Numerical Methods I
Code: 107933Credits: 6
| Degree programme | Type | Course |
|---|---|---|
| Mathematics | OB | 2 |
Contact lecturer
- Name :
- Jose Maria Mondelo Gonzalez
- Email :
- josemaria.mondelo@uab.cat
Teaching staff
- Susana Serna Salichs
Group languages
You can consult this information at the end of the document.
Prerequisites
Students are expected to know basic results on continuity, differentiability and integrability of real functions in one variable, linear algebra, matrix calculus, basic notions about algorithms and the C programming language. This knowledge is part of the the contents of the subjects: Linear algebra I, Linear algebra II, Functions of a real variable I, Functions of a real variable II and Scientific programming, all of the first year of the Mathematics degree.
Objectives
Science and technology are supported by mathematical models of real phenomena, developed for predictive purposes. A minimum of realism gives rise to models that are difficult to solve in a completely analytical way. A way to study them is by calculating approximate solutions. The study of techniques (numerical methods) to obtain these approximations is the goal of numerical analysis, of which this subject is an introduction.
Numerical methods require a computing effort that depends on the complexity of the model and the desired precision. At first, computations were done by hand. Notable examples are: the numerical integrations performed by the Copenhagen School (1910-1925), or the computations performed by "human computers" for the first NASA's human space missions (decades of 1940, 1950). In both cases, computations were done predominantly by women. The need to speed up calculations gave rise to the development of computers. Since computers exist, they are an essential tool in the application of numerical methods.
The subject's goals are two-sided. On the one hand, the course aims to train the students in general mathematics, as the remaining subjects of the degree do. On the other, this course aims to prepare the students to solve the numerical problems that they can find in their professional practice. This implies both the precise knowledge of several methods, together with their suitability in various situations, and the necessary skills in their application to the resolution of specific problems with the help of a computer.
Learning outcomes
- CM21 (Design numerical resolution algorithms to program and effectively apply numerical methods on a computer.) Design numerical resolution algorithms to program and effectively apply numerical methods on a computer.
- CM22 (Estimate the intrinsic limitations of numerical methods such as systematic errors and calculation uncertainties in relation to the real situation being modelled.) Estimate the intrinsic limitations of numerical methods such as systematic errors and calculation uncertainties in relation to the real situation being modelled.
- KM29 (Identify some of the main methods of numerical calculation.) Identify some of the main methods of numerical calculation.
- KM30 (Implement algorithms in a structured programming language.) Implement algorithms in a structured programming language.
- SM27 (Analyse the suitability of one numerical method over another in a specific problem.) Analyse the suitability of one numerical method over another in a specific problem.
- SM28 (Use mathematical formalism in the design and verification of computer programs.) Use mathematical formalism in the design and verification of computer programs.
- SM29 (Critically evaluate the results obtained on concluding a computation process.) Critically evaluate the results obtained on concluding a computation process.
Contents
Error analysis.
Numerical linear algebra.
Numerical solution of non-linear equations.
Learning activities and methodology
| Title | Hours | ECTS | Learning outcomes |
|---|---|---|---|
| Computer sessions | 14 | 0.56 | CM21, CM22, KM30, SM27, SM28, SM29 |
| Problem sessions | 13 | 0.52 | CM22, KM29, SM27, SM29 |
| Personal study | 88.5 | 3.54 | CM21, CM22, KM29, KM30, SM27, SM28, SM29 |
| Theoretical sessions | 23 | 0.92 | CM22, KM29, SM27 |
In the theoretical sessions, the different numerical methods will be deduced and analyzed in the blackboard. Computer simulations will also be carried out, having as goal both the motivation of the analysis of the different methods and the verification that they behave as predicted by their analysis.
The problem sessions will be devoted to problem solving at the blackboard, with the participation of students.
Practical assignments will be given during the course. Students are expected to carry them on both in the computer sessions and by using personal study time. Some of the computer sessions will be evaluated.
Assessment
Continuous assessment activities
| Title | Weight | Hours | ECTS | Learning outcomes |
|---|---|---|---|---|
| Recovery of the evaluated computer session | 18% | 1.25 | 0.05 | CM21, KM30, SM29 |
| Partial exam | 24,5% | 3 | 0.12 | CM22, KM29, SM27, SM29 |
| Delivery of practical assignements | 12% | 0 | 0 | CM21, CM22, KM29, KM30, SM27, SM28, SM29 |
| Recovery exam | 60% | 3 | 0.12 | CM22, KM29, SM27, SM29 |
| Final exam | 45,5% | 3 | 0.12 | CM22, KM29, SM27, SM29 |
| Evaluated computer session | 18% | 1.25 | 0.05 | CM21, KM30, SM29 |
The course will be evaluated from up to four grades:
- Partial exam (EP).
