CSC350F Numerical Algebra and Optimisation

• Aims
• Outline
• References
• Prerequisites
• Schedule for Fall 2013
• Marking scheme for Fall 2013
• Announcements for current students

• 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
• More on matrices
  • Vector and matrix norms
  • Condition number of a matrix
  • Iterative refinement
  • Eigenvalues and eigenvectors
• Non-square linear systems -- Least squares solution
  • Least squares solution
  • Normal equations
  • QR factorization
  • Gram-Schmidt method for matrices
  • Non-square linear systems -- Least squares solution
• Computation of eigenvalues and eigenvectors
  • Power iteration
  • QR factorization
• Non-linear 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, Broyden's
  • Polynomial equations
• Unconstrained optimisation
  • Golden section search method
  • Newton's method
  • Steepest descent method
  • Gradient methods
  • BFGS method

References

General Numerical Analysis
Michael Heath	Scientific Computing: an introductory survey	McGraw-Hill Inc.
Richard L. Burden and J. Douglas Faires	Numerical Analysis	Brooks/Cole
David Kincaid and Ward Cheney	Numerical Analysis	Brooks/Cole
L. W. Johnson and R. D. Riess	Numerical Analysis	Addison Wesley
David Kahaner, Cleve Moler and Stephen Nash	Numerical Methods and Software	Prentice Hall
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
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
Philippe G. Ciarlet	Introduction to numerical linear algebra and optimisation	Cambridge University Press

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. Basic multivariate calculus.
• Linear Algebra: Matrix and vector addition and multiplication, elementary row operations, linear (in)dependence, inverse matrix, etc.
• Programming: knowledge of basic programming constructs, such as for loops and if-then-else statements; manipulation of vectors and matrices; knowledge of (or will to learn) MATLAB, or knowledge of some conventional programming language, such as FORTRAN or C.
• Other Mathematics: induction.

Schedule for Fall 2013

Lecture	Tuesday 1-3 PM	Room BA B024
Tutorial	Thursday 1-2 PM	Room BA B024
Office Hours	Monday 3:30-4:30 PM	Room BA 4226

Tentative marking scheme for Fall 2013

Problem set 1	12%
Problem set 2	12%
Problem set 3	12%
Midterm test	24%
Final exam	40%

The problem sets include computer work.