Short Syllabus

Detailed Syllabus

Approximations in Scientific Computation
  Sources of Approximation
  Absolute Error and Relative Error
  Data Error and Computational Error
  Truncation Error and Rounding Error
  Forward and Backward Error
  Sensitivity and Conditioning
  Stability and Accuracy 

Systems of Linear Equations

  Solving Linear Systems
    Problem Transformations
    Triangular Linear Systems
    Elementary Elimination Matrices
    LU Factorization
    Implementation of LU Factorization
    Complexity of Solving Linear Systems
  Existence and Uniqueness
  Sensitivity and Conditioning
    Vector Norms
    Matrix Norms
    Matrix Condition Number
    Error Bounds
  Solving Modified Problems
  Improving Accuracy 
  Special Types of Linear Systems
    Symmetric Positive Definite Systems
    Banded Systems 
    Iterative Methods for Linear Systems
Overdetermined Linear Systems
  Linear Least Squares Problems
    Existence and Uniqueness
    Normal Equations
    Orthogonality and Orthogonal Projectors 
  Sensitivity and Conditioning
  Problem Transformations
    Normal Equations
    Augmented System Method
    Orthogonal Transformations
    Triangular Least Squares Problems
    QR Factorization 
  Orthogonalization Methods (Overview coverage only)
    Householder Transformations
    Givens Rotations
    Gram-Schmidt Orthogonalization
  Singular Value Decomposition
    Other Applications of SVD 
  Comparison of Methods

Interpolation and Approximation
  Existence, Uniqueness, and Conditioning
  Polynomial Interpolation
    Monomial Basis
    Lagrange Interpolation
    Newton Interpolation
    Orthogonal Polynomials (Cursorly)
    Interpolating Continuous Functions 
  Piecewise Polynomial Interpolation
    Hermite Cubic Interpolation
    Cubic Spline Interpolation

Eigenvalue Problems
  Eigenvalues and Eigenvectors
    Existence and Uniqueness
      Characteristic Polynomial
      Multiplicity and Diagonalizability
      Eigenspaces and Invariant Subspaces
      Properties of Matrices and Eigenvalue Problems
      Localizing Eigenvalues 
  Sensitivity and Conditioning
    Problem Transformations
    Diagonal, Triangular, and Block Triangular Forms 
  Computing Eigenvalues and Eigenvectors
    Power Iteration
    Inverse Iteration
    Simultaneous Iteration
    QR Iteration
    Jacobi Method
    Comparison of Methods 
    Generalized Eigenvalue Problems
  Computing the Singular Value Decomposition

Nonlinear Equations
    Existence and Uniqueness
    Sensitivity and Conditioning
    Convergence Rates and Stopping Criteria
    Nonlinear Equations in One Dimension
      Interval Bisection
      Fixed-Point Iteration
      Newton's Method
      Secant Method
      Safeguarded Methods      
    Systems of Nonlinear Equations
      Fixed-Point Iteration
      Newton's Method
      Secant Updating Methods
      Robust Newton-Like Methods 

  Optimization Problems
  Existence and Uniqueness
    Unconstrained Optimality Conditions
    Constrained Optimality Conditions 
  Sensitivity and Conditioning
  Optimization in One Dimension
    Golden Section Search
    Newton's Method
    Safeguarded Methods 
  Multidimensional Unconstrained Optimization
    Direct Search
    Steepest Descent
    Newton's Method
    Quasi-Newton Methods
    Secant Updating Methods
    Conjugate Gradient Method
    Truncated or Inexact Newton Methods 
  Nonlinear Least Squares
    Gauss-Newton Method
    Levenberg-Marquardt Method 
Numerical Integration and Differentiation
    Existence, Uniqueness, and Conditioning
    Numerical Quadrature
      Newton-Cotes Quadrature
      Gaussian Quadrature
      Composite Quadrature
      Adaptive Quadrature 
    Other Integration Problems
      Tabular Data
      Romberg Integration
      Double Integrals
      Multiple Integrals  
    Transforming Integrals
      Improper Integrals
    Integral Equations

  Numerical Differentiation
    Finite Difference Approximations
    Automatic Differentiation 
    Richardson Extrapolation

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