The official UMD course syllabus for AMSC460 can be found here: https://www-math.umd.edu/offered-courses/420-amsc-460-computational-methods.html
We will mostly follow it. A tentative course outline is the following:
- Computer Arithmetic and Errors
- Sources of error
- Floating point arithmetic
- Roundoff errrors
- Error propagation and condition numbers
2. Solving linear systems of equations
- LU-factorization with and without pivoting
- LU-factorization for band matrices
- Cholesky factorization
3. Interpolation
- Vandermonde interpolation
- Horner's scheme
- Newton interpolation
- Lagrange interpolation
- Chebyshev interplation
- Splines
4. Numerical Integration
- Elementary quadrature rules
- Composite quadrature rules
- Gaussian quadrature rules
- Adaptive quadrature
5. Linear least squares problems
- Normal equations
- QR-factorization
6. Solution of nonlinear systems of equations
- Bisection method
- Newton's method
- Global Newton method
- Multidimensional Newton method
- Nonlinear iteration methods
- Minimization problems (golden section search, steepest descent)
- Nonlinear least squares
7. Numerical solution of ordinary differential equations
- Explicit Runge-Kutta methods
- Implicit ODE solvers
- Comparison explicit/implicit methods
- Stability functions and regions
- Stiff ODEs
- MATLAB ODE solvers