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In this paper, we propose DQMR algorithm for the Drazin-inverse solution of con sistent or inconsistent linear systems of the form Ax=b where is a singular and in general non-hermitian matrix that has an arbitrary index. DQMR algorithm for singular systems is analogous to QMR algorithm for non-singular systems. We compare this algorithm with DGMRES by numerical experiments.

Consider the linear system

where

In [

In [

In the present paper, the Drazin-Quasi-minimal residual algorithm (DQMR here- after) is another implementation of the projection method for singular linear systems is analogues to Lanczos algorithm for non-singular systems. DGMRES algorithm, in prac- tice, cannot afford to run the full algorithm and it is necessary to use restart. For dif- ficult problems, in most cases, this results in extremely slow convergence, While DQMR algorithm can be implemented using only short recurrences and hence it can be com- puted with little work and low storage requirements per iteration.

The outline of this paper is as follows. In Section 2, we will provide a brief of sum- mary of the review of the theorem and projection method in [

The method we are interested in starts with an arbitrary initial vector

where

Let us define

We call

Note that

The condition (6) is due to Eiermann et al. [

For convenience we denote by

Thus, the polynomial

The projection methods of [

where

We see that unique solution for c exists provided det

As we choose different W, we have a different algorithm: for DGMRES, we choose

In this paper, for DQMR, we choose

We will mention several definitions and theorems, which have projection method converge below.

We will denote by

Definition 1 [

The following theorems will ensure the success of projection method.

Theorem 1 [

The following result that is the justification of the above-mentioned projection approach is Theorem 4.2 in [

Theorem 2 Let

Obviously, of Theorem 2 if

Theorem 3 [

In this section, we will introduce a different implementation of projection method. The algorithm is analogous to QMR algorithm. We must note that in spite of the analogy, DQMR seems to be quite different from QMR, which is for non-singular systems.

As

algorithm [

where they are clear that

respectively.

If we define that

then, we can write for

Therefore, it is obvious that we need to determine

where

Moreover, provided that

Note that

Consequently, provided

and since

If the column vectors of

as in GMRES. Therefore, a least, squares solution could be obtained from the krylove space

over

Similar to [

Let us define

where

Since

where, certainly,

Supposed that

From [

Equation (16) can be simplified as follows:

By using the Hadamard product Equation (18) is reduced. For this purpose, we first introduce the concepts of Hadamard matrix product.

Definition 2 Let A and B be

Let us denote

Now, we can be simplified (18) as follows

For solution system

We must reduce the band matrix,

Generally, if we define

where

where

Since if we assume

Consequently,

and

where

or

Thus,

This gives the following algorithm, which we call the Drazin-QMR for Drazin- inverse solution of singular nonsymmetric linear equations.

Algorithm 4.1 DQMR Algorithm

13. EndFor.

16. End Do.

In this section, we will compute the linear system

This linear system has also been computed by Sidi [

Let M be an odd integer, we discretize the Poisson equation on a uniform grid of mesh size

Here, I and 0 denote, respectively, the

The numerical experiment was performed for

It should be noted A is singular with 1D null space spanned by the vector

We first construct a consistent system with the known solution

Consequently the system

meter ^{−2} in our experiments. The solution we intend to obtain is the vector

In

Size of A | 1024 ´ 1024 | 4096 ´ 4096 | ||||
---|---|---|---|---|---|---|

Method | Its | Time | RE | Its | Time | RE |

DQMR | 155 | 0.61 | 5.9674e−09 | 267 | 7.39 | 8.8234e−09 |

DGMRES | 165 | 0.88 | 3.0294e−09 | 307 | 10.89 | 8.1578e−09 |

Size of A | 4096 ´ 4096 | 16384 ´ 16384 | ||||
---|---|---|---|---|---|---|

Method | Its | Time | RE | Its | Time | RE |

DQMR | 155 | 0.53 | 5.9674e−09 | 267 | 6.88 | 8.8234e−09 |

DGMRES | 165 | 0.81 | 3.0294e−09 | 307 | 10.34 | 8.1578e−09 |

required for convergence, the relative error (RE), for the DGMRES and DQMR methods. As shown in

In this paper, we presented a new method, called DQMR, for Drazin-inverse solution of singular nonsymmetric linear systems. The DQMR algorithm for singular systems is analogous to the QMR algorithm for non-singular systems. Numerical experiments indi- cate that the Drazin-inverse solution obtained by this method is reasonably accurate, and its computation time is less than that of solution obtained by the DGMRES method. Thus, we can conclude that the DQMR algorithm is a robust and efficient tool to com- pute the Drazin-inverse solution of singular linear systems.

First, we would like to thank professor F. Toutounian for her comments that improved our results. Second, we thank the editor and the referees for their carefully reading and useful comments.

Ataei, A. (2016) A QMR-Type Algorithm for Drazin-Inverse So- lution of Singular Nonsymmetric Linear Sys- tems. Advances in Linear Algebra & Matrix Theory, 6, 104-115. http://dx.doi.org/10.4236/alamt.2016.64011