Structured condition numbers and backward errors in scalar product spaces

Abstract. Given a class of structured matrices S, we identify pairs of vectors x, b for which there exists a matrix A ∈ S such that Ax = b, and also characterize the set of all matrices A ∈ S mapping x to b. The structured classes we consider are the Lie and Jordan algebras associated with orthosymmetric scalar products. These include (skew-)symmetric, (skew-)Hamiltonian, pseudo (skew-)Hermitian, persymmetric and perskew-symmetric matrices. Structured mappings with extremal properties are also investigated. In particular, structured mappings of minimal rank are identified and shown to be unique when rank-1 is achieved. The structured mapping of minimal Frobenius norm is always unique and explicit formulas for it and its norm are obtained. Finally the set of all structured mappings of minimal 2-norm is characterized. Our results generalize and unify existing work, answer a number of open questions, and provide useful tools for structured backward error investigations.

Section: Structured Mappings Of Minimal Frobenius Normmentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

See 1 more Smart Citation

Structured Mapping Problems for Matrices Associated with Scalar Products. Part I: Lie and Jordan Algebras

Mackey¹,

Mackey²,

Tisseur³

2008

“…There are several results [5,6] on general and structured (normwise) perturbations based on scalar product spaces. Regarding componentwise perturbations, are there results related to the existence of a corresponding multiplicative Lie group?…”

Section: Main Resultmentioning

confidence: 99%

The Componentwise Structured and Unstructured Backward Errors Can be Arbitrarily Far Apart

Rump¹

2015

Given a linear system Ax = b and some vectorx, the backward error characterizes the smallest relative perturbation of A, b such thatx is a solution of the perturbed system. If the input matrix has some structure such as being symmetric or Toeplitz, perturbations may be restricted to perturbations within the same class of structured matrices. For normwise perturbations, the symmetric and the general backward errors are equal, and the question about the relation between the symmetric and general componentwise backward errors arises. In this note we show for a number of common structures in numerical analysis that for componentwise perturbations the structured backward error can be equal to 1, whilst the unstructured backward error is arbitrarily small. Structures cover symmetric, persymmetric, skewsymmetric, Toeplitz, symmetric Toeplitz, Hankel, persymmetric Hankel, and circulant matrices. This is true although the normwise condition number A −1 A is close to 1.

“…For M = C n×n and certain linear structures, computationally tractable characterizations of (1.1) have been derived that allow to decide whether a certain point z ∈ C is contained in Λ ε (A), see [KKK10] and the references therein. Little is known in this direction for nonlinear structures, except for the Lie groups of real orthogonal and unitary matrices [Sun97,TG06].…”

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confidence: 99%

Computing Extremal Points of Symplectic Pseudospectra and Solving Symplectic Matrix Nearness Problems

Guglielmi¹,

Kreßner²,

Lubich³

2014

We study differential equations that lead to extremal points in symplectic pseudospectra. In a two-level approach, where on the inner level we compute extremizers of the symplectic ε-pseudospectrum for a given ε and on the outer level we optimize over ε, this is used to solve symplectic matrix nearness problems such as the following: For a symplectic matrix with eigenvalues of unit modulus, we aim to determine the nearest complex symplectic matrix such that some or all eigenvalues leave the complex unit circle. Conversely, for a symplectic matrix with all eigenvalues lying off the unit circle, we consider the problem of computing the nearest symplectic matrix that has an eigenvalue on the unit circle.