DOCSLIB.ORG

On Hankel-Like Matrices Generated by Orthogonal Polynomials

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
On Hankel-Like Matrices Generated by Orthogonal Polynomials 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.
Load more

Recommended publications
  • Polynomials and Hankel Matrices
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Polynomials and Hankel Matrices Miroslav Fiedler Czechoslovak Academy of Sciences Institute of Mathematics iitnci 25 115 67 Praha 1, Czechoslovakia Submitted by V. Ptak ABSTRACT Compatibility of a Hankel n X n matrix W and a polynomial f of degree m, m < n, is defined. If m = n, compatibility means that HC’ = CfH where Cf is the companion matrix of f With a suitable generalization of Cr, this theorem is gener- alized to the case that m < n. INTRODUCTION By a Hankel matrix [S] we shall mean a square complex matrix which has, if of order n, the form ( ai +k), i, k = 0,. , n - 1. If H = (~y~+~) is a singular n X n Hankel matrix, the H-polynomial (Pi of H was defined 131 as the greatest common divisor of the determinants of all (r + 1) x (r + 1) submatrices~of the matrix where r is the rank of H. In other words, (Pi is that polynomial for which the nX(n+l)matrix I, 0 0 0 %fb) 0 i 0 0 0 1 LINEAR ALGEBRA AND ITS APPLICATIONS 66:235-248(1985) 235 ‘F’Elsevier Science Publishing Co., Inc., 1985 52 Vanderbilt Ave., New York, NY 10017 0024.3795/85/$3.30 236 MIROSLAV FIEDLER is the Smith normal form [6] of H,. It has also been shown [3] that qN is a (nonzero) polynomial of degree at most T. It is known [4] that to a nonsingular n X n Hankel matrix H = ((Y~+~)a linear pencil of polynomials of degree at most n can be assigned as follows: f(x) = fo + f,x + .
    [Show full text]
  • Inverse Problems for Hankel and Toeplitz Matrices
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Inverse Problems for Hankel and Toeplitz Matrices Georg Heinig Vniversitiit Leipzig Sektion Mathematik Leipzig 7010, Germany Submitted by Peter Lancaster ABSTRACT The problem is considered how to construct a Toeplitz or Hankel matrix A from one or two equations of the form Au = g. The general solution is described explicitly. Special cases are inverse spectral problems for Hankel and Toeplitz matrices and problems from the theory of orthogonal polynomials. INTRODUCTION When we speak of inverse problems we have in mind the following type of problems: Given vectors uj E C”, gj E C”’ (j = l,.. .,T), we ask for an m x n matrix A of a certain matrix class such that Auj = gj (j=l,...,?-). (0.1) In the present paper we deal with inverse problems with the additional condition that A is a Hankel matrix [ si +j] or a Toeplitz matrix [ ci _j]. Inverse problems for Hankel and Toeplitz matices occur, for example, in the theory of orthogonal polynomials when a measure p on the real line or the unit circle is wanted such that given polynomials are orthogonal with respect to this measure. The moment matrix of p is just the solution of a certain inverse problem and is Hankel (in the real line case) or Toeplitz (in the unit circle case); here the gj are unit vectors. LINEAR ALGEBRA AND ITS APPLICATIONS 165:1-23 (1992) 1 0 Elsevier Science Publishing Co., Inc., 1992 655 Avenue of tbe Americas, New York, NY 10010 0024-3795/92/$5.00 2 GEORG HEINIG Inverse problems for Toeplitz matrices were considered for the first time in the paper [lo]of M.
