A Class of Numerical Methods for the Computation of Pythagorean Sums

Moler and Morrison have described an iterative algorithm for the computation of the Pythagorean sum (a
²
+ b
²
)
^½
of two real numbers a and b. This algorithm is immune to unwarranted floating-point overflows, has a cubic rate of convergence, and is easily transportable. This paper, which shows that the algorithm is essentially Halley's method applied to the computation of square roots, provides a generalization to any order of convergence. Formulas of orders 2 through 9 are illustrated with numerical examples. The generalization keeps the number of floating-point divisions constant and should be particularly useful for computation in high-precision floating-point arithmetic.