Syllabus

1. Computer and linear algebra basics: computer arithmetic, rounding errors, error propagation, matrix-vector operations, positive definite matrices.
2. Solution of linear systems: Gaussian elimination, LU factorization, elementary matrices, pivoting, Cholesky factorization, Househoulder transforms, QR factorization.
3. Solution of eigenvalue problems for square matrices: Gershgorin Theorem, diagonal dominant matrices, power method, inverse power method, reduction to Hessenberg form, QR method.
4. Solution of systems of nonlinear equations: rate of convergence for iterative methods, bisection method and secant method for scalar equations, fixed point method, Newton's method.
5. Iterative methods for systems of linear equations and their convergence: Jacobi method, Gauss-Seidel method, SOR method, steepest descent method, conjugate gradient method.
6. Polynomial approximation theory: Bernstein polynomials, Weierstrass Theorem, existence and uniqueness of best approximations, alternant set and characterization of best approximations, Remez method for finding best approximations, Chebyshev polynomials of the first kind, best approximation properties for Chebyshev polynomials, Chebyshev expansions, best L² approximations, characterization of best L² approximations, Legendre polynomials.
7. Polynomial interpolation: Lagrange interpolation -- Vandermonde method, Lagrange formula, error formula, Hermite interpolation, spline interpolation, piecewise Lagrange interpolation.
8. Numerical integration: Newton-Cotes methods, error estimates, Gaussian quadratures, orthogonal polynomials.
9. Numerical ODEs: general concepts, linear single step methods, linear multistep methods, stability and convergence theorems, Runge-Kutta methods.

Examinations