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

What do "analytical" and "numerical" mean? How are they different?

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    $\begingroup$ For some reason, I'm irritated that convention has settled on "analytic" instead of "symbolic." "Numerical" usually indicates an approximate solution obtained by methods of numerical analysis. "Analytical" solutions are exact and obtained by methods of symbolic manipulation, derived using analysis. The methods of numerical analysis are themselves derived using (symbolic) analysis. "Analytical" really fails to convey the intended distinction for me, since both approaches seem analytical. $\endgroup$ – Michael E2 Dec 15 '15 at 18:50
  • $\begingroup$ The answers are mostly correct but ... when you do a "numerical solution" you are generally only getting one answer. Whereas analytic/symbolic solutions gives you answers to a whole set of problems. In other words: for every set of parameters the numerical approach has to be recalculated and the analytic approach allows you to have all (well some) solutions are your fingertips. Generically numerical approaches don't give you deep insight but analytic approaches can. Paraphrasing, having a hammer doesn't make everything a nail. $\endgroup$ – rrogers Jun 11 at 21:07
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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.

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  • $\begingroup$ I would consider the first example an algebraic solution. An analytic solution would make use of continuity and sign changes and such to fix a root IMHO. $\endgroup$ – mvw Jun 30 '17 at 22:42
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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.

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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.

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  • $\begingroup$ You write: some differential equations cannot be solved exactly. I think that must be: most differential equations cannot be solved exactly. And we must rely on numerical techniques to solve them. $\endgroup$ – Han de Bruijn Apr 4 at 19:44
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    $\begingroup$ Decent point but “Some” just means an unspecific amount. It doesn’t mean “few” or less than the majority. I actually would be careful with "most" - though surely from a practical, applied perspective this is true. But are the cardinality of the solution sets of closed form vs not different? We’d want to define “closed form” more precisely in this context - but, for example, we know that the set of elementary functions with elementary anti-derivatives are the same as those without so I wouldn’t throw out “most” without a bit more care personally. $\endgroup$ – user115411 Apr 4 at 23:29
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Numerical methods use exact algorithms to present numerical solutions to mathematical problems.

Analytic methods use exact theorems to present formulas that can be used to present numerical solutions to mathematical problems with or without the use of numerical methods.

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Analytical method gives exact solutions, more time consuming and sometimes impossible. Whereas numerical methods give approximate solution with allowable tolerance, less time and possible for most cases

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    $\begingroup$ You have to elaborate on what you mean by "more time" and "less time". Analytical methods can be harder to derive but if derived are typically faster to compute than their computational counterparts. Examples would be solving the heat equation in a homogeneous cylindrical shell. $\endgroup$ – Frenzy Li Aug 28 '16 at 10:24
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Analytical Method

  • When a problem is solved by means of analytical method its solution may be exact.
  • it doesn't follow any algorithm to solve a problem
  • This method provides exact solution to a problem
  • These problems are easy to solve and can be solved with pen and paper
Numerical Method

  • When a problem is solved by mean of numerical method its solution may give an approximate number to a solution
  • It is the subject concerned with the construction, analysis and use of algorithms to solve a probme
  • It provides estimates that are very close to exact solution
  • This method is prone to erro
  • It can't be solved with pen and paper but can be solved via computer tools like FORTRAN or C++
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  • $\begingroup$ Many of your statements are wrong. 1) By definition the solution of a problem is exact. 2) We use an algorithm to compute the exact solution of, say, linear differential equations of 2nd order. 3) This statement contradicts 1). 4) On the contrary, many problems which admit a close form expression are not easy to solve. 7) No, at best the appropriate numerical method can be very accurate. 8) What do you mean? 9) No, Newton's method for computing the square root of 2 can be done by with pen and paper. $\endgroup$ – Carl Christian Oct 6 at 20:46
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The easiest way to understand analytical and numerical approaches is given below: pi=22/7 is the approximate value which is numerical 1/2=0.5 is the exact value means analytic.

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  • $\begingroup$ This is the same as the previous answer. $\endgroup$ – Tengu Oct 22 '17 at 23:39

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