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 , 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 introduced| P| 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 { } 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, { } 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− 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) . { } 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 H∈ 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-  b1 a1 .  N Tm = . pansion by ek(t) , i.e. if x(t) = k=0 xkek(t), then .. .. { }  . . b  [x(t)]k = xk.  m  P  b a  For a given polynomial p(t) R2n[t], let n(p) denote  m m  ∈ R   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) ∇ 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 from∇ R and the first or the last column S 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 ∇ Proof. We have, for x(t) R [t], ∈ 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 ∇ This implies the assertion. T T Rn = ge eng (3) ∇ 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) = ∈ I and R (t) = T .{ Furthermore,} 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 ” • ∇ 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. { k } 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. Since the mapping and l0(t) = p(t). p(t) R (p) is one-to-one for p(t) R [t], the dimen- The recursions can easily be translated into vector lan- −→ n ∈ 2n sion of the space of matrices Rn(p) equals 2n+1. This leads guage using the fact that multiplying a polynomial in an ex- to the main result of this section. pansion by ek(t) is equivalent to the multiplication by T2n of the corresponding{ } coeffcient vector. Theorem 1 For an (n + 1) (n + 1) matrix R, the fol- × The algorithm emerging from the theorem is a hybrid lowing are equivalent: Levinson-Schur-type algorithm. It is in particular convenient for parallel computation and has O(n) complexity if n pro- 1. The matrix R is of the form R = ET H E for some n n n n n cessors are available. Hankel matrix H. It is possible to calculate only the columns of the upper n+1 factor U , and the quantities l for 0 j i 1 as some 2. The commutator Rn satisfies (3) for some g R . n ij ≤ − ≤ ∇ ∈ inner products of rows of Rn and the uk. This leads to a 3. For some polynomial p(t) R2n[t], Rn = Rn(p). Levinson-type algorithm. It is also possible to calculate only ∈ the lower factor L, which is results in a pure Schur-type al- T If Rn is given in the form Rn = En HEn with H = gorithm. In this case the solution of a system Rnx = b will n [hi+j]i,j=0, then the coefficient vector p of p(t) with re- be obtained by backward substitution. T spect to the basis ek(t) is given by p = E2ns, where h = [h ]2n . If R{ is} given by (2) then the coefficients k k=0 n 4 Fast polynomial multiplication of p(t) are the numbers ri0 (i = 0,..., 2n). In order to speed up the algorithm described in the previous 3 Algorithms for LU-factorization section we need an algorithm for fast multiplication of poly- nomials in OP-expansions, i.e. we assume that polynomials In this and the next sections we consider only strongly non- x(t) and y(t) are given in the form n singular OP-Hankel matrices Rn = Rn(p) = [ rij ]i,j=0. That means we assume that the principal submatrices n m k x(t) = xkek(t) and y(t) = ykek(t). [ rij ]i,j=0 are nonsingular for k = 0, . . . , n. This covers, k=0 k=0 in particular, the case when Rn is positive definite. X X We seek fast algorithms for the LU-factorization of R n We want to compute the expansion of x(t)y(t) by ek(t) . and its inverse. More precisely, we are looking for an up- For this we can use the approximative algorithms from{ [1],} n per Un = [ uij ]i,j=0 and a lower triangular but we can also use the exact algorithms described in [14]. In n matrix Ln = [ lij ]i,j=0 satisfying the latter paper (2N + 1) (N + 1) matrices of the form × RnUn = Ln and uii = 1 (i = 0, . . . , n). (1) 2N,N VN = [ek(cj)]j=0, k=0

In polynomial language this can be written in the form jπ with cj = cos N are considered and algorithms are pre- T sented that multiply a vector by VN or by V with complex- p(t)uk(t) = lk(t), (2) N ity O(N log2 N) and resonable accuracy. where We need the following property, which is mentioned in [14]. k 2n uk(t) = uikei(t), lk(t) = likei(t). Proposition 4 Let w be the first column of the inverse of i=0 i=k 2N X X the matrix VN = [ ek(cj)]j,k=0, and let Dw be the 2N . Then Theorem 2 The columns of U and L in (1) can be com- Dw = diag [wj]j=0 n n e puted via the recursion T VN DwVN = IN+1. bk+1uk+1(t) = (t αk)uk(t) βkuk 1(t) − − − Once the weight vector w is precomputed, it is clear how to multiply two polynomials. First we chose N > 2(m + n) bk+1lk+1(t) = (t αk)lk(t) βklk 1(t), − − − and multiply the matrix VN by coefficient vectors of x(t) where k = 0, . . . , n 1, and y(t), which means that the values of x(t) and y(t) at c − j are computed. Then the computed values are multiplied by bklkk bklkklk 1,k bk+1lk,k+1lk 1,k 1 T βk = , αk = − − − − . each other and by wj, and the VN is applied to obtain the lk 1,k 1 lkklk 1,k 1 coefficient vector of the product in the expansion by e (t) . − − − − { k } 5 Superfast algorithm 6 Other approaches

