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

I don’t have much (good) math education beyond some basic university-level calculus.

What do “analytical” and “numerical” mean? How are they different?


Analytical approach example:

Find the root of f(x)=x5.

Analytical solution: f(x)=x5=0, add +5 to both sides to get the answer x=5

Numerical solution:

let’s guess x=1: f(1)=15=4. A negative number. Let’s guess x=6: f(6)=65=1. A positive number.

The answer must be between them. Let’s try x=6+12: f(72)<0

So it must be between 72 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 x64x5+sin(x)ex+71x=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.

Source : Link , Question Author : jbrennan , Answer Author : J. W. Tanner

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