Bibliography for Graeffe's Method

unabridged

 

  1. On zeros of polynomial and vector solutions of associated polynomial system from Vieta theorem.  
    Wu, Xinyuan
    Appl. Numer. Math.  44  (2003),  no. 3, 415--423, MathSciNet.  
  2. Jacobi-Free and Complex-Free Method for Finding Simultaneously All Zeros of Polynomials Having only Real Zeros
    Wu, Xingyuan; Shao, Rong; Yiran, Z.H.U.  
    Computers and Mathematics with Applications, v 46, n 8-9, October/November, 2003, p 1387-1395, Compendex.  
  3. Tangent Graeffe iteration.    
    Malajovich, Gregorio; Zubelli, Jorge P.    
    Numer. Math. 89 (2001), no. 4, 749--782, MathSciNet.  
  4. Iterative methods for roots of polynomials  
    Mekwi, W.R.
    Masters thesis,  (2001), University of Oxford, U. K.
  5. On the Geometry of Graeffe Iteration
    Malajovich, G.; Zubelli, J. P.
    Journal of Complexity, 2001, vol. 17, no. 3, pp. 541-573, Ingenta.
  6. A generalized Graeffe's iteration for evaluating polynomials and rational functions  
    Gemignani, Luca  
    Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC, 2001, p 143-149, Compendex.
  7. Tangent Graeffe Iteration  
    Malajovich, G. and Zubelli, J. P.
    Cornell University Library, MSRI 1999-047, 27 Aug 1999
  8. Solving Polynomials with Computers
    Victor Y. Pan  
    American Scientist, 86, 62-69, (January-February 1998).
  9. Fast parallel algorithms for Graeffe's root squaring technique
    Jana, P.K.; Sinha, B.P.  
    Computers & Mathematics with Applications, v 35, n 3, Feb, 1998, p 71-80, Compendex.
  10. Solving a Polynomial Equation: Some History and Recent Progress  
    Victor Y. Pan  
    SIAM Review, Vol. 39, No. 2. (Jun., 1997), pp. 187-220, Jstor.
  11. Graeffe's, Chebyshev-like, and Cardinal's processes for splitting a polynomial into factors.    
    Bini, Dario; Pan, Victor Y.    
    Special issue for the Foundations of Computational Mathematics Conference (Rio de Janeiro, 1997). J. Complexity 12 (1996), no. 4, 492--511, MathSciNet.  
  12. Calcul numérique de la mesure de Mahler d'un polynôme par itérations de Graeffe. (French)
    Dégot, Jérôme; Hohl, Jean-Christophe; Jenvrin, Odile
    [Computing the Mahler measure of a polynomial using Graeffe iterations]
    C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 3, 269--272, MathSciNet.  
  13. A universal version of Lobachevski's method for determining the roots of polynomials  
    A.A. Belanov  
    U.S.S.R. Comput. Math. and Math. Phys. 33 (1993) 1661--1663.
  14. A note on the computation of double zeroes of polynomials via Vieta's equations.  
    Hoy, Annegret
    Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe  40  (1991),  no. 3, 143--146, MathSciNet.  
  15. A highly parallel algorithm for root extraction  
    L.H. Jamieson and T.A. Rice  
    IEEE Trans. Comput. 38 (1989) 443--449.
  16. Root squaring using level-index arithmetic  
    C.W. Clenshaw and P.R. Turner  
    Computing 43 (1989) 171--185.
  17. À propos de la méthode de Dandelin-Graeffe. (French)   
    [Zeros of polynomials and the Dandelin-Graeffe method]
    Dedieu, Jean-Pierre    
    C. R. Acad. Sci. Paris Sér. I Math. 309 (1989), no. 20, 1019--1022, MathSciNet.  
  18. A propos de la methods de Dandelin--Graeffe  
    J.-P. Dedieu  
    C.R. Acad. Sci. Paris 309 (1989) 1019--1022.
  19. On the Graeffe-Lobachevskii method for solving algebraic equations. (Romanian)
    Pascu, L.
    Bul. Inst. Politehn. Bucure\c sti Ser. Transport. Aeronave 48 (1986), 11--17, MathSciNet.  
