NUMERICAL ANALYSIS SYLLABUS

First semester

1. Approximation and Interpolation
   • Minimax Polynomial Approximation
   • Lagrange Interpolation
   • Least Squares Polynomial Approximation
   • Piecewise polynomial approximation and interpolation
   • The Fast Fourier Transform
2. Numerical Quadrature
   • Basic quadrature
   • The Peano Kernel Theorem
   • Richardson Extrapolation
   • Asymptotic error expansions
   • Romberg Integration
   • Gaussian Quadrature
   • Adaptive quadrature
3. Direct Methods of Numerical Linear Algebra
   • Triangular systems
   • Gaussian elimination and LU decomposition
   • Pivoting
   • Backward error analysis
   • Conditioning
4. Numerical solution of nonlinear systems and optimization
   • One-point iteration
   • Newton's method
   • Quasi-Newton methods
   • Broyden's method
   • Unconstrained minimization
   • Newton's method
   • Line search methods
   • Conjugate gradients

Second semester

1. Numerical Solution of Ordinary Differential Equations
   • Euler's Method
   • Linear multistep methods
   • One step methods
   • Stiffness
2. Numerical Solution of Partial Differential Equations
   • BVPs for 2nd order elliptic PDEs
   • The five-point discretization of the Laplacian
   • Finite element methods
   • Difference methods for the heat equation
   • Difference methods for hyperbolic equations
   • Hyperbolic conservation laws
3. Some Iterative Methods of Numerical Linear Algebra
   • Classical iterations
   • Multigrid methods