Portal de Periódicos da CAPES

Associated Polynomials and Uniform Methods for the Solution of Linear Problems

1966; Society for Industrial and Applied Mathematics; Volume: 8; Issue: 3 Linguagem: Inglês

10.1137/1008061

1095-7200

J. F. Traub,

Iterative Methods for Nonlinear Equations

Next article Associated Polynomials and Uniform Methods for the Solution of Linear ProblemsJ. F. TraubJ. F. Traubhttps://doi.org/10.1137/1008061PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] L. R. Bragg, A matrix approach to numerical integration, Amer. Math. Monthly, 71 (1964), 391–398 MR0185813 0251.65016 CrossrefISIGoogle Scholar[2] F. Brioschi, Intorno ad alcune questioni d'algebra superiore, Ann. di Sci. Matem. Fis., (1854), 301–312 Google Scholar[3] F. Cajori, Horner's method of approximation anticipated by Ruffini, Bull. Amer. Math. Soc., 17 (1911), 409–414 CrossrefGoogle Scholar[4] A. L. Cauchy, Sur la determination des constantes arbitraires renfermées dans la intégrales des équations différentielles linéares, Oeuvres ser. 2, Vol. 7, Académie des Sciences, Paris, Google Scholar[5] A. De Morgan, A Budget of Paradoxes, Vol. II, London, 1872 Google Scholar[6] Google Scholar[7A] Walter Gautschi, On inverses of Vandermonde and confluent Vandermonde matrices, Numer. Math., 4 (1962), 117–123 10.1007/BF01386302 MR0139627 0108.12501 CrossrefGoogle Scholar[7B] Walter Gautschi, On inverses of Vandermonde and confluent Vandermonde matrices. II, Numer. Math., 5 (1963), 425–430 10.1007/BF01385906 MR0164437 0115.34303 CrossrefGoogle Scholar[8] R. W. Hamming, Numerical methods for scientists and engineers, International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., Inc., New York, 1962xvii+411 MR0137279 0952.65500 Google Scholar[9] W. G. Horner, A new method of solving numerical equations of all orders by continuous approximation, Philos. Trans. Roy. Soc. London, (1819), 308–335 CrossrefGoogle Scholar[10] W. G. Horner, On algebraic transformation, The Mathematician, 1 (1845), 108–112 Google Scholar[11] C. Jordan, Calculus of Finite Differences, Chelsea, New York, 1950 0041.05401 Google Scholar[12] A. Klinger, The Vandermonde matrix, Report, P3201, RAND Corporation, Santa Monica, California, 1965 Google Scholar[13] G. Kowalewski, Interpolation und genäherte Quadratur, Teubner, Berlin, 1932 0004.05605 Google Scholar[14] J. L. Lagrange, Sur les suites recurrences, Nouveau Mémoires de l'Academie Royale de Berlin, 6 (1775), 183–195 Google Scholar[15] E. Laguerre, Théorie des équations numériques, J. Math. Pures Appl., 9 (1883), 99–, 3 série Google Scholar[16] J. Legras, Précis d'analyse numérique, Dunod, Paris, 1963x+257 MR0154388 0119.12002 Google Scholar[17] N. Macon and , A. Spitzbart, Inverses of Vandermonde matrices, Amer. Math. Monthly, 65 (1958), 95–100 MR0103357 0081.01503 CrossrefGoogle Scholar[18] M. Martone, La Funzione Alef di Hoëne Wronski, Catanzaro, 1891 Google Scholar[19] E. Netto, Vorlesungen über Algebra, vol. I, Leipzig, 1896 Google Scholar[20] Nikola Obreschkoff, Verteilung und Berechnung der Nullstellen reeller Polynome, VEB Deutscher Verlag der Wissenschaften, Berlin, 1963viii+298 MR0164003 0156.28202 Google Scholar[21] F. W. J. Olver, The evaluation of zeros of high-degree polynomials, Philos. Trans. Roy. Soc. London. Ser. A., 244 (1952), 385–415 MR0049652 0047.36302 CrossrefISIGoogle Scholar[22] A. Ostrowski, On two problems in abstract algebra connected with Horner's ruleStudies in mathematics and mechanics presented to Richard von Mises, Academic Press Inc., New York, 1954, 40–48 MR0066030 0056.35204 Google Scholar[23] F. D. Parker, Mathematical Notes: Inverses of Vandermonde Matrices, Amer. Math. Monthly, 71 (1964), 410–411 MR1532639 CrossrefISIGoogle Scholar[24] E. Schröder, Ueber unendlich viele Algorithmen zur Auflösung der Gleichungen, Math. Ann., 2 (1870), 317–365 MR1509664 CrossrefGoogle Scholar[25] Mirko Stojaković, Solution du problème d'inversion d'une classe importante de matrices, C. R. Acad. Sci. Paris, 246 (1958), 1133–1135 MR0092210 0081.01502 Google Scholar[26] J. F. Traub, Iterative methods for the solution of equations, Prentice-Hall Series in Automatic Computation, Prentice-Hall Inc., Englewood Cliffs, N.J., 1964xviii+310 MR0169356 0121.11204 Google Scholar[27] J. F. Traub, Generalized sequences with applications to the discrete calculus, Math. Comp., 19 (1965), 177–200 MR0179489 0138.37302 CrossrefISIGoogle Scholar[28] J. F. Traub, Solution of linear difference and differential equations, Bull. Amer. Math. Soc., 71 (1965), 538–541 MR0174881 0135.13004 CrossrefISIGoogle Scholar[29] H. Weber, Lehrbuch der Algebra, vol. I, Chelsea, New York, , (English transl. 