Extension of Range of MINRES-CN Algorithm

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


Introduction and Preliminaries
Suppose that one needs to solve the system of linear equations Ax b  with a conjugate-normal n × n-matrix A. In the context of this paper, conjugate-normality means that 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 segments of the sequence * * , , , , , , , 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 definition of conspectrum in [2]) of A belongs to an algebraic 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].

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 Symmetric matrices.
Let us first recall that any square complex matrix A can be uniquely represented in the form (see [3]) , , We consider the class of conjugate-normal matrices A distinguished by the condition, 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 exceed k + 1.Hence, MINRES-CN is applicable to matrices of this type.

Numerical Results
Therefore, we can apply MINRES-CN to solving systems with conjugate normal coefficient matrices satisfying conditions (4) and (5).The efficiency of the method is illustrated by several examples where band systems were solved.The performance of MINRES-CN2 (which is a specialization of MINRES-CN for conjugate normal matrices whose conspectra belong to a second-degree curve) in these examples is compared with that of the Matlab library program implementing GMRES.
In examples, we used the Matlab library function gmres for GMRES and a specially designed Matlab procedure for MINRES-CN2.The same stopping criterion was used for both methods; namely, 2 r   (6) where r is the current residual, while a positive scalar should be given by the user.For the example under discussion, we set .
 8 10    In all of our experiments, the order of systems was 2000.The right hand sides were generated as pseudorandom vectors with components distributed uniformly on (0, 1).The calculations were performed on a 2 Duo E630 OEM 1.86 GHz PC with core memory of 1024 Mb.
Example 3.1.Suppose that, where for i and (where A = S + K and ), another entries of matrix A are zero.The conspectrum is located on the coordinate axes; i.e., it belong to the second-degree curve, ( ) , It follows that a system with the matrix A can be processed by MINRES-CN2.
Example 3.2.Suppose that where for i and (where A = S + K and ), another entries of matrix A are zero.The conspectrum is located on the real axis and the line x = 2; i.e., it belong to the second-degree curve, ( ) , It follows that a system with the matrix A can be processed by MINRES-CN2.
MINRES-CN2 needs 10 steps and t = 0.02 s, while GMRES requires 12 steps and the time 0.09 s.
Example ), another entries of matrix A are zero.The conspectrum is located on the coordinate axes; i.e., it belongs to the second-degree curve (7).It follows that a system with the matrix A can be processed by MINRES-CN2.7 iteration steps and t = 0.02 s for MINRES-CN2 against 12 steps and t = 0.08 s for GMRES.