Charles Van Loan

Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later

In principle, the exponential of a matrix could be computed in many ways. Methods involv- ing approximation theory, dierential equations, the matrix eigenvalues, and the matrix characteristic polynomial have been proposed. In practice, consideration of computational stability and eciency indicates that some of the methods are preferable to others, but that none are completely satisfactory. Most of this paper was originally published in 1978. An update, with a separate bibliog- raphy, describes a few recent developments.

Nineteen Dubious Ways to Compute the Exponential of a Matrix

In principle, the exponential of a matrix could be computed in many ways. Methods involving approximation theory, differential equations, the matrix eigenvalues, and the matrix characteristic polyn...

Computational frameworks for the fast Fourier transform

1. The Radix-2 Frameworks. Matrix Notation and Algorithms The FFT Idea The Cooley-Tukey Factorization Weight and Butterfly Computations Bit Reversal and Transposition The Cooley-Tukey Framework The Stockham Autosort Frameworks The Pease Framework Decimation in Frequency and Inverse FFTs 2. General Radix Frameworks. The General Radix Ideas Index Reversal and Transposition Mixed-Radix Factorizations Radix-4 and Radix-8 Frameworks The Split-Radix Frameworks 3. High Performance Frameworks. The Multiple DFT Problem Matrix Transposition The Large Single-Vector FFT Problem Multi-Dimensional FFT Problem Distributed Memory FFTs Shared Memory FFTs 4. Selected Topics. Prime Factor FFTs Convolution FFTs of Real Data Cosine and Sine Transforms Fast Poisson Solvers Bibliography Index.

The ubiquitous Kronecker product

Abstract The Kronecker product has a rich and very pleasing algebra that supports a wide range of fast, elegant, and practical algorithms. Several trends in scientific computing suggest that this important matrix operation will have an increasingly greater role to play in the future. First, the application areas where Kronecker products abound are all thriving. These include signal processing, image processing, semidefinite programming, and quantum computing. Second, sparse factorizations and Kronecker products are proving to be a very effective way to look at fast linear transforms. Researchers have taken the Kronecker methodology as developed for the fast Fourier transform and used it to build exciting alternatives. Third, as computers get more powerful, researchers are more willing to entertain problems of high dimension and this leads to Kronecker products whenever low-dimension techniques are “tensored” together.

Generalizing the Singular Value Decomposition

Two generalizations of the singular value decomposition are given. These generalizations provided a unified way of regarding certain matrix problems and the numerical techniques which are used to s...

Approximation with Kronecker Products

Let A be an m-by-n matrix with m=m1m2 and n=n1n2. We consider the problem of finding (mathematical formula omitted) so that (mathematical formula omitted) is minimized. This problem can be solved by computing the largest singular value and associated singular vectors of a permuted version of A. If A is symmetric, definite, non-negative, or banded, then the minimizing B and C are similarly structured. The idea of using Kronecker product preconditioners is briefly discussed.

The Sensitivity of the Matrix Exponential

In this paper we examine how the matrix exponential

A Schur decomposition for Hamiltonian matrices

e^{At}

A Symplectic Method for Approximating All the Eigenvalues of a Hamiltonian Matrix

is affected by perturbations in A. Elementary techniques using

GEMM-based level 3 BLAS: high-performance model implementations and performance evaluation benchmark

\log