Abstract and Applied Analysis

Volume 2014 (2014), Article ID 340803, 9 pages

http://dx.doi.org/10.1155/2014/340803

Research Article

Fast Algorithms for Solving FLS -Factor Block Circulant Linear Systems and Inverse Problem of

Zhaolin Jiang

Department of Mathematics, Linyi University, Linyi, Shandong 276005, China

Received 13 April 2014; Accepted 25 May 2014; Published 17 June 2014

Academic Editor: Tongxing Li

Copyright © 2014 Zhaolin Jiang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Block circulant and circulant matrices have already become an ideal research area for solving various differential equations. In this paper, we give the definition and the basic properties of FLS -factor block circulant (retrocirculant) matrix over field . Fast algorithms for solving systems of linear equations involving these matrices are presented by the fast algorithm for computing matrix polynomials. The unique solution is obtained when such matrix over a field is nonsingular. Fast algorithms for solving the unique solution of the inverse problem of in the class of the level-2 FLS -circulant(retrocirculant) matrix of type over field are given by the right largest common factor of the matrix polynomial. Numerical examples show the effectiveness of the algorithms.

1. Introduction

It is well known that block circulant and circulant matrices may play a crucial role in solving various differential equations such as bi-Hamiltonian partial differential equations, discretized partial differential equations, Hyperbolic-Parabolic partial differential equations, delay differential equations, undamped matrix differential equations, fractional diffusion equations, and Wiener-Hopf equations. By the radial properties of the fundamental solution and radial symmetry of the solution domain, Chen et al. [1] showed the circulant or block circulant features of the coefficient matrices for problems under pure Dirichlet or Neumann boundary condition. Using circulant matrix, Karasözen and Şimşek [2] considered periodic boundary conditions such that no additional boundary terms will appear after semidiscretization. In [3], the resulting dense linear system exhibits a special structure which can be solved very efficiently by a circulant preconditioned conjugate gradient method. Meyer and Rjasanow [4] have given an effective direct solution method for certain boundary element equations in 3D. The main theory of circulant dynamics considered in [5] is about circulant matrix. Ruiz-Claeyssen et al. [6] discussed factor block circulant and periodic solutions of undamped matrix differential equations. Wilde [7] developed a theory for the solution of ordinary and partial differential equations whose structure involves the algebra of circulants. Using circulant contraction of boundary, Chow and Milnes [8] got a numerical solution of a class of Hyperbolic-Parabolic partial differential equations. The Strang-type preconditioner was also used to solve linear systems from differential algebraic equations and delay differential equations; see [9–11].

Circulant matrices arise in many applications in mathematics, physics, and other applied sciences in problems possessing a periodicity property [12–19] and they have been put on a firm basis with the work of Davis [20] and Jiang and Zhou [21]. The circulant matrices, long a fruitful subject of research [20, 21], have in recent years been extended in many directions [22–26]. Factor block circulant matrices and -circulants are other natural extensions of this well-studied class and can be found in [12, 13].

Algorithms for solving systems of linear equations involving matrices with the circulant or factor circulant or -circulant structure were introduced in [27–32].

The problem of finding a real matrix of order , satisfying , for given -dimension real vectors and , is called the inverse problem of the linear system . The applications of this problem come from the study of absolute stability of a class of direct control systems [33]. Many authors have studied this problem for some special structured matrices: Peng and Hu [34] for reflexive and antireflexive matrices and Don [35], Chu [36], and Dai [37] for symmetric matrices.

The fast algorithms presented in this paper avoid the problems of error and efficiency produced by computing a great number of triangular functions by means of other general fast algorithms. There is only error of approximation when the fast algorithm is realized by computers, so the result of the computation is accurate in theory. Specially, the result computed by a computer is accurate over the rational number field.

Definition 1. Let be square matrices each of order . We assume that commutes with each of the ’s. A FLS -factor block circulant matrix of type over field , denoted by , is meant to be a square matrix of the form

A FLS -factor block circulant matrix of type will be referred to as a scalar FLS -circulant matrix [38–40]. In this case the matrix reduces to a nonzero scalar that we will denote by . When is the identity matrix , we drop the word “factor” in the above definition. This kind of matrices is just FLS block circulant. In particular, when are all FLS -circulant matrices, this kind of matrix is called level-2 FLS -circulant matrix of type .