    [Show full text]
  • Hankel Matrix Rank Minimization with Applications to System Identification and Realization ∗
    HANKEL MATRIX RANK MINIMIZATION WITH APPLICATIONS TO SYSTEM IDENTIFICATION AND REALIZATION ∗ y z x { MARYAM FAZEL , TING KEI PONG , DEFENG SUN , AND PAUL TSENG In honor of Professor Paul Tseng, who went missing while on a kayak trip on the Jinsha river, China, on August 13, 2009, for his contributions to the theory and algorithms for large-scale optimization. Abstract. In this paper, we introduce a flexible optimization framework for nuclear norm minimization of matrices with linear structure, including Hankel, Toeplitz and moment structures, and catalog applications from diverse fields under this framework. We discuss various first-order methods for solving the resulting optimization problem, including alternating direction methods, proximal point algorithm and gradient projection methods. We perform computational experiments to compare these methods on system identification problem and system realization problem. For the system identification problem, the gradient projection method (accelerated by Nesterov's extrapola- tion techniques) and the proximal point algorithm usually outperform other first-order methods in terms of CPU time on both real and simulated data, for small and large regularization parameters respectively; while for the system realization problem, the alternating direction method, as applied to a certain primal reformulation, usually outperforms other first-order methods in terms of CPU time. Key words. Rank minimization, nuclear norm, Hankel matrix, first-order method, system identification, system realization 1. Introduction. The matrix rank minimization problem, or minimizing the rank of a matrix subject to convex constraints, has recently attracted much renewed interest. This problem arises in many engineering and statistical modeling application- s, where notions of order, dimensionality, or complexity of a model can be expressed by the rank of an appropriate matrix.
    [Show full text]
  • Exponential Decomposition and Hankel Matrix
    Exponential Decomposition and Hankel Matrix Franklin T. Luk Department of Computer Science and Engineering, Chinese University of Hong Kong, Shatin, NT, Hong Kong. [email protected] Sanzheng Qiao Department of Computing and Software, McMaster University, Hamilton, Ontario L8S 4L7 Canada. [email protected] David Vandevoorde Edison Design Group, Belmont, CA 94002-3112 USA. [email protected] 1 Introduction We consider an important problem in mathematical modeling: Exponential Decomposition. Given a finite sequence of signal values {s1,s2,...,sn}, determine a minimal positive integer r, complex coefficients {c1,c2,...,cr}, and distinct complex knots {z1,z2,...,zr}, so that the following exponential decomposition signal model is satisfied: r − s = c zk 1, for k =1,...,n. (1.1) k X i i i=1 We usually solve the above problem in three steps. First, determine the rank r of the signal sequence. Second, find the knots zi. Third, calculate the coefficients {ci} by solving an r × r Vandermonde system: W (r)c = s(r), (1.2) where 1 1 · · · 1 c1 s1 z1 z2 · · · zr c2 s2 (r) (r) W = . , c = . and s = . . . .. . . r−1 r−1 r−1 z1 z2 · · · zr cr sr A square Vandermonde system (1.2) can be solved efficiently and stably (see [1]). 275 276 Exponential Decomposition and Hankel Matrix There are two well known methods for solving our problem, namely, Prony [8] and Kung [6]. Vandevoorde [9] presented a single matrix pencil based framework that includes [8] and [6] as special cases. By adapting an implicit Lanczos pro- cess in our matrix pencil framework, we present an exponential decomposition algorithm which requires only O(n2) operations and O(n) storage, compared to O(n3) operations and O(n2) storage required by previous methods.
