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Abstract and Applied Analysis
Volume 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

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 [911].

Circulant matrices arise in many applications in mathematics, physics, and other applied sciences in problems possessing a periodicity property [1219] 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 [2226]. 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 [2732].

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 [3840]. 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.

References

  1. W. Chen, J. Lin, and C. S. Chen, “The method of fundamental solutions for solving exterior axisymmetric Helmholtz problems with high wave-number,” Advances in Applied Mathematics and Mechanics, vol. 5, no. 4, pp. 477–493, 2013. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. B. Karasözen and G. Şimşek, “Energy preserving integration of bi-Hamiltonian partial differential equations,” Applied Mathematics Letters, vol. 26, no. 12, pp. 1125–1133, 2013. View at Google Scholar
  3. E. W. Sachs and A. K. Strauss, “Efficient solution of a partial integro-differential equation in finance,” Applied Numerical Mathematics, vol. 58, no. 11, pp. 1687–1703, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. A. Meyer and S. Rjasanow, “An effective direct solution method for certain boundary element equations in 3D,” Mathematical Methods in the Applied Sciences, vol. 13, no. 1, pp. 43–53, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
  5. S. E. Cohn and D. P. Dee, “Observability of discretized partial differential equations,” SIAM Journal on Numerical Analysis, vol. 25, no. 3, pp. 586–617, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. J. Ruiz-Claeyssen, M. Davila, and T. Tsukazan, “Factor block circulant and periodic solutions of undamped matrix differential equations,” Matemática Aplicada e Computacional, vol. 3, no. 1, 1983. View at Google Scholar
  7. A. C. Wilde, “Differential equations involving circulant matrices,” The Rocky Mountain Journal of Mathematics, vol. 13, no. 1, pp. 1–13, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. T. Chow and H. W. Milnes, “Numerical solution of a class of hyperbolic-parabolic partial differential equations by boundary contraction,” Journal of the Society for Industrial and Applied Mathematics, vol. 10, no. 1, pp. 124–148, 1962. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. R. H. Chan, X.-Q. Jin, and Y.-H. Tam, “Strang-type preconditioners for solving systems of ODEs by boundary value methods,” Electronic Journal of Mathematical and Physical Sciences, vol. 1, no. 1, pp. 14–46, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. R. D. Nussbaum, “Circulant matrices and differential-delay equations,” Journal of Differential Equations, vol. 60, no. 2, pp. 201–217, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. Z.-J. Bai, X.-Q. Jin, and L.-L. Song, “Strang-type preconditioners for solving linear systems from neutral delay differential equations,” Calcolo, vol. 40, no. 1, pp. 21–31, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. J. C. Ruiz-Claeyssen and L. A. dos Santos Leal, “Diagonalization and spectral decomposition of factor block circulant matrices,” Linear Algebra and Its Applications, vol. 99, pp. 41–61, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. D. Chillag, “Regular representations of semisimple algebras, separable field extensions, group characters, generalized circulants, and generalized cyclic codes,” Linear Algebra and Its Applications, vol. 218, pp. 147–183, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. R. E. Blahut, Fast Algorithm for Digltal Signal Processing, Addison-Wesley, Reading, Mass, USA, 1984.
  15. T.-K. Ku and C.-C. J. Kuo, “Design and analysis of Toeplitz preconditioners,” IEEE Transactions on Signal Processing, vol. 40, no. 1, pp. 129–141, 1992. View at Publisher · View at Google Scholar · View at Scopus
  16. P. A. Leonard, “Cyclic relative difference sets,” The American Mathematical Monthly, vol. 93, no. 2, pp. 106–111, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. P. Dean, “Atomic vibrations in solids,” Journal of the Institute of Mathematics and Its Applications, vol. 3, pp. 98–165, 1967. View at Google Scholar
  18. F. J. MacWilliams and N. J. A. Sloane, The Theory Error-Correcting Codes, North-Holland, Amsterdam, The Netherlands, 1981.
