A block algorithm for computing rank-revealing QR factorizations

We present a block algorithm for computing rank-revealing QR factorizations (RRQR factorizations) of rank-deficient matrices. The algorithm is a block generalization of the RRQR-algorithm of Foster and Chan. While the unblocked algorithm reveals the rank by peeling off small singular values one by one, our algorithm identifies groups of small singular values. In our block algorithm, we use incremental condition estimation to compute approximations to the nullvectors of the matrix. By applying another (in essence also rank-revealing) orthogonal factorization to the nullspace matrix thus created, we can then generate triangular blocks with small norm in the lower right part ofR. This scheme is applied in an iterative fashion until the rank has been revealed in the (updated) QR factorization. We show that the algorithm produces the correct solution, under very weak assumptions for the orthogonal factorization used for the nullspace matrix. We then discuss issues concerning an efficient implementation of the algorithm and present some numerical experiments. Our experiments show that the block algorithm is reliable and successfully captures several small singular values, in particular in the initial block steps. Our experiments confirm the reliability of our algorithm and show that the block algorithm greatly reduces the number of triangular solves and increases the computational granularity of the RRQR computation.This work was supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research, US Department of Energy, under Contract W-31-109-Eng-38. The second author was also sponsored by a travel grant from the Knud Højgaards Fond.This work was partially completed while the author was visiting the IBM Scientific Center in Heidelberg, Germany.     

Iterative method for computing the Moore-Penrose inverse based on Penrose equations

An iterative algorithm for estimating the Moore-Penrose generalized inverse is developed. The main motive for the construction of the algorithm is simultaneous usage of Penrose equations (2) and (4). Convergence properties of the introduced method as well as their first-order and second-order error terms are considered. Numerical experiment is also presented.     

Adaptive Lanczos methods for recursive condition estimation

Estimates for the condition number of a matrix are useful in many areas of scientific computing, including: recursive least squares computations, optimization, eigenanalysis, and general nonlinear problems solved by linearization techniques where matrix modification techniques are used. The purpose of this paper is to propose anadaptiveLanczosestimator scheme, which we callale, for tracking the condition number of the modified matrix over time. Applications to recursive least squares (RLS) computations using the covariance method with sliding data windows are considered.ale is fast for relatively smalln-parameter problems arising in RLS methods in control and signal processing, and is adaptive over time, i.e., estimates at timet are used to produce estimates at timet+1. Comparisons are made with other adaptive and non-adaptive condition estimators for recursive least squares problems. Numerical experiments are reported indicating thatale yields a very accurate recursive condition estimator.Research supported by the US Air Force under grant no. AFOSR-88-0285.Research supported by the US Army under grant no. DAAL03-90-G-105.Research supported by the US Air Force under grant no. AFOSR-88-0285.     

A Newton iteration process for inverse eigenvalue problems

Summary We suppose an inverse eigenvalue problem which includes the classical additive and multiplicative inverse eigenvalue problems as special cases. For the numerical solution of this problem we propose a Newton iteration process and compare it with a known method. Finally we apply it to a numerical example.     

Perturbations and expressions for generalized inverses in Banach spaces and Moore-Penrose inverses in Hilbert spaces of closed linear operators

The problems of perturbation and expression for the generalized inverses of closed linear operators in Banach spaces and for the Moore-Penrose inverses of closed linear operators in Hilbert spaces are studied. We first provide some stability characterizations of generalized inverses of closed linear operators under T-bounded perturbation in Banach spaces, which are exactly equivalent to that the generalized inverse of the perturbed operator has the simplest expression T⁺(I+δTT⁺)^-1. Utilizing these results, we investigate the expression for the Moore-Penrose inverse of the perturbed operator in Hilbert spaces and provide a unified approach to deal with the range preserving or null space preserving perturbation. An explicit representation for the Moore-Penrose inverse of the perturbation is also given. Moreover, we give an equivalent condition for the Moore-Penrose inverse to have the simplest expression T^†(I+δTT^†)^-1. The results obtained in this paper extend and improve many recent results in this area.     

