Additive preconditioning for matrix computations

Our randomized additive preconditioners are readily available and regularly facilitate the solution of linear systems of equations and eigen-solving for a very large class of input matrices. We study the generation of such preconditioners and their impact on the rank and the condition number of a matrix. We also propose some techniques for their refinement and two alternative versions of randomized preprocessing. Our analysis and experiments show the power of our approach even where we employ weak randomization, that is generate sparse and structured preconditioners, defined by a small number of random parameters.     

Using FFT-based techniques in polynomial and matrix computations: recent advances and applicatons

Computations with univariate polynomials, like the evaluation of product, quotient, remainder, greatest common divisor, etc, are closely related to linear algebra computations performed with structured matrices having the Toeplitz-like or the Hankel-like structures.

The discrete Fourier transform, and the FFT algorithms for its computation, constitute a powerful tool for the design and analysis of fast algorithms for solving problems involving polynomials and structured matrices.

We recall the main correlations between polynomial and matrix computations and present two recent results in this field: in particular, we show how Fourier methods can speed up the solution of a wide class of problems arising in queueing theory where certain Markov chains, defined by infinite block Toeplitz matrices in generalized Hessenberg form, must be solved. Moreover, we present a new method for the numerical factorization of polynomials based on a matrix generalization of Koenig's theorem. This method, that relies on the evaluation/interpolation technique at the Fourier points, reduces the problem of polynomial factorization to the computation of the LU decomposition of a banded Toeplitz matrix with its rows and columns suitably permuted. Numerical experiments that show the effectiveness of our algorithms are presented     

<Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-minimization methods for Hamilton–Jacobi equations: the one-dimensional case

A new approximation technique based on L ¹-minimization is introduced. It is proven that the approximate solution converges to the viscosity solution in the case of one-dimensional stationary Hamilton–Jacobi equation with convex Hamiltonian. This material is based upon work supported by the National Science Foundation grant DMS-0510650. J.-L. Guermond is on leave from LIMSI, UPRR 3251 CNRS, BP 133, 91403 Orsay Cedex, France.     

Accurate computations with Said-Ball-Vandermonde matrices

A generalization of the Vandermonde matrices which arise when the power basis is replaced by the Said-Ball basis is considered. When the nodes are inside the interval (0,1), then those matrices are strictly totally positive. An algorithm for computing the bidiagonal decomposition of those Said-Ball-Vandermonde matrices is presented, which allows us to use known algorithms for totally positive matrices represented by their bidiagonal decomposition. The algorithm is shown to be fast and to guarantee high relative accuracy. Some numerical experiments which illustrate the good behaviour of the algorithm are included.     

The distance from a matrix polynomial to matrix polynomials with a prescribed multiple eigenvalue

For a matrix polynomial P(λ) and a given complex number μ, we introduce a (spectral norm) distance from P(λ) to the matrix polynomials that have μ as an eigenvalue of geometric multiplicity at least κ, and a distance from P(λ) to the matrix polynomials that have μ as a multiple eigenvalue. Then we compute the first distance and obtain bounds for the second one, constructing associated perturbations of P(λ).     

Matrix decomposition MFS algorithms for elasticity and thermo-elasticity problems in axisymmetric domains

In this work, we propose an efficient matrix decomposition algorithm for the Method of Fundamental Solutions when applied to three-dimensional boundary value problems governed by elliptic systems of partial differential equations. In particular, we consider problems arising in linear elasticity in axisymmetric domains. The proposed algorithm exploits the block circulant structure of the coefficient matrices and makes use of fast Fourier transforms. The algorithm is also applied to problems in thermo-elasticity. Several numerical experiments are carried out.     

Some remarks on Dvorcuk's root-finding method

Some properties of a method for factorization of a polynomial into quadratic factors and some remarks on Dvorcuk root-finding method are discussed. The results can be applied to the consideration of the global convergence of iterative processes.This work is partially supported by the Bulgarian Ministry of Science and Education, Contract MM-208/92.     

High-resolution boosted deconvolution of spectroscopic data

The quality of the analysis of spectrometric data consists in correct identification of the existence of peaks and subsequently in good estimation of their positions. In this paper we present high-resolution deconvolution algorithms. We propose an improvement in the efficiency by introducing a boosting operation into the deconvolution process.     

