Fast Algorithms for Solving FLSR-Factor Block Circulant Linear Systems and Inverse Problem ofAX=b

Zhaolin Jiang

doi:10.1155/2014/340803

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 FLSR-factor block circulant (retrocirculant) matrix over fieldF. 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 fieldFis nonsingular. Fast algorithms for solving the unique solution of the inverse problem ofAX=bin the class of the level-2 FLS(R,r)-circulant(retrocirculant) matrix of type(m,n)over fieldFare given by the right largest common factor of the matrix polynomial. Numerical examples show the effectiveness of the algorithms.

Highlights

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, HyperbolicParabolic partial differential equations, delay differential equations, undamped matrix differential equations, fractional diffusion equations, and Wiener-Hopf equations
By Theorem 12, we know that the inverse problem of AX = b has a unique solution in the class of the level-2 FLS (R, r)-circulant matrices of type (m, n) if and only if XY = b has a unique solution, if and only if X is nonsingular, if and only if In is the right largest common factor of the matrix polynomial D(x) and G(x) by Theorem 6, where X is given in Theorem 12
Since KX = X󸀠, where X is given in Theorem 12 and K is given in [3], the inverse problem of CX󸀠 = b has a unique solution in the class of the level-2 FLS (R, r)retrocirculant matrices of type (m, n) if and only if the inverse problem of CKX = b in a variable CK has a unique solution in the class of the level-2 FLS (R, r)-circulant matrices of type (m, n)

Summary

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, HyperbolicParabolic partial differential equations, delay differential equations, undamped matrix differential equations, fractional diffusion equations, and Wiener-Hopf equations. The problem of finding a real matrix A of order n, satisfying AX = b, for given n-dimension real vectors X and b, is called the inverse problem of the linear system AX = b The applications of this problem come from the study of Abstract and Applied Analysis absolute stability of a class of direct control systems [33]. A FLS R-factor block circulant matrix of type (m, n) over field F , denoted by FLScircR(A0, A1, . When R is the identity matrix I, we drop the word “factor” in the above definition This kind of matrices is just FLS block circulant. A FLS R-factor block retrocirculant matrix of type (m, n) over field F , denoted by FLSretrocircR(A0, A1, . Am−1, R are all FLS r-circulant matrices, this kind of matrix is called level-2 FLS (R, r)-retrocirculant matrix of type (m, n). The matrices, vectors, and polynomials considered in the following are always over any field F

The Properties of FLS R-Factor Block Circulant Matrix

Numerical Examples

Conclusion