Tutorials will start on Tuesday, January 13, 2009, in BA B025.

- Aims
- Outline
- References
- Prerequisites
- Schedule for Spring 2009
- Marking scheme for Spring 2009
- Announcements for current students

- Formulate numerical methods for approximation, integration and ODEs.
- Evaluate numerical methods with respect to their convergence, stability, and efficiency.
- Develop and practice computer skills in implementing numerical methods efficiently on the computer.
- Use high level software for studying numerical methods.

- Interpolation
- Introduction
- Polynomial interpolation - Weierstrass theorem
- Monomial basis - Vandermonde matrix
- Lagrange basis
- Newton's basis - Newton's divided differences
- Proof of existence and uniqueness of interpolating polynomial
- Evaluation of a polynomial - Horner's rule
- Error of polynomial interpolation
- Polynomial interpolation with derivative data - Hermite interpolation
- Langrange basis
- Newton's basis - Newton's divided differences
- Proof of existence and uniqueness of Hermite interpolating polynomial
- Error of Hermite interpolation
- Problems with polynomial interpolation - Runge's function
- Piecewise polynomial interpolation - splines
- Piecewise linear interpolation - error formula
- Hermite piecewise cubic interpolation
- Cubic spline interpolation - choice of end-conditions
- B-spline interpolation

- Orthogonal Polynomials and Least Squares Approximation
- Weighted least squares problems - discrete and continuous
- Inner products and norms of functions
- Orthogonal and orthonormal polynomials
- Gram-Schmidt algorithm
- Best least squares approximation
- Tchebychev polynomials

- Quadrature Rules
- Introduction, interpolatory rules, Newton-Cotes rules, polynomial degree of a rule, linearity
- Midpoint rule and error
- Trapezoidal rule and error
- Simpson's rule and error
- Corrected trapezoidal rule and error
- Gauss rules and use of tables, transformation
- Compound quadrature rules, introduction
- Compound midpoint rule and error
- Compound trapezoid rule and error
- Compound Simpson's rule and error
- Romberg integration
- Adaptive quadrature rules
- (Semi-)infinite integrals (truncation and translation), singularities (change of variables)

- Ordinary Differential Equations
- Introduction to ODEs
- Stability of ODEs and systems of ODEs, Jacobian, stiff ODEs
- Introduction to num. methods for ODEs
- Euler's method
- Implicit methods, backward Euler's (BE) and trapezoidal method (TM)
- Runge-Kutta (RK) methods
- Taylor's series methods
- Linear Multistep Methods (LMMs)

General Numerical Analysis |
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Michael Heath | Scientific Computing: an introductory survey | McGraw-Hill Inc. 2002 |

David Kincaid and Ward Cheney | Numerical Analysis | Brooks/Cole 2002 (1996) |

Richard L. Burden and J. Douglas Faires | Numerical Analysis | Brooks/Cole 2001 (1997) |

James Epperson | An introduction to Numerical Methods and Analysis | Wiley 2003 |

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 1974 |

J. Stoer and R. Bulirsch | Introduction to Numerical Analysis | Springer Verlag 1993 (1980) |

Charles Van Loan | Introduction to Scientific Computing | Prentice Hall 2000 |

Numerical Linear Algebra |
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William W. Hager | Applied Numerical Linear Algebra | Prentice Hall 1988 |

Gene Golub and Charles Van Loan | Matrix computations | John Hopkins University Press 1996 |

- 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, multivariate differentiation, etc.
- Linear Algebra: Matrix and vector addition and multiplication, elementary row operations, linear (in)dependence, inverse matrix, etc.
- Numerical Linear Algebra (included in CSC350): Linear solvers for banded and sparse matrices, sparse matrix storage schemes, Discrete least-squares approximation, Nonlinear equations solvers.
- Computational methods: Understanding of round-off error, computer arithmetic, etc.
- Programming: knowledge of some programming language, such as FORTRAN, C or MATLAB.
- Other Mathematics: induction.

Lectures | Thursday 3-5 | (Room GB 220) New room BA 1210 |

Tutorial | Tuesday 3-4 | Room BA B025 |

Office Hours | Tuesday 4-5 | Room BA 4226 |

** Tentative marking scheme for Spring 2009 **

Problem set 1 | 8% |

Problem set 2 | 8% |

Problem set 3 | 8% |

Problem set 4 | 8% |

Midterm test | 24% |

Final exam | 44% |