On neighbouring matrices with quadratic elementary divisors

Summary Algorithms are presented which compute theQR factorization of an order-n Toeplitz matrix inO(n ²) operations. The first algorithm computes onlyR explicitly, and the second computes bothQ andR. The algorithms are derived from a well-known procedure for performing the rank-1 update ofQR factors, using the shift-invariance property of the Toeplitz matrix. The algorithms can be used to solve the Toeplitz least-squares problem, and can be modified to solve Toeplitz systems inO(n) space.     

Perturbation and error analyses for block downdating of a Cholesky decomposition

A new perturbation result is presented for the problem of block downdating a Cholesky decompositionX ^T X = R ^T R. Then, a condition number for block downdating is proposed and compared to other downdating condition numbers presented in literature recently. This new condition number is shown to give a tighter bound in many cases. Using the perturbation theory, an error analysis is presented for the block downdating algorithms based on the LINPACK downdating algorithm and stabilized hyperbolic transformations. An error analysis is also given for block downdating using Corrected Seminormal Equations (CSNE), and it is shown that for ill-conditioned downdates this method gives more accurate results than the algorithms based on the LINPACK downdating algorithm or hyperbolic transformations. We classify the problems for which the CSNE downdating method produces a downdated upper triangular matrix which is comparable in accuracy to the upper triangular factor obtained from the QR decomposition by Householder transformations on the data matrix with the row block deleted.Dedicated to Ji-guang Sun in honour of his 60th birthdayThe work of the second author was supported in part by the National Science Foundation grant CCR-9209726.     

Computing Mixed Discriminants, Mixed Volumes, and Permanents

We construct a probabilistic polynomial time algorithm that computes the mixed discriminant of given n positive definite matrices within a 2 ^O(n) factor. As a corollary, we show that the permanent of an nonnegative matrix and the mixed volume of n ellipsoids in R ⁿ can be computed within a 2 ^O(n) factor by probabilistic polynomial time algorithms. Since every convex body can be approximated by an ellipsoid, the last algorithm can be used for approximating in polynomial time the mixed volume of n convex bodies in R ⁿ within a factor n ^O(n) . Received July 10, 1995, and in revised form May 20, 1996.     

Computation of orthonormal factors for fundamental solution matrices

Summary. In this work, we introduce and analyze two new techniques for obtaining the Q factor in the QR factorization of some (or all) columns of a fundamental solution matrix Y of a linear differential system. These techniques are based on elementary Householder and Givens transformations. We implement and compare these new techniques with existing approaches on some examples. Received October 27, 1997 / Revised version received September 21, 1998 / Published online August 19, 1999     

Factoring wavelet transforms into lifting steps

This article is essentially tutorial in nature. We show how any discrete wavelet transform or two band subband filtering with finite filters can be decomposed into a finite sequence of simple filtering steps, which we call lifting steps but that are also known as ladder structures. This decomposition corresponds to a factorization of the polyphase matrix of the wavelet or subband filters into elementary matrices. That such a factorization is possible is well-known to algebraists (and expressed by the formulaSL(n;R[z, z⁻¹])=E(n;R[z, z⁻¹])); it is also used in linear systems theory in the electrical engineering community. We present here a self-contained derivation, building the decomposition from basic principles such as the Euclidean algorithm, with a focus on applying it to wavelet filtering. This factorization provides an alternative for the lattice factorization, with the advantage that it can also be used in the biorthogonal, i.e., non-unitary case. Like the lattice factorization, the decomposition presented here asymptotically reduces the computational complexity of the transform by a factor two. It has other applications, such as the possibility of defining a wavelet-like transform that maps integers to integers. Research Tutorial Acknowledgements and Notes. Page 264.     

A Deterministic Algorithm for Approximating the Mixed Discriminant and Mixed Volume, and a Combinatorial Corollary

Abstract. We present a deterministic polynomial-time algorithm that computes the mixed discriminant of an n -tuple of positive semidefinite matrices to within an exponential multiplicative factor. To this end we extend the notion of doubly stochastic matrix scaling to a larger class of n -tuples of positive semidefinite matrices, and provide a polynomial-time algorithm for this scaling. As a corollary, we obtain a deterministic polynomial algorithm that computes the mixed volume of n convex bodies in R ⁿ to within an error which depends only on the dimension. This answers a question of Dyer, Gritzmann and Hufnagel. A ``side benefit' is a generalization of Rado's theorem on the existence of a linearly independent transversal.     

Four algorithms for the the efficient computation of truncated pivoted QR approximations to a sparse matrix

Summary. In this paper we propose four algorithms to compute truncated pivoted QR approximations to a sparse matrix. Three are based on the Gram–Schmidt algorithm and the other on Householder triangularization. All four algorithms leave the original matrix unchanged, and the only additional storage requirements are arrays to contain the factorization itself. Thus, the algorithms are particularly suited to determining low-rank approximations to a sparse matrix. Received February 23, 1998 / Revised version received April 16, 1998     

Riesz's Theorem for Orthogonal Matrix Polynomials

We describe the image through the Stieltjes transform of the set of solutions V of a matrix moment problem. We extend Riesz's theorem to the matrix setting, proving that those matrices of measures of V for which the matrix polynomials are dense in the corresponding ² space are precisely those whose Stieltjes transform is an extremal point (in the sense of convexity) of the image set. May 20, 1997. Date revised: January 8, 1998.     

Computing an Eigenvector of a Monge Matrix in Max-Plus Algebra

The problem of finding one eigenvector of a given Monge matrix A in a max-plus algebra is considered. For a general matrix, the problem can be solved in O(n ³) time by computing one column of the corresponding metric matrix Δ(A _λ), where λ is the eigenvalue of A. An algorithm is presented, which computes an eigenvector of a Monge matrix in O(n ²) time.     

