### CSC351S Numerical Approximation, Integration and Ordinary Differential Equations

#### The first meeting of CSC351 is Thursday, January 8, 2009, 3-5 p.m., in GB 220. Note the room number! Tutorials will start on Tuesday, January 13, 2009, in BA B025. Note the room number! Also note the room number has been corrected!

Aims
• 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.
Outline
• 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
• 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
• (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)

References
 General Numerical Analysis 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 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.

Schedule for Spring 2009
 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%
The problem sets include computer work.