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We analyze when it is possible to compute the singular values and singular vectors of a matrix with high relative accuracy. This means that each computed singular value is guaranteed to have some correct digits, even if the singular values have widely varying magnitudes. This is in contrast to the absolute accuracy provided by conventional backward stable algorithms, whichin general only guarantee correct digits in the singular values with large enough magnitudes. It is of interest to compute the tiniest singular values with several correct digits, because in some cases, such as #nite element problems and quantum mechanics, it is the smallest singular values that havephysical meaning, and should be determined accurately by the data. Many recent papers have identi#ed special classes of matrices where high relative accuracy is possible, since it is not possible in general. The perturbation theory and algorithms for these matrix classes have been quite di#erent, motivating us to seek a co...

. Correlation matrices---symmetric positive semidefinite matrices with unit diagonal--- are important in statistics and in numerical linear algebra. For simulation and testing it is desirable to be able to generate random correlation matrices with specified eigenvalues (which must be nonnegative and sum to the dimension of the matrix). A popular algorithm of Bendel and Mickey takes a matrix having the specified eigenvalues and uses a finite sequence of Given rotations to introduce 1s on the diagonal. We give improved formulae for computing the rotations and prove that the resulting algorithm is numerically stable. We show by example that the formulae originally proposed, which are used in certain existing Fortran implementations, can lead to serious instability. We also show how to modify the algorithm to generate a rectangular matrix with columns of unit 2-norm. Such a matrix represents a correlation matrix in factored form, which can be preferable to representing the matrix itself, ...

this paper we introduce certain numbers, called collinearity indices, which are useful in detecting near collinearities in regression problems. The coefficients enter adversely into formulas concerning significance testing and the effects of errors in the regression variables. Thus they provide simple regression diagnostics, suitable for incorporation in regression packages. Keywords and phrases: collinearity, ill-conditioning, linear regression, errors in the variables, regression diagnostics. 1 Introduction

This paper gives perturbation analyses for Q 1 and R in the QR factorization A = Q 1 R, Q T 1 Q 1 = I, for a given real m \Theta n matrix A of rank n. The analyses more accurately reflect the sensitivity of the problem than previous normwise results. The condition numbers here are altered by any column pivoting used in AP = Q1R, and the condition numbers for R are bounded for a fixed n when the standard column pivoting strategy is used. This strategy tends to improve the condition of Q 1 , so the computed Q 1 and R will probably both have greatest accuracy when we use the standard column pivoting strategy. First order normwise perturbation analyses are given for both Q 1 and R. It is seen that the analysis for R may be approached in two ways --- a detailed "matrix--vector equation" analysis which provides tight bounds and resulting true condition numbers, which unfortunately are costly to compute and not very intuitive, and a perhaps simpler "matrix equation" analysis which provides results that are usually weaker but easier to interpret, and which allow efficient computation of a satisfactory estimate for the true condition number. Key Words. QR factorization, perturbation analysis, condition estimation, matrix equations, pivoting AMS Subject Classifications: 15A23, 65F35 1.

. Perturbation bounds in numerical linear algebra are traditionally derived and expressed using norms. Norm bounds cannot reflect the scaling or sparsity of a problem and its perturbation, and so can be unduly weak. If the problem data and its perturbation are measured componentwise, much smaller and more revealing bounds can be obtained. A survey is given of componentwise perturbation theory in numerical linear algebra, covering linear systems, the matrix inverse, matrix factorizations, the least squares problem, and the eigenvalue and singular value problems. Most of the results described have been published in the last five years. Our hero is the intrepid, yet sensitive matrix A. Our villain is E, who keeps perturbing A. When A is perturbed he puts on a crumpled hat: e A = A+E. G. W. Stewart and J.-G. Sun, Matrix Perturbation Theory (1990) 1. Introduction Matrix analysis would not have developed into the vast subject it is today without the concept of representing a matrix by ...

The 3-term Lanczos process leads, for a symmetric matrix, to bases for Krylov subspaces of increasing dimension. The Lanczos basis, together with the recurrence coefficients, can be used for the solution of symmetric indefinite linear systems, by solving the reduced system in one way or another. This leads to well-known methods: MINRES, GMRES, and SYMMLQ. We will discuss in what way and to what extent these approaches differ in their sensitivity to rounding errors. In our analysis we will assume that the Lanczos basis is generated in exactly the same way for the different methods, and we will not consider the errors in the Lanczos process itself. We will show that the method of solution may lead, under certain circumstances, to large additional errors, that are not corrected by continuing the iteration process. Our findings are supported and illustrated by numerical examples. 1 Introduction We will consider iterative methods for the construction of approximate solutions, starting with...

Various methods have been developed for computing the correlation matrix nearest in the Frobenius norm to a given matrix. We focus on a quadratically convergent Newton algorithm recently derived by Qi and Sun. Various improvements to the efficiency and reliability of the algorithm are introduced. Several of these relate to the linear algebra: the Newton equations are solved by minres instead of the conjugate gradient method, as it more quickly satisfies the inexact Newton condition; we apply a Jacobi preconditioner, which can be computed efficiently even though the coefficient matrix is not explicitly available; an efficient choice of eigensolver is identified; and a final scaling step is introduced to ensure that the returned matrix has unit diagonal. Potential difficulties caused by rounding errors in the Armijo line search are avoided by altering the step selection strategy. These and other improvements lead to a significant speedup over the original algorithm and allow the solution of problems of dimension a few thousand in a few tens of minutes.

The contribution of this paper is to describe a general technique to solve some classes of large but sparse semidefinite problems via a robust primal-dual interior-point technique which uses an inexact Gauss-Newton approach with a matrix free preconditioned conjugate gradient method. This approach avoids the ill-conditioning pitfalls that result from symmetrization and from forming the so-called normal equations, while maintaining the primal-dual framework.