On Hankel-like matrices generated by orthogonal polynomials Georg Heinig Kuwait University, Dept.of Math.&Comp.Sci. P.O.Box 5969, Safat 13060, Kuwait, [email protected] Keywords: Hankel matrix, orthogonal polynomials, fast OP-Hankel matrices appear in least square problems of the algorithm following kind: Given a function f(t) on some interval; find n 1 the coefficients ck such that f(t) k=0− ckek(t) becomes 2 k − 2 k Abstract minimal, where g(t) = g(t) dσ. In this connection OP-Hankel matricesk werek introducedj Pj and studied in [9]. R A class of matrices is studied that could be considered as In [2] OP-Hankel matrices were used for preconditioning Hankel matrices with respect to a system of orthogonal poly- of ill-conditioned Hankel systems. This is based on the re- nomials. It is shown that they have similar properties like markable fact that positve definite OP-Hankel matrices can classical Hankel matrices. Main attention is paid to the con- be well conditioned whereas positive definite Hankel matri- struction of fast and superfast algorithms. The main tool is ces are always ill-conditioned. the fact that these matrices are matrix representation of mul- Finally, in [7] it is shown that symmetric Toeplitz ma- tiplication operators. trices and, more general, centrosymmetric Toeplitz-plus- Hankel matrices are unitarily equivalent to a direct sum of 1 Introduction two special OP-Hankel matrices, namely two Chebyshev- Hankel matrices, which are OP-Hankel matrices for Cheby- shev polynomials. Let ak; bk (k = 0; 1;::: ) be given real numbers, where b0 = 0 and b = 0 for k > 0, and e = 0, e (t) = 1. Then the An inversion algorithm for n n OP-Hankel matrices with k 1 0 × three-term6 recursion − complexity O(n2) was, as far as we know, first presented in [9]. More algorithms with this complexity are contained bj+1ej+1(t) = (t aj)ej(t) bjej 1(t) (1) in [4]. The algorithms in [9] and [4] are Levinson-type al- − − − gorithms, which means that they compute, in principle, a defines a system of polynomials ej(t) LU-factorization of the inverse matrix. The disadvantage of Levinson-type algorithm compared with Schur-type algo- j rithms, in which an LU-factorization of the matrix itself is e (t) = e ti (j = 0; 1;::: ); j ij constructed, is that they cannot fully parallelized and speeded i=0 X up to superfast algorithms. As a rule, Levinson-type algo- where e = 0. We introduce the matrices E = [e ]n rithms are also less stable than Schur-type algorithms. jj 6 n ij i;j=0 with eij = 0 for i > j. In [3] an algorithm is proposed that is based on the fact In this paper we consider (n + 1) (n + 1) matrices that positive definite OP-Hankel matrices can be represented × T that can be represented in the form Rn = En HnEn where as the product of two “OP-Vandermonde” matrices and a di- n H = [ hj+k]j;k=0 is a Hankel matrix. We call these matri- agonal matrix between them. For the inversion of the OP- ces OP-Hankel matrices, where “OP” stand for “orthogonal Vandermonde matrices fast algorithms do exist. But this ap- polynomials”. proach requires first to obtain the Vandermonde representa- Let us mention some instances where OP-Hankel matrices tion, which is a separate problem. appear. The most familiar one seems to be matrices in mod- A Schur-type algorithm and the corresponding super- 2 ified moment problems. In fact, let ej(t) be a sequence fast O(n log n) complexity solver for the special case of f g of orthogonal polynomials on the real line and σ some (not Chebyshev-Hankel matrices is presented in [7]. This also necessarilly positive) measure on R (not related to ej (t) ). leads to a superfast solver for symmetric Toeplitz and, f g Then a matrix with entries more general, centrosymmetric Toeplitz-plus-Hankel matri- ces based on real arithmetics. rij = ei(t)ej(t)dσ (2) In this paper the approach from [7] is generalized to ar- R Z bitrary OP-Hankel matrices. The basic fact of our approach is an OP-Hankel matrix. Some general references for this is that OP-Hankel matrices