  20. Some consequences of the application of the Lobachevskii-Graeffe method to homogeneous linear systems of degree N. (Romanian)   
    Stoian, Z.
    Bul. Inst. Politehn. Bucure\cedla sti Ser. Mec. 43 (1981), no. 2, 11--14, MathSciNet.  
  21. Numerical solution of algebraic equations with improved Graeffes method. (Japanese)    
    Awaya, Kiyoshi; Kajima, Shizuko    
    Sci. Rep. Res. Inst. Engrg. Kanagawa Univ. No. 3 (1980), 41--49, MathSciNet.  
  22. An Exact Method for Finding the Roots of a Complex Polynomial
    James R. Pinkert       
    ACM Transactions on Mathematical Software (TOMS), Volume 2 ,  Issue 4  (December 1976), pp. 351 - 363.  
  23. Introduction to Numerical Analysis, Section 10-17, Graeffe's Root-Squaring Technique   
    F. B. Hildebrand   
    McGraw-Hill Book Co., Inc., (1974), New York, pp. 602-608.  
  24. Optimale Abschätzungen von Polynomnullstellen und Polynomkoeffizienten mit Anwendungen auf die Verfahren von Graeffe und Nickel. (German)
    Schröder, Gerhard
    Gesellschaft für Mathematik und Datenverarbeitung Bonn, Ber. No. 89. Gesellschaft für Mathematik und Datenverarbeitung, Bonn, 1974. 58 pp., MathSciNet.  
  25. Osservazioni sull'applicazione del metodo di Graeffe all'elaboratore
    Ricca, Giuseppe; Arrigo, Michelangelo
    IBM 1130. II. (Italian) Atti Accad. Peloritana Pericolanti Cl. Sci. Fis. Mat. Natur. 51 (1971), 215--229, MathSciNet.  
  26. Osservazioni sull'applicazione del metodo di Graeffe all'elaboratore 1130 I.B.M. I. (Italian)
    Ricca, G.; Arrigo, M.
    Atti Accad. Peloritana Pericolanti Cl. Sci. Fis. Mat. Natur. 51 (1971), 69--81, MathSciNet.  
  27. Graeffe's method for eigenvalues.
    Pólya, G.
    Numer. Math. 11 1968, 315--319, MathSciNet.  
  28. Algorithm 340: roots of polynomials by a root-squaring and resultant routine  
    Albert Noltemeier     
    Communications of the ACM, Volume 11 ,  Issue 11  (November 1968), pp. 779 - 782.   
  29. zur Programmierung geeignete Modifikation des Graeffe-Verfahrens  
    D. Jurksch, Eine
    Z. Angew. Math. Mech. 46 (1966) 161--166.
  30. The Root-Squaring Method of Dandelin, Lobachevsky, and Graeffe, §54  
    Whittaker, E. T. and Robinson, G.
    In The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 106-112, 1967.
  31. Remark on algorithm 256: modified Graeffe method  
    G. Stern
    Communications of the ACM, Volume 9 ,  Issue 9  (September 1966), p. 687  
  32. Modified Graeffe method (Algorithm 256)  
    A. A. Grau  
    Comm. ACM 8 (1965) 379--380.
  33. Finding zeros of a polynomial by the Q-D algorithm  
    P. Henrici and Bruce O. Watkins
    Communications of the ACM, Volume 8 ,  Issue 9  (September 1965), 570 - 574.  
  34. A First Course in Numerical Analysis, Section 8.10-2, Graeffe's Root-Squaring Method   
    Anthony Ralston    
    McGraw-Hill Book Co., Inc., (1965), New York, pp. 359-364.  
  35. The Complete Root-Squaring Method  
    D. H. Lehmer  
    Journal of the Society for Industrial and Applied Mathematics, Vol. 11, No. 3. (Sep., 1963), pp. 705-717, Jstor.  
  36. On the Reduction of Number Range in the Use of the Graeffe Process   
    A. A. Grau  
    J. Assoc. Comput. Mach. 10 (1963) 538--544.
  37. The accuracy of roots that are obtained in the numerical solution of algebraic equations by the Lobacevskii -Graeffe method. (Ukrainian)
    Kostovsprimekii, O. M.
