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Numerical methods are used to find solutions to problems when purely analytical methods fail.
Numerical Analysis - an area of mathematics that uses various numerical methods to find numerical approximations to mathematical problems, while also analysing those methods to see if there is any way to reduce the numerical error involved in using them, thus resulting in more reliable numerical methods that give more accurate approximations than previously.
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There is no single word. Mathematical methods, alone, could involve analytical methods, statistical methods of estimation or numerical methods for approximation.
Short answer; there isn't any.Long/picky answer; numerical methods tend to look at, surprisingly, numerical methods on solving certain problems such as finding answers to equations (finding fixed points), calculating errors and really just doing calculations using these methods.Numerical analysis on the other hand, does all of this but also looks deeper into why error occurs from these methods and looks into ways of adjusting these methods or developing better ones that reduce the errors given so as to obtain much more accurate approximations to the solution you are trying to find for a given problem.