Flexible GPBi-CG Method for Nonsymmetric Linear Systems *

We present a flexible version of GPBi-CG algorithm which allows for the use of a different preconditioner at each step of the algorithm. In particular, a result of the flexibility of the variable preconditioner is to use any iterative method. For example, the standard GPBi-CG algorithm itself can be used as a preconditioner, as can other Krylov subspace methods or splitting methods. Numerical experiments are conducted for flexible GPBi-CG for a few matrices including some nonsymmetric matrices. These experiments illustrate the convergence and robustness of the flexible iterative method.


Introduction
Krylov subspace methods are the iterative choice for solving linear system of the form where the matrix A is assumed to be nonsingular.The strength of Krylov subspace methods are most apparent when combined with a preconditioner.We only consider right preconditioning in the paper.Thus one solves the equivalent linear system The preconditioner M is selected to be close to the matrix A. And the matrix 1 M  A is never formed explicitly.Instead, when 1 M v z   is needed, one solves the corresponding system In this paper, we present a flexible version of GPBi-CG, which allows the preconditioner M vary from one iteration to another.Let us denote the matrix n the preconditioner used in the nth iteration.The need to allow for a variable preconditioner arises when the solution of (2) is not obtained exactly (say, by a direct method), but is approximated by a second (inner) iterative method.

M
In recent years, several flexible variants of Krylov subspace methods have been established successfully.They include flexible CG, which is applied on a symmetric positive definite matrix [1], flexible GMRES [2], flexible QMR [3], variable preconditioned GCR [4], flexible BiCG and flexible Bi-CGSTAB [5].Preconditioning as this form is called flexible preconditioning, also known as variable or inexact preconditioning.
The paper is organized as follows.In the next section, we design the FGPBi-CG algorithm, which is a flexible version of GPBi-CG [6].In Section 3, some numerical experiments will be conducted to illustrate the convergence of the algorithm.Furthermore, in some cases, it is shown that FGPBi-CG can achieve convergence to a tolerance when GPBi-CG is not convergent or even FBi-CGSTAB suffers stagnation.Finally we make some concluding remarks in Section 4.
Throughout the paper, 0 x is the initial approximation, 0 0 r b Ax   is the initial residual, and the norm used is 2-norm.

Flexible GPBi-CG Method
We describe the basic idea of variable preconditioning and how it is incorporated with the algorithm GPBi-CG in this section.
The expression is calculated at each iteration of the conventional preconditioned Krylov subspace methods.The object of preconditioning is to change the original coefficient matrix A into another matrix close to identity, i.e.AM I   .Consequently, the following  .Thus, we consider obtaining an approximation of That is, the following system (4) is roughly solved by an iterative method to a certain degree of accuracy that is not sufficient.
Here, an approximation for the system (4) does not need to be solved at the same precision at each iteration.A stopping criteria has been established to make the preconditioner to be changed at each iteration.Different inner-loop can be applied to the system (4) including Krylov subspace methods and stationery iterative methods.
The GPBi-CG algorithm proposed by Zhang [6], uses an unified way to derive a class generalizations of Bi-CG.By choosing different coefficients, namely, the following  and  , the GPBi-CG algorithm will be reduced to other methods based on Bi-CG includes the well known CGS, Bi-CGSTAB, Bi-CGSTAB2.
Next we present a flexible version of GPBi-CG, which needs only some small modification of the GPBi-CG code.
ALGORITHM (FGP-BiCG with right preconditioner) 0 x is an initial guess, 0 ; is an arbitrary vector, such that  , e.g., ; and set , 0 Enddo at if we replace Noted th n M with M, a fixed precondi erical experiments to tioner, the above algorithm e reduced to the standard GPBi-CG method with right preconditioner.

Numerical Experiments will b
In this section, we report some num show the convergence behaviors of FGPBi-CG.In all cases the iteration was started with   0 0, 0, , 0 x   , and the outer-loop is stopped when the relative residual norm

Examples for Toeplitz Matrix
In the first example, we consider a Toeplitz matrix of order 200 with a parameter  .
In this experiment, we choose  to be 3.79 and the inner iteration stopping criteria to be the maximum iteration is max 50 N  and relative residuals range from When the standard GPBi-CG algorithm performs well, the flexible version of GPBi-CG is also convergent, but it need more computation.The results can be seen in Table 1.In the table, "FG(B)" denotes FGPBi-CG with preconditioning Bi-CGSTAB, and so on, while "MV" represents the number of matrix-vector multiplication, "OIt" denotes the number of outer iteration.
From Figure 1 and Table 1, we can see that for the problem t i-CG hat GPB and Bi-CGSTAB method can convergent fast, FGPBi-CG and FBi-CGSTAB will not gain too much.While FGPBi-CG(GPBi-CG) and FBi-CGSTAB(GPBi-CG) will be faster than FGPBi-CG(Bi-CGSTAB) and FBi-CGSTAB(Bi-CGSTAB) re- In this experiment, we choose  e th to be 1.9 and the inner iteration stopping criteria to b e maximum iteratio CG is c gen hile Bi-CGSTAB is not.As a result, w We consider the finite difference discretization of the par- and relative residuals range from  2 that GPBionver t w see from hen Bi-CGSTAB is used as an inner iteration, the number of matrix vector multiplication is not influenced by the inner relative residual stopping criteria, i.e., the inner iteration is not terminated until it reaches N max (the largest iterative number of inner loop).For this reason, GPBi-CG used as the inner iteration performs better.tial diffe l at [3 ]) rentia equ ion ( ,5,7

Examples for Model Problem
at the exact solution to the discretized equation on a unit square, where f is such th .The mesh is chosen of equal size in both dimension (32 nodes), and the corresponding matrix is thus of order 1024.
In the first example, we take 10   , 100   and the inner iteration stopping criteria is N ranges from 30 , the FBi-CGSTAB(Bi-CGSTAB) needs 316 matrix vector multiplications to reach the prescribed tolerance, faster than that of Bi-CGSTAB.
In   4 that FGPBi-CG and FBi-CGSTAB with flexible precondition converges, but the cost of FGPBi-CG is about one and a half that of FB

Conclusion
We have formulated a flexible version of GPBi-CG for the large sparse nonsymmetric linear systems.The preconditioning is carried out by roughly solving Az = v by an iterative metho our proposal, the i according to satisfy a certain accuracy of approximation or the maximum number of iterations, so the preconditioner is changed at each outer iteration.Our numerical experiments show that FGPBi-CG is a viable alternative to GPBi-CG.And some examples show that FGPBi-CG is convergent when GPBi-CG suffers from stagnation.
following examples, we use stopping criterion for inner-loop as: the l-th approximation when comt puting Az v  at k-th steps of the outer-loop.
We can see from the Figure1that GPBi-CG converges faster than that of Bi-CGSTAB.

Figure 2
and Table
relative residuals is 10 For this choice, Figure3and Table show that both Bi-CGSTAB and GPBi-CG perform q well for this ex uite ample, flexible versions of these algorithms are also convergent.If the inner-loop stopping criterion is N max =