    Computational Mathematics and Techniques (Ukrainian), 1963 , pp. 6--17 Vidavn. Akad. Nauk Ukraïn. RSR, Kiev, MathSciNet.  
  38. On a modification of the QD-algorithm with Graeffe-type convergence.    
    Rutishauser, Heinz    
    Z. Angew. Math. Phys. 13 1962 493--496.
  39. Generalized transformation formulas in the Lobatchevsky-Graeffe method of determining the roots of entire and holomorphic functions. (Russian)  
    Kostovskiui, A. N.
    Z. Vy cisl. Mat. i Mat. Fiz. 1 1961, 345--349, MathSciNet.  
  40. Resultant Procedure and the Mechanization of the Graeffe Process      
    Bareiss, Erwin H.    
    J. Assoc. Comput. Mach. 7 1960 346--386.
  41. Truncation Error in the Graeffe Root-Squaring Method  
    Weeg, Gerard P.    
    J. Assoc. Comput. Mach. 7  (1960) 69--71.
  42. Dandelin, Lobacevskii, or Graeffe  
    Alston S. Householder  
    The American Mathematical Monthly, Vol. 66, No. 6. (Jun. - Jul., 1959), pp. 464-466, Jstor.  
  43. On modern matrix iteration processes of Bernoulli and Graeffe type.
    Bauer, F. L.
    J. Assoc. Comput. Mach. 5 1958 246--257, MathSciNet.  
  44. Detection and evaluation of a certain type of complex roots by Graeffe's root-squaring method.    
    Ghosh, P. K.    
    Bull. Calcutta Math. Soc. 49 1957 43--46.
  45. Praktische Auswertung von Ausnahmefällen beim Graeffeschen Verfahren  
    A. Hirschleber  
    Z. Angew. Math. Mech. 37 (1957) 257--259.
  46. Ausnamefälle des Graeffeschen Verfahrens  
    A. Hirschleber  
    Z. Angew. Math. Mech. 36 (1956) 254--255.
  47. Sur la methode de Graeffe
    C. Orloff  
    C.R. Acad. Sci. Paris 243 (1956) 1269--1270.
  48. Simplification de la méthode de Graeffe au moyen des spectres mathématiques. (French)
    Orloff, Constantin
    Bull. Soc. Math. Phys. Serbie 8 1956 39--46, MathSciNet.  
  49. Principles of Numerical Analysis, Section 3.1, The Graeffe Process.
    Alston S. Householder  
    McGraw-Hill Book Co., Inc., (1953), New York, pp. 106-114.
  50. Sur une adaptation de la méthode de Graeffe au calcul automatique. (French)
    Ceschino, Francis
    C. R. Acad. Sci. Paris 236, (1953). 1945--1947, MathSciNet.  
  51. Rechenkontrollen und Rechenschemata zum Graeffeschen Verfahren  
    K.H. Bachmann
    Wiss. Z. Tech. Hochsch. Dresden 2 (1952--1953) 327--332.
  52. Notes on the Graeffe Method of Root Squaring (in Mathematical Notes)  
    G. C. Best  
    American Mathematical Monthly, Vol. 56, No. 2. (Feb., 1949), pp. 91-94, Jstor.  
  53. Über die Methode von Graeffe. (German)
    Bodewig, E.
    Z. Angew. Math. Mech. 29, (1949). 91--93, MathSciNet.  
  54. The Determination of the Complex Zeros of A Polynomial  
    G. C. Best
    The American Mathematical Monthly, Vol. 54, No. 5. (May, 1947), pp. 269-273, Jstor.  
  55. The accuracy of the root-squaring method for solving equations  
    P.G. Hoel and D.D. Wall  
    J. Math. and Phys. 26 (1947) 156--164.
  56. The Graeffe Process  
    K. Mitchell  
    Math. Tab. Aids Comp. 2 (1946/47) 57--59.
  57. On Graeffe's method for solving algebraic equations.   
    Bodewig, E.
    Quart. Appl. Math. 4, (1946). 177--190, MathSciNet.  
