Download PDF by Nicholas J. Higham: Accuracy and Stability of Numerical Algorithms

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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6944 . . , which is clearly wrong given the fact that 0 < f ( x ) < 1/2 for all x 0. A 10-significant-figure approximation to cos a; is therefore not sufficient to yield a value of f(x) with even one correct figure. The problem is that 1 — c has only 1 significant figure. The subtraction 1 — c is exact, but this subtraction produces a result of the same size as the error in c. In other words, the subtraction elevates the importance of the earlier error. In this particular example it is easy to rewrite f(x) to avoid the cancellation.

3) suffers massive cancellation. This is damaging cancellation because one of the arguments, fl b2 — 4ac), is inexact, so the subtraction brings into prominence the earlier rounding errors. How to avoid the cancellation is well known: obtain the larger root (in absolute value), x1, from and the other from the equation 0:1X2 = c/a. Unfortunately, there is a more pernicious source of cancellation: the subtraction b2 — 4ac. Accuracy is lost here when 62 4ac (the case of nearly equal roots), and no algebraic rearrangement can avoid the cancellation.

This book contains perturbation theory for most of the problems considered, for example, in Chapters 7 (linear systems), 20 (the least squares problem), 21 (underdetermined systems), and 25 (nonlinear systems). Analysing truncation errors, or discretization errors, is one of the major tasks of the numerical analyst. Many standard numerical methods (for example, the trapezium rule for quadrature, Euler's method for differential equations, and Newton's method for nonlinear equations) can be derived by taking finitely many terms of a Taylor series.

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