i agree with the people who say you should study some numerical methods concerning this ... newton raphson methods, traipezoidal methods are probably the ones mostly used in simulators. read about their deficiencies.
a simple example will be that you use a numerical method eg simpson's 1/3 rule to solve an integration problem, you can get the error to be 5% from the real value (that's when you solve the integration completely without numerical methods). if you use a more accurate method like the traipezoidal rule, your error will reduce to maybe 1% from the actual number. so while simpson's rule might be faster, it might not necessarily be more accurate. it all depends on how accurate you want the result.