Dra. Marlliny Monsalve

The inexact Newton-Kleinman method is an iterative scheme for numerically solving large scale algebraic Riccati equations. At each iteration, the approximate solution of a Lyapunov linear equation is required. Specifically designed projection of the Riccati equation onto an iteratively generated approximation space provides a possible alternative. Our numerical experiments with enriched approximation spaces seem to indicate that this latter approach is superior to Newton-type strategies on realistic problems, thus giving experimental ground for recent developments in this direction. As part of an explanation of why this is so, we derive several matrix relations between the iterates produced by the same projection approach applied to both the (quadratic) Riccati equation and its linear counterpart, the Lyapunov equation.

Nonlinear matrix equations arise in different scientific topics, such as applied statistics, control theory, and financial mathematics, among others. As in many other scientific areas, Newton's method has played an important role when solving these matrix problems. Under standard assumptions, the specialized Newton's methods that have been developed for specific problems exhibit local and q-quadratic convergence and require a suitable initial guess. They also require, as usual, a significant amount of computational work per iteration, that in this case involve several matrix factorizations per iterations. As expected, whenever a Newton's method can be developed, a secant method can also be developed. Indeed, more recently, secant methods for solving specific nonlinear matrix problems have been developed opening a new line of research. As in previous scenarios, these specialized secant methods exhibit local and q-superlinear convergence, also require a suitable initial guess, and avoid the use of derivatives in the formulation of the schemes. In this review we start by recalling the presence of Newton's method and the secant methods, and also their classical relationship, in different and sometimes unexpected scenarios for vector problems. Then we present and describe the state of the art in the use of Newton's method and also the secant method in the space of matrices. A second objective is to present a unified approach for describing the features of these classical schemes, that in the space of matrices represent an interesting research area with special features to be explored.

Nonlinear matrix equations arise in different scientific topics, such as applied statistics and control theory, among others. Standard approaches to solve them include and combine some variations of Newton's method, matrix factorizations, and reduction to generalized eigenvalue problems. fIn this paper we explore the use of secant methods in the space of matrices, that represent a new approach with interesting features.
For the special problem of computing the inverse or the pseudoinverse of a given matrix, we propose a specialized secant method for which we establish stability and q-superlinear convergence, and for which we also present some numerical results. In addition, for solving quadratic matrix equations, we discuss several issues, and present preliminary and encouraging numerical experiments.

Our purpose is to solve Riccati equations which arise in many applications in Control Theory. It is standard to seek low-rank solutions (for storage issues). Since this is a nonlinear matrix equation, Newton’s method is a successful strategy; it requires the solution of a Lyapunov equation at each iteration. In this work in progress, we propose to use an Inexact Newton method to solve the Riccati equation and a Krylov projection method to solve the Lyapunov equation with a recycling process.

Motivated by the classical Newton-Schulz method for finding the inverse of a nonsingular matrix, we develop a new inversion-free method for obtaining the minimal Hermitian positive definite solution of the matrix rational equation $X + A^*X^{-1}A = I$, where $I$ is the identity matrix and $A$ is a given nonsingular matrix. We present convergence results and discuss stability properties when the method starts from the available matrix $AA^*$. We also present numerical results to compare our proposal with some previously developed inversion-free techniques for solving the same rational matrix equation.

A secant method has been recently proposed for solving nonlinear matrix problems with interesting features, including low computational cost and q-superlinear convergence for special cases. In this work, we analyze this secant method for the special problem of computing the sign of a given matrix. The global and $q$-superlinear convergence of the proposed secant method are proved from specialized initial guesses, and the numerical stability is also established. We complement our analysis with several numerical experiments that show the advantages of using the secant method over the well-known variant of Newton's method for the same problem.

We discuss different variants of Newton's method for computing the p-th root of a given matrix. A suitable implementation is presented for solving the Sylvester equation, that appears at every Newton's iteration, via Kronecker products. This approach is quadratically convergent and stable, but too expensive in computational cost. In contrast we propose and analyze some specialized versions that exploit the commutation of the iterates with the given matrix. These versions are relatively inexpensive but have either stability problems or stagnation problems when good precision is required. Hybrid versions are presented to take advantage of the best features in both approaches. Preliminary and encouraging numerical results are presented for p=3 and p=5.

We present and analyze new iterative schemes for solving the large-scale Sylvester equation AX-XB=C where X \in \R^{n x p}, and A,B and C are given matrices. These new schemes are based on fixed point iterations and some recently developed methods for solving block linear systems of equations. Our schemes are flexible in the sense that for solving the block linear system, at each iteration, any available method (direct or iterative) can be used. We present a convergence analysis under some hypothesis on the matrices A and B. We also present encouraging numerical results for large-scale problems. In particular, the new schemes are compare favorably with schemes based on using block Krylov subspace method directly on the Sylvester equation and with recently developed method based in the construction of a low rank approximation of the matrix C.

We extend and improve recently proposed algorithms to solve the problem of minimizing the distance from a given matrix to the cone of symmetric and diagonally dominant matrices with positive diagonal (SDD^{+}). We present a variety of criteria to select a subset of the supporting hyperplanes of the faces of SDD^{+}, and also of the polar cone (SDD^+)^o, to then apply Dykstra's alternating projection method. These selections reduce the number of projections and therefore reduce the required computational work. In all our new algorithms, the symmetry and the diagonal dominance of the obtained matrix are guaranteed. Preliminary numerical experiments indicate that some of the selection criteria produce a significant reduction in CPU time.