What’s the difference between analytical and numerical approaches to problems?

Analytical approach example:

Find the root of $f(x)=x-5$.

Analytical solution: $f(x)=x-5=0$, add $+5$ to both sides to get the answer $x=5$

Numerical solution:

let's guess $x=1$: $f(1)=1-5=-4$. A negative number. Let's guess $x=6$: $f(6)=6-5=1$. A positive number.

The answer must be between them. Let's try $x=\frac{6+1}{2}$: $f(\frac{7}{2})<0$

So it must be between $\frac{7}{2}$ and $6$...etc.

This is called bisection method.

Numerical solutions are extremely abundant. The main reason is that sometimes we either don't have an analytical approach (try to solve $x^6-4x^5+\sin (x)-e^x+7-\frac{1}{x} =0$) or that the analytical solution is too slow and instead of computing for 15 hours and getting an exact solution, we rather compute for 15 seconds and get a good approximation.

The simplest breakdown would be this:

• Analytical solutions can be obtained exactly with pencil and paper;
• Numerical solutions cannot be obtained exactly in finite time and typically cannot be solved using pencil and paper.

These distinctions, however, can vary. There are increasingly many theorems and equations that can only be solved using a computer; however, the computer doesn't do any approximations, it simply can do more steps than any human can ever hope to do without error. This is the realm of "symbolic computation" and its cousin, "automatic theorem proving." There is substantial debate as to the validity of these solutions -- checking them is difficult, and one cannot always be sure the source code is error-free. Some folks argue that computer-assisted proofs should not be accepted.

Nevertheless, symbolic computing differs from numerical computing. In numerical computing, we specify a problem, and then shove numbers down its throat in a very well-defined, carefully-constructed order. If we are very careful about the way in which we shove numbers down the problem's throat, we can guarantee that the result is only a little bit inaccurate, and usually close enough for whatever purposes we need.

Numerical solutions very rarely can contribute to proofs of new ideas. Analytic solutions are generally considered to be "stronger". The thinking goes that if we can get an analytic solution, it is exact, and then if we need a number at the end of the day, we can just shove numbers into the analytic solution. Therefore, there is always great interest in discovering methods for analytic solutions. However, even if analytic solutions can be found, they might not be able to be computed quickly. As a result, numerical approximation will never go away, and both approaches contribute holistically to the fields of mathematics and quantitative sciences.

Analytical is exact; numerical is approximate.

For example, some differential equations cannot be solved exactly (analytic or closed form solution) and we must rely on numerical techniques to solve them.