Introduction to Numerical Analysis

by Arnold Neumaier

• ISBN13: 9780521333238
• ISBN10: 0521333237
• Format: Hardcover
• Copyright: 2001-10-01
• Publisher: Cambridge University Press

Summary

Numerical analysis is an increasingly important link between pure mathematics and its application in science and technology. This textbook provides an introduction to the justification and development of constructive methods that provide sufficiently accurate approximations to the solution of numerical problems, and the analysis of the influence that errors in data, finite-precision calculations, and approximation formulas have on results, problem formulation and the choice of method. It also serves as an introduction to scientific programming in MATLAB, including many simple and difficult, theoretical and computational exercises. A unique feature of this book is the consequent development of interval analysis as a tool for rigorous computation and computer assisted proofs, along with the traditional material.

Author Biography

Arnold Neumaier is a Professor of Computational Mathematics at the University of Wien

Table of Contents

Preface	vii
The Numerical Evaluation of Expressions	1	(60)
Arithmetic Expressions and Automatic Differentiation	2	(12)
Numbers, Operations, and Elementary Functions	14	(9)
Numerical Stability	23	(10)
Error Propagation and Condition	33	(5)
Interval Arithmetic	38	(15)
Exercises	53	(8)
Linear Systems of Equations	61	(69)
Gaussian Elimination	63	(10)
Variations on a Theme	73	(9)
Rounding Errors, Equilibration, and Pivot Search	82	(12)
Vector and Matrix Norms	94	(5)
Condition Numbers and Data Perturbations	99	(7)
Iterative Refinement	106	(6)
Error Bounds for Solutions of Linear Systems	112	(7)
Exercises	119	(11)
Interpolation and Numerical Differentiation	130	(49)
Interpolation by Polynomials	131	(14)
Extrapolation and Numerical Differentiation	145	(8)
Cubic Splines	153	(12)
Approximation by Splines	165	(5)
Radial Basis Functions	170	(12)
Exercises	182
Numerical Integration	179	(54)
The Accuracy of Quadrature Formulas	179	(8)
Gaussian Quadrature Formulas	187	(9)
The Trapezoidal Rule	196	(7)
Adaptive Integration	203	(7)
Solving Ordinary Differential Equations	210	(9)
Step Size and Order Control	219	(6)
Exercises	225	(8)
Univariate Nonlinear Equations	233	(68)
The Secant Method	234	(7)
Bisection Methods	241	(9)
Spectral Bisection Methods for Eigenvalues	250	(6)
Convergence Order	256	(9)
Error Analysis	265	(8)
Complex Zeros	273	(8)
Methods Using Derivative Information	281	(12)
Exercises	293	(8)
Systems of Nonlinear Equations	301	(44)
Preliminaries	302	(9)
Newton's Method and Its Variants	311	(11)
Error Analysis	322	(7)
Further Techniques for Nonlinear Systems	329	(11)
Exercises	340	(5)
References	345	(6)
Index	351