Structured Matrices, Continued Fractions, and Root Localization of Polynomials

We give a detailed account of various connections between several classes of objects: Hankel, Hurwitz, Toeplitz, Vandermonde, and other structured matrices, Stietjes- and Jacobi-type continued fractions, Cauchy indices, moment problems, total positivity, and root localization of univariate polynomials. Along with a survey of many classical facts, we provide a number of new results.

[1]  Albert Edrei,et al.  On the generating functions of totally positive sequences II , 1952 .

[2]  Julius Borcea,et al.  Hyperbolic polynomials and spectral order , 2003 .

[3]  E. I. Jury.  Remembering Four Stability Theory Pioneers of the Nineteenth Century , 1996 .

[4]  Vladimir Derkach.  On Indefinite Moment Problems and Resolvent Matrices of Hermitian Operators in Kreîn Spaces , 1996 .

[5]  W. Burnside,et al.  Theory of equations , 1886 .

[6]  I. J. Schoenberg.  Zur Abzählung der reellen Wurzeln algebraischer Gleichungen , 1934 .

[7]  F. R. Gantmakher.  The Theory of Matrices , 1984 .

[8]  Yves Genin,et al.  Euclid algorithm, orthogonal polynomials, and generalized Routh-Hurwitz algorithm , 1996 .

[9]  P. Henrici.  Applied and Computational Complex Analysis, Volume 2: Special Functions, Integral Transforms, Asymptotics, Continued Fractions , 1977 .

[10]  I. S. Kats.  SPECTRAL FUNCTIONS OF A STRING , 1983 .

[11]  Stefan Rolewicz,et al.  On a problem of moments , 1968 .

[12]  Joseph W. Rogers.  LOCATIONS OF ROOTS OF POLYNOMIALS , 1983 .

[13]  N. Piskunov,et al.  Differential and integral calculus, vol. II / N. Piskunov; translated from the Russian by George Yankovsky , 1981 .

[14]  J. Grommer,et al.  Ganze transzendente Funktionen mit lauter reellen Nullstellen. , 1914 .

[15]  M. Kreĭn,et al.  The Markov Moment Problem and Extremal Problems , 1977 .

[16]  G. Pólya,et al.  Theory of functions, zeros, polynomials, determinants, number theory, geometry , 1977 .

[17]  F. V. Atkinson,et al.  Discrete and Continuous Boundary Problems , 1964 .

[18]  J. H. B. Kemperman,et al.  A Hurwitz Matrix is Totally Positive , 1982 .

[19]  C. Hermite,et al.  On the number of roots of an algebraic equation contained between given limits , 1977 .

[20]  L. Orlando,et al.  Sul problema di Hurwitz relativo alle parti reali delle radici di un' equazione algebrica , 1911 .

[21]  Vyacheslav Pivovarchik.  Symmetric Hermite-Biehler Polynomials with Defect , 2007 .

[22]  T. A. Brown,et al.  Theory of Equations. , 1950, The Mathematical Gazette.

[23]  T. Stieltjes.  Recherches sur les fractions continues , 1995 .

[24]  Dario Bini,et al.  Numerical Methods for Structured Matrices and Applications , 2010 .

[25]  A. T. Fuller,et al.  Aperiodicity determinants expressed in terms of roots , 1988 .

[26]  Peter Lancaster,et al.  The theory of matrices , 1969 .

[27]  Q. I. Rahman,et al.  Analytic theory of polynomials , 2002 .

[28]  Jr. Bernard A. Asner,et al.  On the Total Nonnegativity of the Hurwitz Matrix , 1970 .

[29]  W. J. Thron,et al.  Continued Fractions: Analytic Theory and Applications , 1984 .

[30]  L. J. Lange.  Continued fraction applications to zero location , 1986 .

[31]  E. I. Jury,et al.  On aperiodicity robustness , 1998 .

[32]  Peter Henrici.  Special functions, integral transforms, asymptotics, continued fractions , 1977 .

[33]  H. Wall,et al.  Analytic Theory of Continued Fractions , 2000 .

[34]  William B. Jones,et al.  Survey of continued fraction methods of solving moment problems and related topics , 1982 .

[35]  I. J. Schoenberg.  On the Zeros of the Generating Functions of Multiply Positive Sequences and Functions , 1955 .

[36]  G. Frobenius,et al.  Ueber das Trägheitsgesetz der quadratischen Formen. , 1895 .

[37]  Biswa Nath Datta.  Application of Hankel matrix to the root location problem , 1976 .

[38]  Thomas C. Craven,et al.  The Fox–Wright functions and Laguerre multiplier sequences , 2006 .

[39]  M. Kreĭn,et al.  The Markov moment problem and extremal problems : ideas and problems of P.L. Čebyšev and A.A. Markov and their further development , 1977 .

[40]  C. Hermite,et al.  Extrait d'une lettre de Mr. Ch. Hermite de Paris à Mr. Borchardt de Berlin sur le nombre des racines d'une équation algébrique comprises entre des limites données. , 1856 .

[41]  L. Hesselholt,et al.  Higher Algebra , 1937, Nature.

[42]  A. Fuller,et al.  On redundance in aperiodicity criteria , 1979 .

[43]  Olga Holtz.  Hermite–Biehler, Routh–Hurwitz, and total positivity , 2003, math/0512591.

