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Computational methods and numerical analysis

The mathematical methods needed for computations in engineering and the sciences must be transformed from the continuous to the discrete in order to be carried out on a computer. For example, the computer integration of a function over an interval is accomplished not by applying integral calculus to the function expressed as a formula but rather by approximating the area under the function graph by a sum of geometric areas obtained from evaluating the function at discrete points. Similarly, the solution of a differential equation is obtained as a sequence of discrete points determined, in simplistic terms, by approximating the true solution curve by a sequence of tangential line segments. When discretized in this way, many problems can be recast in the form of an equation involving a matrix (a rectangular array of numbers) that is solvable with techniques from linear algebra. Numerical analysis is the study of such computational methods. Several factors must be considered when applying numerical methods: (1) the conditions under which the method yields a solution, (2) the accuracy of the solution, and, since many methods are iterative, (3) whether the iteration is stable (in the sense of not exhibiting eventual error growth), and (4) how long (in terms of the number of steps) it will generally take to obtain a solution of the desired accuracy.

The need to study ever-larger systems of equations, combined with the development of large and powerful multiprocessors (supercomputers) that allow many operations to proceed in parallel by assigning them to separate processing elements, has sparked much interest in the design and analysis of parallel computational methods that may be carried out on such parallel machines.

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