Numerical analysis

Course objectives

The course intends to present numerical methods of approximation for the solution of several mathematical problems that occur in many applications and in mathematical modeling. In particular, the following topics will be treated both from a theoretical and an algorithmic point of view: 1. Solution of systems of linear equations 2. Solution of    systems of non-linear equations 3. Approximation of functions and experimental data 4. Numerical quadrature 5. Numerical methods for ordinary differential equations The course includes some Lab sessions to develop codes in    MATLAB. 1. Knowledge and understanding The students who have passed the exam will know the most important numerical techniques on the topics presented in the course. 2. Applied knowledge and understanding The students who have passed the exam will be able to deal with data storage correctly and to decide which type of numerical method should be used to solve their problem. Moreover, they will be able to code the algorithms in    MATLAB. 3. Making judgments The students will be able to evaluate the results produced by their programs and to produce tests and simulations. 4. Communication skills The students will be able to present and explain the proposed solution of some problems and exercises both at the blackboard and using a computer. 5. Learning skills The acquired knowledge will construct the basis to study more specialized aspects of numerical analysis, scientific computing and simulations. The students will become familiar with different concepts and techniques related to the topics presented in the course.

Channel 1
SILVIA NOSCHESE Lecturers' profile

Program - Frequency - Exams

Course program
1. Linear and nonlinear systems of equations Conditioning analysis of the problem. Direct and iterative methods. Stability of the algorithms. Iterative methods for linear systems. Convergence and theorem of the spectral radius. Some iterative methods: Gauss-Seidel, Jacobi, relaxation. Direct methods for some classes of matrices: Thomas, Cholesky. Comparison of the methods. Newton methods in R^n. 2. Interpolation of functions and data Lagrange and Hermite polynomial interpolation. Divided differences and Newton form. Chebyshev nodes and polynomials. Conditioning and stability of the algorithms. Lebesgue function and constant. Splines. Convergence properties. Numerical derivation via finite differences. Approximation of derivatives via splines functions. 

 3. Numerical integration Newton-Cotes formulas. Gaussian formulas. Convergence and error estimates. 

 4. Numerical methods for ordinary differential equations Numerical methods for the approximation of the Cauchy problem. Explicit and implicit schemes. One step and multistep methods. One step methods: consistency or order p, stability and convergence. Local and global error. Error analysis. Some one-step methods: Euler, Heun, modified Euler, Crank-Nicolson. Runge-Kutta methods. 
The course includes a laboratory activity to develop a number of codes (in MATLAB) corresponding to the algorithms presented during the lectures.
Prerequisites
The course requires a basic knowledge of Mathematical Analysis and Linear Algebra, these notions are tipically given in the courses Calcolo 1, Analisi Matematica 1, Analisi Matematica 2 and Algebra Lineare. Moreover the student must have a good knowledge of a programming language (C, C++ or MATLAB), this corresponds to the course Laboratorio di Programmazione e Calcolo and to a course of Abilità Informatiche (MATLAB).
Books
A. Quarteroni, R. Sacco, F. Saleri, P. Gervasio, “Matematica numerica”, Springer, 2014
Teaching mode
The organization of this course is the following: Lectures (70%), Exercise and lab sessions (30%). All the materials for this course will be distributed via the platform e-learning Sapienza.
Frequency
strongly recommended
Exam mode
The exam aims to evaluate learning through a written test and an oral test. 
In the written test the student will be asked to answer theoretical questions and solve problems quite similar to those discussed in class. 
The oral exam will consist in the discussion of methods illustrated in the course, their properties and their implementation in MATLAB. 
To pass the exam it is necessary:
 - to have delivered the exercise sheets; - to have a sufficient mark in the written test
; - to have passed the oral exam.
Bibliography
W. Gautschi, “Numerical Analysis”, Birkhauser, 2012
Lesson mode
The organization of this course is the following: Lectures (75%), Exercise and lab sessions (25%). All the materials for this course will be distributed via the platform e-learning Sapienza.
  • Lesson code1010982
  • Academic year2025/2026
  • CourseMathematics
  • CurriculumMatematica per le applicazioni
  • Year3rd year
  • Semester1st semester
  • SSDMAT/08
  • CFU6