CSC350F Numerical Algebra and Optimisation

• Aims
• Outline
• References
• Prerequisites
• Schedule for Fall 99
• Marking scheme for Fall 99

• Formulate numerical methods for solving linear and nonlinear algebraic equations and systems.
• Evaluate numerical methods with respect to their accuracy and efficiency.
• Develop and practice computer skills in implementing numerical methods efficiently on the computer.
• Use high level software for studying numerical methods.

• Introduction
  • Computer representation of numbers
  • Computer arithmetic
  • Round-off error, machine epsilon, underflow, overflow
  • Error propagation
  • Conditioning and stability
• Square linear systems - direct methods
  • Gauss elimination
  • LU factorisation
  • Pivoting, scaling
  • Forward and back substitutions
  • Symmetric and symmetric positive definite matrices, Choleski factorisation
  • Special cases: tridiagonal, banded, general sparse matrices
• Discussion on matrices
  • Vector and matrix norms
  • Condition number of a matrix
  • Iterative refinement
  • Eigenvalues and eigenvectors
• Square linear systems - iterative methods
  • Jacobi
  • Gauss-Seidel
  • SOR, SSOR
  • Acceleration, Conjugate Gradient method
• Non-square linear systems
  • Least squares solution
  • Normal equations
  • QR factorization
  • Gram-Schmidt method for matrices
• Nonlinear equations and systems
  • Bisection
  • Secant
  • Fixed point iteration, Newton's method
  • Convergence
  • Newton's method for systems, Jacobian matrix
  • Modifications to Newton's method, approximations to Jacobian
  • Quasi-Newton methods
  • Polynomial equations
• Unconstrained optimisation
  • Steepest descent method
  • Gradient methods

References

General Numerical Analysis
Richard L. Burden and J. Douglas Faires	Numerical Analysis	Brooks/Cole 1997, 6th edition
David Kincaid and Ward Cheney	Numerical Analysis	Brooks/Cole 1996
Michael Heath	Scientific Computing: an introductory survey	McGraw-Hill Inc. 1997
L. W. Johnson and R. D. Riess	Numerical Analysis	Addison Wesley
David Kahaner, Cleve Moler and Stephen Nash	Numerical Methods and Software	Prentice Hall 1989
S. D. Conte and Carl de Boor	Elementary Numerical Analysis	McGraw-Hill Inc.
G. Dahlquist and A. Bjorck (trans. N. Anderson)	Numerical Methods	Prentice Hall 1974
J. Stoer and R. Bulirsch	Introduction to Numerical Analysis	Springer Verlag 1980, 1993
Numerical Linear Algebra
William W. Hager	Applied Numerical Linear Algebra	Prentice Hall 1988
Gene Golub and Charles Van Loan	Matrix computations	John Hopkins University Press 1996
Philippe G. Ciarlet	Introduction to numerical linear algebra and optimisation	Cambridge University Press
C. W. Groetsch and J. Thomas King	Matrix methods and applications	Prentice Hall 1988

Prerequisites

• General: Ability to handle notation and to do algebraic manipulation.
• Calculus: Differentiation and integration of polynomial, trigonometric, exponential, logarithmic and rational functions, continuity, limits, graphs of functions, Taylor series, Rolle's theorem, mean-value theorem, de l' Hospital's rule, etc.
• Linear Algebra: Matrix and vector addition and multiplication, elementary row operations, linear (in)dependence, inverse matrix, etc.
• Programming: knowledge of some conventional programming language, preferably FORTRAN or C; knowledge of (or will to learn) MATLAB.
• Other Mathematics: induction.

Schedule for Fall 99

Lecture	Monday 7-9pm	Room LM 158
Tutorial	Monday 6-7pm	Rooms ?? and ??
Office Hours	Monday 3:30-4:30pm	Room SF 3304

Tentative marking scheme for Fall 99

Problem set 1	8%
Problem set 2	8%
Problem set 3	8%
Problem set 4	8%
Midterm test	20%
Final exam	48%

The problem sets include computer work.