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.     

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     

Numerical enclosure for each eigenvalue in generalized eigenvalue problem

An algorithm for enclosing all eigenvalues in generalized eigenvalue problem Ax=λBx is proposed. This algorithm is applicable even if A∈C^n×n is not Hermitian and/or B∈C^n×n is not Hermitian positive definite, and supplies nerror bounds while the algorithm previously developed by the author supplies a single error bound. It is proved that the error bounds obtained by the proposed algorithm are equal or smaller than that by the previous algorithm. Computational cost for the proposed algorithm is similar to that for the previous algorithm. Numerical results show the property of the proposed algorithm.     

Error analysis of efficient evaluation algorithms for tensor product surfaces

Backward stability of the Casteljau algorithm and two more efficient algorithms for polynomial tensor product surfaces with interest in CAGD is shown. The conditioning of the corresponding bases are compared. These algorithms are also compared with the corresponding Horner algorithm and their higher accuracy is shown. A running error analysis of the algorithms is also carried out providing algorithms which calculate “a posteriori” sharp error bounds simultaneously to the evaluation of the surface without increasing significantly the computational cost.     

Automatic,guaranteed integration of analytic functions

Davis introduced a method for estimating linear functionals of analytic functions by using Cauchy's Integral Formula. This is used to construct methods for numerical integration which give rigorous error bounds. By combining these bounds with strategies for order and subinterval adaptation, a program is developed for automatic integration of analytic functions. Interval analysis is used to validate the bounds.     

Running error for the evaluation of rational Bézier surfaces

Error analysis of the usual method to evaluate rational Bézier surfaces is performed. The corresponding running error analysis is also carried out and the sharpness of our running error bounds is shown. We also modify the evaluation algorithm to include such error bounds without increasing significantly its computational cost.     

Solution of finite systems of equations by interval iteration

In actual practice, iteration methods applied to the solution of finite systems of equations yield inconclusive results as to the existence or nonexistence of solutions and the accuracy of any approximate solutions obtained. On the other hand, construction of interval extensions of ordinary iteration operators permits one to carry out interval iteration computationally, with results which can give rigorous guarantees of existence or nonexistence of solutions, and error bounds for approximate solutions. Examples are given of the solution of a nonlinear system of equations and the calculation of eigenvalues and eigenvectors of a matrix by interval iteration. Several ways to obtain lower and upper bounds for eigenvalues are given.Sponsored by the United States Army under Contract No. DAAG29-80-C-0041.     

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.     

Forward error analysis of Gaussian elimination

Summary Part I of this work deals with the forward error analysis of Gaussian elimination for general linear algebraic systems. The error analysis is based on a linearization method which determines first order approximations of the absolute errors exactly. Superposition and cancellation of error effects, structure and sparsity of the coefficient matrices are completely taken into account by this method. The most important results of the paper are new condition numbers and associated optimal component-wise error and residual estimates for the solutions of linear algebraic systems under data perturbations and perturbations by rounding erros in the arithmetic floating-point operations. The estimates do not use vector or matrix norms. The relative data and rounding condition numbers as well as the associated backward and residual stability constants are scaling-invariant. The condition numbers can be computed approximately from the input data, the intermediate results, and the solution of the linear system. Numerical examples show that by these means realistic bounds of the errors and the residuals of approximate solutions can be obtained. Using the forward error analysis, also typical results of backward error analysis are deduced. Stability theorems and a priori error estimates for special classes of linear systems are proved in Part II of this work.     

Finding all solutions of separable systems of piecewise-linear equations using integer programming

Finding all solutions of nonlinear or piecewise-linear equations is an important problem which is widely encountered in science and engineering. Various algorithms have been proposed for this problem. However, the implementation of these algorithms are generally difficult for non-experts or beginners. In this paper, an efficient method is proposed for finding all solutions of separable systems of piecewise-linear equations using integer programming. In this method, we formulate the problem of finding all solutions by a mixed integer programming problem, and solve it by a high-performance integer programming software such as GLPK, SCIP, or CPLEX. It is shown that the proposed method can be easily implemented without making complicated programs. It is also confirmed by numerical examples that the proposed method can find all solutions of medium-scale systems of piecewise-linear equations in practical computation time.     

