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Exponential time integrators are well‐established discretization methods for time semilinear systems of ordinary differential equations. These methods use φ−functions, which are matrix functions related to the exponential. This work introduces an algorithm to speed up the computation of the φ−function action over vectors for two‐dimensional (2D) matrices expressed as a Kronecker sum. For that, we present an auxiliary exponential‐related matrix function that we express using Kronecker products of one‐dimensional matrices. We exploit state‐of‐the‐art implementations of φ−functions to compute this auxiliary function's action and then recover the original φ−action by solving a Sylvester equation system. Our approach allows us to save memory and solve exponential integrators of 2D+time problems in a fraction of the time traditional methods need. We analyze the method's performance considering different linear operators and with the nonlinear 2D+time Allen–Cahn equation. [ABSTRACT FROM AUTHOR]

We discuss the use of a matrix‐oriented approach for numerically solving the dense matrix equation AX + XAT + M1XN1 + ... + MℓXNℓ = F, with ℓ ≥ 1, and Mi, Ni, i = 1, ... , ℓ of low rank. The approach relies on the Sherman–Morrison–Woodbury formula formally defined in the vectorized form of the problem, but applied in the matrix setting. This allows one to solve medium size dense problems with computational costs and memory requirements dramatically lower than with a Kronecker formulation. Application problems leading to medium size equations of this form are illustrated and the performance of the matrix‐oriented method is reported. The application of the procedure as the core step in the solution of the large‐scale problem is also shown. In addition, a new explicit method for linear tensor equations is proposed, that uses the discussed matrix equation procedure as a key building block. [ABSTRACT FROM AUTHOR]

We consider the numerical solution of large scale singular (continuous‐time) Lyapunov equations of the form AX + XA⊤ + BB⊤ = 0, where A is semistable, that is, its spectrum is contained in the left half plane, with the exception of a few semisimple eigenvalues at zero. We also consider the case of a few semisimple eigenvalues on the imaginary axis. We assume that we know these few eigenvalues (zero or imaginary), and that we have or can compute the corresponding invariant subspaces. We use this information to build an appropriate newly proposed subspace on which to project the Lyapunov equations, and then compute a low‐rank approximation to the least squares solution. Selected illustrative numerical examples are provided. [ABSTRACT FROM AUTHOR]

In various applications, for instance, in the detection of a Hopf bifurcation or in solving separable boundary value problems using the two-parameter eigenvalue problem, one has to solve a generalized eigenvalue problem with 2 × 2 operator determinants of the form (B1 ⊗ A2 - A1 ⊗ B2)z = µ B1 ⊗ C2 - C1 ⊗ B2)z. We present efficient methods that can be used to compute a small subset of the eigenvalues. For full matrices of moderate size, we propose either the standard implicitly restarted Arnoldi or Krylov-Schur iteration with shift-and-invert transformation, performed efficiently by solving a Sylvester equation. For large problems, it is more efficient to use subspace iteration based on low-rank approximations of the solution of the Sylvester equation combined with a Krylov-Schur method for the projected problems. [ABSTRACT FROM AUTHOR]