Ateneo - UPO

The course provides notions on error analysis, finite precision number system and arithmetic, efficient methods for the solution of systems of linear equation. The course also approaches the main issues related to function approximation and numerical integration.

Teacher's notes available on the DIR platform.

The course, corresponding to 3 CFU, is delivered through expository teaching (DE) activities consisting of blackboard-based lectures. It aims to provide students with the fundamental knowledge of the main basic numerical methods and their properties, with particular reference to the solution of mathematical problems of interest in applied physics. The lectures will be complemented by computer laboratory sessions (10 hours), during which the mathematical tools introduced will be directly applied through the use of computers.

The course aims to develop students’ ability to understand the meaning of the numerical algorithms introduced, analyze their accuracy, and evaluate their effectiveness in relation to the problem under consideration. Students will also acquire the ability to use appropriate mathematical language in the description of numerical methods and their related results.

At the end of the course, students will be able to:

- understand the principles underlying the main numerical methods for solving elementary mathematical problems;
- correctly apply basic numerical algorithms to solve standard problems;
- analyze approximation errors and evaluate the accuracy of the results obtained;
- compare different numerical approaches according to the required precision and computational cost.

The main educational objective is to provide mathematical and computational tools that enable students to approach quantitative problems in applied physics in a conscious and effective way, developing critical analysis skills and the ability to select the most appropriate method.

Knowledge of the fundamental concepts provided in previous mathematics courses. More specifically: functions and sequences, limits, differential calculus of one variable, Taylor expansions, and integral calculus of one variable. Vector spaces, linear systems, and matrix algebra.

The course will be delivered through blackboard-based lectures, aimed at presenting the theoretical concepts and numerical methods covered in the course. The lectures will be complemented by computer laboratory sessions, during which the mathematical tools introduced will be directly applied through the use of computers.

The laboratory activities will involve the active participation of students, with the aim of consolidating the understanding of the topics covered, developing practical skills in the implementation of numerical algorithms, and promoting the critical analysis of the results obtained.

The concepts addressed during the course will be discussed and further explored collectively in the classroom and applied during practical exercises, in order to foster independent judgement, the ability to interpret results, and the conscious use of numerical methods.

Attendance: recommended.

Students with physical disabilities, Learning Disabilities or Special Education Needs can request specific services and tools via the Staff Sviluppo e Coordinamento Carriere e Servizi alle Studentesse e agli Studenti, consulting the University webpage:
https://www.uniupo.it/en/services/servicesstudents-physical-or-learning-disabilities

Students with disabilities, learning disabilities or special education needs, once they have contacted the University Staff, can refer to the tutor in charge of the course to define the examination modalities, concerning academic aspects.

The examination consists of an oral test on the topics covered during the theoretical lectures and the practical activities carried out in the computer laboratory.

During the oral examination, students must demonstrate that they have acquired the knowledge related to the course contents, using appropriate mathematical tools, technical language, and terminology. The examination may include a discussion of the numerical methods studied, their properties, error analysis, and problem conditioning, as well as aspects related to the implementation of algorithms in the Octave computing environment.

The assessment takes into account the level of understanding of the theoretical concepts, the ability to correctly apply the acquired numerical methods, the interpretation of the results obtained, and the ability to independently address new problems.

Grading criteria:

18–20/30: the student demonstrates an essential knowledge of the course topics and a basic understanding of the main numerical methods. The student is able to describe the fundamental procedures and apply them to standard problems, although with possible inaccuracies in explaining the steps or interpreting the results.
21–23/30: the student demonstrates a fair knowledge of the course contents, understands the functioning of numerical algorithms, and is able to correctly apply the introduced methods, providing an adequate description of the procedures and results.
24–26/30: the student demonstrates a good knowledge of the topics, the ability to analyze the characteristics of numerical methods, evaluate error and problem conditioning, and critically interpret the results obtained.
27–29/30: the student demonstrates a very good and in-depth knowledge of the course contents, mastery of mathematical language, autonomy in problem analysis, and the ability to compare different numerical approaches in relation to accuracy and computational cost.
30/30 and honours: the student demonstrates excellent mastery of the theoretical and practical contents of the course, the ability to contextualize numerical methods within applied physics, independently select appropriate solution strategies, and critically discuss the advantages, limitations, and reliability of the solutions obtained.

