Given the square matrices $A, B, D, E$ and the matrix $C$ of conforming dimensions, we consider the linear matrix equation $A{\mathbf X} E+D{\mathbf X} B = C$ in the unknown matrix ${\mathbf X}$. Our aim is to provide an overview of the major algorithmic developments that have taken place over the past few decades in the numerical solution of this and related problems, which are producing reliable numerical tools in the formulation and solution of advanced mathematical models in engineering and scientific computing.

/pdf/computational-methods-for-linear-matrix-equations-29c8rmvfvt.pdf

Computational Methods for Linear Matrix Equations

Nonlinear eigenvalue problems arise in a variety of science and engineering applications and in the past ten years there have been numerous breakthroughs in the development of numerical methods. This article surveys nonlinear eigenvalue problems associated with matrix-valued functions which depend nonlinearly on a single scalar parameter, with a particular emphasis on their mathematical properties and available numerical solution techniques. Solvers based on Newton's method, contour integration, and sampling via rational interpolation are reviewed. Problems of selecting the appropriate parameters for each of the solver classes are discussed and illustrated with numerical examples. This survey also contains numerous MATLAB code snippets that can be used for interactive exploration of the discussed methods.

/pdf/the-nonlinear-eigenvalue-problem-316eav07xe.pdf

The nonlinear eigenvalue problem

A novel parallel algorithm for the integration of linear initial-value problems is proposed. This algorithm is based on the simple observation that homogeneous problems can typically be integrated much faster than inhomogeneous problems. An overlapping time-domain decomposition is utilized to obtain decoupled inhomogeneous and homogeneous subproblems, and a near-optimal Krylov method is used for the fast exponential integration of the homogeneous subproblems. We present an error analysis and discuss the parallel scaling of our algorithm. The efficiency of this approach is demonstrated with numerical examples.

/pdf/paraexp-a-parallel-integrator-for-linear-initial-value-24icr9w59n.pdf

PARAEXP: A Parallel Integrator for Linear Initial-Value Problems

When using the Arnoldi method for approximating $f(A){\mathbf b}$, the action of a matrix function on a vector, the maximum number of iterations that can be performed is often limited by the storage requirements of the full Arnoldi basis. As a remedy, different restarting algorithms have been proposed in the literature, none of which has been universally applicable, efficient, and stable at the same time. We utilize an integral representation for the error of the iterates in the Arnoldi method which then allows us to develop an efficient quadrature-based restarting algorithm suitable for a large class of functions, including the so-called Stieltjes functions and the exponential function. Our method is applicable for functions of Hermitian and non-Hermitian matrices, requires no a priori spectral information, and runs with essentially constant computational work per restart cycle. We comment on the relation of this new restarting approach to other existing algorithms and illustrate its efficiency and numer...

/pdf/efficient-and-stable-arnoldi-restarts-for-matrix-functions-3ps3j6j7s2.pdf

Efficient and stable Arnoldi restarts for matrix functions based on quadrature

The RKFIT algorithm outlined in [M. Berljafa and S. Guettel, Generalized rational Krylov decompositions with an application to rational approximation, SIAM J. Matrix Anal. Appl., 2015] is a Krylov-based approach for solving nonlinear rational least squares problems. This paper puts RKFIT into a general framework, allowing for its extension to nondiagonal rational approximants and a family of approximants sharing a common denominator. Furthermore, we derive a strategy for the degree reduction of the approximants, as well as methods for their conversion to partial fraction form, for the efficient evaluation, and root-finding. We also discuss commons and differences of RKFIT and the popular vector fitting algorithm. A MATLAB implementation of RKFIT is provided and numerical experiments, including the fitting of a MIMO dynamical system and an optimization problem related to exponential integration, demonstrate its applicability.

/pdf/the-rkfit-algorithm-for-nonlinear-rational-approximation-50dr5sekbz.pdf

The RKFIT algorithm for nonlinear rational approximation

Matrix functions are a central topic of linear algebra, and problems of their numerical approximation appear increasingly often in scientific computing. We review various rational Krylov methods for the computation of large-scale matrix functions. Emphasis is put on the rational Arnoldi method and variants thereof, namely, the extended Krylov subspace method and the shift-and-invert Arnoldi method, but we also discuss the nonorthogonal generalized Leja point (or PAIN) method. The issue of optimal pole selection for rational Krylov methods applied for approximating the resolvent and exponential function, and functions of Markov type, is treated in some detail.

/pdf/rational-krylov-approximation-of-matrix-functions-numerical-4sb2wro9al.pdf

Rational Krylov approximation of matrix functions: Numerical methods and optimal pole selection

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Rational Krylov approximation of matrix functions: Numerical methods and optimal pole selection