    [Show full text]
  • Analysis of a Fast Hankel Eigenvalue Algorithm
    Analysis of a Fast Hankel Eigenvalue Algorithm a b Franklin T Luk and Sanzheng Qiao a Department of Computer Science Rensselaer Polytechnic Institute Troy New York USA b Department of Computing and Software McMaster University Hamilton Ontario LS L Canada ABSTRACT This pap er analyzes the imp ortant steps of an O n log n algorithm for nding the eigenvalues of a complex Hankel matrix The three key steps are a Lanczostype tridiagonalization algorithm a fast FFTbased Hankel matrixvector pro duct pro cedure and a QR eigenvalue metho d based on complexorthogonal transformations In this pap er we present an error analysis of the three steps as well as results from numerical exp eriments Keywords Hankel matrix eigenvalue decomp osition Lanczos tridiagonalization Hankel matrixvector multipli cation complexorthogonal transformations error analysis INTRODUCTION The eigenvalue decomp osition of a structured matrix has imp ortant applications in signal pro cessing In this pap er nn we consider a complex Hankel matrix H C h h h h n n h h h h B C n n B C B C H B C A h h h h n n n n h h h h n n n n The authors prop osed a fast algorithm for nding the eigenvalues of H The key step is a fast Lanczos tridiago nalization algorithm which employs a fast Hankel matrixvector multiplication based on the Fast Fourier transform FFT Then the algorithm p erforms a QRlike pro cedure using the complexorthogonal transformations in the di agonalization to nd the eigenvalues In this pap er we present an error analysis and discuss
    [Show full text]
  • Paperfolding and Catalan Numbers Roland Bacher
    Paperfolding and Catalan numbers Roland Bacher To cite this version: Roland Bacher. Paperfolding and Catalan numbers. 2004. hal-00002104 HAL Id: hal-00002104 https://hal.archives-ouvertes.fr/hal-00002104 Preprint submitted on 17 Jun 2004 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Paperfolding and Catalan numbers Roland Bacher June 17, 2004 Abstract1: This paper reproves a few results concerning paperfolding sequences using properties of Catalan numbers modulo 2. The guiding prin- ciple can be described by: Paperfolding = Catalan modulo 2 + signs given by 2−automatic sequences. 1 Main results In this paper, a continued fraction is a formal expression a b + 1 = b + a b + a b + . 0 b + a2 0 1 1 2 2 1 b2+... with b0, b1,...,a1, a2,... ∈ C[x] two sequences of complex rational functions. We denote such a continued fraction by a1| a2| ak| b0 + + + . + + . |b1 |b2 |bk and call it convergent if the coefficients of the (formal) Laurent-series ex- pansions of its partial convergents P0 b0 P1 b0b1 + a1 Pk a1| a2| ak| = , = ,..., = b0 + + + . + ,... Q0 1 Q1 b1 Qk |b1 |b2 |bk k C 1 are ultimately constant.
    [Show full text]
  • Matrix Theory
    Matrix Theory Xingzhi Zhan +VEHYEXI7XYHMIW MR1EXLIQEXMGW :SPYQI %QIVMGER1EXLIQEXMGEP7SGMIX] Matrix Theory https://doi.org/10.1090//gsm/147 Matrix Theory Xingzhi Zhan Graduate Studies in Mathematics Volume 147 American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE David Cox (Chair) Daniel S. Freed Rafe Mazzeo Gigliola Staffilani 2010 Mathematics Subject Classification. Primary 15-01, 15A18, 15A21, 15A60, 15A83, 15A99, 15B35, 05B20, 47A63. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-147 Library of Congress Cataloging-in-Publication Data Zhan, Xingzhi, 1965– Matrix theory / Xingzhi Zhan. pages cm — (Graduate studies in mathematics ; volume 147) Includes bibliographical references and index. ISBN 978-0-8218-9491-0 (alk. paper) 1. Matrices. 2. Algebras, Linear. I. Title. QA188.Z43 2013 512.9434—dc23 2013001353 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2013 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government.
    [Show full text]
  • Forward-Backward Hankel Matrix Fitting for Spectral Super-Resolution
    Forward-Backward Hankel Matrix Fitting for Spectral Super-Resolution Zai Yang and Xunmeng Wu School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China [email protected] Abstract—Hankel-based approaches form an important class which eventually results in simple algorithm design [6]–[8]. of methods for line spectral estimation within the recent spec- However, this causes a fundamental limitation of Hankel- tral super-resolution framework. However, they suffer from the based methods, to be specific, the estimated poles do not fundamental limitation that their estimated signal poles do not lie on the unit circle in general, causing difficulties of physical lie on the unit circle in general, resulting in difficulties of interpretation and performance loss. In this paper, we present physical interpretation and potential performance loss (see the a modified Hankel approach called forward-backward Hankel main context). This paper aims at resolving this fundamental matrix fitting (FB-Hankel) that can be implemented by simply problem. modifying the existing algorithms. We show analytically that In this paper, by using the classic forward-backward tech- the new approach has great potential to restrict the estimated poles on the unit circle. Numerical results are provided that nique [9], we propose a modified Hankel-based approach for corroborate our analysis and demonstrate the advantage of FB- line spectral estimation, named as forward-backward Hankel Hankel in improving the estimation accuracy. 1 matrix fitting (FB-Hankel). The core of FB-Hankel is introduc- Index Terms—Forward-backward Hankel matrix fitting, line ing a (conjugated) backward Hankel matrix which is composed spectral estimation, spectral super-resolution, Vandermonde de- of the same spectral modes as and processed jointly with composition, Kronecker’s theorem.
