MCS 471 Numerical Analysis --- Summer 2007

List of Topics by Lecture

Detailed List of Topics by Lecture: (during summer - 1 week of lectures covers 2 weeks in the list below)

Section Numbers (<number>):

• Week 1: Chapter 0 Fundamentals - evaluating a polynomial (0.1); floating point representation, IEEE standard, machine epsilon (0.3); significant digits; catastrophic cancellation (0.4); review of Calculus and algorithms (0.5).
• Week 2: Chapter 1 Solving Equations - Bisection method, accuracy (1.1); fixed-point Iteration, cobweb diagrams, convergence (1.2); forward and backward error, multiple roots (1.3)
• Week 3: Newton's method, quadratic convergence rate (1.4); Secant Method (skip False Position), Muller's Method, hybrid methods (1.5)
• Week 4: Chapter 2 Solving Equations - Forward Gaussian Elimination and Backward Substitution, computational complexity of FGE & BS (2.1); LU Decomposition (2.2);
• Week 5: sources of error, residuals, small pivots, norms, condition numbers (2.3); LU with virtual row partial pivoting and virtual row scaling, (2.4)
• Week 6: continue with partial pivoting, etc., iterative methods (2.5); Nonlinear systems (2.7)
• Week 7: Chapter 3 Interpolation- Lagrange interpolation, Newton's divided differences (3.1); Newton-Gregory polynomial form for equally spaced data; interpolation error (3.2)
• Week 8: Chebyshev interpolation (3.3); splines (3.4); Chapter 4 Least Squares - least squares approximation (4.1)
• Week 9: Chapter 5 - Numerical Calculus - approximation of derivatives, forward, backward and central finite differences (5.1); numerical integration, Newton-Cotes Rules and local errors: (5.2);
• Week 10: composite rules, global errors (5.2); Romberg integration (5.3); adaptive integration (5.4) Gaussian quadrature (5.5)
• Week 11: Chapter 6 ODE's - Review ODEs, Euler's method (6.1); more numerical solutions of ODEs, local theoretical error, modified Euler's method (6.2); systems of ODE's (6.3);
• Week 12: Runge-Kutta Method [RK4] (6.4); predictor-corrector multistep methods (Adams-Moulton Method) (6.7);
• Week 13: Chapter 7 Boundary Value Problems and Shooting Method - shooting method (7.1); finite differences (7.2);
• Week 14: Chapter 12 Eigenvalue Problems - power method (12.1); catch-up
• Week 15: Exams and Review