Real Fast Structure-Preserving Algorithm for Eigenproblem of Complex Hermitian Matrices

Jiangzhou Lai,Linzhang Lu

doi:10.1155/2013/438320

Abstract

It is well known that the flops for complex operations are usually 4 times of real cases. In the paper, using real operations instead of complex, a real fast structure-preserving algorithm for eigenproblem of complex Hermitian matrices is given. We make use of the real symmetric and skew-Hamiltonian structure transformed by Wilkinson's way, focus on symplectic orthogonal similarity transformations and their structure-preserving property, and then reduce it into a two-by-two block tridiagonal symmetric matrix. Finally a real algorithm can be quickly obtained for eigenvalue problems of the original Hermitian matrix. Numerical experiments show that the fast algorithm can solve real complex Hermitian matrix efficiently, stably, and with high precision.

Highlights

We consider eigenvalue problems of the complex Hermitian matrix of the formHx = λx, H ∈ Cn×n, x ∈ Cn, (1)where λ is a real scalar and H ∈ Cn×n is a complex Hermitian matrix of the form H = A + iB, (2)where A is a real symmetric matrix; that is, AT = A ∈ Rn and B is a real skew-symmetric nonzero matrix, that is, B = −BT ≠ 0 ∈ Rn, given by A = HT + 2 H, B
In the following discussion, inspired by the algorithms for the eigenproblem of Hamiltonian matrix [2–11], we devise an algorithm by taking full advantage of the special structure of (6)
In this paper we first prove that orthogonal symplectic similarity transformations preserve the symmetry and skew-Hamiltonian structure of (6), and eigenvectors of (6), since the eigenvectors of (6) have special structure; that is, it is orthogonal symplectic. By these transformations, a real stable, accurate, and fast method is devised to reduce (6) into a block symmetry tridiagonal matrix, and we get the eigenproblem of the original Hermitian matrix

Summary

Introduction

In the following discussion, inspired by the algorithms for the eigenproblem of Hamiltonian (skew-Hamiltonian) matrix [2–11], we devise an algorithm by taking full advantage of the special structure of (6). In this paper we first prove that orthogonal symplectic similarity transformations preserve the symmetry and skew-Hamiltonian structure of (6), and eigenvectors of (6), since the eigenvectors of (6) have special structure; that is, it is orthogonal symplectic. By these transformations, a real stable, accurate, and fast method is devised to reduce (6) into a block symmetry tridiagonal matrix, and we get the eigenproblem of the original Hermitian matrix.

Basic Orthogonal Symplectic Matrix and Its Algorithm

The Reduction of the Symmetry and Skew-Hamiltonian Matrix S

Implementation Aspects

Numerical Experiments