International Journal of Algebra, Vol. 5, 2011, no. 9, 403 - 411 Fast Algorithms for Solving the Inverse Problem of AX = b in the Class of the ULS r-Circulant (Retrocirculant) Matrices Zhiping Tian Department of Mathematics Shandong Vocational College of Science and Technology Weifang Shandong, 261053, China [email protected] Abstract In this paper, fast algorithms for solving the unique solution of the inverse problem of AX = b in the class of the ULS r-circulant(retrocirculant) matrices over a field F are given by the largest common factor of poly- nomial. Examples show the effectiveness of the algorithm. Mathematics Subject Classification: 15A29; 65F15 Keywords: inverse problem, ULS r-circulant matrix 1 Introduction Circulant matrices have important applications in various disciplines including signal processing[1-4], communications[5-6], image processing[7-8], computer vision[9], encoding[10], physics, probability and statistics, number theory, ge- ometry, and the numerical solution of ordinary and partial differential equa- tions, and they have been put on firm basis with the work of P. Davis[11] and Z. L. Jiang[12]. The circulant matrices, long a fruitful subject of research [11, 12], have in recent years been extended in many directions [13-18]. The f(x)-circulant matrices are another natural extension of this well-studied class, and can be found in [16]. The f(x)-circulant matrix has a wide application, especially on the gen- eralized cyclic codes[16], where f(x) ∈ F[x] is a monic polynomial with no repeated roots in its splitting field over a field F. The properties and struc- tures of the xn − rx − r-circulant matrices, which are called ULS r-circulant 404 Zhiping Tian matrices, are better than those of the general f(x)-circulant matrices, so there are good algorithms for solving the inverse problem of AX = b in the class of the ULS r-circulant (retrocirculant) matrices over a field F. The fast al- gorithms presented 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, and only the elements in the first row of the ULS r-circulant(retrocirculant) matrix and the constant term are used by the fast algorithm, so the result of the computation is accurate in theory. Specially, the result computed by a computer is accurate over the rational field. 2 Definition and Lemma Definition 2.1. A sum of the upper and the lower(ULS) r-circulant matrix over a field F, denoted by ULScircr(a0,a1,...,an−1), is meant a square matrix of the form ⎡ ⎤ a0 a1 ... an−1 ⎢ ⎥ ⎢ ⎥ ⎢ ran−1 a0 + ran−1 ... an−2 ⎥ ⎢ ⎥ ⎢ ran−2 ran−1 + ran−2 ... an−3 ⎥ ⎢ ⎥ ⎢ . ⎥ . ⎢ . ⎥ ⎢ ⎥ ⎣ ra2 ra3 + ra2 ... a1 ⎦ ra1 ra2 + ra1 ... a0 + ran−1 n×n We define Θr as the basic ULS r-circulant matrix over F, that is, ⎡ ⎤ 010... 00 ⎢ ⎥ ⎢ ⎥ ⎢ 001... 00⎥ ⎢ ⎥ ⎢ . ⎥ Θr = ⎢ . ⎥ . ⎢ ⎥ ⎣ 000... 01⎦ rr0 ... 00 n×n In the following, let that g(x)=xn − rx − r has no repeated roots in its splitting field over F. It is easily verified that the polynomial g(x)=xn −rx−r is both the minimal polynomial and the characteristic polynomial of the matrix Θr. In addition, Θr is nonderogatory and n Θr = rIn + rΘr. In view of the structure of the powers of the basic ULS r-circulant matrix Θr over F, it is clear that n−1 i A = ULScircr(a0,a1,...,an−1)= aiΘr (1) i=0 Fast algorithms for solving the inverse problem of AX = b 405 Thus, A is a ULS r-circulant matrix over F if and only A = f(Θr) for some n−1 i polynomial f(x) over F. The polynomial f(x)= i=0 aix will be called the representer of the ULS r-circulant matrix A over F. By Definition 2.1 and Equation (1), it is clear that A is a ULS r-circulant matrix over F if and only if A commutes with Θr, that is, AΘr =ΘrA. In addition to the algebraic properties that can be easily derived from the representation (1), we mention that ULS r-circulant matrices have very nice structure. The product of two ULS r-circulant matrices is a ULS r- circulant matrix and A−1 is a ULS r-circulant matrix, too. Furthermore, let F[Θr]={A|A = f(Θr),f(x) ∈ F[x]}. It is a routine to prove that F[Θr]isa commutative ring with the matrix addition and multiplication. Definition 2.2. A sum of the upper and the lower(ULS) r-retrocirculant matrix over a field F, denoted by ULSretrocircr(a0,a1,...,an−1), is meant a square matrix of the form ⎡ ⎤ a0 ... an−2 an−1 ⎢ ⎥ ⎢ ⎥ ⎢ a1 ... an−1 + ra0 ra0 ⎥ ⎢ ⎥ ⎢ a2 ... ra0 + ra1 ra1 ⎥ ⎢ ⎥ ⎢ . ⎥ . ⎢ . ⎥ ⎢ ⎥ ⎣ an−2 ... ran−4 + ran−3 ran−3 ⎦ an−1 + ra0 ... ran−3 + ran−2 ran−2 n×n Lemma 2.3. Let A = ULScircr(a0,a1,...,an−1) be a ULS r-circulant matrix over F and B = ULSretrocircr(an−1,an−2,...,a1,a0) an ULS r-retrocirculant matrix over F. Then BK = A or B = AK, where ⎛ ⎞ 00... 01 ⎜ ⎟ ⎜ 00... 10⎟ K = ⎜ ⎟ (2) ⎝ ... ... ... ... ⎠ 10000 Lemma 2.4[18]Let F[x] be the polynomial ring over F, and let f(x),g(x) ∈ F[x]. f(x)10 Suppose that the polynomial matrix is changed into the g(x)01 d(x) u(x) v(x) polynomial matrix by a series of elementary row opera- 0 s(x) t(x) tions, then (f(x),g(x)) = d(x) and f(x)u(x)+g(x)v(x)=d(x). n ∼ Lemma 2.5. F[x]/x − rx − r = F[Θr]. ProofConsider the following F-algebra homomorphism ϕ : F[x] → F[Θr] 406 Zhiping Tian f(x) → A = f(Θr) for f(x) ∈ F[x]. It is clear that ϕ is an F-algebra epimorphism. So we have ∼ F[x]/kerϕ = F[Θr]. Since F[x] is a principal ideal integral domain, there is a monic polynomial p(x) ∈ F[x] such that kerϕ = p(x). Since xn−rx−r is the minimal polynomial n of Θr, then p(x)=x − rx − r. By Lemma 2.5, we have the following lemma. Lemma 2.6. Let A = ULScircr(a0,a1,...,an−1) be an ULS r-circulant matrix over F. Then A is nonsingular if and only if (f(x),g(x)) = 1, where n−1 i n f(x)= i=0 aix is the representer of A and g(x)=x − rx − r. Proof A is nonsingular if and only if f(x)+xn − rx − r is an invertible element in F[x]/xn − rx − r. By Lemma 2.5, if and only if there exists u(x)+xn − rx − r∈F[x]/xn − rx − r such that f(x)u(x)+xn − rx − r =1+xn − rx − r if and only if there exist u(x),v(x) ∈ F[x] such that f(x)u(x)+(xn − rx − r)v(x) = 1 if and only if (f(x),xn − rx − r)=1. 3 Fast algorithm for solving the inverse prob- lem In this section, Consider the linear system AX = b (3) T T Where X =(x0,x1,...,xn−1) and b =(b0,b1,...,bn−1) are given. The key of the inverse problem is how to find A, for this purpose, we first prove the following results. T T Theorem 3.1. Let X =(x0,x1,...,xn−1) , b =(b0,b1,...,bn−1) , D = ULScircr(xn−1−rx0,xn−2,...,x1,x0) and B = ULScircr(bn−1−rb0,bn−2,...,b1,b0). Then the inverse problem of AX = b has a unique solution in the class of the ULS r-circulant matrices of order n if and only if DY = b has a unique solution. Proof If the inverse problem of AX = b has a unique solution in the class of the ULS r-circulant matrices of order n, then there exists a unique ULS r- circulant matrix A = ULScircr(a0,a1,...,an−1) of order n such that AX = b, i.e. T T ULScircr(a0,a1,...,an−1)(x0,x1,...,xn−1) =(b0,b1,...,bn−1) (4) T Let β =(an−1,...,a1,a0 + ran−1) , we know that β is the unique solution of DY = b, by Equation (4). Conversely, if DY = b has a unique solution T Y =(an−1,...,a1,a0) and let A = ULScircr(a0 − ran−1,a1,...,an−1) and Fast algorithms for solving the inverse problem of AX = b 407 B = ULScircr(bn−1 −rb0,bn−2,...,b1,b0), then DA = B has a unique solution A = D−1B. Since D−1X =(0,...,0, 1)T , then AX = D−1BX = BD−1X = b. So A = D−1B is the unique solution of the inverse problem of AX = b in the class of the ULS r-circulant matrices of order n. Theorem 3.2. The inverse problem of AX = b has a unique solution in the class of the ULS r-circulant matrices of order n if and only if (d(x),g(x)) = 1, n−1 i n where d(x)=(xn−1 − rx0)+ i=1 xn−1−ix ,g(x)=x − rx − r, X and b are given in Theorem 3.1. Proof By Theorem 3.1, we know that the inverse problem of AX = b has a unique solution in the class of the ULS r-circulant matrices of order n if and only if DY = b has a unique solution if and only if D is nonsingular if and only if (d(x),g(x)) = 1 by Lemma 2.6, where D is given in Theorem 3.1.