Nothing is more important for numerics than a veey good grasp of analysis.
But there's understanding and there's understanding. If all you have is a piece of paper then there's no way around analytical approach. People used to be unbelievably good at it, including the applied crowd, like engineers.
Having matlab at hand makes it possible to save years of analytical tinkering and simplifications and approximations.
It is useful to understand what happens exactly but we no longer need 3-4 years of calculus depths... I mean, this used to be THE subject in most engineering programs round the world. These days it's more like bootstrapping the intuition.
And they use poor and imprecise approximations that they can't debug, because they just plug in numbers and hope for answer.
What point are you trying to make?
Yep, what you want is a good and predictable approximation.
> What point are you trying to make?
In practice most Matlab solutions you are describing are awful, in terms of performance and in terms of accuracy due to ignorance about underlying theory.
If you want to calculate a numerical solution to a novel analytic problem you absolutely need a good understanding of analysis. Being able to plug things into Matlab is only useful if Matlab implements a good solver for that problem.
>It is useful to understand what happens exactly but we no longer need 3-4 years of calculus depths.
Indeed. We need 5+ years of analysis now.