Complements of Mathematics and Calculus

The polynomial interpolation problem. Proof of the uniqueness theorem. Construction of the interpolation polynomil 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. Computation of the weights of trapezoidal and Cavalieri-Simpson rules.

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 and chords' 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. Linearly dependent and independent vectors.

Recalls on matrices in R(mxn): Diagonal and triangular matrices, symmetric matrices, diagonally dominant. Norms of matrices: definition, properties, norm-1 and infinity-norm.
Product, determinant, eigenvalues of diagonal, tridiagonal and triangular matrices. Determinant of a 2x2 and 3x3 matrix (Sarrus rule). Properties of determinants. Definition of inverse matrix. Linear systems Ax=b 2x2, 3x3, an overview of nxn systems. Homogeneous systems. Compatible (determined and undetermined) and incompatible (impossible) systems. Condition number of a square matrix. Condition number of a linear system in the particular case of perturbation of the vector of constant terms. Square submatrices, minor and rank of a matrix, augmented matrix [A|b]. Rouché-Capelli theorem (in the case A(nxn), without proof). Examples of determined, undetermined, impossible 2x2 and 3x3 systems. Degrees of freedom. Norm-2 of a matrix and condition number in norm-2. Gauss elimination method and LU factorization. Jacobi and Gauss-Seidel iterative methods.