By Nicholas J. Higham
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Additional resources for Accuracy and Stability of Numerical Algorithms, Second Edition
Example text
This example illustrates the principle that we should recognize divided differences (which may be in disguise) and take care in their evaluation when the arguments are close. 2. QR Factorization Any matrix A E IR mxn , m ~ n, has a QR factorization A = QR, where Q E IR mxn has orthonormal columns and R E IRnxn is upper trapezoidal (rij = 0 for i > j). 22 PRINCIPLES OF FINITE PRECISION COMPUTATION One way of computing the QR factorization is to premultiply A by a sequence of Givens rotations-orthogonal matrices G that differ from the identity matrix only in a 2 x 2 principal submatrix, which has the form [ COS 0 -sinO sinO] cosO' With Al := A, a sequence of matrices Ak satisfying Ak = GkA k- l is generated.
The only way to guarantee accurate computed roots is to use extended precision (or some trick tantamount to the use of extended precision) in the evaluation of b2 - 4ac. Another potential difficulty is underflow and overflow. 1020 = 0 then overflow occurs, since the maximum floating point number is of order 1038 ; the roots, however, are innocuous: X = 1 and x = 2. 9 11 COMPUTING THE SAMPLE VARIANCE are 1020 and 2 . 10 20 . In the latter equation we need to scale the variable: defining = 10 20 y gives 10 20 y2 - 3 .
With the notation above, and for the 2-norm, . {IILlAI12 p(y) = mm ~: (A Proof. If (A giving + LlA)y = b then + LlA)y = b } r := b - Ay = LlAy, so IILlAI12 > IIrl12 _ IIAI12 - IIAI1211Yl12 - p(y). 7) is attainable. 0 . 1 says that p(y) measures how much A (but not b) must be perturbed in order for y to be the exact solution to the perturbed system, that is, p(y) equals a normwise relative backward error. , p(y) = O(u)) then the approximate solution y must be regarded as very satisfactory. 2). To illustrate these concepts we consider two specific linear equation solvers: Gaussian elimination with partial pivoting (GEPP) and Cramer's rule.