Concise Numerical Mathematics

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American Mathematical Soc., 2003 - Mathematics - 453 pages
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This book succinctly covers the key topics of numerical methods. While it is basically a survey of the subject, it has enough depth for the student to walk away with the ability to implement the methods by writing computer programs or by applying them to problems in physics or engineering. The author manages to cover the essentials while avoiding redundancies and using well-chosen examples and exercises.The exposition is supplemented by numerous figures. Work estimates and pseudo codes are provided for many algorithms, which can be easily converted to computer programs. Topics covered include interpolation, the fast Fourier transform, iterative methods for solving systems of linear and nonlinear equations, numerical methods for solving ODEs, numerical methods for matrix eigenvalue problems, approximation theory, and computer arithmetic. In general, the author assumes only a knowledge of calculus and linear algebra. The book is suitable as a text for a first course in numerical methods for mathematics students or students in neighboring fields, such as engineering, physics, and computer science.
 

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Contents

Interpolation by Polynomials
1
Spline Functions
23
The Discrete Fourier Transform and Its Applications
41
Solution of Linear Systems of Equations
59
Nonlinear Systems of Equations
105
The Numerical Integration of Functions
123
Explicit OneStep Methods for Initial Value Problems in Ordinary Differential Equations
161
Multistep Methods for Initial Value Problems of Ordinary Differential Equations
189
The Conjugate Gradient and GMRES Methods
311
Eigenvalue Problems
339
Numerical Methods for Eigenvalue Problems
355
Peanos Error Representation
393
Approximation Theory
401
Computer Arithmetic
423
Bibliography
443
Index
449

Boundary Value Problems for Ordinary Differential Equations
247
Jacobi GaussSeidel and Relaxation Methods for the Solution of Linear Systems of Equations
281

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Page xiv - The paper deals with some new methods for the numerical solution of initial value problems for ordinary differential equations.
Page xiii - THE NUMERICAL SOLUTION OF INITIAL AND BOUNDARY VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS.

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