A concise introduction to numerical analysis

✍ Scribed by Faul, Anita C

• Publisher: Chapman and Hall/CRC
• Year: 2016
• Tongue: English
• Leaves: 292
• Category: Library

✦ Synopsis

This textbook provides an accessible and concise introduction to numerical analysis for upper undergraduate and beginning graduate students from various backgrounds. It was developed from the lecture notes of four successful courses on numerical analysis taught within the MPhil of Scientific Computing at the University of Cambridge. The book is easily accessible, even to those with limited knowledge of mathematics.

Students will get a concise, but thorough introduction to numerical analysis. In addition the algorithmic principles are emphasized to encourage a deeper understanding of why an algorithm is suitable, and sometimes unsuitable, for a particular problem.

A Concise Introduction to Numerical Analysis strikes a balance between being mathematically comprehensive, but not overwhelming with mathematical detail. In some places where further detail was felt to be out of scope of the book, the reader is referred to further reading.

The book uses MATLAB^® implementations to demonstrate the workings of the method and thus MATLAB's own implementations are avoided, unless they are used as building blocks of an algorithm. In some cases the listings are printed in the book, but all are available online on the book’s page at www.crcpress.com.

Most implementations are in the form of functions returning the outcome of the algorithm. Also, examples for the use of the functions are given. Exercises are included in line with the text where appropriate, and each chapter ends with a selection of revision exercises. Solutions to odd-numbered exercises are also provided on the book’s page at www.crcpress.com.

This textbook is also an ideal resource for graduate students coming from other subjects who will use numerical techniques extensively in their graduate studies.

✦ Table of Contents

Content: Fundamentals Floating Point Arithmetic Overflow and Underflow Absolute, Relative Error, Machine Epsilon Forward and Backward Error Analysis Loss of Significance Robustness Error Testing and Order of Convergence Computational Complexity Condition Revision Exercises Linear Systems Simultaneous Linear Equations Gaussian Elimination and Pivoting LU Factorization Cholesky Factorization QR Factorization The Gram-Schmidt Algorithm Givens Rotations Householder Reflections Linear Least Squares Singular Value Decomposition Iterative Schemes and Splitting Jacobi and Gauss-Seidel Iterations Relaxation Steepest Descent Method Conjugate Gradients Krylov Subspaces and Pre-Conditioning Eigenvalues and Eigenvectors The Power Method Inverse Iteration Deflation Revision Exercises Interpolation and Approximation Theory Lagrange Form of Polynomial Interpolation Newton Form of Polynomial Interpolation Polynomial Best Approximations Orthogonal polynomials Least-Squares Polynomial Fitting The Peano Kernel Theorem Splines B-Spline Revision Exercises Non-Linear Systems Bisection, Regula Falsi, and Secant Method Newton's Method Broyden's Method Householder Methods Muller's Method Inverse Quadratic Interpolation Fixed Point Iteration Theory Mixed Methods Revision Exercises Numerical Integration Mid-Point and Trapezium Rule The Peano Kernel Theorem Simpson's Rule Newton-Cotes Rules Gaussian Quadrature Composite Rules Multi-Dimensional Integration Monte Carlo Methods Revision Exercises ODEs One-Step Methods Multistep Methods, Order, and Consistency Order Conditions Stiffness and A-Stability Adams Methods Backward Differentiation Formulae The Milne and Zadunaisky Device Rational Methods Runge-Kutta Methods Revision Exercises Numerical Differentiation Finite Differences Differentiation of Incomplete or Inexact Data PDEs Classification of PDEs Parabolic PDEs Elliptic PDEs Parabolic PDEs in Two Dimensions Hyperbolic PDEs Spectral Methods Finite Element Method Revision Exercises

✦ Subjects

Numerical analysis;Textbooks.;Numerical analysis.