1898 ed.) Google Scholar Next article FiguresRelatedReferencesCited ByDetails Predictor-Feedback Prescribed-Time Stabilization of LTI Systems With Input DelayIEEE Transactions on Automatic Control, Vol. 67, No. 6 | 1 Jun 2022 Cross Ref On the Optimal Recovery Threshold of Coded Matrix MultiplicationIEEE Transactions on Information Theory, Vol. 66, No. 1 | 1 Jan 2020 Cross Ref Calculation of the transformation kernels for the functional transformation method2017 10th International Workshop on Multidimensional (nD) Systems (nDS) | 1 Sep 2017 Cross Ref The numerical rank of Krylov matricesLinear Algebra and its Applications, Vol. 528 | 1 Sep 2017 Cross Ref Solving some generalized Vandermonde systems and inverse of their associate matrices via new approaches for the Binet formulaApplied Mathematics and Computation, Vol. 290 | 1 Nov 2016 Cross Ref Special Linear SystemsParallelism in Matrix Computations | 26 July 2015 Cross Ref Numerical conditioningWalter Gautschi, Volume 1 | 4 September 2013 Cross Ref On Numerical Aspects of Pseudo-Complex Powers in ℝ3Computational Science and Its Applications – ICCSA 2014 | 1 Jan 2014 Cross Ref Special Types of MatricesHandbook of Linear Algebra, Second Edition | 26 November 2013 Cross Ref Fast Algorithms for Structured Matrix ComputationsHandbook of Linear Algebra, Second Edition | 26 November 2013 Cross Ref Threshold Encryption into Multiple CiphertextsFoundations and Practice of Security | 1 Jan 2012 Cross Ref Signal flow graph approach to inversion of (H,m) -quasiseparable-Vandermonde matrices and new filter structuresLinear Algebra and its Applications, Vol. 432, No. 8 | 1 Apr 2010 Cross Ref A Traub-like Algorithm for Hessenbergquasiseparable- Vandermonde Matrices of Arbitrary OrderNumerical Methods for Structured Matrices and Applications | 1 Jan 2010 Cross Ref Computations with quasiseparable polynomials and matricesTheoretical Computer Science, Vol. 409, No. 2 | 1 Dec 2008 Cross Ref A Björck–Pereyra-type algorithm for Szegö–Vandermonde matrices based on properties of unitary Hessenberg matricesLinear Algebra and its Applications, Vol. 420, No. 2-3 | 1 Jan 2007 Cross Ref Generalized associated polynomials and functions of second kindJournal of Computational and Applied Mathematics, Vol. 178, No. 1-2 | 1 Jun 2005 Cross Ref The accurate inversion of vandermonde matricesComputers & Mathematics with Applications, Vol. 47, No. 6-7 | 1 Mar 2004 Cross Ref A fast parallel Björck–Pereyra-type algorithm for solving Cauchy linear equationsLinear Algebra and its Applications, Vol. 302-303 | 1 Dec 1999 Cross Ref Vandermonde matrices on Chebyshev pointsLinear Algebra and its Applications, Vol. 283, No. 1-3 | 1 Nov 1998 Cross Ref Hermite's formula for vector polynomial interpolation with applications to structured matricesApplicable Analysis, Vol. 70, No. 3-4 | 1 Sep 1998 Cross Ref A Fast Algorithm for Inversion of Confluent Vandermonde-Like Matrices Involving Polynomials that Satisfy a Three-Term Recurrence RelationSIAM Journal on Matrix Analysis and Applications, Vol. 19, No. 3 | 31 July 2006AbstractPDF (113 KB)LANCZOS, GAUSS AND THE PROBLEM OF MOMENTSQuaestiones Mathematicae, Vol. 21, No. 1-2 | 1 Mar 1998 Cross Ref Lagrange's formula for tangential interpolation with application to structured matricesIntegral Equations and Operator Theory, Vol. 30, No. 1 | 1 Mar 1998 Cross Ref Generalized cauchy-vandermonde matricesLinear Algebra and its Applications, Vol. 270, No. 1-3 | 1 Feb 1998 Cross Ref A Generalized Hilbert Matrix Problem and Confluent Chebyshev--Vandermonde SystemsSIAM Journal on Matrix Analysis and Applications, Vol. 19, No. 1 | 31 July 2006AbstractPDF (393 KB)Displacement-structure approach to polynomial Vandermonde and related matricesLinear Algebra and its Applications, Vol. 261, No. 1-3 | 1 Aug 1997 Cross