Computation of Best Simultaneous Approximations with a Singularity

Simultaneous approximation errors are generally discontinuous when the function to be approximated contains a zero in its domain of definition. In this article we indicate how the presence of such a zero (or, equivalently, the resulting singularity in the error expression) affects the computational schemata for finding all the best approximations. In particular, we develop an algorithm and show that its convergence rate is “best possible expected” in the sense that it is quadratic, as in the case for continuous errors. Numerical examples are provided.