A block Cholesky-LU-based QR factorization for rectangular matrices

The Householder method provides a stable algorithm to compute the full QR factorization of a general matrix. The standard version of the algorithm uses a sequence of orthogonal reflections to transform the matrix into upper triangular form column by column. In order to exploit (level 3 BLAS or structured matrix) computational advantages for block-partitioned algorithms, we develop a block algorithm for the QR factorization. It is based on a well-known block version of the Householder method which recursively divides a matrix columnwise into two smaller matrices. However, instead of continuing the recursion down to single matrix columns, we introduce a novel way to compute the QR factors in implicit Householder representation for a larger block of several matrix columns, that is, we start the recursion at a block level instead of a single column. Numerical experiments illustrate to what extent the novel approach trades some of the stability of Householder's method for the computational efficiency of block methods.     

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     

Rank-two update algorithms for the minimum volume enclosing ellipsoid problem

We consider the problem of computing a (1+ε)-approximation to the minimum volume enclosing ellipsoid (MVEE) of a given set of m points in R ⁿ . Based on the idea of sequential minimal optimization (SMO) method, we develop a rank-two update algorithm. This algorithm computes an approximate solution to the dual optimization formulation of the MVEE problem, which updates only two weights of the dual variable at each iteration. We establish that this algorithm computes a (1+ε)-approximation to MVEE in O(mn ³/ε) operations and returns a core set of size O(n ²/ε) for ε∈(0,1). In addition, we give an extension of this rank-two update algorithm. Computational experiments show the proposed algorithms are very efficient for solving large-scale problem with a high accuracy.     

Fast Wavelet Transform for Toeplitz Matrices and Property Analysis

Fast wavelet transform algorithms for Toeplitz matrices are proposed in this paper. Distinctive from the well known discrete trigonometric transforms, such as the discrete cosine transform （DCT） and the discrete Fourier transform （DFT） for Toeplitz matrices, the new algorithms are achieved by compactly supported wavelet that preserve the character of a Toeplitz matrix after transform, which is quite useful in many applications involving a Toeplitz matrix. Results of numerical experiments show that the proposed method has good compression performance similar to using wavelet in the digital image coding. Since the proposed algorithms turn a dense Toeplitz matrix into a band-limited form, the arithmetic operations required by the new algorithms are O（N） that are reduced greatly compared with O（N log N） by the classical trigonometric transforms.     

Equivalence between modified symplectic Gram-Schmidt and Householder SR algorithms

The SR factorization for a given matrix A is a QR-like factorization A=SR, where the matrix S is symplectic and R is J-upper triangular. This factorization is fundamental for some important structure-preserving methods in linear algebra and is usually implemented via the symplectic Gram-Schmidt algorithm (SGS). There exist two versions of SGS, the classical (CSGS) and the modified (MSGS). Both are equivalent in exact arithmetic, but have very different numerical behaviors. The MSGS is more stable. Recently, the symplectic Householder SR algorithm has been introduced, for computing efficiently the SR factorization. In this paper, we show two new and important results. The first is that the SR factorization of a matrix A via the MSGS is mathematically equivalent to the SR factorization via Householder SR algorithm of an embedded matrix. The later is obtained from A by adding two blocks of zeros in the top of the first half and in the top of the second half of the matrix A. The second result is that MSGS is also numerically equivalent to Householder SR algorithm applied to the mentioned embedded matrix.     

Surrogate methods for linear inequalities

Three algorithms are developed and validated for finding a pointx inR ⁿ that satisfies a given system of inequalities,Axb. A andb are a given matrix and a given vector inR ^m×n andR ^m , respectively, with the rows ofA assumed normalized. The algorithms are iterative and are based upon the orthogonal projection of an infeasible point onto the manifold of the bounding hyperplanes of some of the given constraints. The choice of the active constraints and the actual projection are accomplished through the use of surrogate constraints.This work was supported in part by the City University of New York Research Center. The author thanks Professor D. Goldfarb for suggesting the problem and also for his valuable literature information and time. The word surrogate was borrowed from one of his works.     

Tight Multiwavelet Frames with Different Matrix Dilations and Matrix Translations

The characterization of tight multiwavelet frames with different matrix dilations and matrix translations for L ²(R ^d ) is established. The result contains and further extends the generalizations that have appeared in the literature. Two sufficient conditions for affine frames are also presented.     

A note on the error analysis of classical Gram–Schmidt

An error analysis result is given for classical Gram–Schmidt factorization of a full rank matrix A into A = QR where Q is left orthogonal (has orthonormal columns) and R is upper triangular. The work presented here shows that the computed R satisfies R ^T R = A ^T A + E where E is an appropriately small backward error, but only if the diagonals of R are computed in a manner similar to Cholesky factorization of the normal equations matrix. At the end of the article, implications for classical Gram–Schmidt with reorthogonalization are noted.A similar result is stated in Giraud et al. (Numer Math 101(1):87–100, 2005). However, for that result to hold, the diagonals of R must be computed in the manner recommended in this work.Jesse Barlow’s research was supported by the National Science Foundation under grant no. CCF-0429481.     

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.     

Spherical submanifolds in a Euclidean space

In this paper, we are interested in extending the study of spherical curves in R ³ to the submanifolds in the Euclidean space R ^n+p . More precisely, we are interested in obtaining conditions under which an n-dimensional compact submanifold M of a Euclidean space R ^n+p lies on the hypersphere S ^n+p−1(c) (standardly imbedded sphere in R ^n+p of constant curvature c). As a by-product we also get an estimate on the first nonzero eigenvalue of the Laplacian operator Δ of the submanifold (cf. Theorem 3.5) as well as a characterization for an n-dimensional sphere S ⁿ (c) (cf. Theorem 4.1).