Math 471   Introduction to Numerical Methods (Section 002)

Instructor Manabu Machida
Office 3836 East Hall
Office hours  MWTh 11am–12pm (and by appointment)

Meeting times  MWF 8am–9am at 4088 EH

Prerequisites:  Differential equations (Math 216, 256, 286, or 316), Linear algebra (214, 217, 417, or 419), and a working knowledge of one high-level computer language (Matlab). No credit granted to those who have completed or are enrolled in Math 371 or 472.

Background and Goals:  This is a survey of the basic numerical methods which are used to solve scientific problems. The emphasis is evenly divided between the analysis of the methods and their practical applications. Some convergence theorems and error bounds are proven. The course also provides an introduction to MATLAB, an interactive program for numerical linear algebra, as well as practice in computer programming. One goal of the course is to show how calculus and linear algebra are used in numerical analysis. There is software that can be used as a black box, but in this course we will look under the hood and see how the methods work.

Content:  Topics may include computer arithmetic, Newton's method for non-linear equations, polynomial interpolation, numerical integration, systems of linear equations, initial value problems for ordinary differential equations, quadrature, partial pivoting, spline approximations, partial differential equations, Monte Carlo methods, 2-point boundary value problems, and the Dirichlet problem for the Laplace equation.

Alternatives:  Math 371 is a less sophisticated version intended principally for sophomore and junior engineering students.

Subsequent Courses:  The sequence Math 571–572 is mainly taken by graduate students, but should be considered by strong undergraduates. Math 471 is good preparation for Math 571 and 572, although it is not prerequisite to these courses.

A Friendly Introduction to Numerical Analysis
by Brian Bradie
Pearson Prentice Hall, 1st ed., 2005
ISBN10: 0130130540, ISBN13: 9780130130549


Grading Policy:

Student Data Form:  FormFAQ

Lecture Notes:  (see also Prof. Krasny's lecture notes)


1:  Wed,Sep4   Floating point arithmetic  (first day handout)
2:  Fri,Sep6   Finite difference approximation of a derivative
3:  Mon,Sep9   First Matlab computation  (supplement)
4:  Wed,Sep11   The bisection method  (supplement)
5:  Fri,Sep13   Fixed-point iteration  Homework Set 1 Due  (problems)
6:  Mon,Sep16   Newton's method
7:  Wed,Sep18   Review of linear algebra  Quiz
8:  Fri,Sep20   Gaussian elimination
9:  Mon,Sep23   Pivoting
10:  Wed,Sep25   Vector and matrix norms
11:  Fri,Sep27   Error analysis  Homework Set 2 Due  (problems)
12:  Mon,Sep30   LU factorization
13:  Wed,Oct2   Two-point boundary value problem
14:  Fri,Oct4   Two-point boundary value problem  (supplement)  Homework Set 3 Due  (problems)
15:  Mon,Oct7   Iterative methods: Jacobi method and Gauss-Seidel method
16:  Wed,Oct9   Convergence of iterations
17:  Fri,Oct11   Operation counts
Mon,Oct14   Fall Study Break
18:  Wed,Oct16   SOR
19:  Fri,Oct18   SOR  Homework Set 4 Due  (problems)
20:  Mon,Oct21   Two-dimensional boundary-value problems  (supplement)
21:  Wed,Oct23   Review
Fri,Oct25   Midterm Exam  (solutions)
22:  Mon,Oct28   Rayleigh quotient
23:  Wed,Oct30   The power method and inverse power method
24:  Fri,Nov1   Polynomial approximation  Homework Set 5 Due  (problems)  Project 1 Due  (problems)
25:  Mon,Nov4   Polynomial interpolation
26:  Wed,Nov6   Newton's form
27:  Fri,Nov8   Optimal interpolation points  (supplement)
28:  Mon,Nov11   Error analysis
29:  Wed,Nov13   Error analysis
30:  Fri,Nov15   Piecewise linear interpolation  Homework Set 6 Due  (problems)
31:  Mon,Nov18   Spline interpolation
32:  Wed,Nov20   Numerical integration
33:  Fri,Nov22   Richardson extrapolation
34:  Mon,Nov25   Orthogonal polynomials
35:  Wed,Nov27   Gaussian quadrature  Homework Set 7 Due  (problems)  Project 2 Due  (problems)
Fri,Nov29   Thanksgiving recess
36:  Mon,Dec2   Gaussian quadrature
37:  Wed,Dec4   Euler's method
38:  Fri,Dec6   Second-order Runge-Kutta method  Homework Set 8 Due  (problems)
39:  Mon,Dec9   Fourth-order Runge-Kutta method
40:  Wed,Dec11   Review  Homework Set 9 Due  (problems)
Thu,Dec19   Final Exam  (solutions)

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