DATE = 09/24/09

LIST OF TOPICS

These are the topics to be covered this semester. Some remarks:

• The list is not chronologically ordered.
• Cross-listing with the texts is eleshere.
• The level of detail (or depth of coverage) will vary.

1. Introduction + MATLAB
2. Floating Point Number System
   • Standard notation; mantissa and exponent
   • Binary/Hex Representation
   • Relative and Absolute Errors
   • Catastrophic Cancellation
3. Root Finding
   • Scalar: Bisection, Newton, and Secant Methods
   • System: Newton, TBD
   • Fixed Point Iterations
   • Rate of Convergence
4. Numerical Differentiation
   • Taylor series derivation
   • Total Error = Truncation Error + Roundoff Error ; optimal tradeoff
   • Richardson Extrapolation
5. Polynomial Interpolation
   • The Vandermonde basis and matrix
   • Runge function example + Chebyshev points
   • Theorem on the error in polynomial interpolation ; examples
   • Horner's algorithm , Newton basis , Lagrange basis
   • Implementation: Software `v = interp(x,y,u)' (see p.95 Moler for the 1st use)
6. Piecewise (PW) Polynomial Interpolation
   • PW linear interpolation ; error bounds
   • PW cubic Hermite interpolation
   • Cubic Splines
   • TBD
7. Quadrature
   • Newton-Cotes Rules (including Trapezoid Rule + Simpson Rule)
   • Romberg integration
   • Error Estimates
   • Gaussian Quadrature
   • Implementation: especially Adaptive Quadrature
   • TBD: singularities, 2D, high dimensional integrals, ...
8. Linear Equations
   • Elimination as A = LU
   • Gaussian Elimination (GE) with Partial Pivoting , PA = LU
   • Implementation: `DECOMP' , FE , BS ; multiple RHS's
   • Theory: Vector Norms , Matrix Norms , Condition Number
   • Theory: Relative Error vs. Relative Residual
   • Tridiagonal Matrices and Applications
   • GE for Block Matrices ; Update Algorithms
9. Least Squares
   • Models , Basis Functions , Normal Equations
   • QR Decomposition: General Ideas , Gram-Schmidt
   • QR (cont): (practical) Implementation using Householder reflections (or (TBD) Givens Rotations)
10. ODE's ; time permitting
11. Eigenvalue Problems ; time permitting