On condition numbers and algorithms for determining a rigid body movement

Using a set of landmarks to represent a rigid body, a rotation of the body can be determined in least-squares sense as the solution of an orthogonal Procrustes problem. We discuss some geometrical properties of the condition number for the problem of determining the orthogonal matrix representing the rotation. It is shown that the condition number critically depends on the configuration of the landmarks. The problem is also reformulated as an unconstrained nonlinear least-squares problem and the condition number is related to the geometry of such problems. In the common 3-D case, the movement can be represented by using a screw axis. Also the condition numbers for the problem of determining the screw axis representation are shown to closely depend on the configuration of the landmarks. The condition numbers are finally used to show that the used algorithms are stable.     

Laguerre-Galerkin method for nonlinear partial differential equations on a semi-infinite interval

Summary. A Laguerre-Galerkin method is proposed and analyzed for the Burgers equation and Benjamin-Bona-Mahony (BBM) equation on a semi-infinite interval. By reformulating these equations with suitable functional transforms, it is shown that the Laguerre-Galerkin approximations are convergent on a semi-infinite interval with spectral accuracy. An efficient and accurate algorithm based on the Laguerre-Galerkin approximations to the transformed equations is developed and implemented. Numerical results indicating the high accuracy and effectiveness of this algorithm are presented. Received October 6, 1997 / Revised version received July 22, 1999 / Published online June 21, 2000     

Iterative refinement for constrained and weighted linear least squares

We present an algorithm for mixed precision iterative refinement on the constrained and weighted linear least squares problem, the CWLSQ problem. The approximate solution is obtained by solving the CWLSQ problem with the weightedQR factorization [6]. With backward errors for the weightedQR decomposition together with perturbation bounds for the CWLSQ problem we analyze the convergence behaviour of the iterative refinement procedure.In the unweighted case the initial convergence rate of the error of the iteratively refined solution is determined essentially by the condition number. For the CWLSQ problem the initial convergence behaviour is more complicated. The analysis shows that the initial convergence is dependent both on the condition of the problem related to the solution,x, and the vector =Wr, whereW is the weight matrix andr is the residual.We test our algorithm on two examples where the solution is known and the condition number of the problem can be varied. The computational test confirms the theoretical results and verifies that mixed precision iterative refinement, using the system matrix and the weightedQR decomposition, is an effective way of improving an approximate solution to the CWLSQ problem.     

Theory and numerical evaluation of oddoids and evenoids: Oscillatory cuspoid integrals with odd and even polynomial phase functions

The properties of oscillating cuspoid integrals whose phase functions are odd and even polynomials are investigated. These integrals are called oddoids and evenoids, respectively (and collectively, oddenoids). We have studied in detail oddenoids whose phase functions contain up to three real parameters. For each oddenoid, we have obtained its Maclaurin series representation and investigated its relation to Airy–Hardy integrals and Bessel functions of fractional orders. We have used techniques from singularity theory to characterise the caustic (or bifurcation set) associated with each oddenoid, including the occurrence of complex whiskers. Plots and short tables of numerical values for the oddenoids are presented. The numerical calculations used the software package CUSPINT [N.P. Kirk, J.N.L. Connor, C.A. Hobbs, An adaptive contour code for the numerical evaluation of the oscillatory cuspoid canonical integrals and their derivatives, Comput. Phys. Commun. 132 (2000) 142–165].     

Perturbation analysis of generalized saddle point systems

We discuss the perturbation analysis of generalized saddle point systems in this paper. We give the nonlinear perturbation bounds, then derive the condition numbers, and analyze the sensitivity of the computed solutions.     

Finding minimum height elimination trees for interval graphs in polynomial time

The elimination tree plays an important role in many aspects of sparse matrix factorization. The height of the elimination tree presents a rough, but usually effective, measure of the time needed to perform parallel elimination. Finding orderings that produce low elimination is therefore important. As the problem of finding minimum height elimination tree orderings is NP-hard, it is interesting to concentrate on limited classes of graphs and find minimum height elimination trees for these efficiently. In this paper, we use clique trees to find an efficient algorithm for interval graphs which make an important subclass of chordal graphs. We first illustrate this method through an algorithm that finds minimum height elimination for chordal graphs. This algorithm, although of exponential time complexity, is conceptionally simple and leads to a polynomial-time algorithm for finding minimum height elimination trees for interval graphs.This work was supported by grants from the Norwegian Research Council.     

