NUMERICAL METHODS FOR LINEAR ALGEBRA AND FUNCTIONAL ANALYSIS METODI NUMERICI PER L'ALGEBRA LINEARE E L'ANALISI FUNZIONALE

A.Y. Credits
2022/2023 9
Lecturer Email Office hours for students
Giovanni Stabile Tuesday 14-16 or by appointment
Teaching in foreign languages
Course with optional materials in a foreign language English
This course is entirely taught in Italian. Study materials can be provided in the foreign language and the final exam can be taken in the foreign language.

Assigned to the Degree Course

Applied Informatics (LM-18)
Curriculum: PERCORSO COMUNE
Date Time Classroom / Location
Date Time Classroom / Location

Learning Objectives

The scope of the course is to provide basic concepts of numerical methods for linear algebra and functional analysis

Program

1 I Principles of Numerical Mathematics
1.1 Well-posedness and Condition Number of a Problem
1.2 Stability of Numerical Methods
1.3 A Priori And A Posteriori Analysis
1.4 Sources of Error in Computational Models

2 Solution of Linear Systems
2.1 Direct Methods
2.1.1 The Gaussian Elimination Method
2.1.2 LU Factorization
2.2 Computing the Inverse of a Matrix
2.3 Iterative Methods for Solving Linear Systems
2.3.1 Jacobi Method
2.3.2 Gauss-Seidel Method
2.3.3 Stationary and Nonstationary Iterative Methods
2.3.3 Convergence Analysis of the Richardson Method
2.3.4 Relaxation Method
2.3.7 Stopping Criteria For Iterative Methods
2.4 Undetermined Systems
2.5 Regression
2.5.1 Least-squares method

3 Rootfinding for Nonlinear Equations and systems of nonlinear Equations
3.1 The Bisection Method
3.2 Metodi sul termine lineare e/o sulla sua derivata
3.2.1 The Methods of Chord
3.2.2 The Methods of Secant
3.2.3 The Newton's method
3.2.4 Fixed Point Methods
3.3 Stopping Criteria
3.4 Aitken’s Acceleration
3.3 Rootfinding for Systems of Nonlinear Equations
3.3.1 The Newton's Method
3.3.2 The Quasi Newton's Method
3.3.3 The Broyden's Method
3.3.4 Fixed Point Methods

4 Function Interpolation
4.1 Polynomial Interpolation
4.1.1 Interpolation with Lagrance Polynomials
4.2 Stability of Polynomial Interpolation

5 Numerical Integration
5.1.1 The Midpoint or Rectangle Formula
5.1.2 The Trapezoidal Formula
5.1.3 The Cavalieri-Simpson Formula
5.2 Newton-Cotes Formulae
5.3 Error Analysis of Numerical Integration
5.4 Gauss Points

6 Approximation of Eigenvalues and Eigenvectors
6.1 The Eigenvalue Problem
6.2 The Power Method
6.2.1 Convergence of Power Method
6.2.2 Stopping Criteria
6.3 Inverse Power Method
6.4 Geometrical Location of the Eigenvalues
6.4.1 Theorem of the Gershgorin circles
6.5 The QR Iteration

7 Numerical Solution of Ordinary Differential Equations
7.1 The Cauchy Problem
7.2 Concept of Stability
7.3 Numerical Approximation of Cauchy Problem
7.3.1 Euler Explicit
7.3.2 Euler Implicit
7.3.3 Crank-Nicholson
7.3.4 Heun's Method
7.4 Analysis of One-Step Methods
7.5 Multistep Methods
7.6 BDF Methods
7.7 Runge-Kutta Methods

No prerequisites

Learning Achievements (Dublin Descriptors)

Knowledge and understanding.  Learn the techniques for the numerical programming of numerical methods for linear algebra and functional analysis. At the end of the course, the student will have acquired a good knowledge of the mathematical topics covered in the classes.

Applying knowledge and understanding. Acquiring the ability to implement numerical methods for linear algebra and functional analysis. Developing the ability to program, testing interpreting the results correctly. Acquiring the ability to solve mathematical problems using problem solving environment.

Making judgments. acquiring the ability to find the most suitable numerical method for the solution of linear algebra or differential problems.

Communication skills. acquiring the ability to rigorously define the mathematical problem studied in the course and to expose its numerical methods, outlining its fundamental properties

Learning skills. ability to study and solve problems similar but not necessarily the same as those dealt with during lessons.

Teaching Material

The teaching material prepared by the lecturer in addition to recommended textbooks (such as for instance slides, lecture notes, exercises, bibliography) and communications from the lecturer specific to the course can be found inside the Moodle platform › blended.uniurb.it

Not available

Teaching, Attendance, Course Books and Assessment

Teaching

- Frontal lessons

- Examples and exercises using Python in the computer laboratory

- The whole material uploaded in the Moodle platform http://blended.uniurb.it

Attendance

Not compulsory

Course books

Quarteroni, A., Sacco, R., & Saleri, F. (2007). Numerical Mathematics. In Texts in Applied Mathematics. Springer New York. https://doi.org/10.1007/b98885

Assessment

A laboratory exercise in python da prepared at home and sent before the oral examination

Oral examination consisting into:

Two questions on the topics covered in the classes

The final mark is the average of the mark of the exercise and the oral examination

Teaching

- Use of the textbook

- Weekly check in the Moodle platform http://blended.uniurb.it

Attendance

Not compulsory

Course books

The same as attending students

Assessment

The same as attending students

Notes

None

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