Definition 2. Let be square matrices each of order . We assume that commutes with each of the ’s. A FLS -factor block retrocirculant matrix of type over field , denoted by , is meant to be a square matrix of the form

A FLS -factor block retrocirculant matrix of type will be referred to as scalar FLS -retrocirculant. In this case, the matrix reduces to a nonzero scalar that we will denote by . When is the identity matrix , we drop the word “factor” in the above definition. This kind of matrices is just FLS block retrocirculant. In particular, when are all FLS -circulant matrices, this kind of matrix is called level-2 FLS -retrocirculant matrix of type .

For the convenience of application, we give the obvious results in the following lemmas.

Lemma 3. Let be a FLS -factor block circulant matrix over and a FLS -factor block retrocirculant matrix over . Then or , where

Lemma 4 (see [31]). Suppose that the partitioned polynomial matrix is changed into the partitioned polynomial matrix by a series of elementary row operations, then is the right largest common factor of the matrix polynomial and , and .

The matrices, vectors, and polynomials considered in the following are always over any field .

2. The Properties of FLS -Factor Block Circulant Matrix

We define as the basic FLS -factor block circulant matrix over ; that is,

It is easily verified that the matrix polynomial is the form characteristic polynomial of the matrix . In addition, .

In view of the structure of the powers of the basic FLS -factor block circulant matrix over , it is clear that

Thus, is a FLS -factor block circulant matrix over if and only if for some matrix polynomial over . The matrix polynomial will be called the representer of the FLS -factor block circulant matrix over .

By Definition 1 and (5), it is clear that is a FLS -factor block circulant matrix over if and only if commutes with ; that is,

In addition to the algebraic properties that can be easily derived from the representation (5), we mention the following. The product of two FLS -factor block circulant matrices is a FLS -factor block circulant matrix of the same type.

Furthermore, two FLS -factor block circulant matrices,

commute if the ’s commute with the ’s. Since FLS -circulant matrices commute under multiplication, then level-2 FLS -circulant matrices commute under multiplication.

Theorem 5. The inverse matrix of a nonsingular FLS -factor block circulant matrix over is also a FLS -factor block circulant matrix of the same type.

Proof. From representation (5), we have and is also a FLS -factor block circulant matrix of the same type if and only if there exist over such that

where are square matrices each of order .

Since and , then

if and only if

if and only if

Since is nonsingular, so

By the above system of (12), the existence of in the system of (8) has been proved.

Theorem 6. Let    be a FLS -factor block circulant matrix of type over . Then is nonsingular if and only if is the right largest common factor of the matrix polynomial and , where and .

Proof. Let be the right largest common factor of the matrix polynomial and . Then there exists matrix polynomial ,  ,   such that

Substituting by in the equation , we have . Since is nonsingular, then is nonsingular. By Theorem 5, we know that there exists matrix polynomial such that ; then

So is the right largest common factor of the matrix polynomial and .

Conversely, if is the right largest common factor of the matrix polynomial and , then there exists matrix polynomial ,   such that

Substituting by in the above matrix equations, we have

Since and , then

By (17), we know that is nonsingular.

Theorem 7. Let    be a FLS -factor block retrocirculant matrix of type over . Then is nonsingular if and only if is the right largest common factor of the matrix polynomial and , where and .

Proof. Since is nonsingular, by Lemma 3, is nonsingular if and only if is nonsingular. By Theorem 6, we know that is nonsingular if and only if is the right largest common factor of the matrix polynomial and . Then is nonsingular if and only if is the right largest common factor of the matrix polynomial and .

Theorem 8. Let    be a nonsingular FLS -factor block circulant matrix of type over . Then there exists matrix polynomial such that .

Proof. Since matrix is nonsingular, we can change the partitioned polynomial matrix into the partitioned polynomial matrix by a series of elementary row operations.

By Lemma 4, we have

Substituting by in the above matrix equations, we have

Since and , then

By (20), we know that .

Theorem 9. Let    be a nonsingular FLS -factor block retrocirculant matrix of type over . Then there exists matrix polynomial such that .

Proof. By Lemma 3, we know that . Since both and are nonsingular, then is nonsingular and . By Theorem 8, there exists matrix polynomial such that . Then .

By Theorems 8 and 9, we can get the fast algorithm for finding the inverse of the FLS -factor block circulant matrix or the inverse of the FLS -factor block retrocirculant matrix.

Step  1. From the matrix     (or ), we get the matrix polynomial .

Step  2. Change the partitioned polynomial matrix into the partitioned polynomial matrix by a series of elementary row operations.

Step  3. If , then the matrix (or ) is nonsingular and     (or ).

3. Fast Algorithms for Solving a FLS -Factor Block Circulant (or Retrocirculant) Linear System

Consider the FLS -factor block circulant (or retrocirculant) linear system

where is a FLS -factor block circulant (or retrocirculant) matrix of type over ,  , and .

