Lin and Bairstow methods

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A.C. Aitken, Note on the acceleration of Lin's process of iterated penultimate remainder, Quart. J. Mech. Appl. Math. 8 (1955) 251--258.

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A.C. Aitken, Studies in Practical Mathematics VIII. On the iterative methods of Lin and Friedman for factorizing polynomials, Proc. Roy. Soc. Edinburgh 64 (1956) 190--199.

A.C. Aitken, Studies in Practical Mathematics VI. On the factorization of polynomials by Iterative methods, Proc. Roy. Soc. Edinburgh 63 (1951) 174--191.

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D.W. Arthur, The use of interval arithmetic to bound the zeros of real polynomials, J. Inst. Math. Appl. 10 (1972) 231--237.

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L. Bairstow, Investigations relating to the stability of the aeroplane, Reports and Memoranda No. 154 of Advisory Committee for Aeronautics (1914) 51--64.

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L. Berg, On the simultaneous calculation of two zeros, Computing 24 (1980) 87--91.

C.M. Birtwistle and D.J. Evans, On the generalization of Bairstow's method, BIT 7 (1967) 175--190.

A.D. Booth, Numerical Methods (Butterworth, London, 1955).

D.W. Boyd, Nonconvergence in Bairstow's method, SIAM J. Numer. Anal. 14 (1977) 571--574.

K.W. Brodlie, On Bairstow's method for the solution of polynomial equations, Math. Comp. 29 (1975) 816--826.

O. Brudaru, Systolic arrays to solve non-linear equations by a non-iterative method, Analele Stiintifice Universitatea "Al I Cuza", Matematics S 33 (4) (1987) 305--310.

R.A. Buckingham, Numerical Methods (Pitman, London, 1957) 251--304.

B. Carnahan, H.A. Luther and J.O. Wilkes, Applied Numerical Methods (Wiley, New York, 1969) 141--209.

F.M. Carrano, A modified Bairstow method for multiple zeros of a polynomial, Math. Comp. 27 (1973) 781--792.

L. Cerlienco, G. Delogu and F. Piras, The search for quadratic divisors of a polynomial by the method of linear recurrent sequences, Rend. Mat. (7) 1 (1981) 623--631.

F. Ceschino, Sur la résolution déquations par la méthode de Lin, Ann. Soc. Sci. Bruxelles Sér. I 67 (1953) 77--82.

C.F. Chen and M.M. Chen, Performing Lin's method via Routh-type algorithms or Hurwitz-type determinants, Proc. IEEE 68 (1980) 1447--1449.

C.F. Chen and M.H. Lin, A generalization of Lin's method for polynomial factorization, J. Franklin Inst. 326 (1989) 849--860.

C.L. Chen and M.H. Lin, A generalization of Lin's method for polynomial factorization, J. Franklin Inst. 326 (1989) 849--860.

K.J. Cohen, Certification of Algorithm 30, Comm. ACM 5 (1962) 50.

I.E. Durand, Solutions Numérique des Équations Algébriques. Tome I: Équations du Type F(x)=0; Racines d'une Polynôme (Masson, Paris, 1960) 279--281.

R. Dussaud, Sur les singularités relatives à la méthode de Bairstow classique ou généralisée, C.R. Acad. Sci. Paris 260 (1965) 5449--5452.

R. Dussaud, Sur une généralisation de la méthode de Bairstow, C.R. Acad. Sci. Paris 258 (1964) 4907--4909.

J. Dvorcuk, Factorization of a polynomial into quadratic factors by Newton's method, Apl. Mat. 14 (1969) 54--80.

J. Dvorchuk, Factorization of a polynomial into quadratic factors by Newton's method, Apl. Mat. 14 (1969) 54--80.

K.W. Ellenberger, Algorithm 30: Numerical solution of the polynomial equation, Comm. ACM 3 (1960) 643.

K.W. Ellenberger, On programming the numerical solution of polynomial equations, Comm. ACM 3 (1960) 644--647.

T. Fiala and A. Krebsz, On the convergence and divergence of Bairstow's method, Numer. Math. 50 (1987) 477--482.

C.E. Fröberg, Introduction to Numerical Analysis (Addison-Wesley, Reading, MA, 1969).

R.A. Frazer and W.J. Duncan, On the numerical solution of equations with complex roots, Proc. Roy. Soc. London 125 (1929) 68--82.

