642:574 Numerical Analysis: Course Syllabus

4: Numerical Solution of Systems of Linear Equations

4.1 Gaussian elimination
4.2 Pivoting and scaling in Gaussian elimination
4.3 Variants of Gaussian elimination
4.4 Error analysis 
4.5 Basic iterative methods: Jacobi, Gauss-Seidel, SOR
4.6 General theory of iterative methods
4.7 Conjugate gradient method

5: The Matrix Eigenvalue Problem

5.1 Eigenvalue location, error, and stability results
5.2 The power and inverse power methods
5.3 The QR method
5.4 The calculation of eigenvectors and inverse iteration

6: Numerical Solution of Nonlinear Equations

6.1 Some basic methods: bisection, false position, secant, and Newton's methods
6.2 Convergence theory for one-point iteration methods
6.3 Systems of nonlinear equations: Newton's method
6.4 Quasi-Newton methods
6.5 Unconstrained optimization

7: Two Point Boundary Value Problems: Finite Difference and Finite Element Methods

7.1 Derivation of finite difference methods
7.2 Error estimates for finite difference methods
7.3 Derivation of finite element methods
7.4 Error estimates for finite element methods

8: Finite Difference Methods for Model Problems in Partial Differential Equations

8.1 Poisson's equation
8.2 Heat equation