General algorithm of computation of $c$-table and detection of valleys

Abstract

We present a review of all interesting results concerning the c-table obtained by the authors for the last two decades. These results are not widely known because they were presented in publications of limited circulation. We discuss different computational aspects of software producing the $c$-tables in the presence of blocs and their evolution following the evolution of the computer environment: effects of the use of 32-bit arithmetic .≈8 digits), 64-bit arithmetic (double precision, ≈16 digits), and Bailey’s Fortran multiprecision package .32 or 64 digits), competition between the ascending and descending algorithms, relationship between the complexity of computation and precision, overflow and underflow problems, competition between different formulas allowing one to overcome the blocs in the $c$-table, practical simple criterion of detecting numerical zeros in the c-table allowing to identify the blocs, and automatic detection of valleys.