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Course, academic year 2017/2018
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Analysis of Matrix Calculations 1 (M) - NMNM931
Czech title: Analýza maticových výpočtů 1 (M)
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
Is provided by: NMNM331
Guarantor: RNDr. Iveta Hnětynková, Ph.D.
Class: M Mgr. MMIB
M Mgr. MMIB > Povinně volitelné
Classification: Mathematics > Numerical Analysis
Incompatibility : NMNM331, NNUM006
Interchangeability : NMNM331, NNUM006
Is pre-requisite for: NMPG349
In complex pre-requisite: NMNM332
Annotation -
Last update: G_M (19.05.2012)

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, yákladní 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 and tutorials in a lecture hall. Practicals in computer laboratory (Matlab enviroment).

Requirements to the exam -
Last update: G_M (19.05.2012)

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

Syllabus -
Last update: RNDr. Iveta Hnětynková, Ph.D. (10.09.2015)

1. A brief overview of related topics from previous courses (the Schur theorem, orthogonal transformations and the QR decomposition, the Gaussian elimination and the LU decomposition, the spectral decomposition, the singular value decomposition).

2. Orthogonal transformations in the complex field.

3. Numerical evaluation and applications of the singular value decomposition (the rank, kernel and range of a matrix, low rank matrix approximations).

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

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

6. Krylov subspace methods. Comparison of short a long recurrences (loss of orthogonality, stability, prize).

7. The conjugate gradient (CG) method and its connection to the Lanczos method.

8. The generalized minimal residual method (GMRES) and its connection to the Arnoldi method.

9. Overview of other Krylov subspace methods.

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

 
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