Tibor Boros

Pivoting and backward stability of fast algorithms for solving Cauchy linear equations

Abstract Three fast O(n2) algorithms for solving Cauchy linear systems of equations are proposed. A rounding error analysis indicates that the backward stability of these new Cauchy solvers is similar to that of Gaussian elimination, thus suggesting to employ various pivoting techniques to achieve a favorable backward stability. It is shown that Cauchy structure allows one to achieve in O(n2) operations partial pivoting ordering of the rows and several other judicious orderings in advance, without actually performing the elimination. The analysis also shows that for the important class of totally positive Cauchy matrices it is advantageous to avoid pivoting, which yields a remarkable backward stability of the suggested algorithms. It is shown that Vandermonde and Chebyshev–Vandermonde matrices can be efficiently transformed into Cauchy matrices, using Discrete Fourier, Cosine or Sine transforms. This allows us to use the proposed algorithms for Cauchy matrices for rapid and accurate solution of Vandermonde and Chebyshev–Vandermonde linear systems. The analytical results are illustrated by computed examples.

A fast parallel Björck–Pereyra-type algorithm for solving Cauchy linear equations☆

Abstract We propose a new fast O( n 2 ) parallel algorithm for solving Cauchy systems of linear equations. We perform an a priori rounding error analysis and obtain componentwise bounds for the forward, backward and residual errors. These bounds indicate that for the class of totally positive Cauchy matrices the new algorithm is forward and backward stable, producing a remarkably high relative accuracy. In particular, Hilbert linear systems, often considered to be too ill-conditioned to be attacked, can be rapidly solved with high precision. The results indicate a close resemblance between the numerical properties of Cauchy matrices and the much-studied Vandermonde matrices. In fact, our proposed Cauchy solver is an analog of the well-known Bjorck–Pereyra algorithm for Vandermonde systems. As a by-product we obtain favorably backward error bounds for the original Bjorck–Pereyra algorithm. Several computed examples illustrate a connection of high relative accuracy to the concepts of effective well-conditioning and total positivity.

Structured Matrices and Unconstrained Rational Interpolation Problems

We describe a fast recursive algorithm for the solution of an unconstrained rational interpolation problem by exploiting the displacement structure concept. We use the interpolation data to implicitly define a convenient non-Hermitian structured matrix, and then apply a computationally efficient procedure for its triangular factorization. This leads to a transmission-line interpretation that makes evident the interpolation properties. We further discuss connections with the Lagrange interpolating polynomial as well as questions regarding the minimality and the admissible degrees of complexity of the solutions.

A generalized Schur-type algorithm for the joint factorization of a structured matrix and its inverse

A Schur-type algorithm is presented for the simultaneous triangular factorization of a given (non-degenerate) structured matrix and its inverse. The algorithm takes the displacement generator of a Hermitian, strongly regular matrixR as an input, and computes the displacement generator of the inverse matrixR−1 as an output. From these generators we can directly deduce theLD−1L* (lower-diagonal-upper) decomposition ofR, and theUD−1U* (upper-diagonallower) decomposition ofR−1. The computational complexity of the algorithm isO(rn2) operations, wheren andr denote the size and the displacement rank ofR, respectively. Moreover, this method is especially suited for parallel (systolic array) implementations: usingn processors the algorithm can be carried out inO(n) steps.

Analytic interpolation problems and lattice filter design

The so-called analytic interpolation problem is addressed and solved. The objective is to find the family of rational interpolants which are analytic in a certain region of the complex plane. It turns out that the usual linear fractional map cannot be used to describe the solution set conveniently. Instead, an affine parametrization formula is proposed as the natural framework to impose analyticity constraint on the interpolants. All solutions of the interpolation problem are characterized in terms of a generating system, which can be obtained efficiently via a fast recursive algorithm. The recursive procedure can be used to update the solutions whenever a new interpolation constraint is added to the input data set. It is shown that the analytic interpolation problem is solvable if and only if the corresponding unconstrained problem is solvable, i.e., if and only if the interpolation data-set is consistent. The above results have many applications in different areas such as stable lattice filter design, channel identification, and Q-parametrization of stabilizing controllers.<<ETX>>

A recursive method for solving unconstrained tangential interpolation problems

An efficient recursive solution is presented for the one-sided unconstrained tangential interpolation problem. The method relies on the triangular factorization of a certain structured matrix that is implicitly defined by the interpolation data. The recursive procedure admits a physical interpretation in terms of discretized transmission lines. In this framework the generating system is constructed as a cascade of first-order sections. Singular steps occur only when the input data is contradictory, i.e., only when the interpolation problem does not have a solution. Various pivoting schemes can be used to improve numerical accuracy or to impose additional constraints on the interpolants. The algorithm also provides coprime factorizations for all rational interpolants and can be used to solve polynomial interpolation problems such as the general Hermite matrix interpolation problem. A recursive method is proposed to compute a column-reduced generating system that can be used to solve the minimal tangential interpolation problem.

Q-parametrization for unstable plants: an interpolation approach

This paper deals with a tangential interpolation problem which arises in the Q-parametrization of stabilizing controllers for unstable plants. The authors give a fast, recursive algorithm to compute the so-called generating system that can be used to parametrize all stable rational interpolants. The authors method is based on a displacement structure approach.