Complements of Mathematics and Calculus

Why numerical analysis. Floating-Point Representation and errors,
stability of computations. Condition Number and ll-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. Runge counter-example. Interpolation vs extrapolation. Linear regression, least square method in the discrete case.

Numerical derivative: forward and backward approximation, mid-point formula, second derivative's approximation, Taylor series and error formulas.

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. Definitions and examples of absolute stability and instability, 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 and LU factorization. Jacobi and Gauss-Seidel iterative methods. Brief overview of localization and numerical approximation of Eigenvalues and Eigenvectors.