This course aims to illustrate the basic techniques to solve some fundamental mathematical problems numerically.
D1 - Knowledge and understanding
By the end of the course, the student will have to know and be familiar with the basic techniques to solve problems in numerical linear algebra, to approximate functions, to solve nonlinear equations, to compute integrals and to solve ordinary differential equations. The student will also have to be able to design and implement the related computational algorithms.
D2 - Applying knowledge and understanding
The student will have to be able to face and solve exercises, questions and problems, of both theoretical and computational nature, related to the topics treated in the course.
D3 - Making judgements
The student will have to be able to choose suitably the numerical methods that better approximate the solutions of various mathematical problems, within those ones treated in the course or strictly related to them.
D4 - Communication skills
The student will have to be able to describe techniques and issues of basic Numerical Analysis with an appropriate vocabulary.
D5 - Learning skills
The student will have to be able to read and understand books and articles which treat the topics learnt in the course and, also, to be able to look further into them in his/her own way.

Basic knowledge in Geometry, Mathematical Analysis and Computer Science. It is necessary to have passed the following exams: Geometria 1, Informatica, Analisi 2.

REPRESENTATION OF NUMBERS AND MACHINE ARITHMETIC (Representation of numbers in a given basis. Machine numbers: integer and floating point. Overflow and underflow. Truncation and rounding. Machine precision. Machine arithmetic.) CONDITIONING (Conditioning of elementary operations. Stability of problems and algorithms.) PROGRAMMING IN MATLAB (Main commands and instructions). PRIMER OF LINEAR ALGEBRA (Vector and matrix norms. Eigenvalues, eigenvectors, and spectral radius. Relationships between norms and spectral radius. Hermitian and positive definite matrices and their properties.) SOLUTION OF LINEAR SYSTEMS (Conditioning of linear systems. Gaussian elimination method. LU and RR^H factorization. Pivoting strategies. Solution of overdetermined linear systems in the least squares sense.) COMPUTATION OF EIGENVALUES AND EIGENVECTORS (Gerschgorin's theorems. The power and inverse power methods and their variants.) APPROXIMATION OF FUNCTIONS (Finite dimensional spaces of approximating functions. Density of algebraic polynomial and piece-wise polynomial function spaces in C[a,b] and in L_2[a,b]. Hints of the best approximation problem: existence of the best approximation generalized polynomial, uniqueness in the algebraic polynomial case in C[a,b] and in L_2[a,b].) INTERPOLATION BY ALGEBRAIC POLYNOMIALS (Lagrange form. Linear interpolation operator. Lebesgue's numbers. Hints of Chebyshev polynomials. Convergence of interpolation schemes: Faber's, Natanson's, and Jackson's theorems. Runge phenomenon. Finite difference Newton's formula.) SOLUTION OF NONLINEAR EQUATIONS (Bisection method, The chord, secant, and tangent methods. Error bounds and stopping criteria. General theory for iterative methods. Attractive and repulsive fixed points. Order of convergence.) QUADRATURE FORMULAE (Polynomial order. Interpolatory formulae. Hints of Gaussian formulae. Newton-Cotes' formulae. Convergence: general theorem, Kusmin's theorem for Newton-Cotes' formulae. Composite formulae. Trapezium and Simpson's formulae. Adaptive quadrature.) NUMERICAL METHODS FOR INITIAL VALUE ORDINARY DIFFERENTIAL EQUATIONS (Primer of initial value problems. Lipschitz conditions: classical and one-sided. Existence and uniqueness theorems. Continuous dependence of the solution on the initial data. Computation of the Lipschitz constants for linear problems. Explicit and implicit Euler's methods. The concept of stiffness and comparison between the two Euler methods when applied to stiff problems. One-step methods. Order of consistency and convergence. General convergence theorem for variable stepsize. Runge-Kutta methods with hints of the order conditions.)

