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Table of Contents

  1. Introduction
  2. Error Analysis
    1. Precision and Accuracy
    2. Absolute and Relative Error
    3. Significant Digits
  3. Numeric Representations
    1. Decimal Numbers
    2. Binary Numbers
    3. Floating-point Representation
    4. Double-precision Floating-point Numbers
    5. Problems with the Floating-point Representation
  4. Iteration
  5. Linear Algebra
    1. PLU Decomposition
    2. PLU Decomposition of Tri-diagonal Matrices
    3. Positive-definite Matrices
    4. Cholesky Decomposition
    5. QR Decomposition
    6. Eigenvalues
    7. Norms and Condition Numbers
    8. Jacobi Method
    9. Gauss-Seidel Method
    10. Overrelaxation Techniques
  6. Interpolation
    1. Vandermonde Method
    2. Problems with Interpolation
    3. Lagrange Polynomials
    4. Newton Polynomials
    5. Horner's Rule
    6. Multivariate Interpolation
    7. Chebyshev Polynomials
    8. Legendre Polynomials
    9. Cubic Splines
  7. Least Squares
    1. Linear Regression with Linear Functions
    2. General Linear Regression
    3. Lack-of-Fit Tests
    4. Transformations
    5. Extrapolation
  8. Taylor Series
  9. Bracketing
  10. The Five Tools
  11. Root Finding
    1. Bisection Method
    2. False-position Method
    3. Newton's Method
    4. Secant Method
    5. Polynomials
    6. Müller's Method
    7. Newton's Method in Higher Dimensions
  12. Optimization
    1. Golden-mean Search
    2. Newton's Method
    3. Quadratic Optimization
    4. Gradient Descent
    5. Random Brute-force Search
    6. Simulated Annealing
  13. Differentiation
    1. Centred Divided-difference Formulae
    2. Backward Divided-difference Formulae
    3. Richardson Extrapolation
    4. Higher-order Derivatives
  14. Integration
    1. The Trapezoidal Rule
    2. The Composite-trapezoid Rule
    3. Romberg Integration
    4. Newton-Cotes Integration (Simpson's, Simpson's 3/8th, Boole's)
    5. Gaussian Integration
  15. Initial-Value Problems (IVPs)
    1. Euler's Method
    2. Heun's Method
    3. 4th-order Runge-Kutta
    4. Multiple-step Methods
    5. Runge-Kutta-Fehlberg
    6. Stiff ODEs
    7. Systems of IVPs
    8. Higher-order IVPs
  16. Boundary-Value Problems (BVPs)
    1. Finite-difference Methods
    2. Finite-element Methods
    3. Elliptic PDEs
    4. Parabolic PDEs
    5. Hyperbolic PDEs
  17. Linear Programming
    1. Simplex Method
    2. Interior-point Methods