Complements of Mathematics and Numerical Analysis

Why numerical analysis. Floating-Point Representation and errors, stability of computations. Condition Number and ill-conditioning, stability of algorithms and problems.

The interpolating polynomial problem. Proof of the uniqueness theorem. Construction of the interpolating polynomial with Vandermonde method and with Lagrange method. Interpolation error. Chebyshev interpolation points. Runge counter-example. Chebyshev interpolation points. Linear regression, least square method in the discrete case.

Quadrature formulas for the approximation of definite integrals: rectangle, mid-point, trapezoidal and Cavalieri-Simpson rule. Newton-Cotes quadrature formulas: nodes, weights, open and closed formulas, degree of precision. Composite quadrature formulas; error's formulas.

An introduction to the numerical approximation of ordinary differential equations. Explicit and implicit Euler's method: geometrical derivation, with Taylor expansion, using derivatives' approximation schemes, based on quadrature formulas. Crank-Nicolson and Heun methods. Analysis of absolute stability, definition of convergence.

Approximation of roots of functions: Bisection, Newton, chords' and secant method. Order of convergence. Stop test. Global convergence theorem for the Newton method.

Some recalls on vectors in the plane and in the space (R2 and R3): Norms of vectors.

Recalls on matrices in R(mxn): Diagonal and triangular, symmetric, diagonally dominant, symmetric positive definite. Norms of matrices: definition, properties, norm-1, norm-2 and infinity-norm.
Determinant of a 2x2 and 3x3 matrix (Sarrus rule). Condition number of a square matrix. Condition number of a linear system in the particular case of perturbation of the vector of constant terms. Gauss elimination method, A=LU, PA=LU, Cholesky factorizations. Jacobi and Gauss-Seidel iterative methods. Brief overview of localization of Eigenvalues.