocked method, and with Split Grid its performance improves by about $(\tilde n\times \tilde n)$4 even on intra-node MPI; preliminary inter-node results indicate approximately $(\tilde n\times \tilde n)$5 better performance than unblocked Lanczos on $(\tilde n\times \tilde n)$6 nodes with slower inter-node network (Jeong et al., 2022). The implication is that ABLE cannot be judged only by Ritz convergence; communication, memory pressure, and multi-right-hand-side throughput are integral.

In model reduction, ABTL shows the opposite side of the trade-off: expensive shift solves are accepted because adaptive tangential directions sharply target transfer-function error. On FDM40k at reduced order $(\tilde n\times \tilde n)$7, the reported timings and $(\tilde n\times \tilde n)$8 errors are 27.21 s and $(\tilde n\times \tilde n)$9 for ABTL, 269.36 s and $T_n$0 for IRKA, and 38.12 s and $T_n$1 for TRKSM (Kaouane et al., 2019). This places adaptive selection of interpolation points and tangential directions at the center of the method rather than as a secondary tuning step.

For almost continuous spectra, the adaptive terminator changes the quality of the reduced model rather than the Krylov basis itself. The added cost is stated as only $T_n$2, and in the dense-spectrum regime the Kreĭn–Nudelman terminator yields about one order of magnitude smaller error at fixed $T_n$3 than plain Gauss quadrature, while also turning standing-wave artifacts into outgoing propagating waves in wave problems (Zimmerling et al., 11 Feb 2026). That formulation broadens ABLE beyond eigensolvers: adaptation may act on the projected operator rather than on the block basis.

For randomized low-rank approximation, improved theory explains when increasing Krylov depth is worth the cost. With Gaussian start, the block Lanczos method achieves

$T_n$4

with

$T_n$5

whereas the corresponding power method requires

$T_n$6

for the same style of bound (Wang et al., 2015). This gives a precise theoretical basis for adaptive control of Krylov depth and oversampling.

Taken together, these results describe ABLE as a family of block Lanczos methods in which adaptation is introduced wherever the fixed-parameter method exposes a strong trade-off: block size against convergence rate, restart thickness against memory, filtering strength against matvec cost, tangential directions against transfer-function error, terminator strength against standing-wave artifacts, and orthonormalization effort against residual-gap growth. The surveyed literature does not yet provide a single canonical ABLE algorithm, but it supplies the principal structural components, performance regimes, and numerical constraints from which such algorithms are assembled (Šimonová et al., 2021).