Series of logical steps that can be followed and verified as correct (proof of theorem etc).  Example: If you use the quadratic formula to solve for x in a quadratic equation, this is an analytical solution to the problem.

Problems which are solved via other means.  If, instead of using the quadratic formula, you try a lot of values for x and y, this is a numerical solution.

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

Series of logical steps that can be followed and verified as correct (proof of theorem etc).  Example: If you use the quadratic formula to solve for x in a quadratic equation, this is an analytical solution to the problem.

Problems which are solved via other means.  If, instead of using the quadratic formula, you try a lot of values for x and y, this is a numerical solution.

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

I've struggled with this question myself. In one sense, almost all algorithms are numerical when calculated values (intermediate or otherwise) are represented by floating point numbers in a finite precision machine and therefore subject to round off error. By this very restrictive definition, there are relatively few "analytic" (i.e. exact computation) algorithms. There are some though. For example, the work by Morris Newman concerning the exact solution to integral linear systems. See http://dlmf.nist.gov/about/bio/MNewman.

However, I believe this definition to be too restrictive and not useful. It's probably better to distinguish between exact and non-exact algorithms without using the terms "analytic" and "numerical".

So what are analytic algorithms? I don't really know, but I think they should probably include formulas that don't involve approximations in a theoretical sense. For example, I believe that simple expressions involving trigonometric functions or even special functions should be considered analytic, even though the evaluation of such functions likely requires an algorithm that is subject to round off error. Also, expressions that involve (convergent) infinite sums might be considered analytic, even though there will most likely be truncation error when evaluated by a machine.

On the other hand, recursive algorithms are often easily expressed and in the limit don't involve theoretical approximations. However, I think most people would regard an iterative or recursive algorithm as a numerical method. So where does that leave us?

Like certain other ideas in mathematics, an unambiguous and simultaneously useful definition of an analytic method is not easily had. It's like distinguishing between an applied and theoretical mathematician, or between different branches of mathematics. Such distinctions are to some extent in the eye of the beholder.

Dear Sarah,

I posted an answer to such a similar question today. As you have said, the list of methods you indicated are numerical or quasi analytic methods , and they are different from analytic methods, where solutions or objects are exact and in a closed or quadrature forms. Approximate methods give, with a certain error of approximation, approximate value to an exact value or object that is either difficult to use for practical purposes or may not be known exactly. The particular method you mentioned, Adomian method is actually a semi - analytic or simply numerical methods, which uses fast convergent series shown in the article I have attached for you. It may offer a help to your question.

Regards

Dear Sarah,

some analytical methods like Picard or Taylor expansion are useful for numerical solutions of equations/problems, but they are useful for proof of Theorems (e.g. existence of solution for y' = f(x, y) ode) and in some cases they can give the exact solution, not only numerical one.

Some numerical methods like Newton iteration give approximate solutions (which can be exact in some special cases) but can be useful for theoretical concept.

Therefore for me the difference from analytical and numerical is not about the nature of the method itself, but is about the use done and the nature of solutions given by it.

Gianluca

I agree with the comment of Gianluca Argentini. Regards

See this link

https://www.quora.com/What-is-the-difference-between-a-numerical-and-an-analytical-solution

Please refer to similar question within RG

https://www.researchgate.net/post/What_is_the_difference_between_analytical_semi-analytical_mathematical_numerical_and_theoretical_models?_ec=topicPostOverviewFollowedQuestions

I am still saying what I have expressed there. In particular

"The range of meaning is a language problem, and only a custom of usage can

be exposed according to personal feelings."

" Among mathematical models those deserving the adjective analytical are

certainly those which involve functions and operations on them for describing

the evolution and/or distribution of quantities asigned to real amounts, duration,

probabilities, etc. And here many different preferences are given to this notion.

For some people analytical means simply usage of analytical functions. A wider

meaning is that the analytical model is described by elementary functions and

basic operations of calculus ((differentiation, integration, matrix calculus,...)

without having exact form of the solution. More required persons prefer to need

the final formulas given explicitly, be it in a series, or other limit form. Finally, some

agree that the model uses just integers (e.g. like in discrete state and discrete

time models, algorithms of recurrences etc.). Seemingly, the common feature of

analytical model is that they use formulas for describing relation between

numbers representing particular quantities; with an assumption, that this is an

exact model. As long as we accept the model as exact, it is not named numerical."

