NUMERICAL CALCULUS

Course objectives

GENERAL OBJECTIVES This course provides the basic knowledge of some numerical methods for the solution of common mathematical problems in applied sciences and engineering and gives the basic concepts of Matlab programming. The course acts as a link between the basic courses of Calculus I and II and Geometry and the engineering courses of the following years. Particular attention will be devoted to the analysis of the methods and their use in Matlab environment. To this end, the course will consist of theoretical lectures, whose aim is to illustrate the main characteristics of the methods, and practical lectures, where simple application problems will be solved also using Matlab. SPECIFIC OBJECTIVES 1. Knowledge and understanding: the student will know the basic concepts of numerical analysis and the main characteristics of some of the numerical methods commonly used to solve problems that arise in applied sciences. 2. Applying knowledge and understanding: the student will be able to use the numerical methods learned, he will be trained to recognize the class of numerical methods necessary to solve a given problem, to select the most suitable one, to provide an algorithmic solution and to find the solution using Matlab. 3. Making judgments: the student will be able to select a numerical method suitable for solving some test problems, to estimate approximation errors and to analyze its performance through numerical experiments. 4. Communication skills: the student will be trained to rigorously describe the basic mathematical concepts of numerical analysis, the algorithmic formulation of some numerical methods, the results of numerical tests. 5. Learning skills: the student will be able to classify a problem with respect to the class of numerical methods required for its solution, to use some basic numerical methods to solve some application problems even in Matlab environment.

Channel 1
VITTORIA BRUNI Lecturers' profile
VITTORIA BRUNI Lecturers' profile
FRANCESCA PITOLLI Lecturers' profile

Program - Frequency - Exams

Course program
The course consists of 60 hours of instruction, including lectures and practical sessions, and awards 6 ECTS credits. The content is organized around the following key topics: Conditioning of a Problem and Algorithm Stability: Introduction to the concepts of problem conditioning and algorithmic stability, essential for understanding the behavior of numerical methods. Iterative Methods for Solving Nonlinear Equations and Systems: Study of methods such as bisection, Newton’s method, the secant method, and fixed-point iteration. Analysis of convergence properties and stopping criteria. Numerical Linear Algebra: Solution of linear systems using direct methods and their practical applications. Development of iterative methods including Jacobi, Gauss-Seidel, and Successive Over-Relaxation (SOR) methods, along with convergence analysis. Numerical Methods for Initial Value Problems (IVPs) in Ordinary Differential Equations (ODEs): Euler’s method, Runge-Kutta methods, their convergence behavior, and an introduction to implicit methods. Finite Difference Methods for Boundary Value Problems: Application of finite difference schemes to solve linear and nonlinear ODEs, and transport equations. Introduction to MATLAB: Fundamentals of the MATLAB computing environment. Use of MATLAB for solving problems using numerical methods and for visualizing computational results.
Prerequisites
Analisi I e Geometria
Books
L. Gori, Calcolo Numerico, Ed. Kappa, 2006 L. Gori, M.L. Lo Cascio, F. Pitolli, Esercizi di Calcolo Numerico, Ed. Kappa, 2007 Materiale integrativo disponibile sulla pagina di e-learning del corso
Frequency
Attendance is not mandatory, but strongly recommended.
Exam mode
The purpose of the assessment is to verify whether the student has achieved the intended learning objectives. It consists of solving numerical computation exercises. Students are required to identify the most appropriate numerical method for a given problem, describe its main features, and discuss fundamental numerical aspects (such as accuracy, convergence, and stability). They must also carry out numerical tests and critically analyze the results obtained. The evaluation includes a written exam and an optional oral exam.
Bibliography
S.D. Conte, C. de Boor: Elementary Numerical Analysis. An Algorithmic Approach. McGraw-Hill, 1980 A. Quarteroni, F. Saleri, P. Gervasio: Calcolo Scientifico. Springer, 2017 S.C. Chapra, R. P. Canale: Numerical Methods For Engineers. McGraw-Hill, 2010
Lesson mode
The course consists of 60 hours of instruction, including lectures and hands-on lab sessions, and awards 6 ECTS credits. Lectures and lab sessions will be held in Italian.
FRANCESCA PITOLLI Lecturers' profile

Program - Frequency - Exams

Course program
The course consists of 60 hours of instruction, including lectures and practical sessions, and awards 6 ECTS credits. The content is organized around the following key topics: Conditioning of a Problem and Algorithm Stability: Introduction to the concepts of problem conditioning and algorithmic stability, essential for understanding the behavior of numerical methods. Iterative Methods for Solving Nonlinear Equations and Systems: Study of methods such as bisection, Newton’s method, the secant method, and fixed-point iteration. Analysis of convergence properties and stopping criteria. Numerical Linear Algebra: Solution of linear systems using direct methods and their practical applications. Development of iterative methods including Jacobi, Gauss-Seidel, and Successive Over-Relaxation (SOR) methods, along with convergence analysis. Numerical Methods for Initial Value Problems (IVPs) in Ordinary Differential Equations (ODEs): Euler’s method, Runge-Kutta methods, their convergence behavior, and an introduction to implicit methods. Finite Difference Methods for Boundary Value Problems: Application of finite difference schemes to solve linear and nonlinear ODEs, and transport equations. Introduction to MATLAB: Fundamentals of the MATLAB computing environment. Use of MATLAB for solving problems using numerical methods and for visualizing computational results.
Prerequisites
Analisi I e Geometria
Books
L. Gori, Calcolo Numerico, Ed. Kappa, 2006 L. Gori, M.L. Lo Cascio, F. Pitolli, Esercizi di Calcolo Numerico, Ed. Kappa, 2007 Materiale integrativo disponibile sulla pagina di e-learning del corso
Frequency
Attendance is not mandatory, but strongly recommended.
Exam mode
The purpose of the assessment is to verify whether the student has achieved the intended learning objectives. It consists of solving numerical computation exercises. Students are required to identify the most appropriate numerical method for a given problem, describe its main features, and discuss fundamental numerical aspects (such as accuracy, convergence, and stability). They must also carry out numerical tests and critically analyze the results obtained. The evaluation includes a written exam and an optional oral exam.
Bibliography
S.D. Conte, C. de Boor: Elementary Numerical Analysis. An Algorithmic Approach. McGraw-Hill, 1980 A. Quarteroni, F. Saleri, P. Gervasio: Calcolo Scientifico. Springer, 2017 S.C. Chapra, R. P. Canale: Numerical Methods For Engineers. McGraw-Hill, 2010
Lesson mode
The course consists of 60 hours of instruction, including lectures and hands-on lab sessions, and awards 6 ECTS credits. Lectures and lab sessions will be held in Italian.
  • Lesson code1021737
  • Academic year2025/2026
  • CourseMechanical Engineering
  • CurriculumSingle curriculum
  • Year3rd year
  • Semester2nd semester
  • SSDMAT/08
  • CFU6