Conditioning of Problems and Stability of Algorithms
This chapter begins with a discussion of why numerical linear algebra is different from linear algebra and why it is important in engineering and science. The two types of error, backward error and forward error, are defined and examples presented. Backward error is generally more meaningful. An algorithm can be unstable, meaning that there are significantly many cases where data for the algorithm are perfectly good, and the results are in error. Solving the quadratic equation using the classical formula is an unstable algorithm. An analysis of stability is normally done using backward error analysis. A particular problem may be sensitive to perturbations in its data ...
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