The earliest record of numeric approximation is a Babylonian tablet portraying an approximation of the square root of two. However, today 'numerical analysis' usually involves computer algorithms which use far more advanced numerical approximations to derive suitable, if not exact solutions to mathematical problems. This is necessary as infinite precision arithmetic is often impractical.
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Many of these algorithms therefore are iterative and the number of steps they take is not finite as in direct methods. The input given to these algorithmic programs serves as an initial guess which is then taken and improved little by little until these approximations converge to the exact solution only in the limit. One can then run a convergence test to evaluate whether the output is sufficiently accurate as a usable approximation.
These algorithms can be applied with great benefits in practically all fields of engineering and natural sciences, as well as some social sciences.The use of computers to enact these algorithms has helped these fields immeasurably since prior to their invention, the methods had to be followed 'by hand' using hard interpolation in large printed tables of values as opposed to simply running the appropriate program to calculate those interpolation formula functions. The math hasn't changed as drastically, but it goes much faster now as part of a software's algorithm.