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2019/2020

Numerical calculus

Code: 100120 ECTS Credits: 6
Degree Type Year Semester
2500149 Mathematics OT 4 0

Contact

Name:
Magdalena Caubergh
Email:
Magdalena.Caubergh@uab.cat

Use of Languages

Principal working language:
catalan (cat)
Some groups entirely in English:
No
Some groups entirely in Catalan:
Yes
Some groups entirely in Spanish:
No

Other comments on languages

La llengua vehicular de les classes de pràctiques serà majoritàriament l'anglès

Teachers

Sundus Zafar

Prerequisites

It is advisable to have successfully completed all mandatory courses, and also to know a programming language.

Objectives and Contextualisation

Systems of linear and non-linear equations, and ordinary differential equations are present in many mahtematical models of physical processes. In this course we will study numerical techniques for the approximate solution of systems of linear and non-linear equations, and both initial and boundary value problems for ordinary differential equations. We will also study algorithms for the computation of eigenvalues of matrices.

The main goal of the course is that students learn all these methods from their mathematical foundation by studying their convergence properties, ant also that they are able to program them. Computer sessions are a fundamentel part of the course, and they are intended for students to valuebetter understand the features of the different numerical methods.

Competences

  • Assimilate the definition of new mathematical objects, relate them with other contents and deduce their properties.
  • Calculate and reproduce certain mathematical routines and processes with agility.
  • Generate innovative and competitive proposals for research and professional activities.
  • Students must be capable of communicating information, ideas, problems and solutions to both specialised and non-specialised audiences.
  • Students must have and understand knowledge of an area of study built on the basis of general secondary education, and while it relies on some advanced textbooks it also includes some aspects coming from the forefront of its field of study.
  • When faced with real situations of a medium level of complexity, request and analyse relevant data and information, propose and validate models using the adequate mathematical tools in order to draw final conclusions

Learning Outcomes

  1. Control for errors produced by machines when computing.
  2. Generate innovative and competitive proposals for research and professional activities.
  3. Know how to program algorithms for mathematical calculation.
  4. Students must be capable of communicating information, ideas, problems and solutions to both specialised and non-specialised audiences.
  5. Students must have and understand knowledge of an area of study built on the basis of general secondary education, and while it relies on some advanced textbooks it also includes some aspects coming from the forefront of its field of study.
  6. Understand the internal workings of computers and be critical of the results received.

Content

  1. Initial value problems for ordinary differential equations
    • One-step methods: Euler and Taylor.
    • Local discretization error.
    • Runge-Kutta methods.
    • Convergence of one-step methods.
    • Fehlberg step-size control.
    • Comments on multistep methods.
    • Stiff problems.
  2. Numerical solution of systems of non-linear equations
    • Matrix norms.
    • Fixed point methods: convergence and error estimation.
    • Newton's method in several variables.
  3. Boundary value problems for ordinary differential equations
    • Single shooting method.
    • Multiple shooting method.
    • Finite difference methods.
  4. Numerical linear algebra
    • Perturbation analysis of linear systems.
    • QR method for square and over-determined systems.
    • Iterative methods for linear systems. Convergence and error estimation.
    • Power and inverse power methods for the computation of eigenvalues and eigenvectors.
    • QR method for the computation of eigenvalues and eigenvectors.
  5. Approximation of functions
    • Gaussian quadrature.
    • Fast Fourier Transform.

Methodology

The theoretical and problem sessions will be carried out in a classroom. These sessions will be devoted to the presentation of theoretical aspects of numerical methods, their basic properties and the solution of problems, some of them of theoretical nature and some of them requiring the use of a calculator. Problem lists will be supplied along the course.

The seminar sessions with will be carried out in a computer room. In these sessions, studens will solve an applied problem through the implementation in the C programming language of methods studied in the course. These practical sessions will be evaluated from the delivery towards the end of the course (a date will be announced) of the C code and a report.

The gender perspective goes beyond the contents of courses, since it implies also a revision of teaching methodologies and interactions between students and lecturers, both inside and outside the classroom. In this sense, participative teaching methodologies that give rise to an equality environment, less hierarchical in the classroom, avoiding examples stereotyped in gender and sexist vocabulary, are usually more favorable to the full integration and participation of female students in the classroom. Because of this, their effective implementation will be attempted in this course.

Activities

Title Hours ECTS Learning Outcomes
Type: Directed      
Computer sessions 12 0.48 1, 6, 2, 5, 4, 3
Problem classes 8 0.32 2, 5, 4
Theoretical classes 30 1.2 2, 5, 4
Type: Autonomous      
Personal study 50 2 2, 5, 4, 3
Problem solving and computer work 44 1.76 1, 6, 2, 5, 4, 3

Assessment

The course will be graded from three evaluation activities:

  • Parcial exam (PE): an exam that exists in the resolution of problems that deal with the first part of the course (details will be announced at Campus Virtual)
  • Final exam (FE): an exam of all the course, with theoretical questions and problems.
  • Practical work (PR): delivery of C programs and a report

Besides that, the students will have the option of taking an additional recovery exam (RE) with the same format as FE.

Nor the partial exam nor the practical work will be recoverable.

In order to succeed in this course, it is mandatory that max(FE,RE)>=4 and PR>=4.

The final grade will be

0.1*PE+0.5*max(FE,RE)+0.4*PR

Honor grades will be granted at the first complete evaluation. Once given, they will no be withdrawn even if another student obtains a larger grade after consideration of the RE exam.

Assessment Activities

Title Weighting Hours ECTS Learning Outcomes
Final exam 0.50 3 0.12 2, 5
Partial exam 0.10 0 0 2, 5
Practical work 0.40 0 0 1, 6, 2, 5, 4, 3
Recovery exam 0.50 3 0.12 2, 5

Bibliography

General references:

  • J. Stoer and R. Burlisch, Introduction to numerical analysis, 3a ed, Springer, 2002.
  • A. Ralston and P. Rabinowitz, A first curse in numerical analysis, McGraw-Hill, 1988.
  • G. Dahlquist and A. Björck, Numerical methods, Englewood Cliffs (N.J.) : Prentice-Hall, 1974.
  • A. Aubanell, A. Benseny y A. Delshams, Eines bàsiques del càlcul numèric, Manuals de la U.A. B., 1991.
  • A. Quarteroni, R. Sacco and F. Saleri, Numerical Mathematics, TAM, Springer, 2000.

Specialized references:

  • R. L. Burden and J. D. Faires, Análisis Numérico, Grupo Editorial Iberoamérica, México D. F., 1985.
  • G. W. Gear, Numerical initial value problems in ordinary differential equations, Prentice-Hall, 1971.
  • E. Hairer, S.P. Nørsett, G. Wanner, Solving ordinary differential equations. Vol. 1, Springer-Verlag, 1987.
  • E. Hairer, S.P. Nørsett, G. Wanner, Solving ordinary differential equations. Vol. 2, Springer-Verlag, 1991.