New iterative methods for finding matrix sign function: derivation and application
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(2019) 38:53
New iterative methods for finding matrix sign function: derivation and application Mohammad Momenzadeh1 · Taher Lotfi1 Received: 22 September 2018 / Revised: 22 November 2018 / Accepted: 16 January 2019 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019
Abstract The objective of this research was to provide two new methods for the sign function of a matrix. It is discussed that the schemes are novel and present global convergence behaviors. Then, the high convergence speeds of these iterative methods are proved and confirmed for calculating the matrix sign of different types of nonsingular matrices to reveal their applicability over the existing iterative methods of the same type. Keywords Sign function · Matrix iterations · Eigenvalues · High order · Attraction basin Mathematics Subject Classification 65F30 · 41A25 · 65F60
1 Introductory notes The generic matrix function f (A) of a given matrix A ∈ Cn×n is defined formally by the integral representation 1 f (A) = f (ζ )(ζ I − A)−1 dζ, (1) 2πi γ where f : → C is an analytic function, ⊆ C and γ is a closed curve which encircles all eigenvalues of A (it should be contained in the domain of analyticity of f ). The integral representation (1) is known as the Cauchy integral formula (Higham 2008). The integral of a matrix M should be understood as the matrix whose entries are the integrals of the entries of M. However, this mathematically appealing formula for computing the matrix functions is complicated and needs complex analysis to be fully understandable. Hence, several important other strategies for computing the matrix functions have been proposed and investigated, such as the Jordan canonical form and iterative methods for applied numerical problems (see, e.g.
Communicated by Jinyun Yuan.
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Taher Lotfi [email protected]; [email protected] Mohammad Momenzadeh [email protected]
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Department of Mathematics, Hamedan Branch, Islamic Azad University, Hamedan, Iran
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M. Momenzadeh, T. Lotfi
Filbir 1994; Howland 1983). Among wide application of matrix functions in mathematics, we refer the readers to computing the inverse of ill-conditioned matrices that occur when solving financial PDEs using radial basis function (RBF) or finite difference (FD) methods (Company et al. 2016; Soleymani and Zaka Ullah 2018; Soleymani et al. 2018). In 1971, Roberts in Roberts (1980) introduced the matrix sign function as a tool for model reduction and for solving Lyapunov and algebraic Riccati equations. He defined the sign function as a Cauchy integral and obtained the following integral representation of sign(A): 2 ∞ 2 sign(A) = S = (t I + A2 )−1 dt. (2) π 0 The matrix sign function has basic theoretical and algorithmic relations with the matrix square root, the polar decomposition (see e.g., Higham et al. 2004), and from time to time, with the matrix pth roots [see for more Higham (2008, chapter 5)]. For example, a large class of iterations for the matrix square root can be obtained from corresponding
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