What does this research mean for the field?

Explicit error formulas and a posteriori error bounds for block rational Krylov approximations of matrix functions can be derived and practically evaluated without the use of quadratures. Novelty: ClaimNovelty.METHODOLOGICAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

To derive explicit error formulas and computable a posteriori error bounds for block rational Krylov approximations of matrix functions.

August 22, 2026SIAM Journal on Numerical AnalysisOpen Access

Error Formulas for Block Rational Krylov Approximations of Matrix Functions

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Authors

SMStefano Massei LRLeonardo Robol

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Overview

Theoretical analysis derives explicit quadrature-free error formulas for block rational Krylov approximations of matrix functions, indicating streamlined a posteriori error estimation.

Key Points

To derive explicit error formulas and computable a posteriori error bounds for block rational Krylov approximations of matrix functions.
Characterized the residual of the block full orthogonal method using a block generalization of the residual polynomial.
Developed an alternative error formulation based on the block collinearity of residuals.
Constructed a posteriori error bounds from spectral matrix data and tested them across numerical benchmark examples.
Established two explicit error representations for block rational Krylov approximations that avoid the need for numerical quadratures.
Derived computable a posteriori upper bounds that depend directly on the spectral information of the matrix argument.
Demonstrated practical effectiveness and evaluative simplicity across several test problems without numerical integration overhead.

Cite This Study

Massei et al. (2026) studied this question.

synapsesocial.com/papers/6a895e6bca7ade938187c874 https://doi.org/10.1137/25m1751815

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August 22, 2026SIAM Journal on Numerical AnalysisOpen Access

Error Formulas for Block Rational Krylov Approximations of Matrix Functions

View Full Paper

Ask AI

Bookmark

Authors

SMStefano Massei LRLeonardo Robol

Discussion

Member takes

Overview

Theoretical analysis derives explicit quadrature-free error formulas for block rational Krylov approximations of matrix functions, indicating streamlined a posteriori error estimation.

Key Points

To derive explicit error formulas and computable a posteriori error bounds for block rational Krylov approximations of matrix functions.
Characterized the residual of the block full orthogonal method using a block generalization of the residual polynomial.
Developed an alternative error formulation based on the block collinearity of residuals.
Constructed a posteriori error bounds from spectral matrix data and tested them across numerical benchmark examples.
Established two explicit error representations for block rational Krylov approximations that avoid the need for numerical quadratures.
Derived computable a posteriori upper bounds that depend directly on the spectral information of the matrix argument.
Demonstrated practical effectiveness and evaluative simplicity across several test problems without numerical integration overhead.

Cite This Study

Massei et al. (2026) studied this question.

synapsesocial.com/papers/6a895e6bca7ade938187c874 https://doi.org/10.1137/25m1751815

View Full Paper

Ask AI

Bookmark