CMSC/AMSC/MAPL 460 Computational Methods


Class:  TuTh......12:30pm- 1:45pm (CSI 1121)


Office Hours: Monday 10-11:30 and by appointment, in AVW 3365.


Instructor: Ramani Duraiswami  E-mail: ramani AT;    
Office Hours: Monday 10-11:30 and by appointment, in AVW 3365.

Teaching Assistant: Alison Teoh; E-mail: AT  
Office Hours: 1:45 to 3:45 pm on Wednesdays.


Textbook (Required)Numerical Computing with MATLAB, by Cleve Moler, ISBN 0-89871-560-1

Individual Chapters may be downloaded from the author's web site at         

The book may be purchased from the bookstore, or from the web.


Software (required): MATLAB.     
You will need reliable access to MATLAB and a printer for doing homework in this course.

If you already do not have access to Matlab and have a PC, the best option would be to buy a student edition from the bookstore.


You can also get by without buying this copy and using the software which should be accessible from University computers.

Registered students should receive email with details on class accounts on the Grace computers.


Printing: Most homework will call for printing material (graphs, programs and the like off Matlab) and submitting it.  Emailed homework is NOT acceptable.


Prerequisites: Programming, advanced calculus, linear algebra.


Description in the catalog: Basic computational methods for interpolation, least squares, approximation, numerical quadrature, numerical solution of polynomial and transcendental equations, systems of linear equations and initial value problems for ordinary differential equations. Emphasis on methods and their computational properties rather than their analytical aspects.


Homework will be given out periodically, and will be due on the first class in the following  week from the date handed out. No late homework, without prior arrangement. Homework will be posted on this web page.


Collaboration Policy:  You may study together and discuss problems and methods of solution with each other to improve your understanding. You are welcome to discuss assignments in a general way among yourselves, but you may not use other students' written work or programs. Use of external references for your work should be cited. Clear similarities between your work and others will result in a grade reduction for all parties. Flagrant violations will be referred to appropriate university authorities.


You are responsible for checking this page.

Policy: Honor code

Grading: Homework 40%, Mid-Term 25%, Final 35%

Previous versions of this course: (for reference) Fall-2005 Spring-2007




09/02, 2009


Lecture 0

Chapter 1           


Accessing MATLAB on GRACE from a PC

09/04, 2008


Lecture 1

Introduction to the course.

Rules. Introduction to MATLAB

09/09, 2008


Lecture 2


Errors. Well posed problems. Floating point representation.

Keywords: fixed point, floating point, Mantissa, significand, exponent, sign, overflow, underflow, zero, Inf, NaN, float (single precision), double (double precision), IEEE 754


Homework 1

Due 09/18

Matlab: do the following problems in the text: 1.5, 1.6., 1.7, and 1.20

 Floating point representation: 1.34, 1.35, 1.38, 1.39

09/11, 2008


Lecture 3


Recap of the floating point representation; examples of how representation errors can cause problems during calculations; forward and backward error analysis



Lecture 4


Matrices, vectors,



Homework 2


Due 09/25

Do the following problems in the text: 2.7, 2.8, 2.11, 2.16, 2.18

Also for a small extra-credit of one point register for the class forum  and post something

09/18, 2008


Lecture 5


Solving diagonal and triangular systems

Gaussian elimination

LU decomposition

09/23, 2008


Lecture 6


LU decomposition

Permutation Matrices

Matlab tricks, Wrap -up

09/25, 2008


Lecture 7



Polynomial Interpolation

Monomials & Vandermonde matrices

Lagrange & Newton forms

Instability of polynomial interpolation

 Due 10/07

Homework 3



1. Do the following problems            3.3, 3.4, 3.7, 3.9

2. Read section 3.4 of the book, and summarize the shape-preserving piecewise cubic spline algorithm. How would a program to interpolate a spline using this algorithm differ from one using the cubic spline algorithm discussed in class.

09/30, 2008



Lecture 8



Error analysis of Polynomial interpolation

Piecewise Linear Interpolation

10/02, 2008


Lecture 9



Cubic spline interpolation. Tridiagonal system solution.

Horner’s rule.

10/07, 2008



Lecture 10




Zero finding, bisection. Secant method. Newton method.


Due 10/14

Homework 4



1. Do the following problems:  4.3, 4.8, 4.9, 4.15, 4.18

10/09, 2008


Lecture 11 opt

Lecture 11 

Wrap up of zero finding and optimization

Least squares – normal equations

10/14, 2008



Lecture 12


Least Square – QR algorithm. Givens Rotations

10/16, 2008


Lecture 13

Least squares. QR via the Householder transform

Reference: John Kerl’s article on Householder transforms. (local copy)

Due 10/23

Homework 5

Least squares:  Do the following problems from the text:

5.5, 5.7, 5.8, 5.12 


10/21, 2008




Sample exam           Solutions

You are allowed to bring a calculator and a single sheet of paper to the exam with any information you want on it. However, you should prepare the material on the sheet yourself, and submit it with the exam.

10/23, 2008


 Lecture 14

Numerical Integration, Newton-Cotes Formulae

10/28, 2008


 Lecture 15

(notes above)

Adaptive Integration, Richardson extrapolation, Romberg integration

10/30, 2008


Lecture 16

(notes above)

Gaussian Integration




Homework 6

1.     Problems  6.1, 6.3, 6.6

2.     Derive error bounds for the approximation of the integral below via the Simpson 1/3 rule of integration in terms of the size of the domain of integration and derivatives of the function f(t):


11/04, 2008



Lecture 17

Election Day … Go Vote! (if you are eligible to).


Ordinary differential equations; initial value problems, standard form, Euler method, modified Euler Method

11/06, 2008


Lecture 18


matlab: volteratest.m  rabfox.m

11/11, 2008



Lecture 19

multistep methods; implicit methods; Adams-Bashforth and Adams Moulton;

notions of stability and stiffness

matlab: stiff_ode.m

 Due 11/20

Homework 7


11/13, 2008


Lecture 20

Eigenvalues and Eigenvectors (background)

11/18, 2008


Lecture 21

(use link above)

Power Algorithm, Rayleigh quotient, QR

11/20, 2008


Lecture 22

(use link above)

QR algorithm with shifts, Singular value decomposition

Due 12/04

Homework 8


11/25, 2008



Lecture 23

(use link below)

Fourier Analysis

11/27, 2008




12/02, 2008


Lecture 24



Fourier Analysis

12/04, 2008


Lecture 25

(use link above)


12/09, 2008



Lecture 26


Last Class


Sample final

12/19, 2008





Friday, Dec 19 1:30pm-3:30pm in the same classroom

Material: things covered after the mid term, plus basic concepts from throughout the course.

You are allowed to bring a single sheet of paper to the exam with any information you want on it. However, you should prepare the material on the sheet yourself, and submit it with the exam.

Useful Links

Previous versions of 460 offered.

Prof. O'Leary: Fall 2002 (some of my material is adapted from this course)

Prof. Elman: 

 MATLAB resources:

Introductory Tutorials

MATLAB tutorial from University of Utah

MATLAB tutorial from Carnegie Mellon University

MATLAB tutorial from Indiana University

Slightly more advanced Tutorials

More complete references/tutorials/FAQs