can be described in 3 different kind of appearance of OP-Hankel matrices are [6], [5], [4]. ways: T T 1. They are matrices of the form Rn = En HnEn. Proposition 2 An OP-Hankel matrix Rn = En HnEn be- longs to n and (3) holds with 2. They are matrices Rn for which the “displacement” H R T T R has rank 2 and a special structure, where g = (Sn anIn+1)en bnen 1: n n n n − − − T is the− tridiagonal matrix defined below in Section 2. n Since the dimension of the space of all (n + 1) (n + 1) × 3. They are the matrices of restricted multiplication oper- Hankel matrices equals 2n + 1, the space of all (n + 1) (n + 1) OP-Hankel matrices is also equal to 2n + 1. Hence× ators with respect to the basis ek(t) . f g the following is true. The last interpretation allows us to derive immediately Levinson- and Schur-type algorithms for LU- factorization of Corollary 1 Any matrix Rn n admits a representa- tion R = ET H E , where H2 His a (n + 1) (n + 1) strongly nonsingular OP-Hankel matrices and their inverses. n n n n n × The combination of the Levinson and the Schur-type algo- Hankel matrix. rithms can be used to speed up the algorithm to complexity We give now a third characterization of OP-Hankel ma- O(n log3 n), using the fast algorithms from [14]. trices. Let Rn[t] denote the space of all polynomials of de- gree less than or equal to n with real coeffcients and n the N P 2 Displacement structure projection defined for polynomials x(t) = k=0 xkek(t) (N n) by ≥ n P Throughout the paper, let denote the Tm (m + 1) (m + 1) x(t) = x e (t): tridiagonal matrix × Pn k k kX=0 Note that tk = 0 if k > 2n. Furthermore, for a given a0 b1 Pn .. polynomial x(t), let [x(t)]k denote its coefficient in its ex- 2 b1 a1 . 3 N Tm = : pansion by ek(t) , i.e. if x(t) = k=0 xkek(t), then .. .. f g 6 . b 7 [x(t)]k = xk. 6 m 7 P 6 b a 7 For a given polynomial p(t) R2n[t], let n(p) denote 6 m m 7 2 R 4 5 the operator in Rn[t] defined by We consider the commutator (or displacement) transforma- tion R = R T T R . Since all eigenvalues of T are n(p)x(t) = np(t)x(t): (5) r n n n − n n n R P simple the kernel of has dimension n+1. Furthermore, Rn For p(t) = t we set n := n(p). can be reproduced fromr R and the first or the last column S R r n of Rn. In fact, the following is easily checked. Proposition 3 n(p) n n n(p) = bn+1 (g(t)[x(t)]n Proposition 1 Let rk denote the (k + 1) th column of Rn R S − S R − [p(t)x(t)] e (t)) : and tk the (k + 1) th column of Rn (k = 0; : : : ; n + 1), n+1 n then r Proof. We have, for x(t) R [t], 2 n 1 rk+1 = (Tnrk ak bkrk 1 + tk); n n(p)x(t) = ntp(t)x(t) nt np(t)x(t) bk+1 − − − S R P − P Q = tp(t)x(t) b [p(t)x(t)] e (t); Pn − n+1 n+1 n 1 rk 1 = (Tnrk ak bk+1rk+1 + tk); where n = I n, and − bk 1 − − Q − P − n(p)x(t) nx(t) = ntp(t)x(t) np(t) ntx(t) where we put here r 1 = rn+1 = 0. R S P − P Q − = tp(t)x(t) Pn − Let n denote the space of all matrices Rn for which b [x(t)] p(t)e (t): H n+1 nPn n+1 Rn has the form r This implies the assertion. T T Rn = ge eng (3) r n − Let R (p) denote the matrix of the operator (p) with n Rn for some g Rn+1. Obviously, we may assume that the last respect to the basis ek(t) . In particular we have Rn(1) = 2 I and R (t) = T .f Furthermore,g component of g is zero. We shall show that n is just the set n n n of all OP-Hankel matrices corresponding toH the data and ak I . For this we mention first that from the fact that the kernel R (f) = [ I 0 ] p(T ) n bk n n N 0 of has dimension n + 1 and g has n degrees of freedom it r follows that for any N > 2n, and the relation in Proposition 2.2 can be dim 2n + 1: (4) written in the form Hn ≤ Next we observe the following. R (p)T T R (p) = g eT e gT ; (6) n n − n n n n − n n where g is the coefficient vector of g(t) with respect to ex- This theorem can be proved by straightforward verifica- pansion of g(t) by e (t) . tion. f k g That means the matrices (p) belong to the class The initial polynomials u (t), l (t) are given by u (t) = 1 Rn Hn 0 0 0 and are, therefore, OP-Hankel matrices.