Growth factors of pivoting strategies associated with Neville elimination

Neville elimination is a direct method for solving linear systems. Several pivoting strategies for Neville elimination, including pairwise pivoting, are analyzed. Bounds for two different kinds of growth factors are provided. Finally, an approximation of the average normalized growth factor associated with several pivoting strategies is computed and analyzed using random matrices.     

Backward stability of a pivoting strategy for sign-regular linear systems

A matrixA issign-regular if, for each orderk, allk×k submatrices ofA have determinant with the same sign. In this paper, a pivoting strategy ofO(n) operations for the Gaussian elimination of linear systems whose coefficient matrices are sign-regular is proposed. Backward error analysis of this pivoting strategy is performed and small error bounds are obtained. Our results can also be applied to linear systems whose coefficient matrices have sign-regular inverses.     

A note on matrices with maximal growth factor for Neville elimination

Neville elimination is a direct method for the solution of linear systems of equations with advantages for some classes of matrices and in the context of pivoting strategies for parallel implementations. The growth factor is an indicator of the numerical stability of an algorithm. In the literature, bounds for the growth factor of Neville elimination with some pivoting strategies have appeared. In this work, we determine all the matrices such that the minimal upper bound of the growth factor of Neville elimination with those pivoting strategies is reached.     

Structured mixed and componentwise condition numbers of some structured matrices

Structured matrices, such as Cauchy, Vandermonde, Toeplitz, Hankel, and circulant matrices, are considered in this paper. We apply a Kronecker product-based technique to deduce the structured mixed and componentwise condition numbers for the matrix inversion and for the corresponding linear systems.     

Pivoting strategies leading to diagonal dominance by rows

Pivoting strategies for Gaussian elimination leading to upper triangular matrices which are diagonally dominant by rows are studied. Forward error analysis of triangular systems whose coefficient matrices are diagonally dominant by rows is performed. We also obtain small bounds of the backward errors for the pivoting strategies mentioned above. Our examples of matrices include H-matrices and some generalizations of diagonally dominant matrices, and scaled partial pivoting for the 1-norm is an example of these pivoting strategies. In the case of an M-matrix, a pivoting strategy of computational complexity is proposed, which satisfies all the results of the paper. Received June 6, 1997 / Revised version received October 27, 1997     

Perturbation theory and backward error forAX−XB=C

Because of the special structure of the equationsAX–XB=C the usual relation for linear equations backward error = relative residual does not hold, and application of the standard perturbation result forAx=b yields a perturbation bound involving sep (A, B)^–1 that is not always attainable. An expression is derived for the backward error of an approximate solutionY; it shows that the backward error can exceed the relative residual by an arbitrary factor. A sharp perturbation bound is derived and it is shown that the condition number it defines can be arbitrarily smaller than the sep(A, B)^–1-based quantity that is usually used to measure sensitivity. For practical error estimation using the residual of a computed solution an LAPACK-style bound is shown to be efficiently computable and potentially much smaller than a sep-based bound. A Fortran 77 code has been written that solves the Sylvester equation and computes this bound, making use of LAPACK routines.Nuffield Science Research Fellow. This work was carried out while the author was a visitor at the Institute for Mathematics and its Applications, University of Minnesota.     

Solution of augmented linear systems using orthogonal factorizations

We consider solvingx+Ay=b andA ^T x=c for givenb, c andm ×n A of rankn. This is called the augmented system formulation (ASF) of two standard optimization problems, which include as special cases the minimum 2-norm of a linear underdetermined system (b=0) and the linear least squares problem (c=0), as well as more general problems. We examine the numerical stability of methods (for the ASF) based on the QR factorization ofA, whether by Householder transformations, Givens rotations, or the modified Gram-Schmidt (MGS) algorithm, and consider methods which useQ andR, or onlyR. We discuss the meaning of stability of algorithms for the ASF in terms of stability of algorithms for the underlying optimization problems.We prove the backward stability of several methods for the ASF which useQ andR, inclusing a new one based on MGS, and also show under what circumstances they may be regarded as strongly stable. We show why previous methods usingQ from MGS were not backward stable, but illustrate that some of these methods may be acceptable-error stable. We point out that the numerical accuracy of methods that do not useQ does not depend to any significant extent on which of of the above three QR factorizations is used. We then show that the standard methods which do not useQ are not backward stable or even acceptable-error stable for the general ASF problem, and discuss how iterative refinement can be used to counteract these deficiencies.Dedicated to Carl-Eric Fröberg on the occasion of his 75th birthdayThis research was partially supported by NSERC of Canada Grant No. A9236.     

Perturbation results for nearly uncoupled Markov chains with applications to iterative methods

Summary The standard perturbation theory for linear equations states that nearly uncoupled Markov chains (NUMCs) are very sensitive to small changes in the elements. Indeed, some algorithms, such as standard Gaussian elimination, will obtain poor results for such problems. A structured perturbation theory is given that shows that NUMCs usually lead to well conditioned problems. It is shown that with appropriate stopping, criteria, iterative aggregation/disaggregation algorithms will achieve these structured error bounds. A variant of Gaussian elimination due to Grassman, Taksar and Heyman was recently shown by O'Cinneide to achieve such bounds.Supported by the National Science Foundation under grant CCR-9000526 and its renewal, grant CCR-9201692. This research was done in part, during the author's visit to the Institute for Mathematics and its Applications, 514 Vincent Hall, 206 Church St. S.E., University of Minnesota, Minneapolis, MN 55455, USA     

Component-wise perturbation analysis and error bounds for linear least squares solutions

Perturbation bounds for the linear least squares problem min _x Ax –b₂ corresponding tocomponent-wise perturbations in the data are derived. These bounds can be computed using a method of Hager and are often much better than the bounds derived from the standard perturbation analysis. In particular this is true for problems where the rows ofA are of widely different magnitudes. Generalizing a result by Oettli and Prager, we can use the bounds to compute a posteriori error bounds for computed least squares solutions.     

