Project is due November 26 -- 10% off for each day late. Project must be approved by professor by November 1. The project should probably involve writing and testing a computer code. You should explain the method, the types of problems it is good for, its advantages and disadvantages. Show examples of ``good results'' and poor performance. Error analysis and/or operational analysis might be worthwhile.
The presentation of your project will vary but here are a couple of suggested formats/outlines.

For an application project:

Abstract -- very brief explanation of the physical problem and the mathematics you will use to solve it.

Explain the physical problem clearly and the equations that arise. Someone who knows nothing about the topic should understand it. Derive or explain the equations.

Explain the numerical methods you use and how they relate to your problem

Present specific examples that demonstrate that you were successful in solving the applied problem. Explain what your solution means

Explain any difficulties the problem might have in special cases or conditions needed (E.g., Do you need a close starting guess?, etc.)

Conclusions

References

Attach computer code, figures, output of computer code.

Abstract -- very brief explanation of the types of problems the method should be good for and a little bit about the math involved -- this will be very project dependent. If you are comparing methods, you want to mention briefly your conclusions.

Explain the mathematical problem to be solved clearly.

Give a derivation of the method or the math background to understand why it works.

Discuss when the method should work best, convergence, when it shouldn't work, etc.

Present specific examples that demonstrate that you were successful in coding up the method and demonstrate cases where the method slows or fails.

Conclusions

References

Attach computer code, figures, output of computer code.

Nonlinear Equations:
Roots of a polynomial using synthetic division and deflation or Bairstow's Method
Fractional linear root finding method
Muller's Method for finding roots including complex ones and Inverse quadratic interpolation
Quasi-Newton Methods for Nonlinear Systems -- Theory and Implementation of Broyden's Method
Functional iteration for Nonlinear Systems
Horner's rule and Horner's ruls for derivatives
Division using Newton's method
Chord Method (Epperson)
Halley's method (Epperson)
Secant method including error analysis
Hybrid root finding methods (Brent's method)
Interpolation:
Neville's iterated interpolation
Hermite Polynomials
Spline Code (Cubic, Bezier, B-) and error analysis
Using Splines for Integration and Differentiation
Parametric curve interpolation using Hermite cubics (Bezier curves)
Orthogonal Polynomials
Legendre Polynomials and applications
Chebyshev Polynomials and applications
Continuous Least Squares
(Fast) Fourier Transform
Pade Approximants
Linear Algebra:
Solving linear equations with full pivoting and error analysis
Householder's Method
QR algorithm
Singular Value Decomposition
Finding determinants by the definition vs. by Gaussian Elimination (finding eigenvalues by combining techniques)
Eigenvalues, Eigenvectors -- Deflation, Inverse Power Method
Sparse matrix solutions
SOR
Householder's method
Block tridiagonal solvers
Cholsky's method
Careful matrix error analysis and examples
Jacobi and Gauss-Seidel including proof of convergence for diagonally dominant case
Jacobi and Gauss-Seidel including proof convergence is better for GS for symmetric positive definite case
Numerical Integration:
Multiple integration
Gaussian Quadrature
Adaptive Quadrature
Quadrature with non-smooth integrands, or singularities
Monte Carlo Methods
Differential Equations:
Implicit ODE solvers for nonlinear ODEs -- using Newton's or Secant Method
Stiff ODEs and Stiff Systems
Error Control for ODE solvers Runge-Kutta-Fehlberg Method)
Variable step size for multi-step ODE solvers
Nonlinear ODE Boundary Value Problems
Rayleigh-Ritz and Finite Element Ideas
Finite Elements in One Dimension (Variational Principle)
Elliptic PDEs solution - finite difference - diagonally dominant band matrices
Parabolic PDEs solution - forward and backward differences and stability, Crank-Nicolson method
Optimization:
Conjugate gradient method
Steepest Descent Methods
Optimization -- Linear Programming
Simplex Method
Miscellaneous:
Computer arithmetic
Multigrid Methods

DNA mutation probability -- linear algebra -- Markov chains
Protein and Nucleic Acid evolution -- mutation probabliity matrices
Syringe Deflection, Beam deflection - ODE-BVPs, 4th order
Reservoir routing -- Time dependent ODEs
Pipe Networks -- Systems of linear equations, linear independence
Hydraulic Systems and pipe networks -- systems of nonlinear equations
Predator-Prey ODE systems (B&F p. 329)
Glucose uptake and insulin production -- system of ODEs
Free-fall with resistance -- nonlinear root finding, ODE solutions, extensions (B&F p. 65)
Financial modeling -- annuities, mortgages -- root finding (B&F p. 77)
Financial modeling -- options pricing -- PDEs
Sports modeling -- e.g. shutout probablities and extensions, root finding (B&F p. 77)
Position and speed interpolation -- Hermite polynomials (B&F p. 142)
Probabilities using the bell curve -- numerical integration (B&F p. 207)
Modeling particles moving through a fluid -- PDEs or integration (B&F p. 207)
Non-linear pendulum vs. Linear Pendulum -- Systems of ODEs (B&F p. 329)
Population modeling problems -- Markov chains, eigenvalues (B&F pp. 387 and  443)
Random walk type problems and particle motion -- iteration (B&F p. 459)
Least squares for complicated realistic functions, e.g. moth energy (B&F p.486)
Electrostatic potential -- boundary value problems (B&F p. 632)