I think your question is very broad as it depends on context.
For example, the following branches of mathematics study solving polynomial equations:
Would you include others too? Certainly depending on context.
If you are interested in a book in Numerical Analysis, the book Numerical Analysis by Richard L. Burden and J. Douglas Faires is excellent (wow - look at the price). They even have a web site where there is code in SW like Maple and Mathematica.
Of course, there are also some Dover (great prices), like Introduction to Numerical Analysis: Second Edition by F. B. Hildebrand.
You can also find many things online for free like Equation Solving.
You can also find Open Course-ware at place like MIT where you can find notes and video lectures on the topic.
Lastly, you might look into a Computer Algebra System - which include open source freeware and commercial variants. The offer many routines for this which are optimized or you can program your own (like the routines by Burden and Faires) routines to experiment with methods, their pitfalls, error analyses and the like.