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Displacement Structure Approach to Cauchy and Cauchy Journal of Computational and Applied Mathematics 138 (2002) 259–272 www.elsevier.com/locate/cam Displacement structure approach to Cauchy and Cauchy–Vandermonde matrices: inversion formulas and fast algorithms Zheng-Hong Yanga;b; ∗, Yong-Jian Hub aDepartment of Mathematics, P.O. Box 71, East Campus, China Agricultural University, 100083 Beijing, China bDepartment of Mathematics, Beijing Normal University, 100875 Beijing, China Received 2 September 2000; received in revised form 22 February 2001 Abstract The con4uent Cauchy and Cauchy–Vandermonde matrices are considered, which were studied earlier by various authors in di5erent ways. In this paper, we use another way called displacement structure approach to deal with matrices of this kind. We showthat the Cauchy and Cauchy–Vandermonde matrices satisfy some special type of matrix equations. This leads quite naturally to the inversion formulas and fast algorithms for matrices of this kind. c 2002 Elsevier Science B.V. All rights reserved. MSC: 15A57; 65D05 Keywords: Cauchy–Vandermonde matrix; Hermite rational interpolation; Displacement structure; Inversion formula; Fast algorithm 1. Introduction and preliminaries Let c =(c1;:::;cn) and d =(d1;:::;dk ) (1.1) be two arrays of nodes with all ci and dj distinct pairwise. Associated with c and d the (simple) Cauchy matrix is deÿned as 1 n;k C(c; d)= ; (1.2) c − d i j i;j=1 Partially Supported by National Science Foundation of China. ∗ Correspondence address: Department of Mathematics, P.O. Box 71, East Campus, China Agricultural University, 100083 Beijing, China. 0377-0427/02/$ - see front matter c 2002 Elsevier Science B.V. All rights reserved. PII: S 0377-0427(01)00378-8 260 Z.-H. Yang, Y.-J. Hu / Journal of Computational and Applied Mathematics 138 (2002) 259–272 and the Cauchy–Vandermonde (CV) matrix is deÿned as Cm(c; d)=[C(c; d) Vm(c)]; (1.3) where m−1 1 c1 ··· c1 1 c ··· cm−1 2 2 V (c)= ∈ Cn×m (1.4) m . . ··· . m−1 1 cn ··· cn is the (simple) Vandermonde matrix corresponding to c. One of the most important properties of Cauchy–Vandermonde matrices is their application to rational interpolation with ÿxed poles. For a given array of interpolation data (ci;Ái); we want to k ÿnd a proper rational function f(x)=q(x)=p(x) (deg q(x) ¡ deg p(x)) with p(x)= j=1(x − dj) such that f(ci)=Ái; (i =1;:::;n): (1.5) k Let f(x) have a partial fraction decomposition of the form f(x)= j=1 j=(x − dj); then, the above problem is equivalent to solving the following system of equations: C(c; d) = Á; (1.6) T T with =(1;:::;k ) and Á =(Á1;:::;Án) . If f(x) is not proper, then the Euclidean algorithm yields k j m−1 f(x)= + k+1 + k+2x + ···+ k+mx x − dj j=1 and the above interpolation problem is equivalent to solving the linear system of equations Cm(c; d) = Á (1.7) T with =(1;:::;k+m) and Á as above. Cauchy and Cauchy–Vandermonde matrices appeared in a sequence of recent papers [9–11,1,3]. In [10,9] MuhlbachK derived the determinant and inverse representations and the Lagrange–Hermite interpolation formula for Cauchy–Vandermonde matrix. In [11] VavÄrÃNn gave the factorization and inversion formulas for Cauchy and CV matrices in another way. The method in [10] is direct computation and seems to be rather complicated. Although in [11] the relations between Cauchy– Vandermonde matrices and rational interpolation problems were pointed out, the algorithm process for interpolants was not given. In the present paper we make use of another method called displacement structure approach to deal with Cauchy and CV matrices. We point out that con4uent Cauchy and CV matrices satisfy some of the special matrix equations and deriving their inverses is equivalent to solving only two linear systems of equations with Cauchy and CV matrices as coePcient matrices, and we also give the fast algorithms for solving such kinds of linear systems. Z.-H. Yang, Y.