An interpolatory approximation of the matrix exponential based on Faber polynomials

Moret, Igor; Novati, Paolo

doi:10.1016/s0377-0427(00)00261-2

Cited by 45 publications

(47 citation statements)

References 20 publications

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“…Given a real parameter γ > 0, the Shift-Invert Lanczos method constructs the Krylov subspace K m ((I + γA) −1 , v), computes the projection and restriction T m of the shifted and inverted matrix (I + γA) −1 , and then computes an approximation as Q m f (γ −1 (T −1 m − I))e 1 , where the columns of Q m form an orthonormal basis of K m ((I + γA) −1 , v). The method was analyzed in [40] for a class of functions, and in [31] for the exponential function. In [39] a study of the parameter γ was performed, and in the symmetric non-singular case the value γ = 1/ √ λ min λ max was obtained as a quasi-optimal estimate.…”

Section: Numerical Comparisonsmentioning

confidence: 99%

“…For particularly challenging problems, however, an unacceptably large approximation space may be required to obtain a sastisfactory approximation. This difficulty has lead to the study of enhancement techniques that aim either at enriching the approximation space or at making the overall procedure less expensive [4], [16], [18], [22], [33], [39], [40], [41], [46], [54].…”

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confidence: 99%

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A new investigation of the extended Krylov subspace method for matrix function evaluations

Knizhnerman

Simoncini

2010

Numerical Linear Algebra App

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Abstract. For large square matrices A and functions f , the numerical approximation of the action of f (A) to a vector v has received considerable attention in the last two decades. In this paper we investigate the Extended Krylov subspace method, a technique that was recently proposed to approximate f (A)v for A symmetric. We provide a new theoretical analysis of the method, which improves the original result for A symmetric, and gives a new estimate for A nonsymmetric. Numerical experiments confirm that the new error estimates correctly capture the linear asymptotic convergence rate of the approximation. By using recent algorithmic improvements, we also show that the method is computationally competitive with respect to other enhancement techniques.

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Section: Numerical Comparisonsmentioning

confidence: 99%

mentioning

confidence: 99%

A new investigation of the extended Krylov subspace method for matrix function evaluations

Knizhnerman

Simoncini

2010

Numerical Linear Algebra App

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“…This technique has been employed by Knizhnerman (1991) and Moret and Novati (2001) for the exponential function.…”

Section: Chebyshev Methodsmentioning

confidence: 99%

Exponential Integrators

Ostermann

2015

Encyclopedia of Applied and Computational Mathematics

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In this paper we consider the construction, analysis, implementation and application of exponential integrators. The focus will be on two types of stiff problems. The first one is characterized by a Jacobian that possesses eigenvalues with large negative real parts. Parabolic partial differential equations and their spatial discretization are typical examples. The second class consists of highly oscillatory problems with purely imaginary eigenvalues of large modulus. Apart from motivating the construction of exponential integrators for various classes of problems, our main intention in this article is to present the mathematics behind these methods. We will derive error bounds that are independent of stiffness or highest frequencies in the system.Since the implementation of exponential integrators requires the evaluation of the product of a matrix function with a vector, we will briefly discuss some possible approaches as well. The paper concludes with some applications, in which exponential integrators are used.

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“…Other choices of node sequences are explored in [16,18,17]. Note that, in view of Remark 2.4, such a basis choice, which is independent of A and b, will generally destroy the finite termination property.…”

Section: Matrix Functions and Their Krylov Subspace Approximationmentioning

confidence: 99%

“…[15,4]) cannot be used. The standard approach for approximating (1.1) directly is based on a Krylov subspace of A with initial vector b [6,7,23,12,14,3,16]. The advantage of this approach is that it requires A only for computing matrix-vector products and that, for smooth functions such as the exponential, it converges superlinearly [6,26,14].…”

Section: Introduction the Evaluation Of F (A)bmentioning

confidence: 99%

A Restarted Krylov Subspace Method for the Evaluation of Matrix Functions

Eiermann¹,

Ernst²

2006

SIAM J. Numer. Anal.

129

155

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Abstract. We show how the Arnoldi algorithm for approximating a function of a matrix times a vector can be restarted in a manner analogous to restarted Krylov subspace methods for solving linear systems of equations. The resulting restarted algorithm reduces to other known algorithms for the reciprocal and the exponential functions. We further show that the restarted algorithm inherits the superlinear convergence property of its unrestarted counterpart for entire functions and present the results of numerical experiments.

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An interpolatory approximation of the matrix exponential based on Faber polynomials

Abstract: In this paper we introduce a method for the approximation of the matrix exponential obtained by interpolation in zeros of Faber polynomials. In particular, we relate this computation to the solution of linear IVPs. Numerical examples arising from practical problems are examined

Cited by 45 publications

References 20 publications

A new investigation of the extended Krylov subspace method for matrix function evaluations

A new investigation of the extended Krylov subspace method for matrix function evaluations

Exponential Integrators

A Restarted Krylov Subspace Method for the Evaluation of Matrix Functions

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