    [Show full text]
  • A Panorama of Positivity
    A PANORAMA OF POSITIVITY ALEXANDER BELTON, DOMINIQUE GUILLOT, APOORVA KHARE, AND MIHAI PUTINAR Abstract. This survey contains a selection of topics unified by the concept of positive semi-definiteness (of matrices or kernels), reflecting natural constraints imposed on discrete data (graphs or networks) or continuous objects (probability or mass distributions). We put empha- sis on entrywise operations which preserve positivity, in a variety of guises. Techniques from harmonic analysis, function theory, operator theory, statistics, combinatorics, and group representations are invoked. Some partially forgotten classical roots in metric geometry and distance transforms are presented with comments and full bibliographical refer- ences. Modern applications to high-dimensional covariance estimation and regularization are included. Contents 1. Introduction 2 2. From metric geometry to matrix positivity 4 2.1. Distance geometry 4 2.2. Spherical distance geometry 6 2.3. Distance transforms 6 2.4. Altering Euclidean distance 8 2.5. Positive definite functions on homogeneous spaces 11 2.6. Connections to harmonic analysis 15 3. Entrywise functions preserving positivity in all dimensions 18 Date: November 13, 2019. 2010 Mathematics Subject Classification. 15-02, 26-02, 15B48, 51F99, 15B05, 05E05, arXiv:1812.05482v3 [math.CA] 12 Nov 2019 44A60, 15A24, 15A15, 15A45, 15A83, 47B35, 05C50, 30E05, 62J10. Key words and phrases. metric geometry, positive semidefinite matrix, Toeplitz ma- trix, Hankel matrix, positive definite function, completely monotone functions, absolutely monotonic functions, entrywise calculus, generalized Vandermonde matrix, Schur polyno- mials, symmetric function identities, totally positive matrices, totally non-negative matri- ces, totally positive completion problem, sample covariance, covariance estimation, hard / soft thresholding, sparsity pattern, critical exponent of a graph, chordal graph, Loewner monotonicity, convexity, and super-additivity.
    [Show full text]
  • Total Positivity of Toeplitz Matrices of Recursive Hypersequences∗
    ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 17 (2019) 125–139 https://doi.org/10.26493/1855-3974.1526.b8d (Also available at http://amc-journal.eu) Total positivity of Toeplitz matrices of recursive hypersequences∗ Tomislav Dosliˇ c´ Faculty of Civil Engineering, University of Zagreb, Zagreb, Croatia Ivica Martinjak Faculty of Science, University of Zagreb, Zagreb, Croatia Riste Skrekovskiˇ FMF, University of Ljubljana, Ljubljana, Slovenia Faculty of Information Studies, Novo Mesto, Slovenia FAMNIT, University of Primorska, Koper, Slovenia Received 9 November 2017, accepted 12 March 2019, published online 6 September 2019 Abstract We present a new class of totally positive Toeplitz matrices composed of recently in- troduced hyperfibonacci numbers of the r-th generation. As a consequence, we obtain that (r) all sequences Fn of hyperfibonacci numbers of r-th generation are log-concave for r ≥ 1 and large enough n. Keywords: Total positivity, totally positive matrix, Toeplitz matrix, Hankel matrix, hyperfibonacci sequence, log-concavity. Math. Subj. Class.: 15B36, 15A45 1 Introduction and preliminary results A matrix M is totally positive if all its minors are positive real numbers. When it is allowed that minors are zero, then M is said to be totally non-negative. Such matrices appears in many areas having numerous applications including, among other topics, graph theory, ∗Partial support of the Croatian Science Foundation through project BioAmpMode (Grant no. 8481) is grate- fully acknowledged by the first author. All authors gratefully acknowledge partial support of Croatian-Slovenian bilateral project “Modeling adsorption on nanostructures: A graph-theoretical approach” BI-HR/16-17-046.
