KNOWLEDGE AND UNDERSTANDING
At the end of the course the student will know some modern numerical
methodologies in the fields of linear and nonlinear systems,
approximation theory and ordinary differential equations.
APPLYING KNOWLEDGE AND UNDERSTANDING
At the end of the course the student will be able to use the studied
numerical methodologies in order to solve problems coming from the
applications.
MAKING JUDGMENTS
At the end of the course the student will be able to recognize the
fundamental characteristics of the considered problems and will have the
ability to appropriately choose the methods to solve them.
COMMUNICATION SKILLS
At the end of the course the student will be able to express himself
appropriately in the description of the numerical methods studied, with
mastery of language and asservativeness in the presentation.
LEARNING SKILLS
At the end of the course the student will be able to consult texts of
Numerical Analysis of medium-high level in order to complete his/her
knowledge.

Basic knowledge of the various numerical methods.
It is necessary to have passed the following exam: Analisi Numerica 1

Numerical solution of linear systems: background on classical direct
methods based on the LU factorization; iterative methods based on the
splitting of the coefficient matrix and convergence results; Richardson
method; methods of gradient type and Conjugate gradient method;
Krylov projection methods; Krylov methods based on the arnoldi
algorithm and convergence results.
Numerical solution of nonlinear equations: background on the scalar
case, bisection method, fixed-point iteration, Newton and secant method;
convergence order; working hypothesis in the nonscalar case; fixed-point
iteration and convergence criteria; Newton's method; Newton type
methods; Broyden method; convergence theorems.
Minimization of functionals: the transformation of the problem in a nonlinear equation; applicability study of Newton's method and its variants;
descent methods; construction of the descent directions; line-search
strategy; backtraking algorithm; convergence results.
Approximation theory: definition of system of orthogonal polynomials and
main properties; interpolation at zeros of orthogonal polynomials; ErdosTuran's theorem; examples of orthogonal polynomials.
Cubic spline function interpolation: natural, periodic and constrained
splines. Minimum energy property. Computation of the cubic interpolating
splines: the momentum linear system.
Primers on Runge-Kutta methods. Automatic integration with variable
stepsize: local error tolerance and global error proportionality, choice of
the integration stepsize. Strategies for the local error estimate: RungeKutta-Fehlberg and Dormand-Prince pairs.

[(1) Y.Saad (2000). Iterative methods for sparse linear systems. Springer]
[(2) J.E.Dennis, R.B. Schnabel (1996). Numerical methods for
unconstrained optimization and nonlinear equations. SIAM]
[(3) V.Comincioli (1995). Analisi numerica. McGraw-Hill]
[(4) appunti dei docenti]

Numerical solution of linear systems: background on classical direct
methods based on the LU factorization; iterative methods based on the
splitting of the coefficient matrix and convergence results; Richardson
method; methods of gradient type and Conjugate gradient method;
Krylov projection methods; Krylov methods based on the arnoldi
algorithm and convergence results.
Numerical solution of nonlinear equations: background on the scalar
case, bisection method, fixed-point iteration, Newton and secant method;
convergence order; working hypothesis in the nonscalar case; fixed-point
iteration and convergence criteria; Newton's method; Newton type
methods; Broyden method; convergence theorems.
Minimization of functionals: the transformation of the problem in a nonlinear equation; applicability study of Newton's method and its variants;
descent methods; construction of the descent directions; line-search
strategy; backtraking algorithm; convergence results.
Approximation theory: definition of system of orthogonal polynomials and
main properties; interpolation at zeros of orthogonal polynomials; ErdosTuran's theorem; examples of orthogonal polynomials.
Cubic spline function interpolation: natural, periodic and constrained
splines. Minimum energy property. Computation of the cubic interpolating
splines: the momentum linear system.

Lectures, both of theoretical character and aimed to the solution of
exercises, also including the design and the execution of related
computer programs.

At the end of each topic of the course, the teacher will provide a copy of
his own notes.

Oral exam, where the students knowledge of the various numerical
techniques tought in the course, as well as the ability to suitably apply
them to the considered mathematical problems, is checked.