By Nicholas J. Higham

ISBN-10: 0898715210

ISBN-13: 9780898715217

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Relative Error and Significant Digits x Let be an approximation to a real number x. The most useful measures of the accuracy of are its absolute error x Eabs(X) = Ix - xl, and its relative error ~ lx-xl Erel(x) = -Ixl- (which is undefined if x = 0). An equivalent definition of relative error is Erel(X) = Ipl, where = x(1 + p). Some authors omit the absolute values from these definitions. When the sign is important we will simply talk about "the error x - x" . In scientific computation, where answers to problems can vary enormously in magnitude, it is usually the relative error that is of interest, because it is scale independent: scaling x -+ ax and x -+ ax leaves Erel (x) unchanged.

You may never see in practice the extremes of behaviour shown here. Let the examples show you what can happen, but do not let them destroy your confidence in finite precision arithmetic! 1. Notation and Background We describe the notation used in the book and briefly set up definitions needed for this chapter. Generally, we use capital subscripted lower case lower case lower case Greek letters letters letters letters A,B,C,Ll,A for bij , Cij, 8ij , Aij for for x,y,z,c,g,h for 0,(3,"(, Jr aij, e, matrices, matrix elements, vectors, scalars, following the widely used convention originally introduced by Householder [644, 1964].

The dotted line is the unit roundoff level. % Choose a starting vector x. while not converged x:=Ax x := x/llxli oo end The theory says that if A has a unique eigenvalue of largest modulus and x is not deficient in the direction of the corresponding eigenvector v, then the power method converges to a multiple of v (at a linear rate). 161 (correct to the digits shown) and an eigenvector [1,1, I]T corresponding to the eigenvalue zero. If we take [1,1, as the starting vector for the power method then, in principle, the zero vector is produced in one step, and we obtain no indication of the desired dominant eigenvalueeigenvector pair.

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


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