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## Subject: NUMERICAL CALCULUS (A.A. 2021/2022)

### degree course in PHYSICS

Course year 2 6 Unit Calcolo numerico A12 (lesson) TAF: Supplementary compulsory subjects SSD: MAT/08 CFU: 3 A12 (laboratory) TAF: Supplementary compulsory subjects SSD: MAT/08 CFU: 3 Teachers: Silvia BONETTINI oral final vote Italiano
Contents download ### Overview

The course aims to provide the knowledge, skills, and tools necessary to tackle basic mathematical analysis problems through numerical techniques and develop computing codes in the Matlab programming environment.
For a more complete understanding of the training objectives, please refer to the reading of the expected learning outcomes following the completion of this training course.

Basic ideas from integral and differential calculus; elements of linear algebra.

### Course contents

Computer representation of numbers: rounding errors and floating-point arithmetic.
Systems of linear equations: Gaussian elimination, pivoting in Gaussian elimination, LU-decomposition. Iterative methods: Jacobi method, Gauss-Seidel method, convergence analysis, stopping rules. Solution of nonlinear equations: bisection method, Newton's method, convergence analysis for Newton's method,
systems of nonlinear equations.
Data and functions approximation: basic functions for approximation, polynomial interpolation, interpolation by splines, least-squares approximation.
Numerical integration: Newton-Cotes integration formulas, error analysis, composite rules.
Basic statements of the MATLAB computing environment: array features, subprograms, operations on files, advanced statements for matrix operations and graphics, character arrays, structured data.

### Teaching methods

The course is delivered through face-to-face lectures and exercises that are carried out with the aid of a blackboard, audiovisual means (slides), and the computational environment Matlab. Attendance to face-to-face lessons is not compulsory. The course is delivered in Italian.

### Assessment methods

The exam will take place at the end of the course according to the official exam schedule. The exam is oral, lasting about 40 minutes. The exam includes 3 questions including an exercise on the implementation of a series of instructions in Matlab language, and two open questions on the numerical analysis topics seen in the course. These questions are aimed at evaluating: - knowledge and understanding skills; - the application of knowledge and understanding; - communication skills; - autonomy of judgment. The grade reported in the exam is given by the overall evaluation in the light of the answers to the 3 questions. The result will be communicated to the individual student at the end of the oral exam.

### Learning outcomes

1) Knowledge and understanding.
At the end of the course and through classroom lessons and individual study, it is hoped that the students will be able to orient themselves within the main concepts of numerical analysis relating to the machine numbers, solution of linear systems and nonlinear equations, and the problem of data approximation, recognizing and knowing how to rigorously describe the main definitions, properties and theorems seen in class.

2) Applied knowledge and understanding.
At the end of the course and through classroom exercises and individual work, it is hoped that the students will be able to model and solve mathematical problems using the techniques of numerical analysis with accuracy.

3) Autonomy of judgment.
At the end of the course, it is hoped that the students will be able to:
a) verify their degree of learning and understanding of the concepts exposed thanks to the possibility of intervention in class;
b) reorganize the knowledge learned and implement one's own ability to critically and independently evaluate what has been learned;
c) mastering a methodological approach that leads to verifying the statements and methods presented by means of rigorous arguments.

4) Communication skills.
At the end of the course, it is hoped that the students will be able to:
a) express their knowledge correctly and logically, recognizing the required topic and responding in a timely and complete manner to the exam questions.
b) face a dialectical confrontation in a timely and coherent way, arguing with precision.

5) Learning skills
At the end of the course, it is hoped that the students will be able to:
a) acquire computational knowledge as one's own heritage, which can be used at any other moment of one's cultural path;
b) have developed an aptitude for a methodological approach that leads to an improvement of the study method with consequent deepening of the ability to learn.