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