- Final exam (EF), of all the course.
- Recovery exam (ER), of all the course.
- Practical grade (PR).
The EP, EF, ER exams will consist in the solution of problems similar to the ones solved during the problem sessions, plus some theoretical questions. They will be carried out without the aid of any electronic device other than a scientific calculator.
The practical part of the course will be evaluated from the submission of a report, an inteview and an in-person submission of C code and result files. The report submission and the interview will be prior to the in-person one. This in-person submission will be done at a computer room (it will be the "evaluated computer session"), with computers disconnected from the network and without the aid of any other electronic device. In this evaluated session, students will run their code, perform a modification of it and submit the results obtained. The dates of all the submissions will be announced. There will be the possibility to recover the in-person submission. The submission of the report and the interview will not be recoverable.
The "june final grade", or course grade (QC), will be computed as
QC := 0.7 max(0.35 EP + 0.65 EF, EF) + 0.3 PR .
In order to pass the course with the QC grade, it will be required that max(0.35 EP + 0.65 EF, EF)>=3.5, PR>=3.5, and QC>=5.
The students would not pass, as well as the ones who wanted to improve their grade, will be allowed to take the recovery exam ER. Then, in order to pass the course it will be needed that
QT := max( max(0.35 EP + 0.65 EF, EF) , ER ) >= 3.5
in addition to PR>=3.5. The final grade will be
QF := 0.7 QT + 0.3 PR .
Once the QC qualification is available, honor grades will be awarded if possible. Honor grades awarded at that time will not be revoked, independently of the value of the QF grade that other students could get via the ER exam.
Students to be evaluated via the unique assessment modality will go directly to the EF exam. They will need to bring their practical assignments report and code, that will be evaluated via an interview, in which they will be required to perform a modification of their code and produce results with it. In this way, they will get a PR grade. Their QC grade will then be computed as
QC := 0.7 EF + 0.3 PR .
In order to pass, it will be required that EF>=3.5, PR>=3.5 and QC>=5. Students that do not pass will still be allowed to take the ER exam. The final grade will then be
QF := 0.7 max(EF,ER) + 0.3 PR .
In order to pass, it will be necessary that max(EF,ER)>=3.5, PR>=3.5 and QF>=5.
Bibliography
Basic bibliography:
Notes by J.M. Mondelo. Available at UAB's Virtual Campus.
Slides by Lluís Alsedà. Available at UAB's Virtual Campus.
A. Aubanell, A. Benseny, A. Delshams: Eines bàsiques de càlcul numèric, Manuals de la UAB 7, Publ. UAB, 1991.
R. Burden, J.D. Faires: Numerical analysis, 6a ed., Brooks/Cole, 1997.
Other bibliography:
M. Grau, M. Noguera: Càlcul numèric, Edicions UPC, 1993.
D. Kincaid, W. Cheney: Numerical analysis, 2a ed., Brooks/Cole, 1996.
P. Henrici: Elements of numerical analysis, Wiley, 1964.
G. Dahlquist, A Björk: Numerical methods, Prentice Hall, 1964.
E. Isaacson, H.B. Keller: Analysis of numerical methods, Wiley, 1966.
J. Stoer, R. Bulirsch: Introduction to numerical analysis, 2a ed., Springer, 1993.
Programming:
B. Kernighan and D.M. Ritchie: The C programming language, 2a ed., Prentice–Hall 1998. En castellà: El lenguaje de programación C, Prentice–Hall Hispanoamericana, 1991.
B.W. Kernighan, R. Pike: The practice of programming, Addison–Wesley 1999.
Software
The practical work of the course will be carried out by writing programs in the C language. For postprocessing and plotting, the gnuplot plotting software will be used together with several utilities of the Unix command line. The evaluated computer session will be carried out under Linux in computer rooms of the Sciences School. Occasionally, and without effect on course grades, Octave and SageMath/Python will also be used.
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 | second semester | morning-mixed |
| (PAUL) Classroom practices | 1 | Catalan/Spanish | second semester | morning-mixed |
| (PLAB) Practical laboratories | 1 | Catalan | second semester | morning-mixed |
| (PAUL) Classroom practices | 2 | Catalan/Spanish | second semester | morning-mixed |
| (PLAB) Practical laboratories | 2 | Catalan | second semester | morning-mixed |
| (PLAB) Practical laboratories | 3 | Catalan | second semester | morning-mixed |