Improved Newton iteration for integral roots

Author:
Richard F. King

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

MSC:
Primary 65.50

MathSciNet review:
0283981

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Abstract | References | Similar Articles | Additional Information

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 √𝑥*, Comm. ACM**10**(1967), 430–432. MR**0240952****[2]**Richard F. King and David L. Phillips,*The logarithmic error and Newton’s method for the square root*, Comm. ACM**12**(1969), 87–88. MR**0285109****[3]**I. F. Ganžela and C. T. Fike,*Sterbenz, P. H*, Math. Comp.**23**(1969), 313–318. MR**0245199**, 10.1090/S0025-5718-1969-0245199-6**[4]**Ichizo Ninomiya,*Generalized rational Chebyshev approximation*, Math. Comp.**24**(1970), 159–169. MR**0261229**, 10.1090/S0025-5718-1970-0261229-8**[5]**Ichizo Ninomiya,*Best rational starting approximations and improved Newton iteration for the square root*, Math. Comp.**24**(1970), 391–404. MR**0273809**, 10.1090/S0025-5718-1970-0273809-4**[6]**David L. Phillips,*Generalized logarithmic error and Newton’s method for the 𝑚th root.*, Math. Comp.**24**(1970), 383–389. MR**0283982**, 10.1090/S0025-5718-1970-0283982-X**[7]**D. G. Moursund and G. D. Taylor,*Optimal starting values for the Newton-Raphson calculation of inverses of certain functions*, SIAM J. Numer. Anal.**5**(1968), 138–150. MR**0225481****[8]**G. D. Taylor,*Optimal starting approximations for Newton’s method*, J. Approximation Theory**3**(1970), 156–163. MR**0263234**

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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