Tentative lecture topics for CSCC51, Spring 99 
We include textbook sections in brackets.
** Subject to change: done on December 1998**



1. Interpolation [Chapter 3]
   =============

Week of Jan 4

1.1  Approximation and interpolation [3]
1.2  Polynomial approximation - Weierstrass theorem [3]
1.3  Polynomial interpolation using monomial basis.
1.4  Polynomial interpolation using Lagrange basis [3.1]
1.5  Existence and uniqueness of polynomial interpolant [3.1]
1.6  Error of the polynomial interpolant [3.1] (start)

Week of Jan 11

1.6  Error of the polynomial interpolant [3.1]
1.7  Polynomial interpolation using Newton's basis -- Divided 
     differences [3.2]

Week of Jan 18

1.8  Evaluating a polynomial -- Horner's rule (nested multiplication) 
     [2.6 pgs 92-94]
1.9  Polynomial interpolation with derivative data.
     Osculating polynomial interpolation [3.3]
1.10 Hermite polynomial interpolation using monomial basis. 
1.11 Hermite polynomial interpolation using Lagrange basis [3.3]
1.12 Hermite polynomial interpolation using Newton's basis [3.3]
1.13 Existence and uniqueness of Hermite polynomial interpolant [3.3]
1.14 Error of the Hermite polynomial interpolant [3.3]
1.15 Pitfalls with polynomial interpolation [3.4]

Week of Jan 25

1.16 Piecewise polynomials and splines [3.4]
1.17 Linear spline interpolation (Lagrange form).
1.18 Cubic spline interpolation -- choice of end-conditions [3.4]



2. Orthogonal polynomials and least squares approximation [Chapter 8]
   ======================================================

Week of Feb 1

2.1  Least squares approximation [8.2]
2.2  Inner products and norms of functions.
2.3  The normal equations for polynomial least squares approximation [8.2]
2.4  Linear independence and orthogonality of functions [8.2]
2.5  Orthogonal polynomials and least squares approximation.
2.6  Constructing sets of orthogonal polynomials.
     The method of undetermined coefficients.
     The Gram-Schmidt orthogonalisation algorithm for functions.
     The three-term recurrence relation algorithm.
2.7  Constructing the least squares polynomial approximation.

Week of Feb 8

2.8  Tchebyshev (Chebyshev) polynomials [8.3]
     The optimal placing of data points in polynomial interpolation.
Midterm test.

Week of Feb 15

Reading week (no lectures).



3. Numerical Integration [Chapter 4]
   =====================

Week of Feb 22

3.1  Introduction [4.3]
3.2  Midpoint rule and error formula [4.3]
3.3  Trapezoidal rule and error formula; alternative derivation [4.3]
3.4  Transforming quadrature rules to other intervals [in 4.7]
3.5  Simpson's rule and error formula (start) [4.3]

Week of Mar 1

3.5  Simpson's rule and error formula (end) [4.3]
3.6  Corrected trapezoidal rule and error formula.
3.7  Convergence of polynomial interpolatory quadrature rules.
3.8  Newton-Cotes quadrature rules [4.3]

Week of Mar 8

3.9  Composite quadrature rules and error formulae [4.4]
     Composite midpoint rule and error formula.
     Other composite rules and error formulae.
3.10 Error estimators for quadrature rules [4.6]
3.11 Adaptive quadrature [4.6]

Week of Mar 15

3.12 Romberg integration and Richardson extrapolation [4.5]
3.13 Gauss quadrature rules [4.7]



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

Week of Mar 22

4.1  Introduction: DEs, ODEs, PDEs and IVPs [5.1]
4.2  Existence and uniqueness of solution of an IVP for an ODE [5.1]
4.3  Second order ODEs and BVPs.
4.4  nth order ODEs and IVPs for ODEs [5.9]
4.5  Stability of ODEs -- Jacobian.
4.6  Numerical methods for first order IVPs for ODEs.
4.7  Forward Euler's method [5.2] (start)

Week of Mar 29

4.7  Forward Euler's method (end)
     Global and local errors.
     Order of a numerical method for IVPs-ODEs [5.3 pgs 269-270]
     Stability of the numerical method [~5.10]
     Stiff ODEs [~5.11]
4.8  Backward Euler's method - Implicit methods [~5.11 and ex 8 pg 348]

Week of Apr 5

4.9  Runge-Kutta methods [5.4, 5.5]
Exam preparation.