Improved Newton iteration for integral roots

Author:
Richard F. King

Journal:
Math. Comp. **25** (1971), 299-304

MSC:
Primary 65.50

DOI:
https://doi.org/10.1090/S0025-5718-1971-0283981-9

MathSciNet review:
0283981

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Abstract: An improved Newton iteration procedure for computing *p*th roots from best Chebyshev or Moursund initial approximations is developed. It differs from the usual Newton method by a multiplicative factor at each step. This multiplier halves the relative error by translating the usual one-sided error curve into a two-sided one, and then adjusting to make a Moursund-like fit. The generalized logarithmic error is used in determining this set of factors.

**[1]**David G. Moursund, "Optimal starting values for Newton-Raphson calculation of ,"*Comm. ACM*, v. 10, 1967, pp. 430-432. MR**39**#2297. MR**0240952 (39:2297)****[2]**Richard F. King & David L. Phillips, "The logarithmic error and Newton's method for the square root,"*Comm. ACM*, v. 12, 1969, pp. 87-88. MR**0285109 (44:2333)****[3]**P. H. Sterbenz & C. T. Fike, "Optimal starting approximations for Newton's method,"*Math. Comp.*, v.**23**, 1969, pp. 313-318. MR**39**#6511. MR**0245199 (39:6511)****[4]**Ichizo Ninomiya, "Generalized rational Chebyshev approximation,"*Math. Comp.*, v. 24, 1970, pp. 159-169. MR**0261229 (41:5845)****[5]**Ichizo Ninomiya, "Best rational starting approximations and improved Newton iteration for the square root,"*Math. Comp.*, v. 24, 1970, pp. 391-404. MR**0273809 (42:8685)****[6]**David L. Phillips, "Generalized logarithmic error and Newton's method for the*m*th root,"*Math. Comp.*, v. 24, 1970, pp. 383-389. MR**0283982 (44:1212)****[7]**D. G. Moursund & G. D. Taylor, "Optimal starting values for the Newton-Raphson calculation of inverses of certain functions,"*SIAM I. Numer. Anal.*, v. 5, 1968, pp. 138-150. MR**37**#1074. MR**0225481 (37:1074)****[8]**G. D. Taylor, "Optimal starting approximations for Newton's method,"*J. Approximation Theory*(To appear.) MR**0263234 (41:7839)**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1971-0283981-9

Keywords:
Newton's method,
integral root,
generalized logarithmic error,
one-sided error,
Moursund approximation,
best rational fit,
Chebyshev-like error,
improvement factors,
convergence rate

Article copyright:
© Copyright 1971
American Mathematical Society