By Robert M. Corless
This e-book presents an in depth advent to numerical computing from the point of view of backward mistakes research. The meant viewers contains scholars and researchers in technology, engineering and arithmetic. The method taken is just a little casual because of the wide range of backgrounds of the readers, however the crucial rules of backward errors and sensitivity (conditioning) are systematically emphasised. The ebook is split into 4 components: half I offers the heritage preliminaries together with floating-point mathematics, polynomials and desktop assessment of features; half II covers numerical linear algebra; half III covers interpolation, the FFT and quadrature; and half IV covers numerical suggestions of differential equations together with initial-value difficulties, boundary-value difficulties, hold up differential equations and a quick bankruptcy on partial differential equations.
The booklet includes special illustrations, bankruptcy summaries and quite a few routines in addition a few Matlab codes supplied on-line as supplementary material.
“I rather just like the specialize in backward blunders research and situation. this is often novel in a textbook and a pragmatic strategy that may deliver welcome attention." Lawrence F. Shampine
A Graduate advent to Numerical tools and Backward mistakes research” has been chosen through Computing experiences as a extraordinary e-book in computing in 2013. Computing studies better of 2013 record involves booklet and article nominations from reviewers, CR type editors, the editors-in-chief of journals, and others within the computing community.
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Extra info for A Graduate Introduction to Numerical Methods: From the Viewpoint of Backward Error Analysis
2 Representation and Computation Error Floating-point arithmetic does not operate on real numbers, but rather on floatingpoint numbers. This generates two types of roundoff errors: representation error and arithmetic error. The first type of error we encounter, representation error, comes from the replacement of real numbers by floating-point numbers. If we let x ∈ R and : R → F be an operator for the standard rounding procedure to the nearest floating-point number4 (see Appendix A), then the absolute representation error Δ x is Δx = x − x = xˆ − x .
2 Numerical solution of Eq. 1) . . . . . . . . . . . . . . . . . 511 Plots of the numerical solutions of the Lorenz system. (a) Time history of x(t), y(t), and z(t). (b) Phase portrait for all three components of the solution . . . . . . . . . . . . . . . . . . . 516 Damped harmonic oscillator . . . . . . . . . . . . . . . . . . 518 A vector field with a nearly tangent computed solution .
476 Derivative of a polynomial in the Hermite interpolational basis . . . . 477 Effect of rounding error on the complex formula . . . . . . . . . . 480 Errors in three finite-difference formulæ . . . . . . . . . . . . . 483 Fourth-order uniform-mesh compact finite-difference derivative . . . 488 Fourth-order variable-mesh compact finite-difference maximum error . . . . . . . . . . . . . . . . . . . . . . . . 490 The degree 2 least-squares fit to noisy data, in the Lagrange basis .