Algorithm to evaluate a fully-parenthesized expression:


First check if parentheses are balanced
  [throw exception if not]

create two stacks - one to hold numbers, and one to hold operator types

loop through all the tokens in the expression:

    identify the token (find out what type it is)

    if token is a number: push it on the number stack

    if token is an operator: push its type on the operator stack

    if token is a left parenthesis: ignore it because already checked balance

    if token is a right parenthesis (means time to perform a calculation):
      [throw exception if no operator or less than two numbers on the stacks]

        pop a number - last one pushed, so it is on right side of calculation
        pop an operator type - to know what calculation to perform
        pop a second number - for the left side of the calculation

        perform the calculation, and push the result on the number stack

  [end of loop - throw exception if operator stack not empty, or if number stack
   does not have exactly one number left on it]

Done: result is on top of number stack