By Nicholas J. Higham
Accuracy and balance of Numerical Algorithms supplies an intensive, up to date therapy of the habit of numerical algorithms in finite precision mathematics. It combines algorithmic derivations, perturbation thought, and rounding mistakes research, all enlivened by means of historic viewpoint and informative quotations.
This moment version expands and updates the insurance of the 1st variation (1996) and comprises quite a few advancements to the unique fabric. new chapters deal with symmetric indefinite platforms and skew-symmetric structures, and nonlinear platforms and Newton's technique. Twelve new sections contain insurance of extra mistakes bounds for Gaussian removing, rank revealing LU factorizations, weighted and limited least squares difficulties, and the fused multiply-add operation stumbled on on a few sleek laptop architectures.
An accelerated remedy of Gaussian removing comprises rook pivoting, besides an intensive dialogue of the alternative of pivoting technique and the results of scaling. The book's special descriptions of floating aspect mathematics and of software program matters mirror the truth that IEEE mathematics is now ubiquitous.
Although no longer designed particularly as a textbook, this new version is an acceptable reference for a sophisticated path. it will possibly even be utilized by teachers in any respect degrees as a supplementary textual content from which to attract examples, ancient standpoint, statements of effects, and routines. With its thorough indexes and huge, updated bibliography, the e-book presents a mine of knowledge in a with ease available shape.
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Additional info for Accuracy and Stability of Numerical Algorithms
2. All the results for Algorithm 2 are correct in all the significant figures shown, except for x = 10-15, when the last digit should be 1. On the other hand, Algorithm 1 returns answers that become less and less accurate as x decreases. 00000005 to the significant digits shown. 2. Computed values of (ex - 1)/x from Algorithms 1 and 2. 000000000000000 Algorithm 2 produces a result correct in all but the last digit: Here are the quantities that would be obtained by Algorithm 2 in exact arithmetic (correct to the significant digits shown): We see that Algorithm 2 obtains very inaccurate values of ex - 1 and log e x, but the ratio of the two quantities it computes is very accurate.
The Need for Pivoting Suppose we wish to compute an LU factorization Clearly, u 11 = and u 22 = 1 - l2 1 u12 = In floating point arithmetic, if is sufficiently small then evaluates to Assuming l21 is computed exactly, we then have Thus the computed LU factors fail completely to-reproduce A. Notice that there is no subtraction in the formation of L and U. Furthermore, the matrix A is very well conditioned The problem, of course, is with the choice of as the pivot. The partial pivoting strategy would interchange the two rows of A before factorizing it, resulting in a stable factorization.
N - 1,1), . . , (2,1); (n,2), . . , (3,2); and so on. 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.
Accuracy and Stability of Numerical Algorithms by Nicholas J. Higham