If is nonsingular, then (21) has a unique solution

The key problem is how to find ; for this purpose, we first prove the following results.

Theorem 10. Let    be a nonsingular FLS -factor block circulant matrix of type over and . Then there exists a unique FLS -factor block circulant matrix of type over such that the unique solution of is the last column (or the first column) of the partitioned matrix .

Proof. Since matrix is nonsingular, the representer of is and .

Let the matrix polynomial be constructed by ,,  where   (or or ), the last column (or the first column) of is , and the matrix commutes with the matrix , for .

By Lemma 4 and Theorem 6, we can change the partitioned polynomial matrix into the partitioned polynomial matrix   by a series of elementary row operations. Then,

That is,

Substituting by in the above two equations, respectively, we have

Since and , then

By (26), we know that is a unique inverse of . Let . By (25), we have

Since the last column (or the th column) of the matrix is and the last column (or the th column) of the is the last column (or the first column) of the partitioned matrix ,  by (27), we know that the unique solution of is the last column (or the first column) of the partitioned matrix .

By Theorem 10, we can get the fast algorithm for solving the FLS -factor block circulant linear system , where   ,  ,  and .

Step  1. From the FLS -factor block circulant linear system , we get the matrix polynomial and .

Step  2. Let the matrix polynomial be constructed by , where (or or ), the last column (or the first column) of is , and the matrix commutes with the matrix , for .

Step  3. Change the partitioned polynomial matrix into the partitioned polynomial matrix by a series of elementary row operations.

Step  4. If , then the FLS -factor block circulant linear system has a unique solution. Substituting by in matrix polynomial , we have the FLS -factor block circulant matrix . So the unique solution of is the last column (or the first column) of the partitioned matrix .

Theorem 11. Let    be a nonsingular FLS -factor block retrocirculant matrix of type over and . Then there exists a unique FLS -factor block circulant matrix of type over such that the unique solution of is the last column (or the first column) of the partitioned matrix .

Proof. Since both and are nonsingular, by Lemma 3, we know that is nonsingular and if and only if . By Theorem 10, we know that there exists a unique FLS -factor block circulant matrix of type over such that the unique solution of in a variable is the last column (or the first column) of the partitioned matrix . So the unique solution of is the last column (or the first column) of the partitioned matrix .

By Theorem 11, we can get the fast algorithm for solving the FLS -factor block retrocirculant linear system , where   ,  , ,  and .

Step  1. From the FLS -factor block retrocirculant linear system , we get the matrix polynomial ,  .

Step  2. Let the matrix polynomial be constructed by ,  where (or or ), the last column (or the first column) of is , and the matrix commutes with the matrix , for .

Step  3. Change the partitioned polynomial matrix into the partitioned polynomial matrix by a series of elementary row operations.

Step  4. If , then the FLS -factor block retrocirculant linear system has a unique solution. Substituting by in matrix polynomial , we have the FLS -factor block circulant matrix . So the unique solution of is the last column (or the first column) of the partitioned matrix .

4. Fast Algorithm for Solving the Inverse Problem of

In this section, sufficient and necessary conditions of existence of the unique solution of the inverse problem of in the class of the level-2 FLS -circulant matrices of type over and that of the level-2 FLS -retrocirculant matrices of type over are presented. Fast algorithms for solving the unique solution of the inverse problem of in the class of the level-2 FLS -circulant matrices of type over and that of the level-2 FLS -retrocirculant matrices of type over are given by the right largest common factor of the matrix polynomial.

Theorem 12. Let ,   , ,   ,   , and , where   and   ,   . Then the inverse problem of has a unique solution in the class of the level-2 FLS -circulant matrices of type if and only if has a unique solution.

Proof. If the inverse problem of has a unique solution in the class of the level-2 FLS -circulant matrices of type , then there exists a unique level-2 FLS -circulant matrices of type    such that . Then

where .

Let ,.

By the multiplication of partitioned matrix and level-2 FLS -circulant matrices commute under multiplication and (28), we know that is the unique solution of .

If has a unique solution  , and let and , where and ,  . Then has a unique solution . Since , then . So is the unique solution of the inverse problem of in the class of the level-2 FLS -circulant matrices of type .

Theorem 13. The inverse problem of has a unique solution in the class of the level-2 FLS -circulant matrices of type if and only if is the right largest common factor of the matrix polynomial and , where and and    are given in Theorem 12.

