Verified bounds for singular values, in particular for the spectral norm of a matrix and its inverse

DOI10.1007/s10543-010-0294-0zbMath1226.65028OpenAlexW2018813194MaRDI QIDQ634116

Siegfried Michael Rump

Publication date: 2 August 2011

Published in: BIT (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/s10543-010-0294-0

zbMATH Keywords

verificationsingular value decompositioncondition numbererror boundsspectral norm

Mathematics Subject Classification ID

Numerical computation of eigenvalues and eigenvectors of matrices (65F15) Numerical solutions to overdetermined systems, pseudoinverses (65F20) Eigenvalues, singular values, and eigenvectors (15A18) Norms of matrices, numerical range, applications of functional analysis to matrix theory (15A60) Algorithms with automatic result verification (65G20)

Related Items (20)

On the real stability radius for some classes of matrices ⋮ Norm bound computation for inverses of linear operators in Hilbert spaces ⋮ Verified inclusions for a nearest matrix of specified rank deficiency via a generalization of Wedin's \(\sin (\theta)\) theorem ⋮ Some improvements of invertibility verifications for second-order linear elliptic operators ⋮ A method for verifying the accuracy of numerical solutions of symmetric saddle point linear systems ⋮ Exact symbolic-numeric computation of planar algebraic curves ⋮ Efficient approaches for verifying the existence and bound of inverse of linear operators in Hilbert spaces ⋮ Fast computation of error bounds for all eigenpairs of a Hermitian and all singular pairs of a rectangular matrix with emphasis on eigen- and singular value clusters ⋮ Componentwise enclosure for solutions of least squares problems and underdetermined systems ⋮ A heuristic verification of the degree of the approximate GCD of two univariate polynomials ⋮ Computing Enclosures for the Matrix Exponential ⋮ An alternative approach to norm bound computation for inverses of linear operators in Hilbert spaces ⋮ A numerical verification method for nonlinear functional equations based on infinite-dimensional Newton-like iteration ⋮ Computer-assisted equilibrium validation for the diblock copolymer model ⋮ Verified bounds for all the singular values of matrix ⋮ Verified norm estimation for the inverse of linear elliptic operators using eigenvalue evaluation ⋮ Some Remarks on the Rigorous Estimation of Inverse Linear Elliptic Operators ⋮ Validated Saddle-Node Bifurcations and Applications to Lattice Dynamical Systems ⋮ Time complexity analysis of quantum difference methods for linear high dimensional and multiscale partial differential equations ⋮ Accuracy of the $s$-Step Lanczos Method for the Symmetric Eigenproblem in Finite Precision

Uses Software

• INTLAB
• mctoolbox
• Harwell-Boeing sparse matrix collection

Cites Work

• Fast and parallel interval arithmetic
• A computer-assisted proof for the Kolmogorov flows of incompressible viscous fluid
• A numerical method to verify the invertibility of linear elliptic operators with applications to nonlinear problems
• Verification of positive definiteness
• Verification methods: Rigorous results using floating-point arithmetic
• A numerical approach to the proof of existence of solutions for elliptic problems
• A computer-assisted instability proof for the Orr-Sommerfeld problem with Poiseuille flow
• Interval Methods for Systems of Equations
• Introduction to Numerical Analysis
• Accuracy and Stability of Numerical Algorithms
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