BASIC NUMERICAL METHODS

The course is an introduction to numerical methods for the solution of basic mathematical problems. Addressed to students of the second year of Mathematics, it aims to present some basic ideas of the art numerical approximation, such as accuracy, robustness and efficiency of the methods, and fundamental techniques for the solution of linear systems, interpolation and approximation functions, solution of nonlinear equations and computation of integrals. The course provides an introduction to the use of Matlab for numerical computation. The implementational aspects of the methods are explained in part during the course, and partly during the afternoon supplementary activities (tutoring) related to the course.

In more detail, the following points are considered:

Knowledge and understanding: one of the fundamental objectives of the course is the learning of the basic methodologies for the numerical resolution of some mathematical problems such as the solution of linear systems, the approximation of data and functions, the solution of nonlinear equations, approximation of integrals. The aim is an active and critical understanding of the discipline, not limited to the mere learning of the methodologies, but also and above all a profound understanding of the basic ideas that transversally permeate the techniques relating to the various problems considered.

Ability to apply knowledge and understanding: the computational tools learned must be mastered and applied to concrete problems. The verification of learning and understanding of the methodologies, in fact, can only be carried out through the practical application of the same to various problems which are partly assigned at home and partly carried out in class.

Making judgments: students must be able to compare the different methods learned during the course, and understand which is the most suitable for solving a particular problem, taking into account both the characteristics of the problem itself (or of the problems that you intend to address) and the resources available or that you decide to use (problem of computational efficiency)

Communication skills: the students are invited to clearly explain the topics covered during the oral exam, and to explicitly illustrate the various steps in solving the problems. Even at the programming level, the clarity of the writing of the codes is taken into account in the evaluation of the documents.

Learning skills: learning is stimulated already during the lectures through direct questions from the teacher. These questions have the triple purpose of drawing the learner's attention, forcing him to think quickly about an answer, and probing the level of the class, so as to be able to better regulate the pace of the teacher's presentation, or to identify and address any critical issues together. 

The teaching activity consists mainly of lectures given by the teacher, during which the topics of the course are presented.

The lessons will be face-to-face.

During class the teacher will illustrate Matlab implementations of some of the methods presented in the lectures. The course is complemented by additional exercise sessions, and tutoring activities.

During the exercise sessions, exercises similar to those proposed in the practical test will be carried out, and any additions to the lessons.

Students are invited to clarify any doubts during the lesson or during the office hours. It is better to clarify the doubts before taking the exam than after.

The course consists of 5 credits of lectures, for a total of 35 hours of classroom teaching, and 1 of exercises, for a total of 12 hours of classroom teaching.

Content of the course

Introduction to the use of the computer for numerical calculations. Introduction to Matlab programming language. Variables and elementary instructions. Cycles. Data structures. Graphics.

Floating point representation. Machine numbers. Truncation and rounding. Machine operations. Digital cancellation. Order of accuracy.

Numerical linear algebra. Elements of linear algebra: vectors, matrices, determinants, inverse matrix. Vector and matrix norms. Natural norms and their representation. Eigenvalues. Spectral radius and its properties. Some special matrices. Direct methods for solving linear systems: triangular systems, Gaussian elimination, pivoting. Factorization A = LU and PA = LU. Compact methods, Choleski factorization and their implementation in python. Conditioning of a linear system. Condition number. Sparse matrices and their representation. Eigenvalues and eigenvectors: brief review. Iterative methods for solving linear systems: Jacobi method, Gauss-Seidel method, SOR method. Stopping criteria. Pointwise and block methods (mention). Singular value decomposition (mention). Localization of the eigenvalues: the theorems Gershgorin-Hadamard. Eigenvalues calculation: direct and inverse power methods.

Approximation of functions and data. Polynomial interpolation. Lagrange form. Linear interpolation operator. Calculation of the interpolation polynomials. Newton's formula of divided differences. The error formula (Lagrange form). Chebyshev polynomials: recursive formula, zeros, the property of minimal norm. Weierstrass theorem on approximation of a continuous function by polynomials (statement). Bernstein polynomials. The problem of the convergence of a sequence of interpolators schemes. Interpolation by piecewise-polynomials. Splines. Calculation of cubic splines. Method of least squares and applications. Normal equations and their geometric interpretation.

Solution of nonlinear equations. General concepts. Methods of bisection, secant and Newton. General theory of iterative methods for nonlinear equations and fixed point problems. Convergence order. Stopping criteria. Aitken method for one and two order of convergence of methods.

Quadrature formulas. Weighted integrals. General form of a quadrature formula. Polynomial order (or degree of precision) of a quadrature formula. Interpolatory formulas. Convergence theorem. Newton-Cotes formulas. Gaussian formulas. Composite formulas: trapezoid and Simpson formulae. Romberg method. Adaptive quadrature (notes).

The course is almost self-contained, and the subject can be prepared using the notes taken during the lectures of the professor. 

In any cases, there are several excellent textbooks in English. An old, still very good one, is the classical book by Eugene Isaacson  (Author), Herbert Bishop Keller (Author): 
Analysis of Numerical Methods (Dover Books on Mathematics) Revised Edition.
It can be bought on Amazon for about 15 US$. 

The book can be supplemented by
Introduction to Numerical Analysis: Second Edition (Dover Books on Mathematics) 2nd Edition, by F. B. Hildebrand, at similar price. 

The exam consists of a written test and an oral test.

The oral test can be accessed only after passing the written test.

The written test consists in solving some proposed exercises with the help of the computer within a time set by the teacher and in any case not exceeding three hours.

Passing the written test allows access to the oral test in the same session or even in a subsequent session, within a year.

All tests are graded on a scale from 0 to 30. 30 is the best grade. 18 is the minimum to pass the exam. The minimum grade obtained in the written test recommended for access to the oral exam is 18/30. The final grade of the exam is obtained as a weighted average between the results of the written and oral tests.

Students with certified disabilities have more time to complete the written test.

Evaluation grid.

Below are the minimum requirements for achieving the various scores. Reference is made to the points indicated in the programming.

Grade range 18-22: it is necessary to know the topics at points 1,2,3,4,11,12,19,20,23,25.

Grade range 23-26: it is necessary to know, in addition to those indicated above, also the topics at points 5,6,7,9,13,14,15,17,21,24.

Grade range 27-30L: it is necessary to know, in addition to those indicated above, also the topics at points 8.16.18.22.26.

The written tests assigned in previous exam sessions can be found at the following addresses:

https://www.dropbox.com/scl/fo/bxy7mn9efsccnbfx3s2oq/h?rlkey=da5g75vw4iv3evyzkqufhc7py&dl=0

https://www.dropbox.com/scl/fo/6e8w6udat78d5ojher52k/APiyoXiy7CETBmu-nZEfBwk?rlkey=ptlbkk1pn6owhelkq8d7ylm73&dl=0 

Examples of oral questions are:

Show me that the naive Gaussian elimination procedure is equivalent to an A=LU factorization of the system matrix

Tell me about Lagrange interpolation

What are spline functions and which are the most commonly used?

What is the polynomial order of a quadrature formula and how does it relate to interprolatory quadrature formulas?