Drazin inverse of partitioned matrices in terms of Banachiewicz-Schur forms

Let be a partitioned matrix, where A and D are square matrices. Denote the Drazin inverse of A by A^D. The purpose of this paper is twofold. Firstly, we develop conditions under which the Drazin inverse of M having generalized Schur complement, S=D-CA^DB, group invertible, can be expressed in terms of a matrix in the Banachiewicz-Schur form and its powers. Secondly, we deal with partitioned matrices satisfying rank(M)=rank(A^D)+rank(S^D), and give conditions under which the group inverse of M exists and a formula for its computation.     

Additive and multiplicative perturbation bounds for the Moore-Penrose inverse

In this paper, we obtain the additive and multiplicative perturbation bounds for the Moore-Penrose inverse under the unitarily invariant norm and the Q - norm, which improve the corresponding ones in [P.Å. Wedin, Perturbation theory for pseudo-inverses, BIT 13(1973)217-232].     

Convexity of the inverse and Moore-Penrose inverse

Convexity properties of the inverse of positive definite matrices and the Moore-Penrose inverse of nonnegative definite matrices with respect to the partial ordering induced by nonnegative definiteness are studied. For the positive definite case null-space characterizations are derived, and lead naturally to a concept of strong convexity of a matrix function, extending the conventional concept of strict convexity. The positive definite results are shown to allow for a unified analysis of problems in reproducing kernel Hilbert space theory and inequalities involving matrix means. The main results comprise a detailed study of the convexity properties of the Moore-Penrose inverse, providing extensions and generalizations of all the earlier work in this area.     

Particular formulae for the Moore-Penrose inverse of a columnwise partitioned matrix

An essential part of Cegielski’s [Obtuse cones and Gram matrices with non-negative inverse, Linear Algebra Appl. 335 (2001) 167-181] considerations of some properties of Gram matrices with nonnegative inverses, which are pointed out to be crucial in constructing obtuse cones, consists in developing some particular formulae for the Moore-Penrose inverse of a columnwise partitioned matrix A = (A₁ : A₂) under the assumption that it is of full column rank. In the present paper, these results are generalized and extended. The generalization consists in weakening the assumption mentioned above to the requirement that the ranges of A₁ and A₂ are disjoint, while the extension consists in introducing the conditions referring to the class of all generalized inverses of A.     

Closest normal matrix finally found!

A method of finding the closest normal matrix in the Frobenius matrix norm is developed. It is shown that if a matrix is represented in those coordinates where its closest normal matrix is diagonal, its restriction to any pair of coordinate directions is a multiple of a real diagonal and skew nondiagonal 2×2 matrix. A convergent algorithm to bring an arbitrary matrix into that form is described and results of numerical tests are reported.Dedicated to the memory of Peter Henrici (1923–1987).     

On efficient implementations of Kogbetliantz's algorithm for computing the singular value decomposition

Summary In this paper we compare several implementations of Kogbetliantz's algorithm for computing the SVD on sequential as well as on parallel machines. Comparisons are based on timings and on operation counts. The numerical accuracy of the different methods is also analyzed.     

On the effective computation of the inertia of a non-hermitian matrix

Summary given a complex lower Hessenberg matrixA with unit codiagonal, a hermitian matrixH is constructed such that, ifH is non-singular InA= InH. IfA is real,H is real symmetric. Classical results of Fujiwara on the root-separation problem and of Schwarz on the eigenvalue-separation problem are included as special cases.The authors' research was conducted at the Universidade Estadual de Campinas and supported by the Fundação de Amparo à Pesquisa do Estado de São Paulo, Brasil, under grant n⁰ 78/0490.     

A method for the numerical solution of the one-dimensional inverse Stefan problem

Summary In this paper we suggest the use of complete families of solutions of the heat equation for the numerical solution of the inverse Stefan problem. Our approach leads to linear optimization problems which can be established and solved easily. Convergence results are proved. In a final section the method is applied to some examples.     

