Numerical Analysis II

General references: S.D. Conte & C. de Boor, Elementary Numerical Analysis: An Algorithmic Approach, Third edition, 1981. McGraw-Hill. L.F. Shampine, R.C. Allen, Jr & S. Pruess, Fundamentals of Numerical Computing, 1997. Wiley. This course is concerned with the same problems as Numerical Analysis I, but the mathematical content is greater. In the interests of simplicity and brevity, the use of complex variables is avoided unless necessary. However, it should be noted that many real variable methods can be generalized to complex cases, although these are not always mentioned here. Familiarity with the notes for Numerical Analysis I is assumed. 1. Elementary Approximation Theory Approximation theory is a major field in its own right, but it is also central to numerical analysis. The problems treated in this section are: (i) the approximation of a function of a real variable by a simpler function, and (ii) the approximation of a set of data points by a function of a real variable. 1.1 Preliminaries To obtain a numerical approximation to a function, we first require accurate function values. These may be obtained in a variety of ways: Bessel’s function J0(x) is used here as an example.