We show now how an algorithm with complexity Let us briefly mention some other approaches to solve linear O(n log3 n) can be constructed. We introduce 2 2 matrix systems with a OP-Hankel coefficient matrix. The first one is polynomials × described in [11]. It is based on displacement structure and Schur complements and applicable to matrices R for which uk(t) uk 1(t) the rank of , where and are tridiagonal Uk(t) = − , T1R RT2 T1 T2 lk(t) lk 1(t) − ” − • matrices, is small compared with the size of the matrix R. This approach, however, does not fully use the specifics of 1 t αk bk+1 Θk(t) = − . OP-Hankel matrices. b βk 0 k+1 ” − • The second approach is based on transformation into Then the relation in Theorem 5.1 can be written in the form Cauchy-like matrices (see [8]) or into a tangential interpola- tion problem (as in [7] for Chebyshev-Hankel matrices) and Uk+1(t) = Uk(t)Θk(t). (3) the application of the algorithm described in [13]. For this the eigenvalues and eigenvectors of the matrix have to be We define, for j > k T precomputed.

Θkj (t) = Θk(t)Θk+1(t) ... Θj 1(t). Finally, a convenient basis change transforms a general − OP-Hankel matrix into a Chebyshev-Hankel matrix. The Then, for j > i > k, basis change can be carried out with the help of the algo- rithms described in [14] with O(n log2 n) complexity. For Θkj(t) = Θki(t)Θij(t),Uj(t) = Uk(t)Θkj(t). (4) the resulting Chebyshev-Hankel system one could use the 2 3 O(n log n) complexity algorithm described in [7]. This In order to achieve complexity O(n log n) it is important 2 to carry out the calculations not with the complete polynomi- leads to a O(n log n) complexity algorithm for general OP- als l (t) but only with the relevant part of them. We define Hankel systems. However, it possibly that the change of the k basis transforms increases the condition number of the matrix j essentially so that the numerical application of this approach j lk(t) = lkiei(t). might be restricted. But this has to be checked by numerical i=k examples and further considerations. X It is easily checked that Θki(t) can be computed only from i i lk 1(t) and lk(t). Furthermore, the following is true for k < References i <− j: [1] A. Dutt, V. Rokhlin, Fast Fourier transforms for noneq- j j j j uispaced data, SIAM J. Sci. Comp., 14 (1993), 1368- li (t) li 1(t) = j+1 lk(t) lk 1(t) Θki(t). (5) − P − 1393. h i h i This leads to the following recursive procedure. j j [2] D. Fasino, Preconditioning finite moment problems, J. Input: [ lk(t) lk 1(t)] , Output: Θkj (t) − Comp. Appl. Math., 65 (1995), 145–155. 1. If j = k + 1 then apply Theorem 5.1. [3] H.J. Fischer, Fast solution of linear systems with 2. Otherwise choose i with k < i < j and carry out the generalized Hankel matrices, Preprint, TU Chemnitz, following steps: November 1997.

i i (a) Apply the Procedure for [ lk(t) lk 1(t)]. The [4] L. Gemignani, A fast algorithm for generalized Han- − output is Θki(t). kel matrices arising in finite-moment problems, Linear j j Algebra Appl. 267 (1997), 41-52. (b) Compute [ li (t) li 1(t)] by (5). − j j [5] G. Golub, M. Gutknecht, Modified moments for indefi- (c) Apply the Procedure for [ li (t) li 1(t)]. The out- − put is Θij (t). nite weight functions, Numer. Math. 67 (1994), 71-92. (d) Compute Θkj (t) = Θki(t)Θij (t) using the algo- [6] S.A. Gustafson, On computational applications of the rithms from [14]. theory of moment problems, Rocky Mountain J.Math., It is convenient to choose i close to the average of j and 2 (1974), 227-240. k. Proceeding in this way the problem to compute Θkj(t) [7] G. Heinig, Chebyshev-Hankel matrices and the is reduced to two subproblems of about half the size plus 2 splitting approach for centrosymmetric Toeplitz-plus- O((j k) log (j k)) operations for polynomial multipli- Hankel matrices, Appl. (to appear) cation.− This ends up− with complexity O((j k) log3(j k)). − 3 − In particular, Un(t) can be computed with O(n log n) oper- [8] G. Heinig, A. Bojanczyk, Transformation tech- ations. niques for Toeplitz and Toeplitz-plus-Hankel matrices, I. Transformations, Linear Algebra Appl. 254 (1997), 193-226, II. Algorithms, Linear Algebra Appl. 278 (1998), 11-36.

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