Several new methods for solving equations

Several new methods for solving one nonlinear equation are developed. Most of the methods are of order three and they require the knowledge of f f′ and f″. The methods will be compared to others in the literature. An extensive bibliography is given.

[1]  A note on King's fifth-order family of methods for solving equations , 1981 .

[2]  Joseph Henry Wegstein,et al.  Accelerating convergence of iterative processes , 1958, CACM.

[3]  P. Jarratt.  Some efficient fourth order multipoint methods for solving equations , 1969 .

[4]  P. Jarratt,et al.  A Rational Iteration Function for Solving Equations , 1966, Comput. J..

[5]  Edmund Taylor Whittaker.  A Formula for the Solution of Algebraic or Transcendental Equations , 1917, Proceedings of the Edinburgh Mathematical Society.

[6]  A. C. Aitken.  XXV.—On Bernoulli's Numerical Solution of Algebraic Equations , 1927 .

[7]  Richard P. Brent,et al.  Optimal iterative processes for root-finding , 1972 .

[8]  P. Wynn.  On a cubically convergent process for determining the zeros of certain functions , 1956 .

[9]  Joseph F. Traub,et al.  On a class of iteration formulas and some historical notes , 1961, CACM.

[10]  Curtis F. Gerald.  Applied numerical analysis , 1970 .

[11]  M. L. Chambers,et al.  Use of double sampling for selecting best population , 1964 .

[12]  J. F. Traub,et al.  On functional iteration and the calculation of roots , 1961, ACM National Meeting.

[13]  J. Traub.  Iterative Methods for the Solution of Equations , 1982 .

[14]  D. B. Popovski,et al.  A family of one-point iteration formulae for finding roots , 1980 .

[15]  H. J. Hamilton,et al.  Roots of equations by functional iteration , 1946 .

[16]  H. T. Kung,et al.  Optimal order and efficiency for iterations with two evaluations , 1976 .

[17]  D. Popovski.  Method of parabolic approximation for solving the equation X=F(X) , 1981 .

[18]  Richard F. King.  A fifth-order family of modified Newton methods , 1971 .

[19]  R. F. King,et al.  An improved pegasus method for root finding , 1973 .

[20]  Richard P. Brent,et al.  An Algorithm with Guaranteed Convergence for Finding a Zero of a Function , 1971, Comput. J..

[21]  W. Werner,et al.  Some improvements of classical iterative methods for the solution of nonlinear equations , 1981 .

[22]  D. Popovski.  A note on neta's family of sixth-order methods for solving equations , 1981 .

[23]  D. J. Hofsommer.  Note on the computation of the zeros of functions satisfying a second order differential equation , 1957 .

[24]  M. G. Cox.  A Bracketing Technique for Computing a Zero of a Function , 1970, Comput. J..

[25]  S. Hitotumatu.  A method of successive approximation based on the expansion of second order. , 1962 .

[26]  D. Hartree.  Notes on iterative processes , 1949, Mathematical Proceedings of the Cambridge Philosophical Society.

[27]  M. Dowell,et al.  The “Pegasus” method for computing the root of an equation , 1972 .

[28]  Ll. G. Chambers.  A quadratic formula for finding the root of an equation , 1971 .

[29]  R. F. King.  A Family of Fourth Order Methods for Nonlinear Equations , 1973 .

[30]  R. F. King.  Tangent methods for nonlinear equations , 1971 .

[31]  I. Kiss,et al.  Über eine Verallgemeinerung des Newtonschen Näherungsverfahrens , 1954 .

[32]  H. J. Hamilton.  A Type of Variation on Newton's Method , 1950 .

[33]  J. H. Wilkinson.  Two algorithms based on successive linear interpolation , 1967 .

[34]  J. Kuntzmann,et al.  Solutions Numeriques des Equations Algebriques , 1963 .

[35]  M. Dowell,et al.  A modified regula falsi method for computing the root of an equation , 1971 .

[36]  Luciano Galeone.  Generalizzazione del metodo di Laguerre , 1977 .

[37]  On Certain Formulae for Numerical Approximation , 1944 .

[38]  H. Fine.  On Newton's Method of Approximation. , 1916, Proceedings of the National Academy of Sciences of the United States of America.

