João R. Cardoso

The Moser-Veselov equation

Abstract We study the orthogonal solutions of the matrix equation XJ − JX T = M , where J is symmetric positive definite and M is skew-symmetric. This equation arises in the discrete version of the dynamics of a rigid body, investigated by Moser and Veselov (Commun. Math. Phys. 139 (1991) 217). We show connections between orthogonal solutions of this equation and solutions of a certain algebraic Riccati equation. This will bring out the symplectic geometry of the Moser–Veselov equation and also reduces most computational issues about solutions to finding invariant subspaces of a certain Hamiltonian matrix. Necessary and sufficient conditions for the existence of orthogonal solutions (and methods to compute them) are presented. Our method is contrasted with the Moser–Veselov approach (Commun. Math. Phys. 139 (1991) 217). We also exhibit explicit solutions of a particular case of the Moser–Veselov equation, which appears associated with the continuous version of the dynamics of a rigid body.

Computing the square root and logarithm of real P -orthogonal matrix

A real square matrix A is P-orthogonal if AT PA = P; P is a fixed real nonsingular matrix, but most of the results in this work require that it is symmetric positive definite or PT = P-1, P2 = ±I. The class of P-orthogonal matrices includes, for instance, orthogonal and symplectic matrices as particular cases. We present an efficient iterative method for computing the P-orthogonal factor in the generalized polar decomposition, which generalizes the well-known Newtons method for the standard polar decomposition. A connection between Newtons method for the matrix square root and polar iterates brings out a new iterative method for computing the principal square root of a P-orthogonal matrix. One important feature of this method is that, when P is symmetric positive definite, it allows us to restore the P-orthogonal property of the exact square root by computing the nearest P-orthogonal matrix. We also analyse the problem of finding the nearest P-symmetric and P-skew-symmetric matrices. New bounds and new estimates for the Pade error of the matrix logarithm are given in order to improve the existing Briggs-Pade algorithms and adapt them to P-orthogonal matrices. Special attention will be paid to the orthogonal case.

Theoretical and numerical considerations about Padé approximants for the matrix logarithm

We show that for a vast class of matrix Lie groups, which includes the orthogonal and the symplectic, diagonal Pade approximants of log((1 + x)/(1 − x)) are structure preserving. The conditioning of these approximants is analyzed. We also present a new algorithm for the Briggs–Pade method, based on a strategy for reducing the number of square roots in the inverse scaling and squaring procedure.

Iteration functions for pth roots of complex numbers

A novel way of generating higher-order iteration functions for the computation of pth roots of complex numbers is the main contribution of the present work. The behavior of some of these iteration functions will be analyzed and the conditions on the starting values that guarantee the convergence will be stated. The illustration of the basins of attractions of the pth roots will be carried out by some computer generated plots. In order to compare the performance of the iterations some numerical examples will be considered.

Exponentials of skew-symmetric matrices and logarithms of orthogonal matrices

Two widely used methods for computing matrix exponentials and matrix logarithms are, respectively, the scaling and squaring and the inverse scaling and squaring. Both methods become effective when combined with Pade approximation. This paper deals with the computation of exponentials of skew-symmetric matrices and logarithms of orthogonal matrices. Our main goal is to improve these two methods by exploiting the special structure of skew-symmetric and orthogonal matrices. Geometric features of the matrix exponential and logarithm and extensions to the special Euclidean group of rigid motions are also addressed.

Computing the Inverse Matrix Hyperbolic Sine

We give necessary and sufficient conditions for solvability of the matrix equation sinh X = A in the complex and real cases and present some algorithms for computing one of these solutions. The numerical features of the algorithms are analysed along with some numerical tests.

On the convergence of Schröder iteration functions for pth roots of complex numbers

In this work a condition on the starting values that guarantees the convergence of the Schroder iteration functions of any order to a pth root of a complex number is given. Convergence results are derived from the properties of the Taylor series coefficients of a certain function. The theory is illustrated by some computer generated plots of the basins of attraction.

On a sub‐Stiefel Procrustes problem arising in computer vision

Summary nA sub-Stiefel matrix is a matrix that results from deleting simultaneously the last row and the last column of an orthogonal matrix. In this paper, we consider a Procrustes problem on the set of sub-Stiefel matrices of order n. For n = 2, this problem has arisen in computer vision to solve the surface unfolding problem considered by R. Fereirra, J. Xavier and J. Costeira. An iterative algorithm for computing the solution of the sub-Stiefel Procrustes problem for an arbitrary n is proposed, and some numerical experiments are carried out to illustrate its performance. For these purposes, we investigate the properties of sub-Stiefel matrices. In particular, we derive two necessary and sufficient conditions for a matrix to be sub-Stiefel. We also relate the sub-Stiefel Procrustes problem with the Stiefel Procrustes problem and compare it with the orthogonal Procrustes problem. Copyright

Conditioning of the matrix-matrix exponentiation

If A has no eigenvalues on the closed negative real axis, and B is arbitrary square complex, the matrix-matrix exponentiation is defined as AB := elog(A)B. It arises, for instance, in Von Newmann’s quantum-mechanical entropy, which in turn finds applications in other areas of science and engineering. In this paper, we revisit this function and derive new related results. Particular emphasis is devoted to its Fréchet derivative and conditioning. We propose a new definition of bivariate matrix function and derive some general results on their Fréchet derivatives, which hold, not only to the matrix-matrix exponentiation but also to other known functions, such as means of two matrices, second order Fréchet derivatives and some iteration functions arising in matrix iterative methods. The numerical computation of the Fréchet derivative is discussed and an algorithm for computing the relative condition number of ABis proposed. Some numerical experiments are included.

Iterative methods for finding commuting solutions of the Yang–Baxter-like matrix equation

The main goal of this paper is the numerical computation of solutions of the so-called Yang–Baxter-like matrix equation AXA=XAX, where A is a given complex square matrix. Two novel matrix iterations are proposed, both having second-order convergence. A sign modification in one of the iterations gives rise to a third matrix iteration. Strategies for finding starting approximations are discussed as well as a technique for estimating the relative error. One of the methods involves a very small cost per iteration and is shown to be stable. Numerical experiments are carried out to illustrate the effectiveness of the new methods.