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Course, academic year 2017/2018
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Analysis of Matrix Calculations 1 - NMNM331
Czech title: Analýza maticových výpočtů 1
Guaranteed by: Department of Numerical Mathematics (32-KNM)
Faculty: Faculty of Mathematics and Physics
Actual: from 2013
Semester: winter
E-Credits: 5
Hours per week, examination: winter s.:2/2 C+Ex [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: taught
Language: Czech
Teaching methods: full-time
Guarantor: RNDr. Iveta Hnětynková, Ph.D.
Class: M Bc. OM
M Bc. OM > Zaměření NUMMOD
M Bc. OM > Povinně volitelné
M Bc. OM > Zaměření STOCH
Classification: Mathematics > Numerical Analysis
Incompatibility : NNUM006
Pre-requisite : NMAG101, NMAG102, NMNM201
Interchangeability : NNUM006
Is incompatible with: NMNM931
Is pre-requisite for: NMNM349
Is interchangeable with: NMNM931
In complex pre-requisite: NMNM332
Annotation -
Last update: G_M (16.05.2012)

The course is devoted to fundamentals of numerical linear algebra, with the concetration on methods for solving linear algebraic equations, including least squares, and on eigenvalue problems. The course emphasizes formulation of questions, motivation and interconnections. Recommended for bachelor's program in General Mathematics, specializations Mathematical Modelling and Numerical Analysis, and Stochastics.
Literature -
Last update: RNDr. Iveta Hnětynková, Ph.D. (07.04.2015)

Duintjer Tebbens, J., Hnětynková, I., Plešinger, M., Strakoš, Z., Tichý, P., Analýza metod pro maticové výpočty, základní metody, Matfzypress, Praha 2012.

Watkins, D.S., Fundamentals of Matrix Computations (Second edition), J. Wiley & Sons, New York, 2002

Fiedler, M., Speciální matice a jejich užití. SNTL Praha, l980

Golub, G.H., Van Loan C.F., Matrix Computations (Third edition). J. Hopkins Univ. Press, Baltimore, 1996

Teaching methods -
Last update: RNDr. Iveta Hnětynková, Ph.D. (07.04.2015)

Lectures in a lecture hall. Practicals in computer laboratory (Matlab enviroment).

Requirements to the exam -
Last update: G_M (27.04.2012)

Written and oral part of the exam reflect the content of the course.

Syllabus -
Last update: RNDr. Iveta Hnětynková, Ph.D. (12.09.2017)

1. A brief overview of related topics from previous courses (the Schur decomposition, the QR decomposition, the LU decomposition, the singular value decomposition).

2. Solution of linear approximation problems (the least squares method, the total least squares method, generalizations, idea of regularization methods).

3. Partial eigenvalue problems (the Arnoldi and the Lanczos method, connections with orthogonal polynomials and Jacobi matrices).

4. Krylov subspace methods. Comparison of short a long recurrences (loss of orthogonality, stability, prize), Faber-Manteuffel theorem.

5. The conjugate gradient (CG) method, MINRES method.

6. The generalized minimal residual method (GMRES), FOM method.

7. The biconjugate gradient method (BiCG). Overview of other Krylov subspace methods.

8. Matrix functions (definition, evaluation, apllications).

9. Special matrices (definition of selected matrices of special structure and properties, applications).

 
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