Numerical Calculus
Code: 104390 ECTS Credits: 6| Degree | Type | Year |
|---|---|---|
| Computational Mathematics and Data Analytics | FB | 1 |
Contact
- Name:
- Susana Serna Salichs
- Email:
- susana.serna@uab.cat
Teachers
- Susana Serna Salichs
Teaching groups languages
You can view this information at the end of this document.
Prerequisites
A first course on calculus in one variable, and a first course on linear algebra.
Objectives and Contextualisation
Modeling turns problems in science and engineering into mathematical problems. The complexity of the real world often gives rise to mathematical problems that cannot be addressed from an analytic approach. Or perhaps they can, but the analytic approach may be too complex in the context in which the solution of the problem is required. For instance, solving the problem could be part of a contract, for which limited time is available.
Numerical methods are techniques from which algorithms can be deduced in order to obtain approximate solutions of mathematical problems. Many times, especially when high precision is required, these algorithms demand a large amount of computations. The use of a computer is then mandatory. Computers are most efficient when using finite precision arithmetic (this is, working with a finite number of digits). This means that each operation introduces error, known as round-off. This is not usually a problem, in particular because we are looking for approximate solutions. Nevertheless, it is necessary to know how to avoid situations in which the propagation of round-off error could completely invalidate our computations.
This course is devoted to the analysis of basic numerical methods, related to the solution of the kind of mathematical problems studied in first-year courses. This analysis has as a goal being able to predict both the quality of the approximations produced by the different methods and the computational effort they involve. This course is also an introduction to scientific computing, this is, the set of techniques and skills needed for the implementation in a computer of the numerical solution of a problem.
The numerical methods studied in this course are building stones of numerical methods for the solution of more sophisticated problems, studied from the second year on, like ordinary and partial differential equations.
Learning Outcomes
- CM05 (Competence) Design numerical, probabilistic algorithm and combinatorial algorithm solutions to solve real problems.
- KM02 (Knowledge) Distinguish the objects of calculus with functions and their properties and uses.
- KM03 (Knowledge) Describe the mathematical concepts and objects specific to numerical calculus.
- SM04 (Skill) Relate the concepts of the calculus of a real variable with methods and objects from other fields.
- SM05 (Skill) Develop independent strategies to solve problems specific to numerical calculus, probability and graph theory.
- SM06 (Skill) Solve problems involving integrals (lengths, areas, volumes, etc.).
Content
Error propagation
Numerical solution of non-linear equations
Numerical linear algebra
Interpolation, differentiation, integration
Activities and Methodology
| Title | Hours | ECTS | Learning Outcomes |
|---|---|---|---|
| Type: Directed | |||
| Problem sessions | 15 | 0.6 | CM05, KM02, KM03, SM04, SM05, SM06, CM05 |
| Theoretical sessions | 30 | 1.2 | CM05, KM02, KM03, SM04, SM05, SM06, CM05 |
| Type: Supervised | |||
| Computer sessions | 8 | 0.32 | CM05, KM03, CM05 |
| Type: Autonomous | |||
| Computer work | 16 | 0.64 | CM05, KM03, CM05 |
| Personal study | 76 | 3.04 | CM05, KM02, KM03, SM04, SM05, SM06, CM05 |
The theoretical sessions will be dedicated to the presentation by the lecturer of various methods and their analysis. The presentation of the methods will be accompanied by examples of their behavior aimed at both facilitating the understanding of the method and motivating its analysis.
In the problem-solving sessions, theoretical and computational problems will be solved.
The computer practice sessions will be dedicated to solving problems that will be implemented in a compiled language. In solving these problems, students will gradually build their personal library of routines that implement basic numerical methods.
Annotation: Within the schedule set by the centre or degree programme, 15 minutes of one class will be reserved for students to evaluate their lecturers and their courses or modules through questionnaires.
