- 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

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.

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% |