##
Math 574 Lecture Notes

Lecture 1: Solution of linear systems
by direct methods
(Gaussian elimination and LU factorization.)
Lecture 2: Solution of linear systems
by direct methods
(Choleski decomposition, advantages of partial pivoting, vector
and matrix norms.)
Lecture 3: Perturbation theory
for linear systems of equations
(Estimates for the relative error, condition number of a matrix.)
**Revised 1/29/2016**
Lecture 4: Matrix iterative methods
(Jacobi, Gauss-Seidel, SOR, and convergence results.) **Revised 1/29/2016**
Lecture 5: Optimization methods
(Steepest descent and conjugate gradient methods.) **Revised 2/10/2016**
Lecture 6: Calculation of Eigenvalues and
Eigenvectors
(Canonical forms of matrices, perturbation theory for eigenvalues
and eigenvectors)
Lecture 7: Numerical Methods for Eigenvalues
and Eigenvectors
(Power and inverse power methods, Gershgorin's theorem)
Lecture 8: QR Algorithm
(Reduction to Hessenberg form, QR factorization of a matrix, QR algorithm)
Lecture 9: QR Algorithm
(Convergence of the QR algorithm)
Lecture 10: Solution of Nonlinear Equations
(Bisection, false position, secant, Newton's method; fixed point iteration)
Lecture 11: Solution of Nonlinear Equations
(Local convergence results, order of convergence, combining methods)
Lecture 12: Solution of Nonlinear Systems
of Equations
(Newton's method, Broyden's method, local convergence results, obtaining
good initial approximations)
Lecture 13: Minimization Problems
(Newton and quasi-Newton methods, steepest descent, Levenberg-Marquardt
method)
Lecture 14: Two-Point Boundary Value Problems
(Shooting method, finite difference method)
Lecture 15: Analysis of Finite Difference Methods
(Discrete maximum principle, stability, error estimates)
Lecture 16: Introduction to the Finite Element
Method
(Variational formulation for Dirichlet boundary conditions,
piecewise polynomial approximation), **Revised 4/4/2016**,
**Revised 5/2/2016**
Lecture 17: Finite Element Method II
(discretized equations, existence-uniqueness, quasi-optimal approximation
of the finite element solution), **Revised 4/13/2016**
Lecture 18: Finite Element Method -- III
(error estimates for piecewise polynomial approximation, error
estimates for the finite element solution, other boundary conditions),
**Revised 5/2/2016**
Lecture 19: Finite difference methods
for the heat equation
(approximation schemes, stability, error estimates)
Lecture 20: Finite element methods
for parabolic problems
(continuous time and fully discrete schemes, error estimates),
**Revised 4/22/2016**
Lecture 21: Finite difference methods
for elliptic equations in 2 dimensions
(stability and convergence, curved boundaries, other boundary conditions,
higher order approximations), **Revised 4/22/2016**