By Nicholas J. Higham
A therapy of the behaviour of numerical algorithms in finite precision mathematics that mixes algorithmic derivations, perturbation thought, and rounding blunders research. software program practicalities are emphasised all through, with specific connection with LAPACK and MATLAB.
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Extra info for Accuracy and Stability of Numerical Algorithms
A 10 significant figure approximation to cos is therefore not sufficient to yield a value of 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 to avoid the cancellation. 5, which is correct to 10 significant figures. 8 SOLVING A QUADRATIC EQUATION 11 To gain more insight into the cancellation phenomenon consider the subtraction (in exact arithmetic) where and The terms ∆a and ∆b are relative errors or uncertainties in the data, perhaps attributable to previous computations.
We solved the system in varying precisions with unit roundoffs u = 2 -t , t = 15:40, corresponding to about 4 to 12 decimal places of accuracy. ) The algorithm used was Gaussian elimination (without pivoting), which is perfectly stable for this symmetric positive definite matrix. 3 gives corresponding results for A = P5 + 5I, where P5 is the Pascal matrix of order 5. 55×105 for the shifted Pascal matrix. 3. Forward errors and relative residuals ||b versus precision t = - log2 u on the x axis.
1984], Spellucci [933, 1980], Optimization: Dennis and Schnabel [300, 1983], Fletcher [377, 1986], [379, 19 88], [380, 1993 ], [381, 1994 ]. Gill, Murray, and Wright [447, 19 8 1 ], Gurwitz [490, 1992], Müller-Merbach [783, 1970], and Wolfe [1108, 1965]. Ordinary differential equation initial value problems: Henrici [517, 1962], [518, 1963], [519, 1964], D. J. Higham [525, 1991]. 6]. 8], and Trefethen and Trummer [1020, 1987]. 2], Lyness [717, and Piessens et al. 1]. 21. Notes and References The term “correct significant digits” is rarely defined in textbooks: it is apparently assumed that the definition is obvious.