The general Neville-Aitken-algorithm and some applications

Summary In this note we will present the most general linear form of a Neville-Aitken-algorithm for interpolation of functions by linear combinations of functions forming a ebyev-system. Some applications are given. Expecially we will give simple new proofs of the recurrence formula for generalized divided differences [5] and of the author's generalization of the classical Neville-Aitkena-algorithm[8]applying to complete ebyev-systems. Another application of the general Neville-Aitken-algorithm deals with systems of linear equations. Also a numerical example is given.     

Iterative refinement for constrained and weighted linear least squares

We present an algorithm for mixed precision iterative refinement on the constrained and weighted linear least squares problem, the CWLSQ problem. The approximate solution is obtained by solving the CWLSQ problem with the weightedQR factorization [6]. With backward errors for the weightedQR decomposition together with perturbation bounds for the CWLSQ problem we analyze the convergence behaviour of the iterative refinement procedure.In the unweighted case the initial convergence rate of the error of the iteratively refined solution is determined essentially by the condition number. For the CWLSQ problem the initial convergence behaviour is more complicated. The analysis shows that the initial convergence is dependent both on the condition of the problem related to the solution,x, and the vector =Wr, whereW is the weight matrix andr is the residual.We test our algorithm on two examples where the solution is known and the condition number of the problem can be varied. The computational test confirms the theoretical results and verifies that mixed precision iterative refinement, using the system matrix and the weightedQR decomposition, is an effective way of improving an approximate solution to the CWLSQ problem.     

Perturbation analysis of generalized saddle point systems

We discuss the perturbation analysis of generalized saddle point systems in this paper. We give the nonlinear perturbation bounds, then derive the condition numbers, and analyze the sensitivity of the computed solutions.     

Iterative refinement enhances the stability ofQR factorization methods for solving linear equations

Iterative refinement is a well-known technique for improving the quality of an approximate solution to a linear system. In the traditional usage residuals are computed in extended precision, but more recent work has shown that fixed precision is sufficient to yield benefits for stability. We extend existing results to show that fixed precision iterative refinement renders anarbitrary linear equations solver backward stable in a strong, componentwise sense, under suitable assumptions. Two particular applications involving theQR factorization are discussed in detail: solution of square linear systems and solution of least squares problems. In the former case we show that one step of iterative refinement suffices to produce a small componentwise relative backward error. Our results are weaker for the least squares problem, but again we find that iterative refinement improves a componentwise measure of backward stability. In particular, iterative refinement mitigates the effect of poor row scaling of the coefficient matrix, and so provides an alternative to the use of row interchanges in the HouseholderQR factorization. A further application of the results is described to fast methods for solving Vandermonde-like systems.     

Perturbation analysis of the orthogonal procrustes problem

Given two arbitrary real matricesA andB of the same size, the orthogonal Procrustes problem is to find an orthogonal matrixM such that the Frobenius norm MA – B is minimized. This paper treats the common case when the orthogonal matrixM is required to have a positive determinant. The stability of the problem is studied and supremum results for the perturbation bounds are derived.     

Fast verification of solutions of matrix equations

Summary. In this paper, we are concerned with a matrix equation where A is an real matrix and x and b are n-vectors. Assume that an approximate solution is given together with an approximate LU decomposition. We will present fast algorithms for proving nonsingularity of A and for calculating rigorous error bounds for . The emphasis is on rigour of the bounds. The purpose of this paper is to propose different algorithms, the fastest with flops computational cost for the verification step, the same as for the LU decomposition. The presented algorithms exclusively use library routines for LU decomposition and for all other matrix and vector operations. Received June 16, 1999 / Revised version received January 25, 2001 / Published online June 20, 2001     

Error analysis of the Björck-Pereyra algorithms for solving Vandermonde systems

Summary A forward error analysis is presented for the Björck-Pereyra algorithms used for solving Vandermonde systems of equations. This analysis applies to the case where the points defining the Vandermonde matrix are nonnegative and are arranged in increasing order. It is shown that for a particular class of Vandermonde problems the error bound obtained depends on the dimensionn and on the machine precision only, being independent of the condition number of the coefficient matrix. By comparing appropriate condition numbers for the Vandermonde problem with the forward error bounds it is shown that the Björck-Pereyra algorithms introduce no more uncertainty into the numerical solution than is caused simply by storing the right-hand side vector on the computer. A technique for computing running a posteriori error bounds is derived. Several numerical experiments are presented, and it is observed that the ordering of the points can greatly affect the solution accuracy.     

High-resolution boosted deconvolution of spectroscopic data

The quality of the analysis of spectrometric data consists in correct identification of the existence of peaks and subsequently in good estimation of their positions. In this paper we present high-resolution deconvolution algorithms. We propose an improvement in the efficiency by introducing a boosting operation into the deconvolution process.     

Improving and estimating the accuracy of Strassen's algorithm

Fast matrix multiplication algorithms of Strassen and Winograd are known to have weaker numerical accuracy than usual (inner product) multiplication. In this paper, we show that scaling usually improves accuracy when operands have elements of widely varying magnitude. We also propose estimators for numerical errors, based on samples of the result. All these estimators can be computed in operations. Experiments prove the effectiveness of the scaling idea and of the absolute error estimator. Received February 20, 1996/ Revised version received July 1, 1997