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 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
    Estimating errors and the condition number
    The Cholesky decomposition
    Banded matrices and diagonally dominant matrices


  6. Linear least squares problems
    Least squares and the normal equations
    The QR factorization
    Truncated SVD and regularization


  7. Iterative methods for linear systems
    Stationary iterative methods
    Convergence of stationary iterative methods
    The Conjugate Gradient method
    Preconditioning


  8. Eigenvalues and singular values
    The power method
    The inverse power method


  9. Nonlinear systems
    Newton's method and modifications