Improved Newton iteration for integral roots

King, Richard F.

doi:10.1090/S0025-5718-1971-0283981-9

Improved Newton iteration for integral roots
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Math. Comp. 25 (1971), 299-304 Request permission

Abstract:

An improved Newton iteration procedure for computing pth 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.

References

David G. Moursund, Optimal starting values for Newton-Raphson calculation of $\surd x$, Comm. ACM 10 (1967), 430–432. MR 0240952, DOI 10.1145/363427.363454
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, DOI 10.1145/362848.362861
I. F. Ganžela and C. T. Fike, Sterbenz, P. H, Math. Comp. 23 (1969), 313–318. MR 245199, DOI 10.1090/S0025-5718-1969-0245199-6
Ichizo Ninomiya, Generalized rational Chebyshev approximation, Math. Comp. 24 (1970), 159–169. MR 261229, DOI 10.1090/S0025-5718-1970-0261229-8
Ichizo Ninomiya, Best rational starting approximations and improved Newton iteration for the square root, Math. Comp. 24 (1970), 391–404. MR 273809, DOI 10.1090/S0025-5718-1970-0273809-4
David L. Phillips, Generalized logarithmic error and Newton’s method for the $m$th root, Math. Comp. 24 (1970), 383–389. MR 283982, DOI 10.1090/S0025-5718-1970-0283982-X
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 225481, DOI 10.1137/0705011
G. D. Taylor, Optimal starting approximations for Newton’s method, J. Approximation Theory 3 (1970), 156–163. MR 263234, DOI 10.1016/0021-9045(70)90024-9