*(English)*Zbl 1097.33512

Summary: Numerical calculations of elliptic integrals of the first and second kinds are usually done using algorithms of *R. Bulirsch* [Numer. Math. 7, 353–354 (1965; Zbl 0128.37204)] or *B. C. Carlson* [Numer. Math. 33, No. 1, 1–16 (1979; Zbl 0438.65029)]. These algorithms are based on the descending Landen transformation and the duplication theorem respectively. The algorithms are compared as to the computing time and keeping the prescribed tolerance. The comparison is done in single precision with a given tolerance and leads to a rating concerning the range of the arguments. Bulirsch’s algorithm for calculating the elliptic integral of the second kind,

produces cancellation errors in the range $\{(x,{k}_{c});0<|x|<1,\phantom{\rule{4pt}{0ex}}|{k}_{c}|>1\}$. A numerically stable alternative in this range is presented. The duplication theorem is interpreted both in the sense of Euler’s addition formula for the elliptic integral of the first kind or the Jacobian elliptic function sinus amplitudinis and in the sense of Weierstrass.