Emilio Defez

Some applications of the Hermite matrix polynomials series expansions

This paper deals with Hermite matrix polynomials expansions of some relevant matrix functions appearing in the solution of differential systems. Properties of Hermite matrix polynomials such as the three terms recurrence formula permit an efficient computation of matrix functions avoiding important computational drawbacks of other well-known methods. Results are applied to compute accurate approximations of certain differential systems in terms of Hermite matrix polynomials.

On Hermite matrix polynomials and Hermite matrix functions

In this paper properties of Hermite matrix polynomials and Hermite matrix functions are studied. The concept ot total set with respect to a matrix functional is introduced and the total property of the Hermite matrix polynomials is proved. Asymptotic behaviour of Hermite matrix polynomials is studied and the relationship of Hermite matrix functions with certain matrix differential equations is developed. A new expression of the matrix exponential for a wide class of matrices in terms of Hermite matrix polynomials is proposed.

Orthogonal matrix polynomials with respect to linear matrix moment functionals: Theory and applications

In this paper orthogonal matrix polynomials with respect to a right matrix moment functional are introduced. Basic results, important examples and applications to the approximation of matrix integrals are studied. Error bounds for the proposed matrix quadrature rules are given.

Efficient orthogonal matrix polynomial based method for computing matrix exponential

The matrix exponential plays a fundamental role in the solution of differential systems which appear in different science fields. This paper presents an efficient method for computing matrix exponentials based on Hermite matrix polynomial expansions. Hermite series truncation together with scaling and squaring and the application of floating point arithmetic bounds to the intermediate results provide excellent accuracy results compared with the best acknowledged computational methods. A backward-error analysis of the approximation in exact arithmetic is given. This analysis is used to provide a theoretical estimate for the optimal scaling of matrices. Two algorithms based on this method have been implemented as MATLAB functions. They have been compared with MATLAB functions funm and expm obtaining greater accuracy in the majority of tests. A careful cost comparison analysis with expm is provided showing that the proposed algorithms have lower maximum cost for some matrix norm intervals. Numerical tests show that the application of floating point arithmetic bounds to the intermediate results may reduce considerably computational costs, reaching in numerical tests relative higher average costs than expm of only 4.43% for the final Hermite selected order, and obtaining better accuracy results in the 77.36% of the test matrices. The MATLAB implementation of the best Hermite matrix polynomial based algorithm has been made available online.

Matrix Cubic Splines for Progressive 3D Imaging

Mathematical theory of matrix cubic splines is introduced, then adapted for progressive rendering of images. 2D subsets of a 3D digital object are transmitted progressively under some ordering scheme, and subsequent reconstructions using the matrix cubic spline algorithm provide an evolving 3D rendering. The process can be an effective tool for browsing three dimensional objects, and effectiveness is illustrated with a test data set consisting of 93 CT slices of a human head. The procedure has been implemented on a single processor PC system, to provide a platform for full 3D experimentation; performance is discussed. A web address for the complete, documented Mathematica code is given.

Progressive transmission of images: PC-based computations, using orthogonal matrix polynomials

Two methods for reconstructing a 3-D image as its 2-D parallel slices are transmitted progressively, in some order, are presented and analyzed. In the originating data base, an ordered set of 2-D slices could represent computer tomography (CT), magnetic resonance images (MRI), or cryosection cross-sections of a 3-D object, for example. With this digital formulation, matrix interpolating polynomial machinery renders a progressively-improving image as slices are received. A piecewise matrix polynomial reconstruction is also considered for reducing computational needs.

Accurate matrix exponential computation to solve coupled differential models in engineering

The matrix exponential plays a fundamental role in linear systems arising in engineering, mechanics and control theory. This work presents a new scaling-squaring algorithm for matrix exponential computation. It uses forward and backward error analysis with improved bounds for normal and nonnormal matrices. Applied to the Taylor method, it has presented a lower or similar cost compared to the state-of-the-art Pade algorithms with better accuracy results in the majority of test matrices, avoiding Pades denominator condition problems.

Computing matrix functions arising in engineering models with orthogonal matrix polynomials

Trigonometric matrix functions play a fundamental role in the solution of second order differential equations. Hermite series truncation together with Paterson-Stockmeyer method and the double angle formula technique allow efficient computation of the matrix cosine. A careful error bound analysis of the Hermite approximation is given and a theoretical estimate for the optimal value of its parameters is obtained. Based on the ideas above, an efficient and highly-accurate Hermite algorithm is presented. A MATLAB implementation of this algorithm has also been developed and made available online. This implementation has been compared to other efficient state-ofthe-art implementations on a large class of matrices for different dimensions, obtaining higher accuracy and lower computational costs in the majority of cases.

A connection between Lagurre's and Hermite's matrix polynomials☆

Abstract In this paper, a connection between Laguerres and Hermites matrix polynomials recently introduced in [1,2] is established.

Computing matrix functions solving coupled differential models

In this paper a modification of the method proposed in [E. Defez, L. Jodar, Some applications of Hermite matrix polynomials series expansions, Journal of Computational and Applied Mathematics 99 (1998) 105-117] for computing matrix sine and cosine based on Hermite matrix polynomial expansions is presented. An algorithm and illustrative examples demonstrate the performance of the new proposed method.