Numerical Analysis:
Adventures in Frustration

Jim Peterson

September 27, 2001

• Numerical Linear Algebra
  • A Simple Lower Triangular System:
    • A Lower Triangular Solver:
    • An Upper Triangular Solver:
    • The LU Decomposition of A Without Pivoting:
    • The LU Decomposition of A With Pivoting:
• EigenValues and EigenVectors
  • The MatLab Approach:
  • Finding Eigenvalues and Eigenvectors:
  • Symmetric Arrays:
• Getting Only a Few Eigenvalues
  • Facial Recognition Background:
  • Matlab Away!
    • The eigs Command:
    • A Simple Example:
    • A Covariance Example:
• Solving Least Squares Systems: SVD Approach
  • The Singular Value Decomposition:
  • Using the SVD of A For Least Squares:
  • Doing It In Matlab!:
    • Our MatLab Session:
  • Solving Least Squares: The QR Approach:
  • The Theory:
    • Plugging It Into Matlab:
• Root Finding and Simple Optimization
  • Root Finding:
    • The Bisection Method:
      • The Bisection Matlab Code:
      • Running the Code:
      • Exercises:
    • Newton's Method:
      • When Do We Do A Newton Step?
      • A Global Newton Method:
      • A Run Time Example:
      • Some Exercises:
    • Adding Finite Difference Approximations to the Derivative:
      • A Finite Difference Global Newton Method:
      • A Run Time Example:
      • Some Exercises:
• Data Fitting
  • Interpolation:
    • The Basics:
    • The Vandermonde Approach:
      • VanderMonde Matlab Code:
      • Run Time Output:
      • Evaluating the Interpolating Polynomial:
      • Horner's Method in Matlab:
      • Run Time Output:
    • Cubic Hermite Interpolation:
      • Just One Interval:
      • The General Problem: n Intervals:
      • The Basic Matlab Code:
      • Locating a Point in the Interval:
      • Evaluating a Piecewise Cubic:
      • Some Run Time Results:
      • Exercises:
    • General Cubic Splines:
      • The Basics:
      • EndPoint Conditions:
      • Exercises:
• Integration
  • Numerical Integration:
  • Newton-Cotes Formulae:
    • The Basics:
    • Evaluating S_mk:
    • Exercise:
    • matlab Implementation:
    • Run Time Output:
• Ordinary Differential Equations:
  • Euler's Method:
    • The Matlab Implementation:
    • The RunTime: Just Tables
    • The RunTime: Plots!
  • Runge-Kutta Methods:
    • The Matlab Implementation:
    • The RunTime: Just Tables
    • The RunTime: Plots!
  • The Adams Methods: Theory
    • The Basics:
    • Adams-Bashford Methods:
    • Adams-Moulton Methods:
  • The Adams Bashford Methods in Matlab:
    • The Implementation:
    • The RunTime: Just Tables
    • The RunTime: Plots!
  • The Adams Moulton Methods in Matlab:
    • The Implementation:
    • The RunTime: Just Tables
    • The RunTime: Plots!
  • Systems of Differential Equations:
  • How Do We Solve Systems?
    • Setting Up the Vector Functions:
    • Updating Our Solver Codes:
  • Solving The BVP:
• Loading Images
  • Reading In the Image:
    • Displaying the Image:
  • Manipulating the Image:
• The Covariance Approach to Recognition:
  • The Basic Plan:
    • Collect Images:
    • The Original Pauli Images:
    • Computing the Pauli Average Image:
    • The Pauli Test Image:
  • Image Processing:
    • Compute the Average of All Images:
    • Compute Difference Images:
    • Convert Image Matrices to Vectors:
    • Convert The Difference Images to Base 255 Doubles:
    • Convert the Double Matrices to Vectors:
    • Compute the Covariance Matrix:
    • Compute Eigenvectors of A^T A:
    • Find Feature Vectors:
    • Compute Class Vectors:
    • Compute The Test Feature Vector:
  • Implementing The Basic Plan in Matlab:
  • Using the Functions In a Matlab Session:
  • Using the Build Functions:
• The Neural Network Approach:
  • The Neural Network ToolBox in Matlab:

This document was translated from L^AT_EX by H^EV^EA and H^AC^HA.