Numerical Methods

Outline

• Preface
• Introduction
  • Basics
  • Errors: general considerations
  • Methodological and roundoff errors
• Numerical methods for linear inhomogeneous sets of equations
• Aim of the direct methods: transformation of the matrix of coefficients into a triangular matrix
• LU decomposition
• LUP decomposition
• Iterative improvement of the solution
• Rounding errors in ill-conditioned and singular systems
• Methods for direct solution of systems with special matrix of coefficients
• The Gauss-Seidel method
• Linear Algebra Libraries
• Apps
• Two coupled linear algebraic equations
• Three coupled linear algebraic equations
• Interpolation of point sets
• Linear interpolation
• Trapezoidal rule
• Cubic splines
• Simpson's rule
• Discrete Fourier Transform
• Fourier analysis of real data sets
• Fast Fourier Transform (FFT)
• Important applications of FT
• Least squares approximation
• The basic problem
• Mathematical formulation of the problem
• Statistical analysis of the least squares problem
• Model Functions with Linear Parameters
• Model functions with non linear parameters
• Add-ons
• Apps
• Three parameter curve fitting
• Numerical solution of equations
• Graphical solutions
• Iterative methods
• The Newton Raphson method
• The Regula Falsi
• The bissection method
• Non linear systems
• Numerical Integration
• Numerical integration of point wise integrands
• Trapezoidal rule
• Simpson's rule
• Use of quadrature formulas
• The composite trapezoidal and Simpson formula
• An efficient implementation of the trapezoidal formula
• The programs QTRAP and TRAPZD
• The Romberg method
• The Gaussian quadrature formula
• Numerical calculation of improper integrals
• Variant of the Gaussian quadrature
• Multiple integrals
• Apps
• Numerical integration and differentiation
• Eigenvalues and Eigenvectors of real matrices
• Introduction: general and regular eigenvalue problems
• Numerical solution of regular eigenvalue problems
• The method of von Mises
• The method of Jacobi
• Eigenvalues of generic real matrices
• Apps
• Matrices
• Numerical methods for ordinary differential equations: initial value problems
• General considerations
• Expansion of the solution in Taylor series
• Euler's method
• Runge-Kutta Methods
• Solving a first order differential equation by fourth order Runge-Kutta
• Solving a second order differential equation by fourth order Runge-Kutta
• The programs ODEINT, RKQC and RK4
• The Runge-Kutta-Fehlberg method
• Other numerical methods for initial value problems
• References
• Numerical Solution of Ordinary Differential Equations, R. Sureshkumar, MIT 10.001 Course Notes
• Numerical methods for ordinary differential equations: boundary value problems
• The second-order linear boundary value problem
• Numerical treatment of the inhomogeneous BVP using the difference method
• Numerical solution of the homogeneous BVP using difference method
• The shooting method
• Apps
• Constant source diffusion
• Limited source diffusion
• Vibrations of a beam with fixed ends
• Vibrations of a cantilever beam
• Computer supported measurement techniques
• Bibliography
• Fortran library
• Example apps