CSC336S Numerical Methods

Spring 2008

Course information for current students:

• 21-4-08: Assignment 4 is now marked and available for pickup. Marks are already downloaded in the cdf site.
  I will hold office hours (for assignment pickup and/or questions) Tuesday 22 April 4-5 p.m. and Tuesday 29 April 4-5 p.m.
• 7-4-08: Assignment 4 is to be handed in my office, BA 4226. If I am not in, please slip assignment under the door.
• 17-3-08: On Monday, 24 March 2008, we will do a lecture in place of a tutorial.
• 10-3-08: Regarding assignment 3: the code for secant must use only one function evaluation per iteration.
• 7-3-08: Some clarifications to assignment 3:
  q1: Hand-in a hard-copy of your code, output and plot, as well as your comments.
  q2 (b): In order to comment on the rate of convergence, you need to output the error at each iteration, and possibly some other quantities. I have adjusted the scriptnl.m file to reflect this.
  You should output the error and similar quantities for Newton's and secant, and state what the quantities are.
• 21-2-08: I will hold office hours Friday 22 February 2-3 and Monday 25 February 4-5, for students who have questions, or those who would like to pick assignment 2 ahead of the exam.
  For the midterm exam, the chapters on computer arithmetic and solving square linear systems are included. The chapter on solving non-square linear systems is not included.
  Calculators are the only aids permitted.
  The exam will be a little longer than 1 hour and will start at the start of the tutorial. After the exam, we will do a break and then lecture.
• 4-2-08: Extra office hours Friday 8 February 2008, 2:30-3:30, BA4226.
• 25-1-08: Extra office hours Friday 25 January 2008, 1-2, BA4226.
• 14-1-08: All assignments are to be done individually. Please read How to Avoid Plagiarism and Advice about academic offences.
  All assignments are due at the start of the lecture.
• 21-12-07: The first meeting of CSC336 is Monday, January 7, 2008, 6-9 p.m., in BA 1180.
  Note the start time! (6 p.m.)
  This Monday, there will be no tutorial. We will use two to two and a half hours of lecture.
  Tutorials will start on Monday, January 14, 2008.
• 21-12-07: The textbook for this course is Michael Heath's, Scientific Computing: an introductory survey, McGraw-Hill Inc. 2002.
  I have ordered a custom copy (a reduced version) of the book, which should be available at the UT bookstore for about $60. The custom-made copy is a reduced version of the book, but includes all the chapters we need for CSC336.
  You may want to consider getting the paperback (international?) edition of the book at amazon.ca, which (I believe) costs about $90. This is a complete version of the book.

Important note on the use of bulletin boards: No parts or whole of answers to the assignment problems should be posted to the boards. Any violation of this rule will bring trouble to the poster.
Please use judgement before posting.

A brief introduction to MATLAB, by C. Christara and W. Wai

Textbook web page (see educational modules)

Material already covered in the course (with textbook sections in parentheses)

7-1-2008 (2.5 hours)
0.   What is Scientific Computing? (1.1, 1.2.1)
1.   Computer arithmetic; data and computational errors
1.1  (Human) Representation of nonnegative integers
   - Algorithm for converting base b integers to decimal
   - Algorithm for converting decimal integers to base b
1.2  (Human) Representation of reals
   - Algorithm for converting base b fractions to decimal
   - Algorithm for converting decimal fractions to base b
1.3  Computer representation of numbers (1.3.1-7)
   - floating-point numbers, mantissa, exponent, normalized mantissa,
     significant digits, overflow, underflow, range of representable numbers,
     representable numbers, chopping, rounding
   - The IEEE Standard
1.4  Round-off error (1.3.5)
1.5  Absolute and relative errors (1.2.2)
1.6  Computer arithmetic (1.3.8)
1.7  Machine epsilon (1.3.5)

14-1-2008 (1 hour) tutorial on MATLAB
14-1-2008 (2 hours) handout: assignment 1
1.8  Error propagation in simple arithmetic calculations (1.3.8-9)
   - Multiplication, division, addition/subtraction
1.9  Error propagation in computation: conditioning of problems (1.2.6)
1.10 Error propagation in computation: stability of algorithms (1.2.7)
1.11 Forward and backward errors (1.2.5)