B. Friedman, Note on approximating complex zeros of a polynomial, Comm. Pure Appl. Math. 2 (1949) 195--208.

T.C. Fry, Some numerical methods for locating roots of polynomials, Quart. Appl. Math. 3 (1945) 89--105.

C.F. Gerald, Applied Numerical Analysis (Addison-Wesley, Reading, MA, 1984) 1--79.

G.H. Golub and T.N. Robertson, A generalized Bairstow algorithm, Comm. ACM 10 (1967) 371--373.

A.A. Grau, A generalization of the Bairstow process, J. Soc. Indust. Appl. Math. 11 (1963) 508--519.

D. Greenspan, On popular methods and extant problems in the solution of polynomial equations, Math. Mag. 31 (1957--1958) 239--253.

R.W. Hamming, Introduction to Applied Numerical Analysis (McGraw-Hill, New York, 1971).

R.W. Hamming, Numerical Methods for Scientists and Engineers (McGraw-Hill, New York, 1962) 356--359.

D.H. Hartree, Numerical Analysis (Oxford, 1952) 84; 201--207.

J.W. Head, Widening the applicability of Lin's iteration process for determining the quadratic factors of polynomials, Quart. J. Mech. Appl. Math. 10 (1957) 122--128.

W. Heitzinger, I. Troch and G. Valentin, Praxis Nichtlinearer Gleichungen (Hanser Verlag, München, 1985).

P. Henrici, Quotient-difference algorithms, in: A. Ralston and H.S. Wilf, Eds., Mathematical Methods for Digital Computers, Vol. II (Wiley, New York, 1967) 35--62.

F.B. Hildebrand, Note on S.N. Lin's method of factoring polynomials, J. Math. and Phys. 32 (1953) 164--170.

F.L. Hitchcock, An improvement on the GCD method for complex roots, J. Math. and Phys. 23 (1944) 69--74.

F.L. Hitchcock, Algebraic equations with complex coefficients, J. Math. and Phys. 18 (3) (1939) 202--210.

F.L. Hitchcock, Finding complex roots of algebraic equations, J. Math. and Phys. 17 (1938) 55--58.

A.S. Householder, The Numerical Treatment of a Single Nonlinear Equation (McGraw-Hill, New York, 1970).

I. Ichim and I. Molnar, A Bairstow's type method for trigonometric polynomials, Numer. Math. 67 (1994) 251--259.

E. Isaacson and H.B. Keller, Analysis of Numerical Methods (Wiley, New York, 1966) 85--133.

L.W. Johnson and R.D. Riess, Numerical Analysis (Addison-Wesley, Reading, MA, 1982) 142--201.

W. Krämer, Einschluss eines Paares konjugiert komplexer Nullstellen eines reellen Polynoms, Z. Angew. Math. Mech. 71 (1991) T820--824.

A. Krebsz, On a generalization of Bairstow's method, in: Miscolc, D. Greenspan, Ed., Colloquia Mathematica Societatis Janos Bolyai Numerical Methods 50 (North-Holland, Amsterdam, 1988) 533---538.

K.S. Kunz, Numerical Analysis (McGraw-Hill, New York, 1957) 1--37.

S.S. Kuo, Computer Applications of Numerical Methods (Addison-Wesley, Reading, MA, 1972).

J.A.N. Lee, Numerical Analysis for Computers (Reinhold, New York, 1966) Chapter 9.

S. Lin, A method of successive approximations of evaluating the real and complex roots of cubic and higher-order equations, J. Math. and Phys. 20 (1941) 231--242.

S.N. Lin, A method for finding roots of algebraic equations, J. Math. and Phys. 22 (2) (1943) 60--77.

G.R. Lindfield and J.E.T. Penny, Microcomputers in Numerical Analysis (Wiley, New York, 1989) Chapter 2.

T.N. Lucas, Finding roots by deflated polynomial approximation, J. Franklin Inst. 327 (1990) 819--830.

Y.L. Luke and D. Ufford, On roots of algebraic equations, J. Math. Phys. 30 (1951) 94--101.

H.A. Luther, A class of iterative techniques for the factorization of polynomials, Comm. ACM 7 (1964) 177--179.

H.A. Luther, An iterative factorization technique for polynomials, Comm. ACM 6 (1963) 108--110.

