Course Outline

1. Numerical Algorithms

   Scientific computing

   Numerical algorithms and errors

2. Roundoff errors

   Floating point numbers

   Roundoff error accumulation and cancellation error

   Errors in general floating point representation

   The IEEE standard

3. Nonlinear equations in one variable

   Bisection method

   Fixed point iteration

   Newton's method and secant method

   Minimizating a function in one variable

4. Review of linear algebra

   Review of basic concepts

   Vector and matrix norms

   Special classes of matrices

5. Linear systems: direct methods

   Gaussian elimination and backward substitution

   LU decomposition

   Pivoting strategies

   Efficient implementation

   The Cholesky decomposition

   Banded matrices and diagonally dominant matrices

   Estimating errors and the condition number

6. Linear least squares problems

   Least squares and the normal equations

   The QR factorization

7. Iterative methods for linear systems

   Stationary iterative methods

   Convergence of stationary iterative methods

   The Conjugate Gradient method

8. Eigenvalues and singular values

   The power method

   The inverse power method

   Singular value decomposition

   SVD for the linear least squares problem

9. Nonlinear systems and unconstrained optimization [time permitting]

   Newton's method and modifications

   Unconstrained optimization

   Minimizing quadratic functions