    [Show full text]
  • Positive Definite Hankel Matrices Using Cholesky Factorization
    COMPUTATIONAL METHODS IN APPLIED MATHEMATICS, Vol.9(2009), No.3, pp.221–225 c 2009 Institute of Mathematics of the National Academy of Sciences of Belarus POSITIVE DEFINITE HANKEL MATRICES USING CHOLESKY FACTORIZATION S. AL-HOMIDAN1 AND M. ALSHAHRANI1 Abstract — Real positive definite Hankel matrices arise in many important applica- tions. They have spectral condition numbers which exponentially increase with their orders. We give a structural algorithm for finding positive definite Hankel matrices using the Cholesky factorization, compute it for orders less than or equal to 30, and compare our result with earlier results. 2000 Mathematics Subject Classification: 65F99, 99C25, 65K30. Keywords: Cholesky factorization, Hankel matrices, real positive definite matrices. 1. Introduction An n × n Hankel matrix H is one with constant antidiagonal elements. It has the following structure: h1 h2 h3 ··· ··· hn h h h ··· h h 2 3 4 n n+1 H = . .. .. .. . ··· . h h ··· ··· ··· h − n n+1 2n 1 The general complex-valued elements can be well-conditioned while real positive definite Hankel matrices are known to be very ill-conditioned since their spectral condition numbers increase exponentially with n. The condition number of a positive definite Hankel matrix is bounded below by 3 · 2(n−6) which is very large even for relatively small orders [9]. Becker- mann [6] gave a better bound γn−1/(16n) with γ ≈ 3.210. Here, we try to force the condition number of the constructed matrix to be as small as possible to reach the latter, or at least the former, experimentally. Several other authors have developed algorithms for the factorization or inversion of Hankel matrices [7, 8, 10].
    [Show full text]
  • Hankel Operators Plus Orthogonal Polynomials Yield Combinatorial Identities
    Hankel Operators plus Orthogonal Polynomials Yield Combinatorial Identities E. A. Herman, Grinnell College ∗ Abstract: A Hankel operator H = [hi+j] can be factored as H = MM , where M maps a space of L2 functions to the corresponding moment sequences. Furthermore, a necessary and sufficient condition for a sequence to be in the range of M can be expressed in terms of an expansion in orthogonal polynomials. Combining these two results yields a wealth of combinatorial identities that incorporate both the matrix entries hi+j and the coefficients of the orthogonal polynomials. 2000 Mathematics Subject Classification: 47B35, 33D45, 05A19. Keywords: Hankel operators, Hausdorff moments, orthogonal polynomials, combinatorial identities, Hilbert matrices, Legendre polynomials, Fibonacci numbers. 1. Introduction The context throughout this paper will be the Hilbert spaces l2 of square-summable se- 2 quences and Lα( 1, 1) of square-integrable functions on [ 1, 1] with the distribution dα, α nondecreasing.− Their inner products are the usual dot− product for the former and the usual integral inner product for the latter. The Hankel operators in Section 2 are exactly as in [9]. That is, we study the infinite matrices H = [hi+j]i,j≥0, where 1 j hj = x dα(x), j =0, 1, 2,... Z−1 The following result in [9] will be essential: Widom’s Theorem: The following are equivalent: (a) H represents a bounded operator on l2. (b) h = O(1/j) as j . j →∞ (c) α(1) α(x)= O(1 x) as x 1−, α(x) α( 1) = O(1 + x) as x 1+. − − → − − → − 1 In Theorem 1 we show that H = MM ∗, where M : L2 ( 1, 1) l2 is the bounded operator α − → that maps each function f L2 ( 1, 1) to its sequence of moments: ∈ α − 1 k M(f)k = x f(x) dα(x), k =0, 1, 2,..
    [Show full text]