-J. Hu / Journal of Computational and Applied Mathematics 138 (2002) 259–272 261 The concept of displacement structure was ÿrst introduced by Kailath et al. [5,6] for Toeplitz and Hankel matrices using the Stein-type operator (·) given by (R)=R − FRA: (1.8) For details of displacement structure theory we refer the reader to the survey article of Kailath and Sayed [7]. In general, the generalized displacement operator is deÿned by {;;F;A}(R)=R − FRA; (1.9) where ; ; F and A are speciÿed matrices such that {;;F;A}(R) has lowrank, say r; independent of n. Then R is said to be a structured or a displacement structure matrix with respect to the displacement operator deÿned by (1.9), and r is referred to as the displacement rank of R.A special case of (1.9) will have a more simple form called an equation of Sylvester-type {;A}(R)=R − RA (1.10) which was ÿrst studied by Heinig [4] for Cauchy-like matrices. Indeed, the Cauchy matrix C(c; d) is the unique solution of the following Sylvester-type matrix equation: 1 1 D(c)C(c; d) − C(c; d)D(d)= . [1 1 ··· 1]; (1.11) . 1 n k where D(c) = diag(ci)i=1 and D(d) = diag(dj)j=1 are the diagonal matrices corresponding to c and d; respectively. Eq. (1.11) implies that the Cauchy matrix C(c; d) has displacement rank 1 with respect to the op- −1 erator {D(c);D(d)}(·). When n=k; multiplying Eq. (1.11) by C(c; d) from both left- and right-hand sides we get 1 1 D(d)C(c; d)−1 − C(c; d)−1D(c)=−C(c; d)−1 . [1 1 ··· 1]C(c; d)−1: (1.12) . 1 T T If we denote by u =(u1;:::;un) and v =(v1;:::;vn) the solutions of the following two equations: C(c; d)u =[1 ··· 1]T and vTC(c; d)=[1 ··· 1]; (1.13) then Eq. (1.12) by elementwise comparison shows that u v n C(c; d)−1 = − i j = −D(u)C(d; c)D(v); (1.14) d − c i j i;j=1 262 Z.-H. Yang, Y.-J. Hu / Journal of Computational and Applied Mathematics 138 (2002) 259–272 n n where D(u) = diag(ui)i=1 and D(v) = diag(vj)j=1; which implies that computation of the inverse of the Cauchy matrix C(c; d) is equivalent to solving only two systems of equations and one system in the symmetric case. 2. Conuent Cauchy and Cauchy–Vandermonde matrices: displacement structures and inversion formulas In this section, we shall derive the displacement structures and fast inversion formulas for con4uent Cauchy and CV matrices. Let us begin by recalling the deÿnitions of con4uent Cauchy and Cauchy– Vandermonde matrices. Our deÿnitions are in accordance with those in [11] and di5er slightly from those in [10]. Let c =(c ;:::;c ;c ;:::;c ;:::;c ;:::;c ) (2.1) 1 1 2 2 p p n1 n2 np and d =(d ;:::;d ;d ;:::;d ;:::;d ;:::;d ) (2.2) 1 1 2 2 q q k1 k2 kq p q be two sequences of interpolation nodes with all ci;dj distinct pairwise and i=1 ni =n; j=1 kj =k. With c and d we associate two classes of matrices. Deÿnition 1. We call the n × k block matrix p;q C(c; d)=(Cij)i;j=1 (2.3) con4uent Cauchy matrix corresponding to c and d; where st ni−1;kj−1 ni×kj Cij =(Cij )s=0;t=0 ∈ C and 1 @s+t 1 x=ci s + t (−1)s Cst = = : ij s!t! @xs@yt x − y s (c − d )s+t+1 y=dj i j Deÿnition 2. We call the n × (k + m) block matrix Cm(c; d)=[C(c; d);Vm(c)] (m ¿ 0) (2.4) con4uent Cauchy–Vandermonde matrix corresponding to c and d; where Vm(c1) t ni−1;m−1 V (c)= . ; with V (c )= ct−s ; (2.5) m . m i s i s=0;t=0 Vm(cp) is the con4uent Vandermonde matrix of size n × m corresponding to c. Z.-H. Yang, Y.-J. Hu / Journal of Computational and Applied Mathematics 138 (2002) 259–272 263 When m = 0 we assume that C0(c; d)=C(c; d). As in the simple case, con4uent Cauchy and CV matrices correspond to rational interpolation q kj problems with prescribed possibly repeated poles. Indeed, let p(x)= j=1(x − dj) and k −1 q(x) q j m f(x)= = jt + xj−1: t+1 j p(x) (x − dj) j=1 t=0 j=1 Then the interpolation problem 1 dsf(x) = Á s! dxs is x=ci (i =1;:::;p; s=0;:::;ni − 1) is reduced to solve the following system: Cm(c; d) = Á (2.6) where kj−1 q col(col(jt)t=0 )j=1 ni−1 p = and Á = col[col(Áis) ] : m s=0 i=1 col(j)j=1 Hereafter, col(ai) denotes the column vector with ai as components. 2.1. Displacement structures In this subsection we derive the displacement structures for con4uent Cauchy and CV matrices. Our starting point is the following Lemma. Lemma 3. Suppose that J(#) and J($) are the m × m and n × n lower triangular Jordan blocks with # = $; respectively; and m−1;n−1 % =(%st)s=0;t=0 is any m × n matrix. Then the Sylvester matrix equation J(#)X − XJ($)T = % (2.7) m−1;n−1 has the unique solution X =(Xst)s=0;t=0 given by s t ' + ÿ (−1)' X = % : (2.8) st s−';t−ÿ ' (# − $)'+ÿ+1 '=0 ÿ=0 Note that when # = $, the uniqueness of the solution of Eq.
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