First Course in Numerical Methods

ISBN-10: 0898719976

ISBN-13: 9780898719970

Edition: 2011

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A First Course on Numerical Methods is designed for students and researchers who seek practical knowledge of modern techniques in scientific computing. Avoiding encyclopaedic and heavily theoretical exposition, the book provides an in-depth treatment of fundamental issues and methods, the reasons behind the success and failure of numerical software, and fresh and easy-to-follow approaches and techniques. The authors focus on current methods, issues and software while providing a comprehensive theoretical foundation, enabling those who need to apply the techniques to successfully design solutions to nonstandard problems. The book also illustrates algorithms using the programming environment of MATLAB®, with the expectation that the reader will gradually become proficient in it while learning the material covered in the book. The book takes an algorithmic approach, focusing on techniques that have a high level of applicability to engineering, computer science and industrial mathematics.
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Book details

Copyright year: 2011
Publisher: Society for Industrial and Applied Mathematics
Binding: Paperback
Pages: 580
Size: 6.75" wide x 9.75" long x 1.00" tall
Weight: 2.574
Language: English

Uri Ascher is a Professor of Computer Science at the University of British Columbia in Vancouver, Canada. He has previously co-authored three other books, published by SIAM, as well as many research papers in the general area of numerical methods and their applications. He is a SIAM Fellow and a recipient of the CAIMS Research Prize.

List of Figures
List of Tables
Numerical Algorithms
Scientific computing
Numerical algorithms and errors
Algorithm properties
Additional notes
Roundoff Errors
The essentials
Floating point systems
Roundoff error accumulation
The IEEE standard
Additional notes
Nonlinear Equations in One Variable
Solving nonlinear equations
Bisection method
Fixed point iteration
Newton's method and variants
Minimizing a function in one variable
Additional notes
Linear Algebra Background
Review of basic concepts
Vector and matrix norms
Special classes of matrices
Singular values
Additional notes
Linear Systems: Direct Methods
Gaussian elimination and backward substitution
LU decomposition
Pivoting strategies
Efficient implementation
The Cholesky decomposition
Sparse matrices
Permutations and ordering strategies
Estimating errors and the condition number
Additional notes
Linear Least Squares Problems
Least squares and the normal equations
Orthogonal transformations and QR
Householder transformations and Gram-Schmidt orthogonalization
Additional notes
Linear Systems: Iterative Methods
The need for iterative methods
Stationary iteration and relaxation methods
Convergence of stationary methods
Conjugate gradient method
*Krylov subspace methods
*Multigrid methods
Additional notes
Eigenvalues and Singular Values
The power method and variants
Singular value decomposition
General methods for computing eigenvalues and singular values
Additional notes
Nonlinear Systems and Optimization
Newton's method for nonlinear systems
Unconstrained optimization
*Constrained optimization
Additional notes
Polynomial Interpolation
General approximation and interpolation
Monomial interpolation
Lagrange interpolation
Divided differences and Newton's form
The error in polynomial interpolation
Chebyshev interpolation
Interpolating also derivative values
Additional notes
Piecewise Polynomial Interpolation
The case for piecewise polynomial interpolation
Broken line and piecewise Hermite interpolation
Cubic spline interpolation
Hat functions and B-splines
Parametric curves
*Multidimensional interpolation
Additional notes
Best Approximation
Continuous least squares approximation
Orthogonal basis functions
Weighted least squares
Chebyshev polynomials
Additional notes
Fourier Transform
The Fourier transform
Discrete Fourier transform and trigonometric interpolation
Fast Fourier transform
Additional notes
Numerical Differentiation
Deriving formulas using Taylor series
Richardson extrapolation
Deriving formulas using Lagrange polynomial interpolation
Roundoff and data errors in numerical differentiation
*Differentiation matrices and global derivative approximation
Additional notes
Numerical Integration
Basic quadrature algorithms
Composite numerical integration
Gaussian quadrature
Adaptive quadrature
Romberg integration
*Multidimensional integration
Additional notes
Differential Equations
Initial value ordinary differential equations
Euler's method
Runge-Kutta methods
Multistep methods
Absolute stability and stiffness
Error control and estimation
*Boundary value ODEs
*Partial differential equations
Additional notes
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