[44]  V. N. Sorokin,et al.  Rational Approximations and Orthogonality , 1991 .

[45]  Thomas Kailath,et al.  Fast reliable algorithms for matrices with structure , 1999 .

[46]  John Gonzalez,et al.  Izbrannye trudy , 2011 .

[47]  Peter Borwein,et al.  Location of Zeros , 2002 .

[48]  Heinz H. Bauschke,et al.  Hyperbolic Polynomials and Convex Analysis , 2001, Canadian Journal of Mathematics.

[49]  M. G. Kreĭn,et al.  Some questions in the theory of moments , 1962 .

[50]  Leonid Gurvits,et al.  Hyperbolic polynomials approach to Van der Waerden/Schrijver-Valiant like conjectures: sharper bounds, simpler proofs and algorithmic applications , 2005, STOC '06.

[51]  Alan D. Sokal.  The multivariate Tutte polynomial (alias Potts model) for graphs and matroids , 2005, Surveys in Combinatorics.

[52]  James Renegar,et al.  Hyperbolic Programs, and Their Derivative Relaxations , 2006, Found. Comput. Math..

[53]  Georg Pick,et al.  Über die Beschränkungen analytischer Funktionen, welche durch vorgegebene Funktionswerte bewirkt werden , 1915 .

[54]  J. Borcea,et al.  Applications of stable polynomials to mixed determinants: Johnson's conjectures, unimodality, and symmetrized Fischer products , 2006, math/0607755.

[55]  Harald Woracek,et al.  Shifted Hermite-Biehler Functions and Their Applications , 2007 .

[56]  Albert Edrei,et al.  On the generation function of a doubly infinite, totally positive sequence , 1953 .

[57]  L. Mirsky,et al.  The Theory of Matrices , 1961, The Mathematical Gazette.

[58]  Sergey Khrushchev.  Orthogonal Polynomials and Continued Fractions: From Euler's Point of View , 2008 .

[59]  Petter Brändén,et al.  On linear transformations preserving the Polya frequency property , 2004 .

[60]  Mikhail Tyaglov,et al.  Hurwitz rational functions , 2010, 1010.0784.

[61]  A. Hurwitz.  Ueber die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Theilen besitzt , 1895 .

[62]  Haakon Waadeland,et al.  Continued fractions with applications , 1994 .

[63]  Samuel Karlin,et al.  Oscillation Properties of Generalized Characteristic Polynomials for Totally Positive and Positive Definite Matrices , 1974 .

[64]  Steve Fisk.  Polynomials, roots, and interlacing , 2006 .

[65]  Harry Dym,et al.  Contributions of Issai Schur to Analysis , 2007, 0706.1868.

[66]  Eugene P. Wigner,et al.  On a Class of Analytic Functions from the Quantum Theory of Collisions , 1951 .

[67]  Lu Yang,et al.  Recent advances in automated theorem proving on inequalities , 1999, Journal of Computer Science and Technology.

[68]  M. Naimark,et al.  The method of symmetric and Hermitian forms in the theory of the separation of the roots of algebraic equations , 1981 .

[69]  A. Scott,et al.  The Repulsive Lattice Gas, the Independent-Set Polynomial, and the Lovász Local Lemma , 2003, cond-mat/0309352.

[70]  V. Olshevsky.  Structured Matrices in Mathematics, Computer Science, and Engineering II , 2001 .

[71]  Henry C. Thacher,et al.  Applied and Computational Complex Analysis. , 1988 .

[72]  G. L..  Collected Papers , 1912, Nature.

[73]  Stephen Barnett.  Comments on "Application of Hankel matrix to the root location problem" , 1976 .

[74]  A. Fuller,et al.  Root location criteria for quartic equations , 1981 .

[75]  I. J. Schoenberg.  A note on multiply positive sequences and the Descartes rule of signs , 1955 .

[76]  A. T. Fuller,et al.  On aperiodicity theorems , 2001 .

[77]  O. Perron,et al.  Die Lehre von den Kettenbrüchen , 2013 .

[78]  Olga Holtz.  The inverse eigenvalue problem for symmetric anti-bidiagonal matrices , 2005 .

[79]  Barry Simon,et al.  m-Functions and inverse spectral analysis for finite and semi-infinite Jacobi matrices , 1997 .

[80]  I. J. Schoenberg,et al.  On the generating functions of totally positive sequences I , 1951, Proceedings of the National Academy of Sciences of the United States of America.

[81]  D. Wagner,et al.  Negatively Correlated Random Variables and Mason’s Conjecture for Independent Sets in Matroids , 2006, math/0602648.

[82]  Osman Güler,et al.  Hyperbolic Polynomials and Interior Point Methods for Convex Programming , 1997, Math. Oper. Res..

[83]  Albert Edrei,et al.  Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence , 1953, Canadian Journal of Mathematics.

[84]  F. Gantmacher,et al.  Oscillation matrices and kernels and small vibrations of mechanical systems , 1961 .

[85]  G. Pólya,et al.  Problems and theorems in analysis , 1983 .

[86]  Nikolaj Tschebotareff,et al.  Über die Realität von Nullstellen ganzer transzendenter Funktionen , 1928 .

[87]  Leonid Gurvits.  The Van der Waerden conjecture for mixed discriminants , 2004 .