Analogues of the Brent-Salamin algorithm for evaluating \(\pi\)
The Brent-Salamin algorithm is a method to compute the digits of \(\pi\). The algorithm involves the Gauss Arithmetic-Geometric Mean (AGM) iteration and was proved by Brent and Salamin using identities of elliptic integrals. The author gives an alternative proof of the Brent-Salamin algorithm using identities of the Jacobi theta functions, which are known to be related to elliptic integrals. The author's proof is somewhat involved but as a by-product several new AGM iterations for computing \(\pi\) are derived. Also, some comparisons are made with AGM iterations for computing \(\pi\) due to the Borweins.
scientific article; zbMATH DE number 6495133