Numerical Methods and Optimisation
Code: 104848Credits: 6
| Degree programme | Type | Course |
|---|---|---|
| Applied Statistics | FB | 2 |
Contact lecturer
- Name :
- Joan Torregrosa Arus
- Email :
- joan.torregrosa@uab.cat
Group languages
You can consult this information at the end of the document.
Prerequisites
It is recommended to have passed the following courses: Àlgebra Lineal, Càlcul 1 and Càlcul 2.
Objectives
This course will provide students the basic numerical methods to solve real problems which arise from science and mainly from applied statistics.
The purpose of the course is that the students learn the mathematical foundations of the methods, their range of applicability and the type of errors that should be expected. The student should also be able to recognize the problems whose solution requires the use of a numerical method, and to apply a proper method to get an approximate solution in an efficient way.
The student shoud also be able not only to use some programming languages to implement and test simple algorithms, but to work with the functions provided by the correspondig software.
Learning outcomes
- KM04 (Recognise the mathematical bases of the methods, the conditions of applicability and the types of errors that can appear in the numerical (algorithmic) solution of different types of problems.) Recognise the mathematical bases of the methods, the conditions of applicability and the types of errors that can appear in the numerical (algorithmic) solution of different types of problems.
- SM02 (Implement algorithms using different programming languages (Máxima, R, Python, Julia), working with the programmed functions provided by the software packages used.) Implement algorithms using different programming languages (Máxima, R, Python, Julia), working with the programmed functions provided by the software packages used.
- SM03 (Solve, using numerical methods, optimisation problems, linear algebra and analysis in general that appear in science and, especially, in statistics.) Solve, using numerical methods, optimisation problems, linear algebra and analysis in general that appear in science and, especially, in statistics.
Contents
1. Errors
Floating point arithmetic. Propagation of errors.
Conditioning of a problem.
2. Numerical Linear Algebra
LU decomposition. Perturbation analysis.
QR decomposition. Applications.
Singular value decomposition. Applications.
3. Numerical Solution of Nonlinear Equations
One variable equations: Fixed point methods. Newton-Raphson's method.
Methods for systems of nonlinear equations.
4. Polynomial interpolation
Lagrange polynomial. Divided differences.
Error estimate.
5. Unconstrained Optimitzation
One dimensional minimization.
Line search methods, gradient, Newton.
6. Constrained Optimitzation
The penalty method.
Augmented Lagrangian method.
7. Numerical Integration
Trapezoidal and Simpson's rules. Romberg method.
Learning activities and methodology
| Title | Hours | ECTS | Learning outcomes |
|---|---|---|---|
| Study | 32 | 1.28 | |
| Problems | 14 | 0.56 | |
| Exercises | 35 | 1.4 | |
| Computer work | 21 | 0.84 | |
| Theory | 26 | 1.04 | |
| Computer sessions | 12 | 0.48 |
In the theoretical lectures the teacher will explain the mathematical foundations and basic properties of the numerical methods and will present several illustrative examples.
Different lists of exercises will be proposed so that the student can practice and learn the contents of each topic. In the problem lectures the teacher will work on the lists of exercises, will solve the doubts of the students and will discuss and provide indications for solving the exercises.
Each computer session will have a script associated. In the computer sessions the student will do the work proposed in the correspondig script under the supervison of the teacher. It is convenient that before the session the student reads carefully the script in order to know the goal of the computer session and the numerical methods to be used.
All the course material will be posted on the Virtual Campus.
For this course, the use of Artificial Intelligence (AI) technologies is permitted exclusively for support and clarification tasks related to solving exercises and practical assignments. Students must clearly identify which parts have been generated using this technology, specify the tools used, and include a critical reflection on how these have influenced both the process and the final outcome of the activity. Failure to be transparent about the use of AI in this assessed activity will be considered a breach of academic honesty and may result in a partial or total penalty to the activity grade, or more severe sanctions in serious cases.
Assessment
Continuous assessment activities
| Title | Weight | Hours | ECTS | Learning outcomes |
|---|---|---|---|---|
| Final-term exam | 40% | 2.5 | 0.1 | KM04, SM03 |
| Mid-term exam | 40% | 2.5 | 0.1 | KM04, SM03 |
| Computer work | 20% | 2 | 0.08 | SM02 |
| Recovery Exam | 80% | 3 | 0.12 | KM04, SM03 |
See the Catalan version.
Bibliography
See the Catalan version.
Software
See the Catalan version.
Course groups and languages
The information provided is provisional until November 30. After this date, you will be able to consult the language of each group through this link. To access the information, you will need to enter the course CODE
| Type of teaching | Group | Language | Semester | Shift |
|---|---|---|---|---|
| (TE) Theory | 1 | Catalan | first semester | afternoon |
| (PAUL) Classroom practices | 1 | Catalan | first semester | afternoon |
| (PLAB) Practical laboratories | 1 | Catalan | first semester | afternoon |
| (PLAB) Practical laboratories | 2 | Catalan | first semester | afternoon |