Afternotes on numerical analysis: a series of lectures on by G. W. Stewart

By G. W. Stewart

This can be a good ordinary creation to numerical research, in simple terms simple arithmetic is needed. it's enjoyable and straightforward to learn. this can be a "small" publication; the biggest part (linear equations) being sixty six pages. despite the fact that, it does conceal loads of ground.

Code fragments are in C and FORTRAN. The C code evidently hasn't been verified (abs() rather than fabs() throughout). there are numerous typos within the textual content in addition to within the code fragments.

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Additional resources for Afternotes on numerical analysis: a series of lectures on elementary numerical analysis presented at the University of Maryland at College Park and recorded after the fact

Example text

The figure shows the course of the secant method starting from a bracket [#1,0:2]. The third iterate £3 joins x% to the right of the zero, and because the function is flat there, #4 is large and negative. 22. The trouble with the secant method in this case is that a straight line is not a good approximation to a function that has a vertical asymptote, followed by a zero and then a horizontal asymptote. On the other hand, the function has a vertical asymptote at x — |, a zero at x — a, and a horizontal asymptote at y = b~~l and therefore should provide a better approximation.

No algorithm, stable or otherwise, can be expected to return an accurate solution to an ill-conditioned problem. Only if we are willing to go to extra effort, like reducing the error e(x), can we obtain a more accurate solution. 21. A number that quantifies the degree of ill-conditioning of a problem is called a condition number. 2. Ill- and well-conditioned roots. From the approximation it follows that \f(x}\ < e when \ f ' ( x * ) ( x — x*)| < e. Hence, or equivalently Thus the number \/\f'(x*)\ tells us how much the error is magnified in the solution and serves as a condition number.

Here d is always on the side of x* that is opposite c, and the value of c is not changed by the iteration. This means that although b is converging superlinearly to x*, the length of the bracket converges to a number that is greater than zero — presumably much greater than eps. Thus the algorithm cannot converge until its erratic asymptotic behavior forces some bisection steps. 9. 5*eps. This will usually be sufficient to push s across the zero to the same side as c, which insures that the next bracket will be of length less than eps — just what is needed to meet the convergence criterion.

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