Product integration rules for volterra integral equations of the first kind

The numerical solution of Volterra integral equations of the first kind can be achieved via product integration. This paper establishes the asymptotic error expansions of certain product integration rules. The rectangular rules are found to produce expansions containing all powers ofh, and the midpoint product method is found to produce even powers ofh. Extrapolation to the limit is then applied.     

Block-iterative methods for consistent and inconsistent linear equations

Summary We shall in this paper consider the problem of computing a generalized solution of a given linear system of equations. The matrix will be partitioned by blocks of rows or blocks of columns. The generalized inverses of the blocks are then used as data to Jacobi- and SOR-types of iterative schemes. It is shown that the methods based on partitioning by rows converge towards the minimum norm solution of a consistent linear system. The column methods converge towards a least squares solution of a given system. For the case with two blocks explicit expressions for the optimal values of the iteration parameters are obtained. Finally an application is given to the linear system that arises from reconstruction of a two-dimensional object by its one-dimensional projections.     

The optimal perturbation bounds of the Moore-Penrose inverse under the Frobenius norm

We obtain the optimal perturbation bounds of the Moore-Penrose inverse under the Frobenius norm by using Singular Value Decomposition, which improved the results in the earlier paper [P.-Å. Wedin, Perturbation theory for pseudo-inverses, BIT 13 (1973) 217-232]. In addition, a perturbation bound of the Moore-Penrose inverse under the Frobenius norm in the case of the multiplicative perturbation model is also given.     

On computingINV block preconditionings for the conjugate gradient method

The INV(k) and MINV(k) block preconditionings for the conjugate gradient method require generation of selected elements of the inverses of symmetric matrices of bandwidth 2k+1. Generalizing the previously describedk=1 (tridiagonal) case tok=2, explicit expressions for the inverse elements of a symmetric pentadiagonal matrix in terms of Green's matrix of rank two are given. These expressions are found to be seriously ill-conditioned; hence alternative computational algorithms for the inverse elements must be used. Behavior of thek=1 andk=2 preconditionings are compared for some discretized elliptic partial differential equation test problems in two dimensions.Presented by the first author at the Joint U.S.-Scandinavian Symposium on Scientific Computing and Mathematical Modeling, January 1985, in honor of Germund Dahlquist on the occasion of his 60th birthday. This work was supported in part by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy under contract DE AC03-76SF00098.     

A sharp version of Bauer–Fike’s theorem

In this paper, we present a sharp version of Bauer–Fike’s theorem. We replace the matrix norm with its spectral radius or sign-complex spectral radius for diagonalizable matrices; 1-norm and ∞$\infty$-norm for non-diagonalizable matrices. We also give the applications to the pole placement problem and the singular system.     

Algebraic connections between the least squares and total least squares problems

Summary In this paper the closeness of the total least squares (TLS) and the classical least squares (LS) problem is studied algebraically. Interesting algebraic connections between their solutions, their residuals, their corrections applied to data fitting and their approximate subspaces are proven.All these relationships point out the parameters which mainly determine the equivalences and differences between the two techniques. These parameters also lead to a better understanding of the differences in sensitivity between both approaches with respect to perturbations of the data.In particular, it is shown how the differences between both approaches increase when the equationsAXB become less compatible, when the length ofB orX is growing or whenA tends to be rank-deficient. They are maximal whenB is parallel with the singular vector ofA associated with its smallest singular value. Furthermore, it is shown how TLS leads to a weighted LS problem, and assumptions about the underlying perturbation model of both techniques are deduced. It is shown that many perturbation models correspond with the same TLS solution.Senior Research Assistant of the Belgian N.F.W.O. (National Fund of Scientific Research)     

The Moore-Penrose inverse for sums of matrices under rank additivity conditions

Some results on the Moore-Penrose inverse for sums of matrices under rank additivity conditions are revisited and some new consequences are presented. Their extensions to the weighted Moore-Penrose inverse of sums of matrices under rank additivity conditions are also considered.