Writing a simple equation parser

Question

What sorts of algorithms would be used to do this (as in, this is a string, and I want to find the answer):

((5 + (3 + (7 * 2))) - (8 * 9)) / 72

Answer 1

If the expressions are known to be fully-parenthesized (that is, all possible parentheses are there), then this can easily be done using recursive-descent parsing. Essentially, each expression is either of the form

 number

or of the form

 (expression operator expression)

These two cases can be distinguished by their first token, and so a simple recursive descent suffices. I've actually seen this exact problem given out as a way of testing recursive thinking in introductory programming classes.

If you don't necessarily have this guarantee, then some form of precedence parsing might be a good idea. Many of the other answers to this question discuss various flavors of algorithms for doing this.