Math 344 - Numerical Methods II

Course Outline (Spring, 2008)

This is a tentative outline and will be updated at least 24 hours in advance before each class.

Hour exam are scheduled on February 8, March 7, and April 11.

Lecture	Topic/Section
	Numerical Methods for Solving Ax=b
1.	Linear Algebra Review - (3.0) (1) Matrix operations: A+B,A-B, A*B, A’ (2) Determinant of a matrix: det(A) (3) Inverse Matrix: A^(-1) (4) Conditions for the existence of A^(-1) Lecture Notes Homework Assignment 1 Solutions
2.	Gaussian Elimination (GE) – (3.1) (1) Three elementary row operations: (2) Gaussian Elimination Algorithm (3) Operation counts of GE Lecture Notes
3.	Pivoting Strategies – (3.2) (1) Why pivoting strategies are needed in computation? (2) Three pivoting strategies: (i) partial pivoting (ii) scaled partial pivoting (iii) total pivoting Lecture Notes Homework Assignment 2 Solutions MatLab programs: gaussbs.m gaussbsmr.m
4.	LU Factorization – (3.5)(3.7) (1) LU factorization by Gaussian Elimination (2) Computation of det(A) and solving Ax=b using LU factorization (3) LU factorization of a symmetric matrix Lecture Notes Homework Assignment 3
5.	Vector and Matrix Norms – (3.3) (1) Vector norms (2) Cauchy-Schwartz inequality (3) Matrix norms (4) Spectral radius of a square matrix (5) Properties of matrix norms and spectral radius Lecture Notes
6*.	Errors Estimates and Condition Number – (3.4) (1) Error estimate (2) Condition number of a matrix (3) Perturbation errors in solving systems of linear equations
7.	Iterative Methods for Solving Ax=b – (3.8) (1) Basic concepts (2) Three methods: (i) Jacobi Method (ii) Gauss-Seidel Method (iii) Successive Over-Relaxation (SOR) Method Lecture Notes Homework Assignment 4 Solutions MatLab Commands for the example on solving Ax=b using Jacobi, Gauss-Seidel and SOR methods: ex_Jacobi_GS_SOR.m Hour Exam 1 Solutions Jacobi.m Gauss_Seidel.m SOR.m Homework Assignment 5
	Numerical Methods for Finding Eigenvalues and Eigenvectors of A: Ax=cx
1.	The Power Method – (4.1) (1) Review of Eigenvalues and Eigenvectors of a Square Matrix (2) Properties (3) Power Method: Idea – derivation (4) Algorithm Lecture Notes Homework Assignment 6
2.	The Inverse Power Method – (4.2) (1) Idea – derivation (2) Algorithm PowerMethod.m Sym_PowerMethod.m
3.	Deflation – (4.3) (1) Deflation technique (2) Deflation for a symmetric matrix Lecture Notes lecture4_2_ex1.m
	Numerical Methods for Solving Initial Value Problems of ODE: f(x,y,y’)=0, x>=a, y(a)=y0.
1.	Basic Concepts and Elementary Theory of IVP of ODE - (7.1) (1) One-Step and Multistep Methods – Finite difference formulas for (2) Explicit methods and implicit methods (3) Elementary Theory of IVP: (i) A Sufficient Condition for Lipschitz Condition (ii) A Sufficient Condition for the Uniqueness of Solutions (4) Picard's Method Lecture Notes
2.	Euler method - (7.2) (1) Derivation of Euler Method (2) Algorithm (3) Approximation error and convergence Lecture Notes eulfun.m (create fun.m) Homework 9 Solutions
3.	Higher-order One-Step Methods: Taylor methods – (7.3) (1) Taylor methods of order n (2) Convergence Homework 10 Lecture Notes
4.	Rung-Kutta methods - (7.4) (1) Taylor polynomial and remainder in two variables (2) Runge-Kutta Methods of Order 2 (3) Runge-Kutta Method of Order 4 Homework 11 Solutions Lecture Notes rk2modeul.m, rk2henu.m, rkmid.m, rk4.m
5.	Multistep Methods - (7.5) (1) Explicit multistep methods - Adams-Bashforth Methods Derivation, algorithm and truncation error (2) Implicit multistep methods - Adams-Moulton Multistep Methods Derivation, algorithm and truncation error (3) Predictor-Corrector Schemes (4) Other multistep methods Homework 12 Lecture Notes
*6.	Convergence and Stability Analysis – (7.6) (1) One-step methods: Conditions for stable, consistent and convergent methods (2) multi-step methods; Root condition Definitions and conditions for stable, consistent and convergent methods
	Numerical Methods for Solving Initial Value Problems of BVP: y’’=f(x,y,y’), a<=x<=b, y(a)=y0, y(b)=y1.
1.	The Shooting Method for Linear BVPs – (8.4) (1) Existence and uniqueness of a boundary value problem (2) Linear shorting method Lecture Notes
2.	The Shooting Method for Nonlinear BVPs – (8.5) (1) Shooting method for nonlinear boundary value problems
3.	Linear Problems with Dirichlet Boundary Conditions - (8.1) (1) Finite different approximations (2) Formulation of A, and b (3) Sufficient condition for AY=b to have a unique solution (4) Upwind Method
4.	Linear Problems with Non-Dirichlet Boundary Conditions - (8.2)

back to the beginning