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
• 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)

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

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, 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.