Applied Mathematics, 2011, 2, 1446-1447

doi:10.4236/am.2011.212205 Published Online December 2011 (http://www.SciRP.org/journal/am)

Copyright © 2011 SciRes. AM

Extension of Range of MINRES-CN Algorithm

Mojtaba Ghasemi Kamalvand

Department of Mathematics, Lorestan University, Khorramabad, Iran

E-mail: m_ghasemi98@yahoo.com

Received January 20, 2011; revised May 10, 2011; accepted May 18, 2011

Abstract

MINRES-CN is an iterative method for solving systems of linear equations with conjugate-normal coeffi-

cient matrices whose conspectra are located on algebraic curves of a low degree. This method was proposed

in a previous publication of author and KH. D. Ikramov. In this paper, the range of applicability of MIN-

RES-CN is extended in new direction. These are conjugate normal matrices that are low rank perturbations

of Symmetric matrices. Examples are given that demonstrate a higher efficiency of MINRES-CN for this

class of systems compared to the well-known algorithm GMRES.

Keywords: Conjugate-Normal Matrices, MINRES-CN Algorithm, MINRES-CN2 Algorithm

1. Introduction and Preliminaries

Suppo se that one n eeds to so lve th e syste m of linear equ-

ations

xb (1)

with a conjugate-normal n × n-matrix A. In the context of

this paper, conjug a te-normality means that

**

AAA (2)

A particular example of conjugate-normal matrices are

symmetric matrices.

The method proposed in [1] is a minimum residual

algorithm for the subspaces, which are the finite seg-

ments of the sequence

**

,, ,,,,,

TT

x AxAxAAx AA x AA x AAAx,

T

(3)

Unlike GMRES, this method, called MINRES-CN, is

described by a recursion whose (fixed) length depends

on the degree m of Γ (the conspectrum (you can see defi-

nition of conspectrum in [2]) of A belongs to an alge-

braic curve Γ of a low degree)). For instance, the length

of the recursion is six in the case m = 2, which is given

the most attention in [1].

2. Extension of Range

In this section, the range of applicability of MINRES-CN

is extended in new direction.

We examine the behavior of MINRES-CN for new

class of matrices A that can be considered as low rank

perturbations of Sym metric matrices.

Let us first recall that any square complex matrix A

can be uniquely represented in the form (see [3])

,,

T

SKS SKK (4)

We consider the class of conjugate-normal matrices A

distinguished by the condition,

1

2

k

krankK

(5)

where n is the order of A. The conspectrum of such a

matrix belongs to the union of the real axis and (at most)

k lines that are parallel to the imaginary axis, i.e., to a

degenerate algebraic curve whose degree does not ex-

ceed k + 1. Hence, MINRES-CN is applicable to matri-

ces of this type.

3. Numerical Results

Therefore, we can apply MINRES-CN to solving sys-

tems with conjugate normal coefficient matrices satisfy-

ing conditions (4) and (5).

The efficiency of the method is illustrated by several

examples where band systems were solved. The perfor-

mance of MINRES-CN2 (which is a specialization of

MINRES-CN for conjugate normal matrices whose con-

spectra belong to a second-degree curve) in these exam-

ples is compared with that of the Matlab library program

implementing GM RES.

In examples, we used the Matlab library function

gmres for GMRES and a specially designed Matlab pro-

cedure for MINRES-CN2. The same stopping criterion