By Nicholas J. Higham

ISBN-10: 0898715210

ISBN-13: 9780898715217

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Extra resources for Accuracy and Stability of Numerical Algorithms, Second Edition

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This means that a small relative change in x can produce a much larger relative change in log x for x R:i 1. :1x/x), and that change in log x may be large in a relative sense. 7 9 CANCELLATION When backward error, forward error, and the condition number are defined in a consistent fashion we have the useful rule of thumb that forward error ::s condition number x backward error, with approximate equality possible. One way to interpret this rule of thumb is to say that the computed solution to an ill-conditioned problem can have a large forward error.

Increasing the precision without preserving important properties such as monotonicity of rounding can vitiate an otherwise reliable algorithm. Increasing the precision without maintaining a correct relationship among the precisions in different parts of an algorithm can also be harmful to the accuracy. 14. Cancellation of Rounding Errors It is not unusual for rounding errors to cancel in stable algorithms, with the result that the final computed answer is much more accurate than the intermediate quantities.

The explanation is that the matrix A cannot be stored exactly in binary floating point arithmetic. The computer actually works with A + LlA for a tiny perturbation LlA, and the dominant eigenvalue and eigenvector of A + LlA are very good approximations to those of A. The starting vector [1,1, I]T contains a nonzero (though tiny) component of the dominant eigenvector of A + LlA. 1A, helped by rounding errors in the multiplication, until convergence to the dominant eigenvector is obtained. Perhaps an even more striking example of beneficial effects of rounding errors is in inverse iteration, which is just the power method applied to the shifted and inverted matrix (A - flI)-l.

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Accuracy and Stability of Numerical Algorithms, Second Edition by Nicholas J. Higham

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