Factorization of analytic functions by means of Koenig's theorem and Toeplitz computations

Summary. By providing a matrix version of Koenig's theorem we reduce the problem of evaluating the coefficients of a monic factor r(z) of degree h of a power series f(z) to that of approximating the first h entries in the first column of the inverse of an Toeplitz matrix in block Hessenberg form for sufficiently large values of n. This matrix is reduced to a band matrix if f(z) is a polynomial. We prove that the factorization problem can be also reduced to solving a matrix equation for an matrix X, where is a matrix power series whose coefficients are Toeplitz matrices. The function is reduced to a matrix polynomial of degree 2 if f(z) is a polynomial of degreeN and . These reductions allow us to devise a suitable algorithm, based on cyclic reduction and on the concept of displacement rank, for generating a sequence of vectors that quadratically converges to the vector having as components the coefficients of the factor r(z). In the case of a polynomial f(z) of degree N, the cost of computing the entries of given is arithmetic operations, where is the cost of solving an Toeplitz-like system. In the case of analytic functions the cost depends on the numerical degree of the power series involved in the computation. From the numerical experiments performed with several test polynomials and power series, the algorithm has shown good numerical properties and promises to be a good candidate for implementing polynomial root-finders based on recursive splitting strategies. Applications to solving spectral factorization problems and Markov chains are also shown. Received September 9, 1998 / Revised version received November 14, 1999 / Published online February 5, 2001     

Characterizations of the polynomial numerical hull of degree k

Six characterizations of the polynomial numerical hull of degree k are established for bounded linear operators on a Hilbert space. It is shown how these characterizations provide a natural distinction between interior and boundary points. One of the characterizations is used to prove that the polynomial numerical hull of any fixed degree k for a Toeplitz matrix whose symbol is piecewise continuous approaches all or most of that of the infinite-dimensional Toeplitz operator, as the matrix size goes to infinity.     

On Schröder’s families of root-finding methods

Schröder’s methods of the first and second kind for solving a nonlinear equation f(x)=0, originally derived in 1870, are of great importance in the theory and practice of iteration processes. They were rediscovered several times and expressed in different forms during the last 130 years. It was proved in the paper of Petkovi? and Herceg (1999) [7] that even seven families of iteration methods for solving nonlinear equations are mutually equivalent. In this paper we show that these families are also equivalent to another four families of iteration methods and find that all of them have the origin in Schröder’s generalized method (of the second kind) presented in 1870. In the continuation we consider Smale’s open problem from 1994 about possible link between Schröder’s methods of the first and second kind and state the link in a simple way.     

On the discretization of a partial differential equation in the neighborhood of a periodic orbit

Sumamry This article is concerned with the comparison of the dynamic of a partial differential equation and its time discretization. We restrict our attention to the neighborhood of a hyperbolic periodic orbit. We show that the discretization possesses an invariant closed curve near the periodic orbit and that the trajectories of the semigroups defined by the partial differential equations and its approximation are close in a sense to be precised provided that different data are allowed. This answers partly an open problem posed in [4]. Examples of application to dissipative partial equations are provided.     

Eigenvalue bounds for the Schur complement with a pressure convection–diffusion preconditioner in incompressible flow computations

If the stationary Navier–Stokes system or an implicit time discretization of the evolutionary Navier–Stokes system is linearized by a Picard iteration and discretized in space by a mixed finite element method, there arises a saddle point system which may be solved by a Krylov subspace method or an Uzawa type approach. For each of these resolution methods, it is necessary to precondition the Schur complement associated to the saddle point problem in question. In the work at hand, we give upper and lower bounds of the eigenvalues of this Schur complement under the assumption that it is preconditioned by a pressure convection–diffusion matrix.     

Multi-dimensional option pricing using radial basis functions and the generalized Fourier transform

We show that the generalized Fourier transform can be used for reducing the computational cost and memory requirements of radial basis function methods for multi-dimensional option pricing. We derive a general algorithm, including a transformation of the Black–Scholes equation into the heat equation, that can be used in any number of dimensions. Numerical experiments in two and three dimensions show that the gain is substantial even for small problem sizes. Furthermore, the gain increases with the number of dimensions.