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.

**Read Online or Download 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 PDF**

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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.