Emilio Defez

Universitat Politècnica de València | UPV

This paper presents three different alternatives to evaluate the matrix hyperbolic cosine using Bernoulli matrix polynomials, comparing them from the point of view of accuracy and computational complexity. The first two alternatives are derived from two different Bernoulli series expansions of the matrix hyperbolic cosine, while the third one is ba...

Differential matrix models provide an elementary blueprint for the adequate and efficient treatment of many important applications in science and engineering. In the present work, we suggest a procedure, extending our previous research results, to represent the solutions of nonlinear matrix differential problems of fourth order given in the form Y(...

This paper presents a new series expansion based on Bernoulli matrix polynomials to approximate the matrix cosine function. An approximation based on this series is not a straightforward exercise since there exist different options to implement such a solution. We dive into these options and include a thorough comparative of performance and accurac...

There are, currently, very few implementations to compute the hyperbolic cosine of a matrix. This work tries to fill this gap. To this end, we first introduce both a new rational-polynomial Hermite matrix expansion and a formula for the forward relative error of Hermite approximation in exact arithmetic with a sharp bound for the forward error. Thi...

The action of the matrix exponential on a vector eAtv, A∈Cn×n, v∈Cn, appears in problems that arise in mathematics, physics, and engineering, such as the solution of systems of linear ordinary differential equations with constant coefficients. Nowadays, several state-of-the-art approximations are available for estimating this type of action. In thi...

Matrix differential equations are at the heart of many science and engineering problems. In this paper, a procedure based on higher-order matrix splines is proposed to provide the approximated numerical solution of special nonlinear third-order matrix differential equations, having the form Y(3)(x)=f(x,Y(x)). Some numerical test problems are also i...

The most popular method for computing the matrix logarithm is a combination of the inverse scaling and squaring method in conjunction with a Padé approximation, sometimes accompanied by the Schur decomposition. In this work, we present a Taylor series algorithm, based on the free-transformation approach of the inverse scaling and squaring technique...

In this paper, we introduce two approaches to compute the matrix hyperbolic tangent. While one of them is based on its own definition and uses the matrix exponential, the other one is focused on the expansion of its Taylor series. For this second approximation, we analyse two different alternatives to evaluate the corresponding matrix polynomials....

This work deals with the simulation of a two‐dimensional ideal lattice having simple tetragonal geometry. The harmonic character of the oscillators give rise to a system of second‐order linear differential equations, which can be recast into matrix form. The explicit solutions which govern the dynamics of this system can be expressed in terms of ma...

The computation of matrix trigonometric functions has received remarkable attention in the last decades due to its usefulness in the solution of systems of second order linear differential equations. Recently, several state-of-the-art algorithms have been provided for computing these matrix functions, in particular for the matrix cosine function.

Matrix exponentials are widely used to efficiently tackle systems of linear differential equations. To be able to solve systems of fractional differential equations, the Caputo matrix exponential of the index α > 0 was introduced. It generalizes and adapts the conventional matrix exponential to systems of fractional differential equations with cons...

In this work we introduce a new method to compute the matrix cosine. It is based on recent new matrix polynomial evaluation methods for the Taylor approximation and a mixed forward and backward error analysis. The matrix polynomial evaluation methods allow to evaluate the Taylor polynomial approximation of the matrix cosine function more efficientl...

This paper presents an implementation of one of the most up-to-day algorithms proposed to compute the matrix trigonometric functions sine and cosine. The method used is based on Taylor series approximations which intensively uses matrix multiplications. To accelerate matrix products, our application can use from one to four NVIDIA GPUs by using the...

In this work we introduce new rational-polynomial Hermite matrix expansions which allow us to obtain a new accurate and efficient method for computing the matrix cosine. This method is compared with other state-of-the-art methods for computing the matrix cosine, including a method based on Padé approximants, showing a far superior efficiency, and h...

In this work an accurate and efficient method based on matrix splines for computing matrix exponential is given. An algorithm and a MATLAB im- plementation have been developed and compared with the state-of-the-art algorithms for computing the matrix exponential. We also developed a par- allel implementation for large scale problems. This implement...

In this work we introduce a new method to compute the matrix cosine. It is based on recent new matrix polynomial evaluation methods for the Taylor approximation and forward and backward error analysis. The matrix polynomial evaluation methods allow to evaluate the Taylor polynomial approximation of the cosine function more efficiently than using Pa...

Computational and Mathematical Methods in Science and Engineering,

Differential matrix models are an essential ingredient of many important scientific and engineering applications. In this work, we propose a procedure to approximate the solutions of special linear fourth-order matrix differential problems of the type Y (4)(x) = A(x)Y (x) + B(x) with higher-order matrix splines. An example is included.