A priori error bounds on invariant subspace approximations by block Krylov subspaces

We derive a priori error bounds for the block Krylov subspace methods in terms of “the sine” between the desired invariant subspace and the block Krylov subspace. The obtained results can be seen as the block analogue of the classical a priori estimates for standard projection methods.     

Roundoff error analysis of algorithms based on Krylov subspace methods

We study the roundoff error propagation in an algorithm which computes the orthonormal basis of a Krylov subspace with Householder orthonormal matrices. Moreover, we analyze special implementations of the classical GMRES algorithm, and of the Full Orthogonalization Method. These techniques approximate the solution of a large sparse linear system of equations on a sequence of Krylov subspaces of small dimension. The roundoff error analyses show upper bounds for the error affecting the computed approximated solutions.This work was carried out with the financial contribution of the Human Capital and Mobility Programme of the European Union grant ERB4050PL921378.     

Towards backward perturbation bounds for approximate dual Krylov subspaces

Given a matrix A,n by n, and two subspaces K and L of dimension m, we consider how to determine a backward perturbation E whose norm is as small as possible, such that k and L are Krylov subspaces of A+E and its adjoint, respectively. We first focus on determining a perturbation matrix for a given pair of biorthonormal bases, and then take into account how to choose an appropriate biorthonormal pair and express the Krylov residuals as a perturbation of the matrix A. Specifically, the perturbation matrix is globally optimal when A is Hermitian and K=L. The results show that the norm of the perturbation matrix can be assessed by using the norms of the Krylov residuals and those of the biorthonormal bases. Numerical experiments illustrate the efficiency of our strategy.     

The restarted shift‐and‐invert Krylov method for matrix functions

In this paper, the numerical evaluation of matrix functions expressed in partial fraction form is addressed. The shift‐and‐invert Krylov method is analyzed, with special attention to error estimates. Such estimates give insights into the selection of the shift parameter and lead to a simple and effective restart procedure. Applications to the class of Mittag–Leffler functions are presented. Copyright © 2012 John Wiley & Sons, Ltd.     

Backward errors of periodic invariant subspaces for regular periodic matrix pairs

In this paper, the backward error of periodic invariant subspaces for regular periodic pairs is defined and its explicit expression is derived. In particular, we also present the expression of the backward error of generalized invariant subspaces for the regular matrix pair. The results are illustrated by two numerical examples.     

Backward error analysis of specified eigenpairs for sparse matrix polynomials

This article studies the unstructured and structured backward error analysis of specified eigenpairs for matrix polynomials. The structures we discuss include $T$-symmetric, $T$-skew-symmetric, Hermitian, skew Hermitian, $T$-even, $T$-odd, $H$-even, $H$-odd, $T$-palindromic, $T$-anti-palindromic, $H$-palindromic, and $H$-anti-palindromic matrix polynomials. Minimally structured perturbations are constructed with respect to Frobenius norm such that specified eigenpairs become exact eigenpairs of an appropriately perturbed matrix polynomial that also preserves sparsity. Further, we have used our results to solve various quadratic inverse eigenvalue problems that arise from real-life applications.     

On the convergence of two‐level Krylov methods for singular symmetric systems

We discuss the convergence of a two‐level version of the multilevel Krylov method for solving linear systems of equations with symmetric positive semidefinite matrix of coefficients. The analysis is based on the convergence result of Brown and Walker for the Generalized Minimal Residual method (GMRES), with the left‐ and right‐preconditioning implementation of the method. Numerical results based on diffusion problems are presented to show the convergence.     

A note on Krylov subspace methods for singular systems

Recently, Calvetti et al. have published an interesting paper [Linear Algebra Appl. 316 (2000) 157–169] concerning the least-squares solution of a singular system by using the so-called range restricted GMRES (RRGMRES) method. However, one of the main results (cf. [loc. cit., Theorem 3.3]) seems to be incomplete. As a complement of paper [loc. cit.], in this note we first make an example to show the incompleteness of that theorem, then we give a modified result.     

Shift-and-Invert Krylov Methods for Time-Fractional Wave Equations

The article deals with evolution problems involving time derivatives of fractional order α, with 1 < α ≤2. The solutions are expressed in terms of operator Mittag-Leffler functions. The action of such operator functions is approximated by rational Krylov methods whose convergence features are investigated.     

On the aliasing error in a class of bidimensional wavelet subspaces

This paper addresses the aliasing error in the setting of a class of bidimensional multiresolution analysis associated with a 2 × 2 dilation matrix of determinant ±2. The explicit expression of the Fourier transform of the aliasing error is established, from which we obtain an optimal L 2 (R 2 )-norm estimation of the aliasing error.     

