The logarithmic error and Newton's method for the square root

ABSTRACT

The problem of obtaining optimal starting values for the calculation of the square root using Newton's method is considered. It has been pointed out elsewhere that if relative error is used as the measure of goodness of fit, optimal results are not obtained when the inital approximation is a best fit. It is shown here that if, instead, the so-called logarithmic error is used, then a best initial fit is optimal for both types of error. Moreover, use of the logarithmic error appears to simplify the problem of determining the optimal initial approximation.

REFERENCES

• 1: David G. Moursund, Optimal starting values for Newton-Raphson calculation of x^{1 2} , Communications of the ACM, v.10 n.7, p.430-432, July 1967 [doi>10.1145/363427.363454]
• 2: MAEHLY, HANS J. Approximations for the Control Data 1604, Ch. 1. Approximations for the square root, Proj: N.R. 044.196, Princeton U., Princeton, N.J., Jan. 1960, pp. 3-11.
• 3: W. J. Cody, Double-precision square root for the CDC-3600, Communications of the ACM, v.7 n.12, p.715-718, Dec. 1964 [doi>10.1145/355588.365122]
• 4: KING, RICHARD. On the double-precision square root routine. Comm. ACM 8, 4 (Apr. 1965), 202. (Letter to the Editor)
• 5: MOURSUND, D. G. Computational aspects of Chebyshev approximation using a generalized weight function. SIAM J. Numer. Anal. 5, 1 (Mar. 1968), 126--137.
• 6: AND TAYLOR, G. D. Optimal starting values for the Newton-Raphson calculation of inverses of certain functions. SIAM J. Numer. Anal. 5,1 (Mar, 1968), 138-150.

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS

Additional Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS

General Terms:
Design, Performance, Reliability, Theory, Verification

Keywords:
Newton's method, best fit, error curve, integer root, logarithmic error, maximal error, optimal approximation, recurrence relation, relative error, square root