An Introduction to Numerical Analysis (2nd Edition) by Kendall Atkinson

By Kendall Atkinson

This moment variation of a customary numerical research textual content keeps association of the unique version, yet all sections were revised, a few commonly, and bibliographies were up-to-date. New themes lined comprise optimization, trigonometric interpolation and the quick Fourier remodel, numerical differentiation, the tactic of strains, boundary worth difficulties, the conjugate gradient strategy, and the least squares ideas of structures of linear equations. comprises many difficulties, a few with recommendations.

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Example text

The last class of problems discussed in this chapter are optimization problems. •. , xn) and to find the point (x 1, ••. , xn) at which the optimum is attained. 1 Iterative solution of a- (1/x) = 0. problems can often be reduced to a system of nonlinear equations, but it is usually better to develop special methods to carry out the optimization directly. The area of optimization is well-developed and extensive. We just briefly introduce and survey the subject. 1), we begin with an example. 3) X for a given a > 0.

Define k=l,2, ... 9), depending on whether chopping or rounding, respectively, is used. 1). 7) is also valid. 01, then j = 1, ... 12) [see Forsythe and Moler (1967, p. 92)]. 18) This says nothing about the relative error, since x Ty can be zero even though all x; and Y; are nonzero. 7). Nonetheless, it is often possible to easily and inexpensively reduce this error a great deal further, and this is usually very important in linear algebra problems. Calculate each product xjyj in a higher precision arithmetic, and carry out the summation in this higher precision arithmetic.

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