2.   Solving square linear systems
2.0  Vectors and matrices -- review of terminology (2.1-2)
     (start)
21-1-2008 (1 hour) tutorial on computer arithmetic
21-1-2008 (2 hours)
2.0  Vectors and matrices -- review of terminology (2.1-2)
     (end)
2.1  Gaussian elimination (GE) (2.4.1, 2.4.3-4)
2.2  Back substitution (2.4.2)
2.3  LU factorization (2.4.3-4)
2.4  Forward substitution (2.4.2)
2.5  Partial (row) pivoting (2.4.5)
2.6  Scaled partial pivoting (2.4.5)
2.7  Complete pivoting (2.4.5)
28-1-2008 (1 hour) tutorial on triangular matrices, GE, LU, pivoting
28-1-2008 (2 hours) handout: assignment 2
2.8  A mathematical description of the GE/LU algorithm (2.4.5)
2.9  Symmetric matrices (2.5.1-2)
2.10 Banded matrices (2.5.3)
4-2-2008 (1 hour) tutorial on assignment 1 -- return assignment 1
4-2-2008 (2 hours)
2.11 Vector and matrix norms (2.3, 2.3.1-2)
2.12 Condition number of a matrix (2.3.3)
     Errors and residuals in computed solutions of linear systems (2.3.4-5)

11-2-2008 (1 hour) tutorial on norms and condition numbers
11-2-2008 (2 hours)
3.   Solving non-square linear systems
3.1  Overdetermined systems (3.1)
3.2  The normal equations for overdetermined systems (3.2, 3.2.1)
3.3  Underdetermined systems
3.4  The normal equations for underdetermined systems
3.5  MATLAB and least squares problems
3.6  Data fitting (3.1)

4.   Solving nonlinear equations
4.1  Nonlinear equations and systems -- introduction (5.1-2)
     (start)

25-2-2008 (1+ hours) midterm
25-2-2008 (1+ hours)
4.1  Nonlinear equations and systems -- introduction (5.1-2)
4.2  Why solve nonlinear equations?
4.3  Multiplicity of roots (5.2)
4.4  Fixed points and roots of functions -- contractive functions (5.2)
4.5  Existence and uniqueness of roots and fixed points (5.2)
4.6  Numerical methods for nonlinear equations (5.2-4)

3-3-2008 (1 hour) tutorial on non-square linear system and least squares
                         q4 of midterm
3-3-2008 (2 hours)
4.7  Convergence rate (speed) of iterative methods (5.4)
4.8  The bisection method (5.5.1)
4.9  Fixed point (functional) iteration methods (5.5.2)
     Convergence of fixed point iteration
10-3-2008 (1 hour) tutorial on non-square linear system and least squares
                          nonlinear equations, bisection, fixed point iteration
                          q2a of midterm
10-3-2008 (2 hours)
     Convergence rate of fixed point iteration
4.10 Newton's method (Newton-Raphson method) (5.5.3)
     Convergence of Newton's
4.11 The secant method (5.5.4)
4.12 Newton's method for systems of nonlinear equations (5.6.2)

17-3-2008 (1 hour) tutorial on nonlinear equations, Newton's, secant
                          q2b of midterm
17-3-2008 (2 hours)
5    Interpolation
5.1  Approximation and interpolation - Introduction [7.1]
5.2  Polynomial approximation - Weierstrass theorem
5.3  Evaluating a polynomial -- Horner's rule (nested multiplication) [7.3.1]
5.4  Polynomial interpolation using monomial basis functions [7.3.1]
5.5  Polynomial interpolation using Lagrange basis functions [7.3.2]
5.6  Existence and uniqueness of polynomial interpolant [7.2]
5.7  Polynomial interpolation using Newton's basis functions
     and the Divided Differences Table [7.3.3]
5.8  Comparison of the three bases

24-3-2008 (3 hours)
5.9  Error of the polynomial interpolant [7.3.5]
5.10 Pitfalls of polynomial interpolation [7.3.5]
5.11 Piecewise polynomials and splines [7.4]
5.12 Linear spline interpolation (Lagrange form) [7.4.2]
     Error in linear spline interpolation
5.13 Cubic spline interpolation -- choice of end-conditions [7.4.2]
     Error in cubic spline interpolation

31-3-2008 (1 hour) tutorial on interpolation
31-3-2008 (2 hours)
6    Numerical Integration [Ch 8]
6.1  Introduction [H 8.1]
6.2  Midpoint rule and error formula [H 8.3.1]
6.3  Composite midpoint rule and error formula [H 8.3.5]
6.4  Trapezoidal rule and error formula [H 8.3.1]
6.5  Simpson's rule and error formula [H 8.3.1]
6.6  Composite quadrature rules and error formulae [H 8.3.5]
6.7  Gauss quadrature rules [H 8.3.3]

7-4-2008 (1 hour) tutorial on quadrature
7-4-2008 (2 hours)
6.8  Table of quadrature rules and error formulae
6.9  Transforming quadrature rules to other intervals [H 8.3.3]
x.x  Summary
x.x  Course evaluations

Notes and handouts:
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