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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10. Solving Linear Equations For an approximate solution y to a linear system Ax = b ( A IRn × n , b n IR ) the forward error is defined as ||x-y||/||x||, for some appropriate norm. 14 PRINCIPLES OF FINITE PRECISION COMPUTATION Another measure of the quality of y, more or less important depending on the circumstances, is the size of the residual r = b - Ay . When the linear system comes from an interpolation problem, for example, we are probably more interested in how closely Ay represents b than in the accuracy of y.
For illustration, consider the system Ax = b, where A is the inverse of the 5×5 Hilbert matrix and b i = (-l)i i. ) 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.
A two-pass computation is undesirable for large data sets or when the sample variance is to be computed as the data is generated. 10 SOLVING LINEAR EQUATIONS 13 This formula is very poor in the presence of rounding errors because it computes the sample variance as the difference of two positive numbers, and therefore can suffer severe cancellation that leaves the computed answer dominated by roundoff. In fact, the computed answer can be negative, an event aptly described by Chan, Golub, and LeVeque [194, 1983] as “a blessing in disguise since this at least alerts the programmer that disastrous cancellation has occurred”.