Accuracy and Stability of Numerical Algorithms by Nicholas J. Higham

By Nicholas J. Higham

A therapy of the behaviour of numerical algorithms in finite precision mathematics that mixes algorithmic derivations, perturbation conception, and rounding errors research. software program practicalities are emphasised all through, with specific connection with LAPACK and MATLAB.

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The simplest cure for this inaccuracy is to sum in the opposite order: from smallest to largest. Unfortunately, this requires knowledge of how many terms to take before the summation begins. 6449 3406, which is correct to eight significant digits. For much more on summation, see Chapter 4. 13. Increasing the Precision When the only source of errors is rounding, a common technique for estimating the accuracy of an answer is to recompute it at a higher precision and to see how many digits of the original and the (presumably) more accurate answer agree.

Thus x denotes the computed approximation to x. Definitions are often (but not always) indicated by ":=" or "=:", with the colon next to the object being defined. We make use of the floor and ceiling functions: [x\ is the largest integer less than or equal to x, and \x \ is the smallest integer greater than or equal to x. The normal distribution with mean u and variance 2 is denoted by N ( u , 2). We measure the cost of algorithms in flops. A flop is an elementary floating point operation: +, —, /, or *.

In summary, while the number of correct significant digits provides a useful way in which to think about the accuracy of an approximation, the relative error is a more precise measure (and is base independent). Whenever we give an approximate answer to a problem we should aim to state an estimate or bound for the relative error. When x and x are vectors the relative error is most often defined with a norm, as \\x — x\\f \\x\\. For the commonly used norms ||x||oo := max; \Xi\, \\x\\1 := and \\x\\2 := (xTx}1/2, the inequality \\x — x\\/ \\x\\ < 10-p implies that components Xi with \Xi\ w ||x|| have about p correct significant decimal digits, but for the smaller components the inequality merely bounds the absolute error.

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