**1st module**

**Dott. Paola Zanolli**

**A.A. 1998/99**

**PROGRAMME**

1. Algebraic and Transcendental Systems: Matrices and Linear Systems, Gauss Elimination, Tridiagonal Systems, The Generalized Newton's Method, Remarks on the Generalized Newton's Method, Eigenvalues and Eigenvectors.

2. Approximation: Discrete Functions, Piecewise Linear Interpolation, Piecewise Parabolic Interpolation, Cubic Spline Interpolation, Lagrange Interpolation, Least Squares.

3. Approximate Integration and Differentiation: The Trapezoidal Rule, Simpson's Rule, Romberg Integration, Numerical Differentiation.

4. Initial Value Problems for Ordinary Differential Equations:
Euler's Method, Convergence of Euler's Method, A Runge-Kutta Method, Higher
Order Runge-Kutta Formulas, Kutta's Fourth-Order Method for a System of
Two First-Order Equations, Kutta's Fourth-Order Formulas for Second-Order
Differential Equations, The Method of Taylor Expansions, Instability, Approximation
of Periodic Solutions of Differential Equations.