Numerical Methods

  1. Dr. George L. Trigg
  1. Christina C. Christara and
  2. Kenneth R. Jackson

Published Online: 4 MAY 2006

DOI: 10.1002/3527607773.ch10

Mathematical Tools for Physicists

Mathematical Tools for Physicists

How to Cite

Christara, C. C. and Jackson, K. R. (2005) Numerical Methods, in Mathematical Tools for Physicists (ed G. L. Trigg), Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, FRG. doi: 10.1002/3527607773.ch10

Editor Information

  1. New Paltz, New York, USA

Author Information

  1. Computer Science Department, University of Toronto, Toronto, Ontario, Canada

Publication History

  1. Published Online: 4 MAY 2006
  2. Published Print: 24 AUG 2005

ISBN Information

Print ISBN: 9783527405480

Online ISBN: 9783527607778

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Keywords:

  • numerical methods;
  • floating-point arithmetic;
  • direct solution of linear algebraic systems;
  • iterative solution of linear algebraic systems;
  • eigenvalues and eigenvectors of matrices;
  • nonlinear algebraic equations and systems;
  • unconstrained optimization;
  • approximation;
  • numerical integration – quadrature;
  • ordinary differential equations;
  • parallel computation

Summary

This chapter contains sections titled:

  • Floating-Point Arithmetic

    • The IEEE Standard

    • Rounding Errors

    • The Effects of Inexact Arithmetic: Some Illustrative Examples

  • The Direct Solution of Linear Algebraic Systems

    • Gaussian Elimination

    • Back Substitution

    • The LU Factorization

    • Forward Elimination

    • Scaling and Pivoting

    • The Cholesky Factorization

    • Banded and Sparse Matrices

    • Rounding Errors, Condition Numbers, and Error Bounds

    • Iterative Improvement

  • The Iterative Solution of Linear Algebraic Systems

    • Basic Iterative Methods

    • The Conjugate-Gradient Method

  • Overdetermined and Underdetermined Linear Systems

    • The Normal Equations for Overdetermined Linear Systems

    • The Normal Equations for Underdetermined Linear Systems

    • Householder Transformations and the QR Factorization

    • Using the QR Factorization to Solve Overdetermined Linear Systems

    • Using the QR Factorization to Solve Underdetermined Linear Systems

    • The Gram–Schmidt Orthogonalization Algorithm

    • Using Gram–Schmidt to Solve Overdetermined Linear Systems

    • Using Gram–Schmidt to Solve Underdetermined Linear Systems

  • Eigenvalues and Eigenvectors of Matrices

    • The Power Method

    • The QR Method

    • Transforming a Symmetric Matrix to Tridiagonal Form

    • Inverse Iteration

    • Other Methods

  • Nonlinear Algebraic Equations and Systems

    • Fixed-Point Iteration

    • Newton's Method for Nonlinear Equations

    • The Secant Method

    • The Bisection and Regula Falsi Methods

    • Convergence

    • Rate of Convergence

    • Newton's Method for Systems of Nonlinear Equations

    • Modifications and Alternatives to Newton's Method

    • Polynomial Equations

    • Horner's Rule

  • Unconstrained Optimization

    • Some Definitions and Properties

    • The Fibonacci and Golden-Section Search Methods

    • The Steepest-Descent Method

    • Conjugate-Direction Methods

    • The Conjugate-Gradient Method

    • Newton's Method

    • Quasi-Newton Methods

  • Approximation

    • Polynomial Approximation

    • Polynomial Interpolation

    • Polynomial Interpolation with Derivative Data

    • The Error in Polynomial Interpolation

    • Piecewise Polynomials and Splines

    • Piecewise Polynomial Interpolation

    • Least-Squares Approximation

  • Numerical Integration – Quadrature

    • Simple Quadrature Rules

    • Composite (Compound) Quadrature Rules

    • Adaptive Quadrature

    • Romberg Integration and Error Estimation

    • Infinite Integrals and Singularities

    • Monte-Carlo Methods

  • Ordinary Differential Equations

    • Initial-Value Problems (IVPs)

    • Boundary-Value Problems (BVPs)

  • Partial Differential Equations (PDEs)

    • Classes of Problems and PDEs

    • Classes of Numerical Methods for PDEs

    • Finite-Difference Methods for BVPs

    • Finite-Element Methods for BVPs

    • Finite-Difference Methods for IVPs

    • The Method of Lines

    • Boundary-Element Methods

    • The Multigrid Method

  • Parallel Computation

    • Cyclic Reduction

  • Sources of numerical software