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By Kendall Atkinson

This moment version of a customary numerical research textual content keeps association of the unique version, yet all sections were revised, a few greatly, and bibliographies were up to date. New issues lined contain optimization, trigonometric interpolation and the short Fourier remodel, numerical differentiation, the tactic of strains, boundary price difficulties, the conjugate gradient process, and the least squares strategies of platforms of linear equations. comprises many difficulties, a few with recommendations.

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The perfidious polynomial. erical Analysis, pp. 1-28, Mathematical Association of America. Problems 1. (a) s Assume f(x) is continuous on as x 1 S = n b, and consider the average n L f(xj) j=l with all points xj in the interval [a, b ]. Show that S=f(r) r for some in [a, b]. Hint: Use the intermediate value theorem and consider the range of values of f(x) and thus of S. (b) Generaliz~ part (a) to the sum n S= L:wJ(x) j-1 with all xj in [a, b) and all wj 2. ~ 0. s 0. 'IT 'IT --

41-59) for a statistical analysis of the error in the numerical solution of differ~tial equations. 3 of Chapter 6 of the present textbook. Inner products Given two vectors x, Y. 11) j-1 the inner product of x and y. 13) The latter inequality is called the Cauchy-Schwarz inequality, and it is proved in a more general setting in Chapter 4. 11) occur commonly in linear algebra problems (for example, matrix multiplication). We now consider the numerical computation of such sums. Assume X; and Y;, i = 1, ...

The inverse y- 1 is known analytically, and we can compare it with f- 1. For n = 6, some of the elements of f- 1 differ from the corresponding elements in y- 1 in the first nonzero digit. 34 This makes the calculation of y- 1 an ill-conditioned problem, and it becomes increasingly ·so as n increases. 3). The general area of ill-conditioned problems for linear systems and matrix inverses is considered in greater detail in Chapter 8. Stability of ·numerical algorithms A numerical method for solving a mathematical problem is considered stable if the sensitivity of the numerical answer to the data is no greater than in the original mathematical problem.

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