A globally convergent variant of mid-point method for finding the matrix sign
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A globally convergent variant of mid-point method for finding the matrix sign Nahid Zainali1 · Taher Lotfi1
Received: 11 March 2018 / Revised: 3 May 2018 / Accepted: 19 May 2018 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2018
Abstract In this research, an efficient variant of mid-point iterative method is given for computing the sign of a square complex matrix having no pure imaginary eigenvalues. It is proven that the method is new and has global convergence with high order of convergence seven. To justify the effectiveness of the new scheme, several comparisons for matrices of different sizes are worked out to show that the new method is efficient. Keywords Matrix sign · Mid-point method · Global convergence · Iterative methods · High order Mathematics Subject Classification 65F60
1 Introductory notes The sign function for the scalar case is defined by 1, Re(z) > 0, sign(z) = −1, Re(z) < 0,
(1)
wherein z ∈ C is not located on the imaginary axis. Roberts in 1980 for the first time extended this definition for matrices, which has several important applications in scientific computing, (see e.g., Benner and Quintana-Ortí 1999; Howland 1983; Kenney and Laub 1995) and the references therein. For example, the off-diagonal decay of the matrix sign function is also a well-developed area of study in statistics and statistical physics Hardin et al. (2013). An
Communicated by Jinyun Yuan.
B
Taher Lotfi [email protected]; [email protected] Nahid Zainali [email protected]
1
Department of Mathematics, Hamedan Branch, Islamic Azad University, Hamedan, Iran
123
N. Zainali, T. Lotfi
application of this function has recently been discussed in Soheili et al. (2015) for tackling some special stochastic differential equations. To proceed formally, let us consider that A ∈ Cn×n is a square matrix having no pure imaginary eigenvalues. We consider A = T J T −1 ,
(2)
as a Jordan canonical form written such that J = diag(J1 , J2 ),
(3)
and the eigenvalues of J1 ∈ C p× p are in the open left half-plane, while the eigenvalues of J2 ∈ Cq×q are in the open right half-plane. It is now possible to write Higham (2008) −I p 0 S = sign(A) = T (4) T −1 , 0 Iq wherein p + q = n. Note that sign(A) is defined when A is nonsingular. This definition makes explicit use of the Jordan canonical form and of the associated transformation matrix T . Neither T nor J are unique, but it can be shown that sign(A) as introduced in (4) does not depend on the particular choice of T or J . Here, a simpler interpretation for the sign matrix in the case of Hermitian case (viz, all eigenvalues are real) can be given by
wherein
S = U diag(sign(λ1 ), . . . , sign(λn ))U ∗ ,
(5)
U ∗ AU = diag(λ1 , . . . , λn ),
(6)
is a diagonalization of the square matrix A. The significance of calculating and finding S in (4) is because of the point that the function of sign plays a key role in matrix functions theory, specially for principal matrix roots and the polar decomposition, for more one may refer to Higham (2008),