  19. Th. Chadjipantelis, S. Kounias, and C. Moyssiadis, “Construction of N2 mod 4-optimal designs for γ using block-circulant matrices,” Journal of Combinatorial Theory. Series A, vol. 40, no. 1, pp. 125–135, 1985. View at Publisher · View at Google Scholar · View at MathSciNet
  20. P. J. Davis, Circulant Matrices, John Wiley & Sons, New York, NY, USA, 1979. View at MathSciNet
  21. Z. L. Jiang and Z. X. Zhou, Circulant Matrices, Chengdu Technology University Publishing Company, Chengdu, China, 1999 (Chinese).
  22. T. de Mazancourt and D. Gerlic, “The inverse of a block-circulant matrix,” IEEE Transactions on Antennas and Propagation, vol. 31, no. 5, pp. 808–810, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. J. L. Stuart and J. R. Weaver, “Matrices that commute with a permutation matrix,” Linear Algebra and Its Applications, vol. 150, pp. 255–265, 1991. View at Publisher · View at Google Scholar · View at MathSciNet
  24. J. L. Stuart, “Diagonally scaled permutations and circulant matrices,” Linear Algebra and Its Applications, vol. 212-213, pp. 397–411, 1994. View at Publisher · View at Google Scholar · View at MathSciNet
  25. R. E. Cline, R. J. Plemmons, and G. Worm, “Generalized inverses of certain Toeplitz matrices,” Linear Algebra and Its Applications, vol. 8, pp. 25–33, 1974. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. G. Berman, “Families of generalized weighing matrices,” Canadian Journal of Mathematics, vol. 30, no. 5, pp. 1016–1028, 1978. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. I. Gohberg, T. Kailath, I. Koltracht, and P. Lancaster, “Linear complexity parallel algorithms for linear systems of equations with recursive structure,” Linear Algebra and Its Applications, vol. 88-89, pp. 271–315, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. F. de Hoog, “A new algorithm for solving Toeplitz systems of equations,” Linear Algebra and Its Applications, vol. 88-89, pp. 123–138, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. M. K. Chen, “On the solution of circulant linear systems,” SIAM Journal on Numerical Analysis, vol. 24, no. 3, pp. 668–683, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. E. Linzer, “On the stability of transform-based circular deconvolution,” SIAM Journal on Numerical Analysis, vol. 29, no. 5, pp. 1482–1492, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. C. Y. He, “On the fast algorithm for computing the inverse of γ-block circulant matrices and γ-block circulant matrices linear systems,” Acta Mathematicae Applicatae Sinica, vol. 22, no. 4, pp. 624–627, 1999 (Chinese). View at Google Scholar · View at MathSciNet
  32. C. Y. He, “On the fast solution of r-circulant linear systems,” Journal of Systems Science and Complexity, vol. 21, no. 2, pp. 182–189, 2001 (Chinese). View at Google Scholar · View at MathSciNet
  33. S. Lefschetz, Stability of Nonlinear Control Systems, Academic Press, New York, NY, USA, 1965. View at MathSciNet
  34. Z. Peng and X. Hu, “The reflexive and anti-reflexive solutions of the matrix equation AX=B,” Linear Algebra and Its Applications, vol. 375, pp. 147–155, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  35. F. J. H. Don, “On the symmetric solutions of a linear matrix equation,” Linear Algebra and Its Applications, vol. 93, pp. 1–7, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  36. K.-W. E. Chu, “Symmetric solutions of linear matrix equations by matrix decompositions,” Linear Algebra and Its Applications, vol. 119, pp. 35–50, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  37. H. Dai, “On the symmetric solutions of linear matrix equations,” Linear Algebra and Its Applications, vol. 131, pp. 1–7, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  38. Z.-L. Jiang and Z.-B. Xu, “Efficient algorithm for finding the inverse and the group inverse of FLSr-circulant matrix,” Journal of Applied Mathematics & Computing, vol. 18, no. 1-2, pp. 45–57, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  39. Z. L. Jiang and D. H. Sun, “Fast algorithms for solving the inverse problem of Ax=b,” in Proceedings of the 8th International Conference on Matrix Theory and Its Applications, pp. 121–124, Taiyuan, China, 2008.
  40. Z. L. Jiang, “Fast algorithms for solving FLS r-circulant linear systems,” in Proceedings of the Spring World Congress on Engineering and Technology (SCET '12), pp. 141–144, Xi’an, China, 2012.