Comparison of software for computing the action of the matrix exponential

Caliari, Marco; Kandolf, Peter; Ostermann, Alexander; Rainer, Stefan

doi:10.1007/s10543-013-0446-0

Cited by 55 publications

(60 citation statements)

References 11 publications

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“…Here we can exploit available methods for computing e A v using only matrix-vector products with A (and possibly A * ), such as those of Al-Mohy and Higham [1], Caliari et al [5], or Frommer, Güttel, and Schweitzer [13].…”

Section: C585 the P Largest Entries Of Ementioning

confidence: 99%

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Estimating the Largest Elements of a Matrix

Higham

Relton

2016

SIAM J. Sci. Comput.

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Abstract. We derive an algorithm for estimating the largest p ≥ 1 values a ij or |a ij | for an m × n matrix A, along with their locations in the matrix. The matrix is accessed using only matrixvector or matrix-matrix products. For p = 1 the algorithm estimates the norm A M := max i,j |a ij | or max i,j a ij . The algorithm is based on a power method for mixed subordinate matrix norms and iterates on n × t matrices, where t ≥ p is a parameter. For p = t = 1 we show that the algorithm is essentially equivalent to rook pivoting in Gaussian elimination; we also obtain a bound for the expected number of matrix-vector products for random matrices and give a class of counterexamples. Our numerical experiments show that for p = 1 the algorithm usually converges in just two iterations, requiring the equivalent of 4t matrix-vector products, and for t = 2 the algorithm already provides excellent estimates that are usually within a factor 2 of the largest element and frequently exact. For p > 1 we incorporate deflation to improve the performance of the algorithm. Experiments on real-life datasets show that the algorithm is highly effective in practice.

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Section: C585 the P Largest Entries Of Ementioning

confidence: 99%

“…4 Set prev ind to contain the locations of the nonzero rows in X. 5 % ynorms contains at least one previously unseen large element.…”

Section: Algorithm For P Largest Elementsmentioning

confidence: 99%

Estimating the Largest Elements of a Matrix

Higham

Relton

2016

SIAM J. Sci. Comput.

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“…More sophisticated methods that replace the polynomial subspace (1.1) by a rational Krylov subspace, utilizing the inverse of shifted versions of A, have also been developed (see Güttel [8] for a recent survey). For the matrix exponentialthere are a number of alternatives to using a Krylov subspace, such as the Leja method [3] and methods based upon Taylor series approximation [1], which have been compared in a recent review paper [2]. In any numerical computation it is important to know the condition number of the problem: when the algorithm used is backward stable, we know that the relative error is approximately bounded above by the condition number times the unit roundoff.…”

mentioning

confidence: 99%

A Block Krylov Method to Compute the Action of the Fréchet Derivative of a Matrix Function on a Vector with Applications to Condition Number Estimation

Kandolf¹,

Relton²

2017

SIAM J. Sci. Comput.

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Abstract. We design a block Krylov method to compute the action of the Fréchet derivative of a matrix function on a vector using only matrix-vector products, i.e., the derivative of f (A)b when A is subject to a perturbation in the direction E. The algorithm we derive is especially effective when the direction matrix E in the derivative is of low rank, while there are no such restrictions on A. Our results and experiments are focused mainly on Fréchet derivatives with rank 1 direction matrices. Our analysis applies to all functions with a power series expansion convergent on a subdomain of the complex plane which, in particular, includes the matrix exponential. We perform an a priori error analysis of our algorithm to obtain rigorous stopping criteria. Furthermore, we show how our algorithm can be used to estimate the 2-norm condition number of f (A)b efficiently. Our numerical experiments show that our new algorithm for computing the action of a Fréchet derivative typically requires a small number of iterations to converge and (particularly for single and half precision accuracy) is significantly faster than alternative algorithms. When applied to condition number estimation, our experiments show that the resulting algorithm can detect ill-conditioned problems that are undetected by competing algorithms.

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“…See e.g. [12] for a recent comparison of four common approaches in the large-scale case. In the second category are methods that consider e sA T z as the solution x(s) to the system of ODEṡ…”

Section: A the Affine Subproblemmentioning

confidence: 99%

Low-Rank Second-Order Splitting of Large-Scale Differential Riccati Equations

Stillfjord

2015

IEEE Trans. Automat. Contr.

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Abstract-We apply first-and second-order splitting schemes to the differential Riccati equation. Such equations are very important in e.g. linear quadratic regulator (LQR) problems, where they provide a link between the state of the system and the optimal input. The methods can also be extended to generalized Riccati equations, e.g. arising from LQR problems given in implicit form. In contrast to previously proposed schemes such as BDF or Rosenbrock methods, the splitting schemes exploit the fact that the nonlinear and affine parts of the problem, when considered in isolation, have closed-form solutions. We show that if the solution possesses low-rank structure, which is frequently the case, then this is preserved by the method. This feature is used to implement the methods efficiently for large-scale problems. The proposed methods are expected to be competitive, as they at most require the solution of a small number of linear equation systems per time step. Finally, we apply our low-rank implementations to the Riccati equations arising from two LQR problems. The results show that the rank of the solutions stay low, and the expected orders of convergence are observed.

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Comparison of software for computing the action of the matrix exponential

Cited by 55 publications

References 11 publications

Estimating the Largest Elements of a Matrix

Estimating the Largest Elements of a Matrix

A Block Krylov Method to Compute the Action of the Fréchet Derivative of a Matrix Function on a Vector with Applications to Condition Number Estimation

Low-Rank Second-Order Splitting of Large-Scale Differential Riccati Equations

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