A Parallel Iterative Method for Exponential Propagation

Journal of Computational and Applied Mathematics, 2016

This work presents a new algorithm for matrix exponential computation that significantly simplifies a Taylor scaling and squaring algorithm presented previously by the authors, preserving accuracy. A Matlab version of the new simplified algorithm has been compared with the original algorithm, providing similar results in terms of accuracy, but reducing processing time. It has also been compared with two state-of-the-art implementations based on Padé approximations, one commercial and the other implemented in Matlab, getting better accuracy and processing time results in the majority of cases.

International Journal of Computer Mathematics, 2014

This work gives a new formula for the forward relative error of matrix exponential Taylor approximation and proposes new bounds for it depending on the matrix size and the Taylor approximation order, providing a new efficient scaling and squaring Taylor algorithm for the matrix exponential. A Matlab version of the new algorithm is provided and compared with Padé state-of-the-art algorithms obtaining higher accuracy in the majority of tests at similar or even lower cost.

2010

We present an approximate algorithm to solve only one variable out of a linear system defined by a matrix with off-diagonal exponential decay entries (including the practically most important class of band limited matrices) via a sub-linear system. This approach thus enables us to solve any subset of solution variables. Parallel implementation of such approximate schemes for every variable enables us to solve the linear system with computational time independent of the matrix size.

Linear Algebra and its Applications, 1996

We analyze the Pad& method for computing the exponential of a real matrix.

Journal of Applied Mathematics, 2011

Iterative splitting methods have a huge amount to compute matrix exponential. Here, the acceleration and recovering of higher-order schemes can be achieved. From a theoretical point of view, iterative splitting methods are at least alternating Picards fix-point iteration schemes. For practical applications, it is important to compute very fast matrix exponentials. In this paper, we concentrate on developing fast algorithms to solve the iterative splitting scheme. First, we reformulate the iterative splitting scheme into an integral notation of matrix exponential. In this notation, we consider fast approximation schemes to the integral formulations, also known as -functions. Second, the error analysis is explained and applied to the integral formulations. The novelty is to compute cheaply the decoupled exp-matrices and apply only cheap matrix-vector multiplications for the higher-order terms. In general, we discuss an elegant way of embedding recently survey on methods for computing ...

1997

For the solutions of linear systems of equations with unsymmetric coefficient matrices, we propose an improved version of the quasi-minimal residual (IQMR) method by using the Lanczos process as a major component combining elements of numerical stability and parallel algorithm design. For Lanczos process, stability is obtained by a coupled two-term procedure that generates Lanczos vectors normalized to unit length. The algorithm is derived in such a way that all inner products and matrix-vector multiplications of a single iteration step are independent, subsequently communication time required for inner products can be overlapped efficiently with computation time. Therefore, the cost of global communication on parallel distributed memory computers is significantly reduced. The resulting IQMR algorithm preserves the favorable properties of the Lanczos process without increasing computational costs. The efficiency of this method is demonstrated by numerical experimental results carried out on a massively parallel distributed memory computer, the Parsytec GC/PowerPlus.

2009

The matrix exponential plays a fundamental role in linear systems arising in engineering, mechanics and control theory. In this paper, an efficient Taylor method for computing matrix exponentials is presented. Taylor series truncation together with a modification of the PatersonStockmeyer method avoiding factorial evaluations, and the scaling-squaring technique, allow efficient computation of the matrix exponential approximation. A careful backward-error analysis of the approximation is given and a theoretical estimate for the optimal scaling of matrices is obtained. The modified Paterson-Stockmeyer implementation was compared with the classical implementation and other efficient state of the art methods on dense matrices for different dimensions from 2× 2 to 100× 100. Numerical tests show that it obtains higher precision than all compared methods in the majority of cases. We show that it presents lower computational cost in terms of matrix products than efficient Padé methods for g...

SIAM Journal on Scientific Computing, 2015

The matrix exponential plays a fundamental role in linear differential equations arising in engineering, mechanics, and control theory. The most widely used, and the most generally efficient, technique for calculating the matrix exponential is a combination of "scaling and squaring" with a Padé approximation. For alternative scaling and squaring methods based on Taylor series, we present two modifications that provably reduce the number of matrix multiplications needed to satisfy the required accuracy bounds, and a detailed comparison of the several algorithmic variants is provided.

1997

For the solutions of linear systems of equations with unsymmetric coefficient matrices, we has proposed an improved version of the quasi-minimal residual (IQMR) method by using the Lanczos process as a major component combining elements of numerical stability and parallel algorithm design. For Lanczos process, stability is obtained by a couple two-term procedure that generates Lanczos vectors scaled to unit length. The algorithm is derived such that all inner products and matrix-vector multiplications of a single iteration step are independent and communication time required for inner product can be overlapped efficiently with computation time. Therefore, the cost of global communication on parallel distributed memory computers can be significantly reduced. In this paper, we describe an efficient implementation of this method which is particularly well suited to problems with irregular sparsity pattern. The corresponding communication cost is independent of the sparsity pattern with several performance improvement techniques such as overlapping computation and communication, balancing the computational load. The performance is demonstrated by numerical experimental results carried out on massively parallel distributed memory computer Parsytec GC/Power Plus.