Anderson-Björck for linear sequences

Author:
Richard F. King

Journal:
Math. Comp. **41** (1983), 591-596

MSC:
Primary 65B99

MathSciNet review:
717729

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Abstract: The proposed one-point method for finding the limit of a slowly converging linear sequence features an Anderson-Björck extrapolation step that had previously been applied to the Regula Falsi problem. Convergence is of order 1.839 as compared to for the well-known Aitken-Steffensen -process, and to 1.618 for another one-point extrapolation procedure of King. There are examples for computing a polynomial's mutiple root with Newton's method and for finding a fixed point of a nonlinear function.

**[1]**A. C. Aitken, "On Bernoulli's numerical solution of algebraic equations,"*Proc. Roy. Soc. Edinburgh*, v. 46, 1926, pp. 289-305.**[2]**Ned Anderson and Ȧke Björck,*A new high order method of regula falsi type for computing a root of an equation*, Nordisk Tidskr. Informationsbehandling (BIT)**13**(1973), 253–264. MR**0339474****[3]**H. Esser,*Eine stets quadratisch konvergente Modifikation des Steffensen-Verfahrens*, Computing**14**(1975), no. 4, 367–369. MR**0413468****[4]**A. S. Householder,*The numerical treatment of a single nonlinear equation*, McGraw-Hill Book Co., New York-Düsseldorf-London, 1970. International Series in Pure and Applied Mathematics. MR**0388759****[5]**Richard F. King,*A secant method for multiple roots*, Nordisk Tidskr. Informationsbehandling (BIT)**17**(1977), no. 3, 321–328. MR**0488699****[6]**Richard F. King,*An extrapolation method of order four for linear sequences*, SIAM J. Numer. Anal.**16**(1979), no. 5, 719–725. MR**543964**, 10.1137/0716054**[7]**Richard F. King,*An efficient one-point extrapolation method for linear convergence*, Math. Comp.**35**(1980), no. 152, 1285–1290. MR**583505**, 10.1090/S0025-5718-1980-0583505-8**[8]**David E. Muller,*A method for solving algebraic equations using an automatic computer*, Math. Tables Aids Comput.**10**(1956), 208–215. MR**0083822**, 10.1090/S0025-5718-1956-0083822-0**[9]**A. M. Ostrowski,*Solution of equations and systems of equations*, Second edition. Pure and Applied Mathematics, Vol. 9, Academic Press, New York-London, 1966. MR**0216746****[10]**J. F. Steffensen, "Remarks on iteration,"*Skandinavisk Aktuarietidskrift*, v. 16, 1933, pp. 64-72.**[11]**J. F. Traub,*Iterative methods for the solution of equations*, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR**0169356****[12]**H. Van de Vel,*A method for computing a root of a single nonlinear equation, including its multiplicity*, Computing**14**(1975), no. 1-2, 167–171 (English, with German summary). MR**0403205**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1983-0717729-6

Keywords:
Linear convergence,
extrapolation,
Aitken's -process,
Steffensen,
Anderson-Björck,
efficiency,
nonlinear equation,
order of convergence,
Regula Falsi,
linear sequence

Article copyright:
© Copyright 1983
American Mathematical Society