[1] V. Comincioli: Analisi Numerica, McGraw-Hill, Milano, 1990 (and successive editions) [2] Course notes supplied by the lecturer

REPRESENTATION OF NUMBERS AND MACHINE ARITHMETIC (Representation of numbers in a given basis. Machine numbers: integer and floating point. Overflow and underflow. Truncation and rounding. Machine precision. Machine arithmetic.) CONDITIONING (Conditioning of elementary operations. Stability of problems and algorithms.) PROGRAMMING IN MATLAB (Main commands and instructions). PRIMER OF LINEAR ALGEBRA (Vector and matrix norms. Eigenvalues, eigenvectors and spectral radius. Relationships between norms and spectral radius. Hermitian and positive definite matrices and their properties.) SOLUTION OF LINEAR SYSTEMS (Conditioning of linear systems. Gaussian elimination method. LU and RR^H factorization. Pivoting strategies. Solution of overdetermined linear systems in the least squares sense.) COMPUTATION OF EIGENVALUES AND EIGENVECTORS (Gerschgorin's theorems. The power and inverse power methods and their variants.) APPROXIMATION OF FUNCTIONS (Finite dimensional spaces of approximating functions. Density of algebraic polynomial and piece-wise polynomial function spaces in C[a,b] and in L_2[a,b]. Hints of the best approximation problem: existence of the best approximation generalized polynomial, uniqueness in the algebraic polynomial case in C[a,b] and in L_2[a,b].) INTERPOLATION BY ALGEBRAIC POLYNOMIALS (Lagrange form. Linear interpolation operator. Lebesgue's numbers. Hints of Chebyshev polynomials. Convergence of interpolation schemes: Faber's, Natanson's and Jackson's theorems. Runge phenomenon. Finite difference Newton's formula.) SOLUTION OF NONLINEAR EQUATIONS (Bisection method, The chord, secant and tangent methods. Error bounds and stopping criteria. General theory for iterative methods. Attractive and repulsive fixed points. Order of convergence.) QUADRATURE FORMULAE (Polynomial order. Interpolatory formulae. Hints of Gaussian formulae. Newton-Cotes' formulae. Convergence: general theorem, Kusmin's theorem for Newton-Cotes' formulae. Composite formulae. Trapezium and Simpson's formulae. Adaptive quadrature.) NUMERICAL METHODS FOR INITIAL VALUE ORDINARY DIFFERENTIAL EQUATIONS (Primer of initial value problems. Lipschitz conditions: classical and one-sided. Existence and uniqueness theorems. Continuous dependence of the solution on the initial data. Computation of the Lipschitz constants for linear problems. Explicit and implicit Euler's methods. The concept of stiffness and comparison between the two Euler methods when applied to stiff problems. One-step methods. Order of consistency and convergence. General convergence theorem for variable stepsize. Runge-Kutta methods with hints of the order conditions.)

Lectures, either of theoretical character or aimed at solving exercises, also including the design and the execution of related computer programs. An additional twenty hours of laboratory support, whose attendance is not mandatory, are planned as well. It is devoted to going further into the programming of the algorithms presented during the lectures. Classroom exercises include those proposed in the past years' written exams, involving the students in person, one after the other.

The lecturer distributes the related notes at the end of each topic. Any changes to the methods described here, which become necessary to ensure the application of the safety protocols related to the COVID-19 emergency, will be communicated on the websites of the Department of Mathematics, Informatics and Geoscience - MIGe and of the Study Program in Mathematics.

The exam is designed to assess both the knowledge of numerical methods covered during the course and the ability to apply these methods to solve exercises. It consists of two parts: a written exam followed by an oral exam. In the written exam, which is not graded separately, students must solve three exercises. These exercises closely resemble those assigned in class and those worked on by the lecturer. They are considered to have a moderate level of difficulty and must be completed within 50 minutes (maximum 1 hour). The primary goal of the written exam is to identify any significant gaps in a student's preparation rather than to certify a high level of proficiency. The actual level of understanding is evaluated during the subsequent oral exam. To be admitted to the oral exam, students must correctly solve at least two out of the three exercises. The oral exam involves a discussion on the topics covered throughout the course and typically lasts about 30 minutes. This part of the exam will assess students’ understanding of fundamental principles of the numerical techniques introduced, focusing on their application in solving nonlinear equations, linear systems of equations, least-squares problems, interpolation, numerical integration, and ordinary differential equations. Students are expected to provide clear, formal descriptions of the numerical methods studied, using appropriate terminology and mathematical rigor. Grades are awarded on a 30-point scale, with a minimum passing mark of 18/30. A score of 18/30 indicates that the student has acquired sufficient knowledge of the subject and demonstrates adequate proficiency in the technical language. This grade will be given to students who correctly answer only two-thirds of the questions during the oral exam. A top score of 30/30 (with honors) will be awarded to students who exhibit excellent knowledge of the subject and proficiency in technical language, along with the ability to think critically and analytically. Students with specific learning disabilities (SLD) who wish to receive compensatory tools during the exam are encouraged to contact the Disability Service and the office for students with specific learning disabilities.