  58. Observations on a theorem of Rey Pastor on Graeffe's method. (Spanish)   
    Vigil, Luis
    Publ. Inst. Mat. Univ. Nac. Litoral 6, (1946). 191--193, MathSciNet.  
  59. The Graeffe Process as Applied to Power Series  
    Lehmer, D. H.
    Mathematical Tables and Other Aids to Computation, Vol. 1, No. 10. (Apr., 1945), pp. 377-383, Jstor.  
  60. The method of Graeffe for solving algebraic equations. (Spanish)
    Massera, Jose L.
    Bol. Fac. Ingen. Montevideo 3, (1945). (Ano 10), 1--20.
  61. On the transformation of the method of Graeffe. (Spanish)
    Babini, J.
    Publ. Inst. Mat. Univ. Nac. Litoral 5, (1945). 45--49, MathSciNet.  
  62. À propos du mémoire: "Recherches sur la méthode de Graeffe,... etc." par Alexandre Ostrowski, à Bâle. (French)
    San Juan, R.
    Acta Math. 75, (1943). 187--190, MathSciNet.  
  63. Addition à notre mémoire: "Recherches sur la méthode de Graeffe et les zéros des polynômes et des séries de Laurent." (French)
    Ostrowski, Alexandre
    Acta Math. 75, (1943). 183--186, MathSciNet.  
  64. Recherches sur la methode de Graeffe et les zéros des polynômes et des series de Laurent  
    A. Ostrowski  
    Acta Math. 72 (1940) 99--257.
  65. Recherches sur la méthode de Graeffe et les zéros des polynomes et des séries de Laurent.
    Ostrowski, Alexandre
    Acta Math. 72, (1940). 157--257, MathSciNet.  
  66. Squaring the Roots (Graeffe's Method)  §5.8.C
    Kármán, T. Von and Biot, M.
    In Mathematical Methods in Engineering: an Introduction to the Mathematical Treatment of Engineering Problems. New York: Mcgraw-Hill, pp. 194-196, 1940.
  67. On the Graeffe Method of Solution of Equations  
    L. L. Cronvich  
    American Mathematical Monthly, Vol. 46, No. 4. (Apr., 1939), pp. 185-190. Jstor.  
  68. The Dandelin-Gräffe method    
    F. Cajori  
    In A history of Mathematics (New York, 1938), 364.
  69. Graeffe's Method and Complex Roots (in Questions, Discussions, and Notes)  
    B. A. Hausmann  
    American Mathematical Monthly, Vol. 43, No. 4. (Apr., 1936), pp. 225-229. Jstor.  
  70. On Graeffe's Method for the Numerical Solution of Algebraic Equations  
    C. A. Hutchinson  
    American Mathematical Monthly, Vol. 42, No. 3. (Mar., 1935), pp. 149-161, Jstor.  
  71. Compléments à la méthode de Gräffe pour la resolution des équations algébriques  
    R. San Juan  
    Bull. Sci. Math. 59 (1935) 104--109.
  72. Note on solving algebraic equations by root cubing
    A.C. Aitken
    Math. Gaz. 15 (1931) 490--491.
  73. The Cubic and Biquadratic Equations Vieta's Transformation in the Complex Plane  
    Archibald Henderson; A. W. Hobbs  
    The American Mathematical Monthly, Vol. 37, No. 10. (Dec., 1930), pp. 515-521, Jstor.  
  74. On Graeffe's method for complex roots of algebraic equations  
    S. Brodetsky and G. Smeal  
    Proc. Cambridge Philos. Soc. 22 (1924) 83--87.
  75. The Dandelin-Gräffe method  
    C. Runge  
    In Praxis der Gleichungen (Berlin-Leipzig, 1921), 136-158.
  76. Über das Graeffesche Verfahren  
    G. Polya  
    Z. Math. Phys. 63 (1915) 275--290.
  77. Über das Graeffesche Verfahren  
    G. Polya  
    Z. Math. Phys. 63 (1914) 275--290.
  78. Sur la méthode de Graeffe  
    G. Polya  
    C.R. Acad. Sci. Paris 156 (1913) 1145--1147.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2005