Introduction to Numerical Analysis and Scientific Computing - Nabil Nassif

. .

Résumé :
Designed for a one-semester course, Introduction to Numerical Analysis and Scientific Computing presents fundamental concepts of numerical mathematics and explains how to implement and program numerical methods. The classroom-tested text helps students understand floating point number representations, particularly those pertaining to IEEE simple and double-precision standards as used in scientific computer environments such as MATLAB(R) version 7. Drawing on their years of teaching students in mathematics, engineering, and the sciences, the authors discuss computer arithmetic as a source for generating round-off errors and how to avoid the use of algebraic expression that may lead to loss of significant figures. They cover nonlinear equations, linear algebra concepts, the Lagrange interpolation theorem, numerical differentiation and integration, and ODEs. They also focus on the implementation of the algorithms using MATLAB(R). Each chapter ends with a large number of exercises, with answers to odd-numbered exercises provided at the end of the book. Throughout the seven chapters, several computer projects are proposed. These test the students' understanding of both the mathematics of numerical methods and the art of computer programming.

Sommaire:
Computer Number Systems and Floating Point Arithmetic Introduction Conversion from Base 10 to Base 2 Conversion from Base 2 to Base 10 Normalized Floating Point Systems Floating Point Operations Computing in a Floating Point System Finding Roots of Real Single-Valued Functions Introduction How to Locate the Roots of a Function The Bisection Method Newton's Method The Secant Method Solving Systems of Linear Equations by Gaussian Elimination Mathematical Preliminaries Computer Storage for Matrices. Data Structures Back Substitution for Upper Triangular Systems Gauss Reduction LU Decomposition Polynomial Interpolation and Splines Fitting Definition of Interpolation General Lagrange Polynomial Interpolation Recurrence Formulae Equally Spaced Data: Difference Operators Errors in Polynomial Interpolation Local Interpolation: Spline Functions Concluding Remarks Numerical Differentiation and Integration Introduction Mathematical Prerequisites Numerical Differentiation Richardson extrapolation Richardson Extrapolation in Numerical Differentiation Numerical Integration Romberg Integration Appendix Advanced Numerical Integration Numerical Integration for Nonuniform Partitions Numerical Integration of Functions of Two Variables Monte Carlo Simulations for Numerical Quadrature Numerical Solutions of Ordinary Differential Equations (ODEs) Introduction Analytic Solutions to ODE Mathematical Settings for Numerical Solutions to ODEs Explicit Runge-Kutta Schemes Adams Multistep Methods Multistep Backward Difference Formulae Finite-Difference Approximation to a Two-Points Boundary Value Problem Bibliography Index Exercises and Computer Projects appear at the end of each chapter.