I would like to better my skills in numerical analysis and specifically that part of numerical analysis which deals with approximations of functions of all kind. Since I don't know the nomenclature that well yet I unfortunately cannot formulate my main goal with appropriate terms. But in essence I would like to build up a toolbox which enables me to write excellent approximation algorithms, and to understand their possible limitations.
Concretely I am asking for reference to a book/course which covers approximation algorithms and their error analysis with some major programming language, such as C++. While many such courses do exist, I am asking for the one you would recommend to a friend or a colleague who is just starting to learn the applications of numerical analysis, and would like to have a comprehensive guide for it.