Ateneo - UPO

The course provides notions on error analysis, finite precision number system and arithmetic, numerical approximation of the solution of nonlinear equations, 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 aims to provide students with knowledge of basic numerical methods and an analysis of their main properties. The goal is for students to acquire and be able to use an appropriate mathematical language in relation to the topics covered in the course, and to develop the ability to use the concepts learned correctly and consciously. The educational objective of the course is to develop the ability to implement the mathematical tools introduced in class on a computer. In particular, students will be able to evaluate a numerical scheme both in terms of the accuracy of the approximation it provides and the computational cost of its implementation.

The knowledge of the main notions provided during a basic course of Mathematics is required. In more details:
Functions and sequences, Limits, Differential calculus, Taylor expansion, Integral calculus in one variable, Initial value problems for ordinary differential equations;
Vector spaces, Linear systems, Matrix algebra, Eigenvalues.

Teaching will take place through lectures on the blackboard and exercises in the computer lab. In addition to the theoretical lessons, the teacher will carry out exercises in the computer lab with the active involvement of the students to deepen the topics covered during the theoretical lessons. The concepts covered by the course will be discussed collegially in the classroom and applied directly during laboratory exercises to stimulate students' critical sense and autonomy of judgment.

Attendance: Advised

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.

Oral exam on the topics covered during the theoretical and practical lessons (computer lab). During the oral exam, the student must be able to demonstrate his/her knowledge of the main course contents using the appropriate terminology. He/She must also demonstrate ability of putting into practice, with critical awareness, some of the activities carried out in laboratory during the course.

ERROR ANALYSIS: Absolute error and relative error. Types of errors. Machine numbers. Representation through truncation and rounding. Machine precision. Overflow and Underflow. Finite arithmetic. Conditioning of a problem. Conditioning of the four fundamental algebraic operations.

APPROXIMATION OF FUNCTION ZEROS: The bisection method: stopping criteria, problem conditioning, and convergence order. Newton's method: local convergence and quadratic convergence order for simple roots. The case of multiple roots: linear convergence and restoration of quadratic convergence when exact multiplicity is known a priori. Quasi-Newton methods.

SOLVING LINEAR SYSTEMS: Simple cases. LU factorization: existence, uniqueness, and computational cost. Diagonally dominant matrices. Symmetric and positive-definite matrices: Cholesky method. Pivoting. Problem conditioning. Basic iterative methods. Overdetermined linear systems: least squares solution, normal equations method, and a brief mention of the method based on QR factorization.

FUNCTION APPROXIMATION: Polynomial interpolation: existence and uniqueness of the interpolating polynomial. Lagrange form. Brief analysis of error. The discrete least squares method.

NUMERICAL INTEGRATION: Interpolatory quadrature formulas. Newton-Cotes formulas: the trapezoidal method and the simple Simpson's method. Error analysis. Precision degree of a quadrature formula. Conditioning analysis of a definite integral and a quadrature formula. Composite trapezoidal and Simpson’s methods and their error analysis.

MATLAB COMPUTING ENVIRONMENT: Definition of scalar variables, vectors, and matrices. Variable types. Colon operator. Subvectors and submatrices. Arithmetic, relational, and logical operators. Main built-in functions. Script-type M-files. Input and output commands for data. Commands for controlling the execution flow. Function-type M-files: input and output parameters. Main 2D graphics commands.

At the end of the course the student must have acquired the basic techniques for the development of numerical methods and the study of their main properties. The student must be able to identify the most appropriate methods for solving some specific numerical problems also in relation to the required accuracy and the available computing resources. He/she must therefore be able to pass from the development and analysis of the methods to their implementation, their use and the related correctness and accuracy tests. The student, on the basis of the analysis of the methods studied, must be able to provide an evaluation of the pros and cons of each method in order to make informed choices in solving problems. The student must also be able to evaluate the quality of the results provided by the numerical tests carried out by means of the codes presented by the teacher in the classroom or made by the student himself. The student will have to acquire the ability to describe, for each problem studied, the nature of the problem itself, the difficulties in solving it on the computer and the ways in which these difficulties are faced. He/she will also have to acquire the ability to effectively present the results of his/her numerical experiences. On the basis of the problems, the methods studied and their analysis, the student will have to acquire the ability to independently develop solution strategies to be applied to new problems.