Assessment
Continous Assessment Activities
| Title | Weighting | Hours | ECTS | Learning Outcomes |
|---|---|---|---|---|
| Computer work | 0.25 | 0 | 0 | CM05, SM05, SM06 |
| Final exam | 0.45 | 3 | 0.12 | CM05, KM02, KM03, SM04, SM05, SM06 |
| Partial exam | 0.3 | 2 | 0.08 | CM05, KM02, KM03, SM04, SM05, SM06 |
The evaluation of the course will be carried out through three activities:
• Partial exam (EP): exam on part of the subject, with theoretical questions and problems.
• Final exam (EF): exam on the entire subject, with theoretical questions and problems.
• Computer practice (PR): submission of code and a report. Some practice sessions will be evaluated.
Additionally, students will be able to take a recovery exam (ER) with the same characteristics as the final exam (EF).
It is a requirement to pass the course that max(0.4EP + 0.6EF, EF, ER) ≥ 5 and that PR ≥ 4.
The final grade for the course will be:
0.8 * max(0.4EP + 0.6EF, EF, ER) + 0.2 * PR.
Honors will be awarded in the first complete evaluation of the course. They will not be withdrawn if another student obtains a higher grade after the ER exam.
Single Evaluation
Students who have opted for the single evaluation mode will need to take a final test, which will consist of a theory and problems exam. In order to take this test, they must submit, before the exam begins, all the practical assignments that students of the subject were required to submit during the course, and these will be evaluated in an oral exam.
If the practice grade is less than 4 out of 10 and the exam grade is less than 5 out of 10, the course cannot be passed. The final grade will be the weighted average of the two activities, with the exam accounting for 80% of the grade and the practice for 20%.
If the final grade is below 5,the student has another opportunity to pass the course through a recovery exam, which will be held on a date determined by the program coordination. In this test, up to 80% of the theoretical grade can be recovered. The practical part is not recoverable.
The criterion to obtain a "non-assessable" grade is: all students who submit at least half of the practical assignments or take one of the exams (EP) or (EF) will be considered to have participated.
Use of AI
In this course, the use of Artificial Intelligence (AI) technologies is not allowed at any stage. Any work that includes fragments generated with AI will be considered academic dishonesty and may result in a partial or total penalty on the activity's grade or more severe sanctions in cases of serious violations.
Bibliography
Basic references:
- A. Aubanell, A. Benseny, A. Delshams. Eines bàsiques de càlcul numèric. Manuals de la UAB 7, Publ. UAB, 1991.
- M. Grau, M. Noguera. Càlcul numèric. Edicions UPC, 1993.
- J.D. Faires, R. Burden. Métodos numéricos, 3a ed. Thomson, 2004.
- R. Burden, J.D. Faires. Numerical analysis, 6a ed. Brooks/Cole, 1997.
- G. Hämmerlin, K.-H. Hoffmann. Numerical mathematics. Springer, 1991.
Advanced references:
- E. Isaacson, H.B. Keller. Analysis of numerical methods. Wiley, 1966.
- J. Stoer, R. Bulirsch. Introduction to numerical analysis, 3a ed. Springer, 2002.
- G. Dahlquist, A. Björk. Numerical methods. Prentice Hall, 1964.
- A. Ralston and P. Rabinowitz. A first course in numerical analysis. McGraw-Hill, 1988.
- A. Quarteroni, R. Sacco and F. Saleri. Numerical Mathematics. Springer, 2000.
Software
- Preferably a Linux environment
- code-oriented text editor (e.g. Kate)
- GNU C compiler
- gnuplot
- image manipulation tools (e.g. imagemagick)
- GNU Octave
Groups and Languages
Please note that this information is provisional until 30 November 2025. You can check it through this link. To consult the language you will need to enter the CODE of the subject.
| Name | Group | Language | Semester | Turn |
|---|---|---|---|---|
| (PLAB) Practical laboratories | 1 | Spanish | second semester | morning-mixed |
| (SEM) Seminars | 1 | Spanish | second semester | morning-mixed |
| (TE) Theory | 1 | Catalan/Spanish | second semester | morning-mixed |