Regards, Joachim

I will give three examples, which will help understand the difference between exact analytical solution from numerical solution:

1) One-dimensional convection equation has an exact solution, which is given by the initial condition convecting with the phase speed. Now if you solve this equation ANY numerical method, you will see a very restrictive numerical parameters for which the computed solution matches with the exact solution with finite digit precision. Take a look at:

• Error dynamics: Beyond von Neumann analysis - T.K. Sengupta, A. Dipankar, P. Sagaut, J. Comp. Physics vol. 226(2), pp 1211-1218 (2007).
• High Accuracy Computing Method: Fluid Flows and Wave Phenomenon -- Tapan K Sengupta, Cambridge Univ. Press, 2013.

2) The second example is that of Lamb-Oseen vortex problem convection. If you look at the convective acceleration term of Navier-Stokes equation, then adopting a polar grid only you will see this to be identically equal to zero, as one would expect analytically. If you use any other structured grid for numerical solution, you will see that this term becomes non-zero. So not all numerical methods are equally relevant. Horses for the courses.

3) The third example is that of flow past a circular cylinder at low super-critical Reynolds numbers. If one of the grid point is not placed at the rear stagnation point , then you will always get a nonzero lift, which is incorrect. For the same reason, users of unstructured grids, always use structured grid near the cylinder surface with grid reflecting top-down symmetry of the flow.

Thus, one should think of uncertainty principle of computing!

@ Tapan K. Sengupta and other P.T.Followers

Dear Sir, you have started an often omitted (at least within RG displays) though very important question a bit aside the current question: How to chose the numerical algorithm of solving the proble, especially if there is no ""analytical" solution? Indeed, the researcher supplied by the theory (i.e. exact formulation of some properties of the solutions) can save much of time AND reaching a more exact solution! In my (not great experience) a problem appered to fit a series of points by a curve with given prameters (the form followed just from the shape of the distribution of the measured points). And what happened? Without pre-defined constraines the PROPER CODE has caculated all possible values of the parameters to be fitted by solving a hight degree polynomial equation (to minimize the error of fitting). Wheras a SIMPLE analysis of the data excluded 99% of the ranges, and the old Newton method solved the problem in mcroseconds. I think this remark supports your suggestion in an acceptable way by even the beginners! IT IS NOT ALWAYS A SIMPLE STEP (the restriction/simplification) of the algorithms applied, however it is a necessary custom to TRY, AT LEAST. Let us note, that even multiplying 31*30*29 one can get the result on a scratch by using (30^2-1)*30 =27000-30= 26970, without calculator! Thus the ANALYSIS of the NUMERICAL algorithm allows to state, that the numerical methods are analytical as well:)

Thanks for your attention, Joachim

classifying the methods to two types as analytical or numerical methods, or some times semi-analytical methods is very difficult in most cases, it is still confusing point, since:

1. we define the analytical methods when it give us exact solution, and numerical methods if it give us approximate solutions, and the semi-analytical numerical methods if is it give us exact solutions or/and approximate solutions depending on the type of equations and its linearity. depending in this methods, almost all method are in third type, semi, since it give us exact in some cases and approximate solutions in other cases.

2. in some cases we can find the exact solutions, but we use approximate solutions for many case: to decrease the cost of finding exact, it is very difficult to find exact, the approximate solutions give us a very good solutions when we compare the approximate of proposed method with the exact solution of previous methods, that is the absolute or relative errors are very good.

in my opinion, I think it is very difficult, or sometimes (useless) to classify methods as numerical or analytical.

thanks

Dear Joachim Domsta

Thank you for your illustrative example, making the point made by us more cogent. Also, Salah Abuasad should not be perturbed by trying to classify problems, as has given an algorithm! The main issue is of being able to calibrate and/ or validate your method(s) before using it for any problem. The choice of model problem with exact solution, rests on the users to address the main mathematical physics process is being looked at. For example, I will advice to use convection and convection-diffusion problems in multiple dimensions for the sake of calibration of methods for Euler and Navier-Stokes equation in fluid mechanics. There must be similar model problems for different problems.in different branches of engineering and sciences.

Analytical method given exact solution but numerical method given approximate solution

Tapan K. Sengupta could you please, simplify your comments, I can't understand your idea ???

• Analytical solution gives the exact solution.
• Numerical solution gives the approximate solution.

Example: F(x)=x-7

Analytic=>

say x-7=0 and x=7

Numerical=>

F(8)=8–7=1 and F(6)=6–7=-1

so answer in range 6 to 8.

next find the middle point and iterate the process.

• Analytical solution gives the exact solution.
• Numerical solution gives the approximate solution.

Example: F(x)=x-7

Analytic=>

say x-7=0 and x=7

Numerical=>

F(8)=8–7=1 and F(6)=6–7=-1

so answer in range 6 to 8.

next find the middle point and iterate the process.

some times the above method give the exact solution from the first iteration that why called fanatical method

Thank you all for the good explanations. I have been in great confusion, as a lot of time the two names are used interchangeably.