What's the formal difference between analytical and numerical?

While trying to wrap my head around differential equations in a practical way, I found a quite enlightening phrase about it

Solving a differential equation can be done in three major ways: analytical, qualitative, and numerical.

Now forget about differential equations: what's the formal definition of analytical and numerical ?

For example, another case where the numerical case looks the same as the analytical one is the following, consider the equation for the generic straight line

$$y = mx + q$$

$m$ has the same value generated by the analytical concept of derivative $\frac{dy}{dx}$, infact the value of the first derivative can be generally used to determine the slope of a line . The first solution is numerical, the second is analytical .

Hoping that this parallel example is fitting and correct, and although I think I intuitively get the difference between a numerical and an analytical solution/approach, I would like a formal definition that can help me distinguish between the 2 in any case, with any math tool .

• @Amzoti quite general definitions, for example back to my derivative example, I would like to add a third case to calculate the slope with $tan(x)$, now is this a numerical or an analytical approach ? And why ? – user2485710 Oct 16 '14 at 17:27
• What's the downvote about ? – user2485710 Oct 16 '14 at 18:08
• @Amzoti Indeed, but I still can't associate the trigonometric case to 1 of the 2 choices and motivate that choice. I also would like to improve my question if there is something wrong with it . – user2485710 Oct 16 '14 at 18:15

Referring to differential equations:

1)the analytical approach is about trying to prove the existence, the uniqueness and find an explicit form for the solution.

As finding an explicit form is almost always very difficult (or impossible) one can go through two different ways

2)the numerical approach is about trying to find an approximate solution using algorithms of the numerical analysis.

3)the qualitative approach is about finding some proprieties of the solution without knowing it. For example we can say where is increasing, if is bounded ..etc

• so back to my first post with the example & my comment with the question about $tan(x)$, in your opinion $tan(x)$ is what for that given problem ? An analytical or numerical solution ? Can you offer a brand new example ? – user2485710 Oct 19 '14 at 5:55
• I haven't understand very well your example. What you have and what you want to compute? – foo90 Oct 19 '14 at 8:37
• Anyway I give you another example: Consider the linear system $Ax=b$, $A\in\mathbb{R}^{n\times n},b\in\mathbb{R}^n$. The problem is to find one $x\in\mathbb{R}^n$ which solves the system. The analytical approach is, once proved that the system have a unique solution ($det(A)\neq 0$), find the exact solution $x=A^{-1}b$. As this might take a long time (if $n$ is very big), exist some algorithms for finding an approximate sol: tis is the numerical approach. The qualitative approach is saying some proprieties on the solution without knowing it, like in this case estimate the norm $||x||$. – foo90 Oct 19 '14 at 8:49
• However there are not a formal definitions for analytic/numerical/qualitative approach. They are concept that help you to understood what we are talking about and nothing more. – foo90 Oct 19 '14 at 8:52
• there are not a formal definitions for there are not a formal definitions for analytic/numerical/qualitative approach/numerical/qualitative approach so a mathematician can look at something and say it's numerical, another one can say no it's analytical ? Plus you just gave your definition of analytical, so there is something to talk about . – user2485710 Oct 19 '14 at 11:20