Jacinto Javier Ibáñez

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.

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.

A family of BDF algorithms for solving Differential Matrix Riccati Equations using adaptive techniques

Abstract Differential Matrix Riccati Equations play a fundamental role in control theory, for example, in optimal control, filtering and estimation, decoupling and order reduction, etc. One of the most popular codes to solve stiff Differential Matrix Riccati Equations (DMREs) is based on Backward Differentiation Formula (BDF). In previous papers the authors of this paper showed two algorithms for solving DMREs based on an iterative Generalized Minimum RESidual (GMRES) approach and on a Fixed-Point approach. In this paper we present two contributions to improve the above algorithms. Firstly six variants of previous algorithms are carried out by using one of above algorithms in the first step and another algorithm to carry out the other steps until reaching convergence. Numerous tests on four case studies have been done comparing both precision and computational costs of MATLAB implementations of the above algorithms. Experimental results show that in some cases these algorithms improve on the speed and convergence of the original algorithms. Secondly, using the previous experimental results and since all algorithms have a similar structure and there is no best algorithm to solve all problems, two general-purpose adaptive algorithms have been designed for selecting the most appropriate algorithm, which can be chosen using a parameter that indicates the stiffness of the DMRE to be solved.

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.

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.

Solving Initial Value Problems for Ordinary Differential Equations by two approaches: BDF and piecewise-linearized methods

Abstract Many scientific and engineering problems are described using Ordinary Differential Equations (ODEs), where the analytic solution is unknown. Much research has been done by the scientific community on developing numerical methods which can provide an approximate solution of the original ODE. In this work, two approaches have been considered based on BDF and Piecewise-linearized Methods. The approach based on BDF methods uses a Chord–Shamanskii iteration for computing the nonlinear system which is obtained when the BDF schema is used. Two approaches based on piecewise-linearized methods have also been considered. These approaches are based on a theorem proved in this paper which allows to compute the approximate solution at each time step by means of a block-oriented method based on diagonal Pade approximations. The difference between these implementations is in using or not using the scale and squaring technique. Five algorithms based on these approaches have been developed. MATLAB and Fortran versions of the above algorithms have been developed, comparing both precision and computational costs. BLAS and LAPACK libraries have been used in Fortran implementations. In order to compare in equality of conditions all implementations, algorithms with fixed step have been considered. Four of the five case studies analyzed come from biology and chemical kinetics stiff problems. Experimental results show the advantages of the proposed algorithms, especially when they are integrating stiff problems.

A GMRES-based BDF method for solving differential Riccati equations

Abstract Differential Riccati equations play a fundamental role in control theory, for example, optimal control, filtering and estimation, decoupling and order reduction, etc. The most popular codes to solve stiff differential Riccati equations use backward differentiation formula (BDF) methods. In this paper, a new approach to solve differential Riccati equations by means of a BDF method is described. In each step of these methods an algebraic Riccati equation is obtained, which is solved by means of Newton’s method. In the standard approach, this system is transformed into a Sylvester equation, which could be solved by means of the well-known Bartels–Stewart method. In our code, we obtain a system of linear equations, defined from a Kronecker product of matrices related to coefficient matrices of the differential Riccati equation, that is solved by means of the iterative generalized minimum residual (GMRES) method. We have also implemented an efficient matrix–vector product in order to reduce the computational and storage cost of the GMRES method. The above approach has been applied in the development of an algorithm to solve differential Riccati equations. The accuracy and efficiency of this algorithm has been compared with the BDF algorithm that uses the Bartels–Stewart method. Experimental results show the advantages of the new algorithm.

A fixed point-based BDF method for solving differential Riccati equations

Abstract This paper describes an approach for solving differential Riccati equations (DRE), by means of the backward differentiation formula (BDF) and resolution of the corresponding implicit equation using Newton’s method with a fixed point approach. The role and use of DRE is especially important in several applications such as optimal control, filtering, and estimation. The goodness of this new method is compared with respect to the so called Dieci method [L. Dieci, Numerical integration of the differential Riccati equation and some related issues, SIAM J. Numer. Anal. 29 (3) (1992) 781–815].

High performance computing of the matrix exponential

This work presents a new algorithm for matrix exponential computation that significantly simplifies a Taylor scaling and squaring algorithm presented previously by the authors, preserving accuracy. A Matlab version of the new simplified algorithm has been compared with the original algorithm, providing similar results in terms of accuracy, but reducing processing time. It has also been compared with two state-of-the-art implementations based on Pade approximations, one commercial and the other implemented in Matlab, getting better accuracy and processing time results in the majority of cases.

Efficient and accurate algorithms for computing matrix trigonometric functions

Trigonometric matrix functions play a fundamental role in second order differential equations. This work presents an algorithm based on Taylor series for computing the matrix cosine. It uses a backward error analysis with improved bounds. Numerical experiments show that MATLAB implementations of this algorithm has higher accuracy than other MATLAB implementations of the state of the art in the majority of tests. Furthermore, we have implemented the designed algorithm in language C for general purpose processors, and in CUDA for one and two NVIDIA GPUs. We obtained a very good performance from these implementations thanks to the high computational power of these hardware accelerators and our effort driven to avoid as much communications as possible. All the implemented programs are accessible through the MATLAB environment.