Tentative lecture topics for CSCC51H, Winter 2012
We include textbook sections in brackets.
** Subject to change: revised, December 2011**



1. General Information/Mathematical Background [Chapter 1]
   ===========================================

Week of Jan 9
 
1.1  Administration details
1.2  Review of floating point arithmetic
1.3  Useful theorems from calculus

2. Interpolation and Approximation [Chapter 7]
   ==============================

Week of Jan 16

2.1  Approximation and interpolation -- basic problem [7.1]
2.2  Existence and uniqueness of polynomial interpolant [7.2]
2.3  Representation of interpolating polynomial [7.3]
2.4  Newton basis (divided differences), monomial basis,
     Lagrange basis [7.3]

Week of Jan 23

2.5  Properties of divided differences [7.3]
2.6  Errors in polynomial interpolation [7.3]
2.7  Interpolating with derivative constraints -- Hermite
     interpolation  (or osculatory interpolation) 
2.8  Difficulties with polynomial interpolation [7.3]

Week of Jan 30

2.9  Piecewise polynomials and splines [7.4]
2.10 Linear splines 
2.11 Cubic splines 
2.12 Choice of end-condition 
2.13 Shape preserving splines and 2-D splines [7.4]

3. Least squares polynomial approximation [Chapter 3 - Review from CSCC50H] 
   =======================================

Week of Feb 6

3.1  Least squares approximation --discrete and continuous 
3.2  Normal equations and existence/uniqueness of solution 
3.3  QR and/or Gram Schmidt algorithm 


4. Numerical Quadrature [Chapter 8]
   =====================

Week of Feb 13

4.1  Basic problem and interpolatory rules [8.2]
4.2  Errors in interpolatory rules - general case [8.2]
4.3  Examples: Trapezoidal rule, Simpsons rule [8.3]
Midterm Test

Week of Feb 20

Reading Week (no lectures)

Week of Feb 27

4.4  Gaussian Quadrature [8.3]
4.5  Orthogonal polynomials
4.6  Composite rules [8.3]

Week of March 5

4.7  Errors in composite rules [8.3]
4.8  Error estimates and validity checks 
4.9  Extrapolation -- Romberg Quadrature [8.7]

Week of Mar 12

4.10 Error estimates for Gaussian Quadrature [8.3]
4.11 Special difficulties: infinite integrals, singularities [8.4]
4.12 Extensions to double integrals [8.4]


5. Ordinary Differential Equations [Chapter 9]
   ===============================

Week of Mar 19

5.1  Mathematical problem [9.1]
5.2  Existence and uniqueness of mathematical solution [9.2]
5.3  Systems of equations [9.1]
5.4  Numerical Methods -- Taylor series, Eulers method [9.3]

Week of Mar 26

5.5  Error measures and error control [9.3]
5.6  Numerical stability and convergence [9.3]
5.7  Limitations of classical theory [9.3]
5.8  Runge-Kutta methods - order of accuracy [9.3.6]

Week of April 2

5.9  Derivation of Runge-Kutta methods [9.3.6]
5.10 Error estimates and adaptive control for methods [9.3.6]
5.11 Special difficulties [9.3.6]

Exam preparation.