Computational Mathematics 1000-114bMOBa

1/ Elements of fl rounding error analysis
2/Interpolation
2.1 Polynomial interpolation
- specifying Lagrange interpolation problem
- existence and uniqueness of a solution
- finite differences algorithm
- approximation error estimates

2.2 Spline approximation
- definition of spline spaces
- linear splines
- cubic splines

3/Approximation
3.1 approximation in Hilbert spaces
- existence and uniqueness
- algorithms including the Gram-Schmidt process
- orthogonal polynomials - properties and application to the polynomial approximation problem
-Chebyshev polynomials and their properties
- Linear Least Square problem as a special case of an approximation problem

3.2 Uniform approximation - (optional if time permits)

4/Numerical integration
4.1 Interpolation quadratures
-Gauss quadratures
4.2 Quadrature rules: trapezoidal and Simpson rules

5/Numerical methods of solving system of algebraic equations
- LU decompositions with partial pivoting
- QR orthogonal decomposition: Housholder method
- condition of the matrices and their influence on rounding error analysis of LU decomposition
- an application of QR factorization of M x N matrix to Linear Least Square problems

6/ Roots of a nonlinear equation
- bisection method
- Newton method
- Secant method
- Banach iteration method
- order of convergence

7/Numerical Eigenproblem (optional if time permits)
- Power Method
- Inverse Power Method