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I'm currently doing a numerical analysis course, and it seems many of the techniques were developed by mathematicians, such as Euler and Newton, who lived before the age of the computer. Why were these mathematicians interested in numerical analysis and what use did it have before computers?
I have a vague idea of some uses it may have, but I still struggle to understand why such great mathematicians would choose to research numerical analysis instead of remaining focused on pure mathematics.
I am not a historian but I propose first "Astronomy" as the answer to your question. The study of planetary motions and seasons was essential for a variety of reasons (agriculture and religion come to mind).
Next, "Navigation": the use of accurate tables (particularly multiplication tables) was essential for good navigation, and this entailed laborious calculations (mostly using spherical geometry).
Finally, "Money". The calculation of interest and mortgages is rumoured to be the motivation for Euler introducing "e" as the limit of receiving interest over an ever-larger numbers of ever-smaller intervals.
When you apply mathematics to the real world this requires a lot of calculation. Astronomy is the oldest science which applied mathematics on a large scale. Then comes physics, chemistry, material sciences and all other sciences and engineering applications. They always required a lot of numerical analysis.
The question is actually very strange: it was evidently MORE reasons for mathematicians to work on numerical methods BEFORE the computers spread than after their spread. Because all calculations had to be made by hand. So this was very time and labor consuming. It is exactly for this reason people were working hard to invent effective algorithms.
For example, when Kepler learned about logarithms he said that "this invention increased the life time of astronomers by a factor of 100".
As late as in 1940 Cornelius Lanczos invented the famous algorithm which is called the "Fast Fourier Transform". In the introduction to his paper (joint with his CHEMISTRY PhD student Danielson) he says: "One could use an expensive harmonic analyser, but we found a method which permits to do this calculation by hand..." (Harmonic analyser is a very sophisticated analog computer designed to do just one task: Fourier expansion. These things were custom made and indeed enormously expensive).
Until the late 1970s there was a profession "navigator". Each long distance airplane and each ocean going ship had (at least) one. The main job of this person was calculations (by hand, using tables, and other aids). Invention of electronic calculator did not eliminate the profession but changed it dramatically. Only the spread of satellite navigation systems+computers eliminated it.
I did not even mention military applications, which were second only to astronomy and navigation in importance.
It is striking to me that someone in the 21st century questions why people who had to calculate by hand were motivated to discover numerical algorithms. Let's see, imagine it is the early 1700s and you want to find the value of the sum of $1/n^2$ for $n = 1,2,3,\ldots$ (this was an unsolved problem for about 90 years). Of course it helps to get an idea of what the answer might be before trying to derive it carefully. So you want a numerical estimate, say to 5 decimal places. You can't do that by hand just from summing the series: it takes over 5000 terms in the series to get within .00001 of the value of the full series. But if you are Euler, you develop (what came to be called) the Euler-Maclaurin summation formula that allows you to estimate the series to over 10 decimal places using just the first 10 terms of the series plus a few correction terms in that summation formula. Can you appreciate that this might be a useful achievement?
I find the question a bit bizarre but realize that it is likely due to the culture difference of those who have grown up in a society where computers are used everywhere.
It was not that long ago in past history (~1965) when I was a undergrad physics major faced with a non-linear (transcendental) equation to solve. No analytic solutions possible yet I had to find a value for 'x' (the unknown) up to 3 decimal places.
The solution in my case was Newton's method and a slide rule (which I still have and sits on the bookcase next to me right now). We students of this class (Classical Mechanics) all had to solve the problem as part of a homework assignment.
One of my fellow students though used the campus computer (CDC 3300) and the Fortran language to implement Newton's method and easily come up with an answer to 5 or 6 decimal digits. And, his program was punched into the so-called IBM punched card.
That single event of learning the power of the computer over the drudgery of the slide rule turned me on to computer programming.
As others have said, computers help but they were not the reason for the development of numerical methods. In physics, most real life problems must be solved numerically -- only text book assignments (usually) of limited scope have analytic solutions available to Algebra or Calculus.
The following text is from the introduction to Carl Runge's "Graphical Methods", in which the author, well before the invention of computers, justifies the existence of numerical analysis:
"A great many if not all of the problems in mathematics may be so formulated that they consist in finding from given data the values of certain unknown quantities subject to certain conditions.
We may distinguish different stages in the solution of a problem. The first stage we might say is the proof that the quantities sought for really exist. In many ... cases the first stage of the solution may be so easy, that we immediately pass on to the second stage of finding methods to calculate the unknown quantities sought for. Or even if the first stage of the solution is not so easy, it may be expedient to pass on to the second stage. For if we succeed in finding methods of calculation that determine the unknown quantities, the proof of their existence is included.
There are not a small number of men who believe the task of the mathematician to end here. This, I think, is due to the fact that the pure mathematician as a rule is not in the habit of pushing his investigation so far as to find something out about the real things of this world. He leaves that to the astronomer, to the physicist, to the engineer. These men, on the other hand, take the greatest interest in the actual numerical values that are the outcome of the mathematical methods of calculation. Suppose the mathematician gives them a method of calculation, perfectly logical and conclusive but taking 200 years of incessant numerical work to complete. They would be justified in thinking that this is not much better than no method at all.
So there arises a third stage of the solution of a mathematical problem in which the object is to develop methods for finding the result with as little trouble as possible. I maintain that this third stage is just as much a chapter of mathematics as the first two stages and it will not do to leave it to the astronomer, to the physicist, to the engineer or whoever applies mathematical methods, for this reason that these men are bent on the results and therefore they will be apt to overlook the full generality of the methods they happen to hit on, while in the hands of the mathematician the methods would be developed from a higher standpoint and their bearing on other problems in other scientific inquiries would be more likely to receive the proper attention."