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     

A posteriori error estimates for linear equations

Summary This paper describes upper and lowerp-norm error bounds for approximate solutions of the linear system of equationsAx=b. These bounds imply that the error is proportional to the quantity wherer is the residual andq is the conjugate index top. The constant of proportionality is larger than 1 and lies in a specified range. Similar results are obtained for approximations toA ^–1 and solutions of nonsingular linear equations on general spaces.Research was partially supported by NSF Grant DMS8901477     

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.     

Fast enclosure for the minimum norm least squares solution of the matrix equation AXB = C

Fast algorithms for enclosing the minimum norm least squares solution of the matrix equation AXB = C are proposed. To develop these algorithms, theories for obtaining error bounds of numerical solutions are established. The error bounds obtained by these algorithms are verified in the sense that all the possible rounding errors have been taken into account. Techniques for accelerating the enclosure and obtaining smaller error bounds are introduced. Numerical results show the properties of the proposed algorithms. Copyright © 2015 John Wiley & Sons, Ltd.     

A survey of truncation error analysis for Padé and continued fraction approximants

To compute the value of a functionf(z) in the complex domain by means of a converging sequence of rational approximants {f _n(z)} of a continued fraction and/or Padé table, it is essential to have sharp estimates of the truncation error ¦f(z)–f _n(z)¦. This paper is an expository survey of constructive methods for obtaining such truncation error bounds. For most cases dealt with, {f _n(z)} is the sequence of approximants of a continued fractoin, and eachf _n(z) is a (1-point or 2-point) Padé approximant. To provide a common framework that applies to rational approximantf _n(z) that may or may not be successive approximants of a continued fraction, we introduce linear fractional approximant sequences (LFASs). Truncation error bounds are included for a large number of classes of LFASs, most of which contain representations of important functions and constants used in mathematics, statistics, engineering and the physical sciences. An extensive bibliography is given at the end of the paper.Research supported in part by the U.S. National Science Foundation under Grants INT-9113400 and DMS-9302584.     

The convergence rate of continued fractions representing solutions of a Riccati equation

The aim of this note is to generalize and apply results on matrix continued fractions representing the solution of discrete matrix Riccati equations. Assuming uniform bounds for the norm of the matrix coefficients of the continued fraction, the minimal and maximal solutions of the corresponding algebraic Riccati equation can be accurately enclosed.     

Fehleranalyse für die Gauß-Elimination zur Berechnung der Lösung minimaler Länge

Summary Presented is a realistic, elementwise analysis for the rounding errors of a generalization of Gauss elimination for solving the linear best least squares problem without pivoting. The bounds are suitable to determine the class of well-posed problems to the given method. A mixed error analysis is given and then the effects of errors in the input data are studied. Numerical examples demonstrate the efficiency.

     

Parameter selection rules for path-following with singular embedding

Summary. Singular embedding methods require appropriately adjusted parameters to guarantee the contraction of locally quadratically convergent iterative methods. Firstly some general rule for the parameter selection is proposed and its rate of convergence is analyzed. Secondly, some modifications in the case of polynomial error and contraction bounds are studied. Finally, these results are applied to the embedding of an elliptic boundary value problem with discontinuous nonlinearities into a family of smooth problems. Here the regularization is done in such a way that the solutions of the resulting auxiliary problems require only one step of Newton's method. Received December 2, 1996 / Revised version reveived April 7, 1998     

Convergence analysis of discretization methods for nonlinear first kind Volterra integral equations

Summary An existence and uniqueness result is given for nonlinear Volterra integral equations of the first kind. This permits, by means of analogous discrete manipulations, a general convergence analysis for a wide class of discretization methods for nonlinear first kind Volterra integral equations to be presented. A concept of optimal consistency allows twosided error bounds to be derived.     

Estimates for multigrid methods based on red-black Gauss-Seidel smoothings

Summary TheMGR[v] algorithms of Ries, Trottenberg and Winter, the Algorithms 2.1 and 6.1 of Braess and the Algorithm 4.1 of Verfürth are all multigrid algorithms for the solution of the discrete Poisson equation (with Dirichlet boundary conditions) based on red-black Gauss-Seidel smoothing. Both Braess and Verfürth give explicit numerical upper bounds on the rate of convergence of their methods in convex polygonal domains. In this work we reconsider these problems and obtain improved estimates for theh–2h Algorithm 4.1 as well asW-cycle estimates for both schemes in non-convex polygonal domains. The proofs do not depend on the strengthened Cauchy inequality.Sponsored by the Air Force Office of Scientific Research under Contract No. AFOSR-86-0163