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Large and sparse matrix problems arise for instance in numerical approximation of differential equations, network problems and optimisation. In this course we study numerical techniques for solution of systems of linear equations and eigenvalue problems with this type of matrices. For systems of linear equations we present two classes of methods: iterative and direct. Among the iterative methods we study basic stationary methods like Jacobi and SOR methods, orthogonalising methods like conjugate gradients and multigrid methods. The direct methods are based on Gaussian elimination with different renumberings of the unknowns in order to keep computing time and memory requirement small. The eigenvalue routines presented are based on Lanczos and Arnoldi algorithms with or without spectral transformations. When solving the homework assignments and computer labs, the students get experience in implementation and evaluation of different algorithms for sparse problems. We will also study how to use toolkit for scientific computations PETSc for the solution of sparse linear systems of equations using different preconditioners.

Teachers
Course coordinator: Larisa Beilina, larisa@chalmers.se, room 2089, homepage


Course literature
  1. Demmel, Applied Numerical Linear Algebra, SIAM 1997, selled at Cremona. List of Errata for the Textbook
  2. PETSc libraries which are a suite of data structures and routines for the scalable (parallel) solution of scientific applications; user manual
Programme
Computer labs
Wednesdays 13-15 (except the first week) in computer room MVF24 (Mathematical Sciences in Physics building).
Fridays 13-15 in computer room MVF24 (exept the first week).
Computer labs and Matlab/PETSc excercises will be included in the assignments below. Templates for computer ex. 2:
  • Template for computing of discretized Laplacian
  • Template for the main program in PETSC for LU decomposition
  • Example of Makefile to be able compile PETSc program at Chalmers
  • Template for petsc function which removes zeros from the original matrix. We can work then with resulting matrix without zeros in the same way as with original matrix.
  • Example of the program in Matlab to present contour plot of the eigenvectors for the case of Poisson equation.

  • Additional templates of PETSc-Makefile and main program in PETSc:
  • Example of the main program in PETSC.
  • Example of the Makefile to compile the above main program like that: make hello

  • Reference literature:
    Course outline and requirements
    The following sections in the textbook will be considered in the book.
    Chapter 6, sections 1-6, 9
    mainly repetition and brief revision on basic linear algebra.
    Chapter 7, sections 1 - 4 To each chapter belong an optional homework assignment and a computer exercise. The homework assignments are similar to the questions in the textbook and such you could expect at the examination. In the computer exercises you will be trained in using different algorithms in numerical linear algebra using MATLAB or PETSc libraries. Either you will use already existing MATLAB and PETSC programs or write your own small MATLAB or PETSc codes. The homework assignments and computer labs give bonus credit points for the examination. Since the occasions reserved for the computer exercises are without supervision you may put questions on these exercises to me at the lectures.
    Assignments

    You may work in a group of 2 persons but hand in only one report for the group.

    Wednesdays 13-15 (except the first week) in computer room MVF24 (Mathematical Sciences at Physics building) Fridays 13-15 in computer room MVF24 (exept the first week when it is in room MVF25).

    Examination
    • To pass this course you should pass the written exam and 2 computer assignments, see description of comp. assignments above.
    • The two compulsory home assignments should be handed in before the final exam.

    Written examination
  • Final exam is compulsory, written.
  • You should be able to state and explain all definitions and theorems given in the course and also apply them in problem solving.
  • Grades are set according to the table below.
  • Grades Chalmers Points Grades GU Points
    - <15 -
    3 15-20 G 15-27
    4 21-27 VG >27
    5 >27

    The first exam will place at 14 March, 8.30-12.30, Maskinhuset at Chalmers:

  • Maskinhuset

  • Bring ID and receipt for your student union fee (this is requirement only for Chalmers students. GU students can come on the exam without receipt).

    You will be notified the result of your exam by email from LADOK (This is done automatically as soon as the exams have been marked an the results are registered.)
    The exams will then be kept at the students office in the Mathematical Sciences building.
    Check that the number of points and your grade given on the exam and registered in LADOK coincide.
    Complaints of the marking should be written and handed in at the office. There is a form you can use, ask the person in the office.).

    The following link will tell you all about the examination room rules at Chalmers: Examination room instructions


    Examination procedures
    In Chalmers Student Portal you can read about when exams are given and what rules apply on exams at Chalmers.
    At the link Schedule you can find when exams are given for courses at University of Gothenburg.
    At the exam, you should be able to show valid identification.
    Before the exam, it is important that you report that you want to take the examination. If you study at Chalmers, you will do this by the Chalmers Student Portal, and if you study at University of Gothenburg, so sign up via GU's Student Portal.

    You can see your results in Ladok by logging on to the Student portal.

    At the examination:
    When it is practical a separate review is arranged. The date of the review will be announced here on the course website. Anyone who can not participate in the review may thereafter retrieve and review their exam on Mathematical sciences study expedition, Monday through Friday, from 9:00 to 13:00. Check that you have the right grades and score. Any complaints about the marking must be submitted in writing at the office, where there is a form to fill out.

    At re-examination:
    Exams are reviewed and picked up at the Mathematical sciences study expedition, Monday through Friday, from 9:00 to 13:00. Any complaints about the marking must be submitted in writing at the office, where there is a form to fill out.
    Old exams
    Old exams are given here:
  • Old exams