Ref The Fast Generalized Parker–Traub Algorithm for Inversion of Vandermonde and Related MatricesJournal of Complexity, Vol. 13, No. 2 | 1 Jun 1997 Cross Ref Fast Inversion of Vandermonde and Vandermonde-Like MatricesCommunications, Computation, Control, and Signal Processing | 1 Jan 1997 Cross Ref Solution of Vandermonde-Like Systems and Confluent Vandermonde-Like SystemsSIAM Journal on Matrix Analysis and Applications, Vol. 17, No. 1 | 17 February 2012AbstractPDF (1142 KB)Fast Algorithms for Confluent Vandermonde Linear Systems and Generalized Trummer’s ProblemSIAM Journal on Matrix Analysis and Applications, Vol. 16, No. 2 | 17 July 2006AbstractPDF (1750 KB)Polynomial roots from companion matrix eigenvaluesMathematics of Computation, Vol. 64, No. 210 | 1 January 1995 Cross Ref Fast Solution of Confluent Vandermonde Linear SystemsSIAM Journal on Matrix Analysis and Applications, Vol. 15, No. 4 | 17 July 2006AbstractPDF (1055 KB)Overlapping Stokes smoothings: survival of the error function and canonical catastrophe integralsProceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences, Vol. 444, No. 1920 | 1 January 1997 Cross Ref Fast inversion of vandermonde-like matrices involving orthogonal polynomialsBIT, Vol. 33, No. 3 | 1 Sep 1993 Cross Ref Mixed, Componentwise, and Structured Condition NumbersSIAM Journal on Matrix Analysis and Applications, Vol. 14, No. 3 | 17 July 2006AbstractPDF (1519 KB)An inversion formula and fast algorithms for Cauchy-Vandermonde matricesLinear Algebra and its Applications, Vol. 183 | 1 Apr 1993 Cross Ref The structured sensitivity of Vandermonde-like systemsNumerische Mathematik, Vol. 62, No. 1 | 1 Dec 1992 Cross Ref A divide and conquer method for polynomial zerosJournal of Computational and Applied Mathematics, Vol. 30, No. 1 | 1 Apr 1990 Cross Ref Stability Analysis of Algorithms for Solving Confluent Vandermonde-Like SystemsSIAM Journal on Matrix Analysis and Applications, Vol. 11, No. 1 | 17 July 2006AbstractPDF (1843 KB)Error analysis of the Bj�rck-Pereyra algorithms for solving Vandermonde systemsNumerische Mathematik, Vol. 50, No. 5 | 1 Mar 1987 Cross Ref Improved z plane polynomial interpolative analysis of switched-capacitor networksIEEE Transactions on Circuits and Systems, Vol. 31, No. 7 | 1 Jul 1984 Cross Ref Solution of linear equations with rational Toeplitz matricesMathematics of Computation, Vol. 34, No. 149 | 1 January 1980 Cross Ref Computer generation of difference approximationsMathematics and Computers in Simulation, Vol. 21, No. 1 | 1 Mar 1979 Cross Ref A survey of Toeplitz and related matricesInternational Journal of Systems Science, Vol. 9, No. 8 | 27 April 2007 Cross Ref Redundancy techniques and fast algorithms for a special large linear systemComputing, Vol. 18, No. 4 | 1 Dec 1977 Cross Ref Evaluation of the state transition matrix for linear time invariant systemsInternational Journal of Systems Science, Vol. 6, No. 2 | 1 Feb 1975 Cross Ref Symbolic analysis of networks by partitioned polynomial interpolationIEEE Transactions on Circuits and Systems, Vol. 21, No. 3 | 1 May 1974 Cross Ref On the Number of Multiplications for the Evaluation of a Polynomial and Some of Its DerivativesJournal of the ACM, Vol. 21, No. 1 | 1 Jan 1974 Cross Ref Symbolic analysis for computer-aided circuit design-The interpolative approachIEEE Transactions on Circuit Theory, Vol. 20, No. 6 | 1 Nov 1973 Cross Ref Computation of Hermite polynomialsMathematics of Computation, Vol. 27, No. 123 | 1 January 1973 Cross Ref Rapid computation of general interpolation formulas and mechanical quadrature rulesCommunications of the ACM, Vol. 14, No. 12 | 1 Dec 1971 Cross Ref Sensitivity: Old Questions, Some New AnswersIEEE Transactions on Circuit Theory, Vol. 18, No. 1 | 1 Jan 1971 Cross Ref Solution of Vandermonde systems of equationsMathematics of Computation, Vol. 24, No. 112 | 1 January 1970 Cross Ref Volume 8, Issue 3| 1966SIAM Review277-426 History Submitted:10 June 1965Accepted:20 December 1965Published online:18 July 2006 InformationCopyright © 1966 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1008061Article page range:pp. 277-301ISSN (print):0036-1445ISSN (online):1095-7200Publisher:Society for Industrial and Applied Mathematics