Proof. By Theorem 12, we know that the inverse problem of has a unique solution in the class of the level-2 FLS -circulant matrices of type if and only if has a unique solution, if and only if is nonsingular, if and only if is the right largest common factor of the matrix polynomial and by Theorem 6, where is given in Theorem 12.

By Lemma 4 and Theorems 12 and 13, we have the following fast algorithms for solving the unique solution of the inverse problem of in the class of the level-2 FLS -circulant matrices of type .

Step  1. From and , , we get the matrix polynomial and , where and  ,  .

Step  2. Change the partitioned polynomial matrix into the partitioned polynomial matrix by a series of elementary row operations, where ,  , .

Step  3. If , then the is nonsingular. So the inverse problem of has a unique solution in the class of the level-2 FLS -circulant matrices of type .

Theorem 14. Let . Then the inverse problem of has a unique solution in the class of the level-2 FLS -retrocirculant matrices of type if and only if has a unique solution, where ,   , and are given in Theorem 12.

Proof. Since , where is given in Theorem 12 and is given in (3), the inverse problem of has a unique solution in the class of the level-2 FLS -retrocirculant matrices of type if and only if the inverse problem of in a variable has a unique solution in the class of the level-2 FLS -circulant matrices of type . By Lemma 3, if and only if . By Theorem 12, we know that the inverse problem of has a unique solution in the class of the level-2 FLS -circulant matrices of type if and only if has a unique solution.

Theorem 15. The inverse problem of has a unique solution in the class of the level-2 FLS -retrocirculant matrices of type if and only if is the right largest common factor of the matrix polynomial and , where and are given in Theorem 13 and and are given in Theorem 14.

Proof. From the proof of Theorem 14, we know that the inverse problem of has a unique solution in the class of the level-2 FLS -retrocirculant matrices of type if and only if the inverse problem of has a unique solution in the class of the level-2 FLS -circulant matrices of type . By Theorem 13, the inverse problem of has a unique solution in the class of the level-2 FLS -circulant matrices of type if and only if is the right largest common factor of the matrix polynomial and .

By Lemma 4 and Theorems 14 and 15, we have the following fast algorithms for solving the unique solution of the inverse problem of in the class of the level-2 FLS -retrocirculant matrices of type .

Step  1. From and , we get the matrix polynomial and , where and ,  .

Step  2. Change the partitioned polynomial matrix into the partitioned polynomial matrix by a series of elementary row operations, where and .

Step  3. If , then the is nonsingular. So the inverse problem of has a unique solution in the class of the level-2 FLS -retrocirculant matrices of type .

5. Numerical Examples

Example 1. Solve the FLS -factor block circulant linear system

where , , and .

From , we get the polynomial matrix

respectively.

Let ,

Then,

We can change the above partitioned polynomial matrix into the partitioned polynomial matrix

where , by a series of elementary row operations, where the polynomials lying in need not to be known.

Since , then the FLS -factor block circulant linear system has a unique solution. On the other hand,

where .

Substituting by in the above matrix polynomial , we have the FLS -factor block circulant matrix

where ,  , and . Then partitioned matrix is as follows:

So the unique solution of is the last column of the partitioned matrix ; that is, .

Example 2. Find the solution of the inverse problem of in the class of the level-2 FLS matrices of type , where ,  , and .

From , , and , we get the polynomial matrix

respectively. Then,

We can change the above partitioned polynomial matrix into the partitioned polynomial matrix

where ,    +  ,  , and , by a series of elementary row operations, where the polynomials lying in need not to be known.

Since , then the inverse problem of has a unique solution in the class of the level-2 FLS -circulant matrices of type . On the other hand,

where ,    +  ,  ,  and .

Substituting by in the above matrix polynomial , we know that a unique solution of the inverse problem of in the class of the level-2 FLS -circulant matrices of type is

where  ,  , and .

6. Conclusion

We give the definition and the basic properties of FLS -factor block circulant (retrocirculant) matrix over field . A fast algorithm for solving a FLS -factor block circulant linear system is presented, and extension is made to solve a FLS -factor block retrocirculant linear system by using the relationship between a FLS -factor block circulant matrix and a FLS -factor block retrocirculant matrix. Sufficient and necessary conditions of existence of the unique solution of the inverse problem of in the class of a FLS -factor block circulant (retrocirculant) matrices over field are presented. Fast algorithms for solving the unique solution of the inverse problem of in the class of a FLS -factor block circulant (retrocirculant) matrices over are given by using the right largest common factor of the matrix polynomial.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The research was supported by the Development Project of Science & Technology of Shandong province (Grant no. 2012GGX10115), NSFC (Grant no. 11201212), and the AMEP of Linyi University, China.

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