C. Mack and A. Porter, New methods for the numerical solution of algebraic equations, Philos. Mag. 40 (1949) 578--585.

G. Maess, Simultane Polynomaufspaltung in Quadratfactoren, Rostock. Math. Kolloq. 18 (1981) 89--96.

J.E. Maxfield, An iterative scheme for finding the real zeros of certain polynomials, SIAM Rev. 2 (1960) 148--150.

V.A. McAuley, A method for the real and complex roots of a polynomial, J. Soc. Indust. Appl. Math. 10 (1962) 657 -667.

W.E. Milne, Numerical Calculus (Princeton, 1949).

J. Morris and J.W. Head, A note on Lin's iteration process for the extraction of complex roots of algebraic equations, Quart. J. Mech. Appl. Math. 6 (1953) 391--397.

K.L. Nielsen, Methods in Numerical analysis (McMillan, New York, 1964).

M. Novotny, Remark on algorithm 30: Numerical solution of the polynomial equation, ACM Trans. Math. Software 11 (1985) 183--184.

K. Ohnaka, Y. Isomoto and S. Makinouchi, On a test of programs for numerical zeros of polynomials, Computing 25 (1980) 163--174.

J.-P. Duport and R. Dussaud, Sur une méthode de calcul sur machine de la décomposition en facteurs des polynômes sur les nombres rationnels, C.R. Acad. Sci. Paris A--B 267 (1968) A111--A113.

NATIONAL PHYSICAL LABORATORY, Modern Computing Methods, Her Majesty's Stationary Office, London (1961).

A. Porter and C. Mack, New methods for the numerical solution of algebraic equations, Philos. Mag. Ser. 7 40 (1949) 578--585.

S.B. Presic, Un procédé itératif pour la factorisation des polynômes, C.R. Acad. Sci. Paris 262 (1966) 862--863.

B. Priestley, Factorization of a polynomial with complex conjugate pairs of roots by Bairstow's method, Electronic Engng. 54 (1982) 28--29.

A. Ralston and P. Rabinowitz, A First Course in Numerical Analysis (McGraw-Hill, New York, 2nd ed., 1987) 354.

M.G. Salvadori and M.L. Baron, Numerical Methods in Engineering (Prentice Hall, New York, 1952).

H.E. Salzer, Some extensions of Bairstow's method, Numer. Math. 3 (1961) 120--124.

K. Samelson, Faktorisierung von Polynomen durch funktionale Iteration, Bayer. Akad. Wiss. Math.-Natur. Kl. Abh. 95 (1959) 1--25.

P.A. Samuelson, Iterative computation of complex roots, J. Math. and Phys. 28 (1949) 259--267.

J. Schröder, Factorization of polynomials by generalized Newton procedures, in: B. Dejon and P. Henrici, Eds., Constructive Aspects of the Fundamental Theorem of Algebra (Wiley/Interscience, New York, 1969) 295--320.

G.W. Stewart, Some iterations for factoring a polynomial. II. A generalization of the secant method, Numer. Math. 22 (1973) 33--36.

G.W. Stewart, Some iterations for factoring a polynomial, Numer. Math. 13 (1969) 458--471.

J. Stoer and R. Bulirsch, Introduction to Numerical Analysis (Springer, New York, 1980) 270--299.

I. Tang, Simultaneous determination of quadratic factors by optimization methods, Math. Comput. Simulation 19 (1) (1977) 57--59.

F.F. Timpner, An iterative method for finding the quadratic factors of a fourth-degree polynomial, J. Indust. Math. 6 (1955) 23--26.

C.A. Traenkle, Determination of the root systems of algebraic equations by affinity transforms, Quart. Appl. Math. 13 (1955) 1--18.

J.H. Wilkinson, The evaluation of the zeros of ill-conditioned polynomials. Parts I and II, Numer. Math. 1 (1959) 150--180.

V.L. Zaguskin, Handbook of Numerical Methods for the Solution of Algebraic and Transcendental Equations (Pergamon, Oxford, 1961).

R. Zurmühl, Doppelzeiliges Horner-Schema: Bildung von Ableitung, Z. Angew. Math. Mech. 42 (1962) 359--361.

R. Zurmühl, Zur Arbeit Herbert H. Salzer "Some extensions of Bairstow's method", Numer. Math. 3 (1961) 320.

R. Zurmühl, Zum Graeffe-Verfahren und Horner-Schema bei komplexen Wurzeln, Z. Angew. Math. Mech. 30 (1950) 283--285.