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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The partial pivoting strategy would interchange the two rows of A before factorizing it, resulting in a stable factorization. 2. An Innocuous Calculation? For any x>0 the following computation leaves x unchanged: for i = 1:60 end for i = 1:60 x = x2 end Since the computation involves no subtractions and all the intermediate numbers lie between 1 and x, we might expect it to return an accurate approximation to x in floating point arithmetic. 0. In fact, for any x, the calculator computes, in place of f(x) = x, the function 18 PRINCIPLES OF FINITE PRECISION COMPUTATION The calculator is producing a completely inaccurate approximation to f(x) in just 120 operations on nonnegative numbers.
9). 27)) we conclude that is bounded by a multiple of K2 (A)u. 5 must eventually dip down to the unit roundoff level. P RINCIPLES 26 OF F INITE P RECISION C OMPUTATION We also note that is of order u in this example, as again we can show it must be from perturbation theory. Since Q is a product of Givens rotations, this means that even though some of the intermediate Givens rotations are very inaccurate, their product is highly accurate, so in the formation of Q, too, there is extensive cancellation of rounding errors.
5 plots the relative errors ||Ak - Ak||2/||A||2, where Ak, denotes the matrix computed in single precision arithmetic (u 6 × 10-8). We see that many of the intermediate matrices are very inaccurate, but the final computed has an acceptably small relative error, of order u. Clearly, there is heavy cancellation of errors on the last few stages of the computation. 5. Relative errors ||A k- Â k||2/||A ||2 for Givens QR factorization. The dotted line is the unit roundoff level. following a suggestion of Wilkinson [1100, ||A||2 1 and A 10 has the form 1985], as a full matrix such that Because y is at the roundoff level, the computed is the result of severe subtractive cancellation and so is dominated by rounding errors.