### CSC350F Numerical Algebra and Optimisation

Aims
• 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.
Outline
• 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

• 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.