Optimal a priori error bounds for the Rayleigh-Ritz method

We derive error bounds for the Rayleigh-Ritz method for the approximation to extremal eigenpairs of a symmetric matrix. The bounds are expressed in terms of the eigenvalues of the matrix and the angle between the subspace and the eigenvector. We also present a sharp bound.

     

Backward error and perturbation bounds for high order Sylvester tensor equation

In this article, we investigate the backward error and perturbation bounds for the high order Sylvester tensor equation (STE). The bounds of the backward error and three types of upper bounds for the perturbed STE with or without dropping the second order terms are presented. The classic perturbation results for the Sylvester equation are extended to the high order case.     

Deflation in Krylov subspace methods and distance to uncontrollability

The task of extracting from a Krylov decomposition the approximation to an eigenpair that yields the smallest backward error can be phrased as finding the smallest perturbation which makes an associated matrix pair uncontrollable. Exploiting this relationship, we propose a new deflation criterion, which potentially admits earlier deflations than standard deflation criteria. Along these lines, a new deflation procedure for shift-and-invert Krylov methods is developed. Numerical experiments demonstrate the merits and limitations of this approach. This author has been supported by a DFG Emmy Noether fellowship and in part by the Swedish Foundation for Strategic Research under the Frame Programme Grant A3 02:128.     

A Krylov subspace algorithm for multiquadric interpolation in many dimensions

We consider the version of multiquadric interpolation wherethe interpolation conditions are the equations s(x_i) = f_i, i= 1,2,..., n, and where the interpolant has the form s(x) =_j=1ⁿ _j (||x – x_j ||² + c²)^1/2 + x R^d, subject to theconstraint _j=1ⁿ _j = 0. The points x_i R^d, the right-hand sidesf_i, i = 1,2,...,n, and the constant c are data. The equationsand the constraint define the parameters _j, j = 1,2,...,n, and. The resultant approximation s f is useful in many applications,but the calculation of the parameters by direct methods requiresO (n³) operations, and n may be large. Therefore iterative proceduresfor this calculation have been studied at Cambridge since 1993,the main task of each iteration being the computation of s(x_i),i = 1,2,...,n, for trial values of the required parameters.These procedures are based on approximations to Lagrange functions,and often they perform very well. For example, ten iterationsusually provide enough accuracy in the case d = 2 and c = 0,for general positions of the data points, but the efficiencydeteriorates if d and c are increased. Convergence can be guaranteedby the inclusion of a Krylov subspace technique that employsthe native semi-norm of multiquadric functions. An algorithmof this kind is specified, its convergence is proved, and carefulattention is given to the choice of the operator that definesthe Krylov subspace, which is analogous to pre-conditioningin the conjugate gradient method. Finally, some numerical resultsare presented and discussed, for values of d and n from theintervals [2,40] and [200,10 000], respectively.     

Parallel implementation of a multiblock method with approximate subdomain solution

Solution of large linear systems encountered in computational fluid dynamics often naturally leads to some form of domain decomposition, especially when it is desired to use parallel machines. It has been proposed to use approximate solvers to obtain fast but rough solutions on the separate subdomains. In this paper approximate solutions via (1) an inner preconditioned GMRES iteration to fixed tolerance, and (2) incomplete factorization (RILU, restricted to the diagonal) are considered. Numerical experiments for a fundamental test problem are included which show speedups obtained on a cluster of workstations as well as on a distributed memory parallel computer. Additionally, the parallel implementation of GCR is addressed, with particular focus on communication costs associated with orthogonalization processes. This consideration brings up questions concerning the use of Householder reflections with GCR.     

A Note on Krylov Methods for Fractional Evolution Problems

The solution of linear fractional-order differential problems is addressed. For this purpose, rational approximations obtained by projections on resolvent Krylov subspaces are considered. Their convergence properties in Hilbert spaces are investigated.     

Backward error,sensitivity, and refinement of computed solutions of algebraic Riccati equations

In this paper, a new backward error criterion, together with a sensitivity measure, is presented for assessing solution accuracy of nonsymmetric and symmetric algebraic Riccati equations (AREs). The usual approach to assessing reliability of computed solutions is to employ standard perturbation and sensitivity results for linear systems and to extend them further to AREs. However, such methods are not altogether appropriate since they do not take account of the underlying structure of these matrix equations. The approach considered here is to first compute the backward error of a computed solution X? that measures the amount by which data must be perturbed so that X? is the exact solution of the perturbed original system. Conventional perturbation theory is used to define structured condition numbers that fully respect the special structure of these matrix equations. The new condition number, together with the backward error of computed solutions, provides accurate estimates for the sensitivity of solutions. Optimal perturbations are then used in an iterative refinement procedure to give further more accurate approximations of actual solutions. The results are derived in their most general setting for nonsymmetric and symmetric AREs. This in turn offers a unifying framework through which it is possible to establish similar results for Sylvester equations, Lyapunov equations, linear systems, and matrix inversions.