# What is difference between numerical solution and approximation?

Lets say I have a problem, for example, initial or boundary value problem. I can't solve the problem analitically. From the literature it is known some appropriate techniques to solve the problem.

Question. What is difference in terminology between numerical and approximate techniques?

I have confused because numerical techniques include an approximation of the analitical solution.

• See Numerical analysis: "Numerical analysis is the study of algorithms that use numerical approximation (as opposed to general symbolic manipulations) for the problems of mathematical analysis". – Mauro ALLEGRANZA Dec 12 '17 at 9:08
• You can "approximate" also an "analitic" solution: we know that $\pi$ is blah, blah, but in order to compute it we have to stop at a certain point: the result of the computation nis obviously an approximation of $\pi$. – Mauro ALLEGRANZA Dec 12 '17 at 9:10

## 1 Answer

Numerical solutions can be exact (at least up to accuracy of the number representation). For instance, Gaussian elimination yields a fairly exact solution to a linear system of equations (up to ill-posedness), though it belongs to the field of numerical analysis because it describes an algorithmic procedure.

Approximations can be analytical. For instance, the factorial can be approximated by the Stirling formula, which is only asymptotically exact, but is quite useful in symbolic computation.