The scaling and modified squaring method for matrix functions related to the exponential

Skaflestad, Bård; Wright, William G.

doi:10.1016/j.apnum.2008.03.035

Cited by 50 publications

(44 citation statements)

References 25 publications

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“…The algorithm here has been adapted for computation of the ψ functions by Koikari [25] and Skaflestad and Wright [33]. Al-Mohy and Higham have extended this work in two recent papers.…”

Section: (Ementioning

confidence: 99%

The Scaling and Squaring Method for the Matrix Exponential Revisited

Higham¹

2009

SIAM Rev.

292

369

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“…The algorithm here has been adapted for computation of the ψ functions by Koikari [25] and Skaflestad and Wright [33]. Al-Mohy and Higham have extended this work in two recent papers.…”

Section: (Ementioning

confidence: 99%

The Scaling and Squaring Method for the Matrix Exponential Revisited

Higham¹

2009

SIAM Rev.

292

369

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“…We show that these methods can be implemented by evaluating a single exponential of an augmented matrix of order n + p, where p − 1 is the degree of the polynomial used to approximate the nonlinear part of the system, thus avoiding the need to compute any ϕ functions. In fact, on each integration step the integrator is shown to produce the exact solution of an augmented linear system of ODEs of dimension n + p. The replacement of ϕ functions with the exponential is important because algorithms for the ϕ functions are much less well developed (though see [16], [23], for example) than those for the exponential itself.…”

Section: ] It Is Not Ementioning

confidence: 99%

Computing the Action of the Matrix Exponential, with an Application to Exponential Integrators

Al-Mohy¹,

Higham²

2011

SIAM J. Sci. Comput.

399

439

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Abstract. A new algorithm is developed for computing e tA B, where A is an n × n matrix and B is n × n 0 with n 0 ≪ n. The algorithm works for any A, its computational cost is dominated by the formation of products of A with n × n 0 matrices, and the only input parameter is a backward error tolerance. The algorithm can return a single matrix e tA B or a sequence e t k A B on an equally spaced grid of points t k . It uses the scaling part of the scaling and squaring method together with a truncated Taylor series approximation to the exponential. It determines the amount of scaling and the Taylor degree using the recent analysis of Al-Mohy and Higham [SIAM J. Matrix Anal. Appl. 31 (2009), pp. 970-989], which provides sharp truncation error bounds expressed in terms of the quantities A k 1/k for a few values of k, where the norms are estimated using a matrix norm estimator. Shifting and balancing are used as preprocessing steps to reduce the cost of the algorithm. Numerical experiments show that the algorithm performs in a numerically stable fashion across a wide range of problems, and analysis of rounding errors and of the conditioning of the problem provides theoretical support. Experimental comparisons with two Krylov-based MATLAB codes show the new algorithm to be sometimes much superior in terms of computational cost and accuracy. An important application of the algorithm is to exponential integrators for ordinary differential equations. It is shown that the sums of the form p k=0 ϕ k (A)u k that arise in exponential integrators, where the ϕ k are related to the exponential function, can be expressed in terms of a single exponential of a matrix of dimension n + p built by augmenting A with additional rows and columns, and the algorithm of this paper can therefore be employed.

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“…This situation pertains, for example, to the functions ψ k (z) = ∞ j=0 z j /( j + k)!, k ≥ 0 [12, Sec. 10.7.4], related to the matrix exponential, for which efficient numerical methods are available [16], [25]. The CS approximation is trivial to implement assuming the availability of complex arithmetic.…”

Section: Discussionmentioning

confidence: 99%

The complex step approximation to the Fréchet derivative of a matrix function

Al-Mohy

Higham

2009

Numer Algor

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We show that the Fréchet derivative of a matrix function f at A in the direction E, where A and E are real matrices, can be approximated by Im f (A + ihE)/ h for some suitably small h. This approximation, requiring a single function evaluation at a complex argument, generalizes the complex step approximation known in the scalar case. The approximation is proved to be of second order in h for analytic functions f and also for the matrix sign function. It is shown that it does not suffer the inherent cancellation that limits the accuracy of finite difference approximations in floating point arithmetic. However, cancellation does nevertheless vitiate the approximation when the underlying method for evaluating f employs complex arithmetic. The ease of implementation of the approximation, and its superiority over finite differences, make it attractive when specialized methods for evaluating the Fréchet derivative are not available, and in particular for condition number estimation when used in conjunction with a block 1-norm estimation algorithm.

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The scaling and modified squaring method for matrix functions related to the exponential

Cited by 50 publications

References 25 publications

The Scaling and Squaring Method for the Matrix Exponential Revisited

The Scaling and Squaring Method for the Matrix Exponential Revisited

Computing the Action of the Matrix Exponential, with an Application to Exponential Integrators

The complex step approximation to the Fréchet derivative of a matrix function

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