Achieving a passing grade requires the ability to understand the fundamental concepts and correctly apply the numerical methodologies introduced during the course. Achieving excellence requires, in addition to technical correctness, independent reasoning skills, the ability to establish connections among the topics, critically evaluate results, and contextualize numerical tools in applied problems.

Since this course constitutes a module of the integrated course Mathematics III, the final assessment of the integrated course is determined jointly by the teachers of the different modules, taking into account the grades obtained in the individual parts of the course.

Preparation for the examination is based on the lecture notes and teaching material provided by the instructor and made available on the DIR platform, which represent the main reference for the topics covered during the course. University textbooks on numerical methods and numerical analysis recommended by the teacher may also be consulted as supplementary study material.

ERROR ANALYSIS: absolute error and relative error. Types of errors. Machine numbers. Representation by truncation and rounding. Machine precision. Overflow and underflow. Finite arithmetic. Conditioning of a problem. Conditioning of the four basic algebraic operations.
SOLUTION OF LINEAR SYSTEMS: Simple cases. LU factorization: existence, uniqueness, and computational cost. Pivoting. Conditioning of the problem. Overdetermined linear systems: least-squares solution, normal equations method, and an introduction to the method based on QR factorization.
FUNCTION APPROXIMATION: Polynomial interpolation: existence and uniqueness of the interpolating polynomial. Lagrange form. Introduction to error analysis. The discrete least-squares method.
NUMERICAL INTEGRATION: Interpolatory quadrature formulas. Newton–Cotes formulas: the simple trapezoidal rule and Simpson’s rule. Error analysis. Degree of precision of a quadrature formula. Conditioning analysis of a definite integral and of a quadrature formula. Composite trapezoidal and Simpson’s rules and related error analysis.
OCTAVE COMPUTING ENVIRONMENT: Definition of scalar variables, vectors, and matrices. Variable types. Colon operator. Subvectors and submatrices. Arithmetic, relational, and logical operators. Expressions. Main built-in functions. Script-type M-files. Commands for data input and output. Commands for execution flow control. Function-type M-files: input and output parameters. Main commands for 2D plotting.

At the end of the course, thanks to the educational resources provided by the Degree Programme (lectures, exercises, computer laboratory activities, teaching material, and application examples), students will have acquired fundamental knowledge and skills related to the development, analysis, and use of numerical methods for the solution of mathematical problems of interest in applied physics.

Knowledge and understanding

Students will acquire the fundamental techniques for the development of numerical methods and knowledge of their main properties, with particular reference to error analysis, problem conditioning, accuracy of results, and the computational aspects of algorithms.

Satisfactory level:
Students know the principles underlying the main numerical methods covered in the course, understand the meaning of the approximations introduced, and recognize the main factors affecting the accuracy and reliability of numerical results.

Advanced level:
Students demonstrate an in-depth understanding of the numerical methods studied, are able to analyze their properties, advantages, and limitations, and understand the role of error, conditioning, and computational cost in the selection and use of algorithms.

Applying knowledge and understanding

Students will be able to identify and apply appropriate numerical methods for the solution of specific problems, taking into account the required accuracy and the available computational resources. They will also be able to implement the algorithms studied, verify their correct functioning, and evaluate their performance through appropriate numerical tests.

Satisfactory level:
Students are able to apply standard numerical methods to solve known problems, implement simple algorithms in a computing environment, and verify the correctness of the results obtained through comparisons and error analysis.

Advanced level:
Students are able to independently select the most appropriate numerical method for the problem under consideration, justifying their choice on the basis of accuracy, stability, and computational cost. They are also able to critically analyze the results of numerical simulations and develop solution strategies for new problems.