[39]  Beny Neta,et al.  A sixth-order family of methods for nonlinear equations , 1979 .

[40]  J. K. Stewart.  Another Variation of Newton's Method , 1951 .

[41]  Marshall C. Y. Kuo.  Solution of Nonlinear Equations , 1968, IEEE Transactions on Computers.

[42]  Takahiko Murakami.  Some Fifth Order Multipoint Iterative Formulae for Solving Equations , 1978 .

[43]  Alston S. Householder,et al.  Polynomial iterations to roots of algebraic equations , 1951 .

[44]  P. Jarratt,et al.  A Note on the Asymptotic Error Constant of a Certain Method for Solving Equations , 1967, Comput. J..

[45]  P. Jarratt,et al.  Multipoint Iterative Methods for Solving Certain Equations , 1966, Comput. J..

[46]  H. T. Kung,et al.  Optimal Order of One-Point and Multipoint Iteration , 1974, JACM.

[47]  Shinemin Lin.  A Method for Finding Roots of Algebraic Equations , 1943 .

[48]  Un metodo iterativo non stazionario per la risoluzione di equazioni algebriche , 1978 .

[49]  D. E. Muller.  A method for solving algebraic equations using an automatic computer , 1956 .

[50]  Ernst Schröder,et al.  Ueber unendlich viele Algorithmen zur Auflösung der Gleichungen , 1870 .

[51]  Some efficient algorithms for the solution of a single nonlinear equation , 1981 .

[52]  Beny Neta,et al.  A new family of higher order methods for solving equations , 1983 .

[53]  E. Hansen,et al.  A family of root finding methods , 1976 .

[54]  J. F. Steffensen.  Remarks on iteration. , 1933 .

[55]  L. Foster.  Generalizations of Laguerre’s Method: Higher Order Methods , 1981 .

[56]  S. Kulik.  On the solution of algebraic equations , 1959 .

[57]  A. Ostrowski.  Solution of equations and systems of equations , 1967 .

[58]  E. H. Bateman.  The Solution of Algebraic and Transcendental Equations by Iteration , 1953 .

[59]  A Modification of Newton's Method , 1948 .

[60]  A. W. Nourein,et al.  Root determination by use of Padé approximants , 1976 .

[61]  One More Correction Formula , 1955 .

[62]  J. Rissanen,et al.  On optimum root-finding algorithms , 1971 .

[63]  B. Dawson,et al.  On the global convergence of Halley's iteration formula , 1975 .

[64]  D. Popovski.  Sixth order methods for solving equations , 1982 .

[65]  Billy Ruth LeBouf.  A Generalization of Newton's Method , 1948 .

[66]  Werner L. Frank,et al.  Finding Zeros of Arbitrary Functions , 1958, JACM.

[67]  Solution of Equations by Interpolation , 1948 .

[68]  V. A. Varyukhin,et al.  Iterative methods of determining the roots of equations more accurately , 1969 .

[69]  Anny Haegemans,et al.  A comparison of non-linear equation solvers , 1976 .

[70]  Beny Neta,et al.  A higher order method for multiple zeros of nonlinear functions , 1983 .

[71]  A. Goldstein.  On Newton's method , 1965 .

[72]  T. J. Dekker,et al.  Two Efficient Algorithms with Guaranteed Convergence for Finding a Zero of a Function , 1975, TOMS.

[73]  D. L. Waidelich,et al.  The numerical solution of equations , 1941, Electrical Engineering.

[74]  A. C. Aitken.  XII.—Further Numerical Studies in Algebraic Equations and Matrices , 1932 .

[75]  P. Jarratt.  Some fourth order multipoint iterative methods for solving equations , 1966 .

[76]  A. M. Ostrowski,et al.  On Approximation of Equations by Algebraic Equations , 1964 .

[77]  A. Gleyzal,et al.  Solution of non-linear equations , 1959 .

[78]  Donald Nudds,et al.  The Use of Rational Functions in the Iterative Solution of Equations on a Digital Computer , 1965, Computer/law journal.

[79]  Beny Neta,et al.  On a family of multipoint methods for non-linear equations , 1981 .

[80]  Åke Björck,et al.  A new high order method of regula falsi type for computing a root of an equation , 1973 .