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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.
Last update: G_M (19.05.2012)
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To finish the course successfully, it is required to pass the exam covering all presented topics, see "Requirements to the exam".
To complete successfully the laboratory part of the exam, a student needs to get 3 point for the activity. A point can be obtain in two ways: by solving a given example on the blackboard, or by completing a homework assigned during the semester. A list of homework (including the deadline for their submission) will be available by the end of October. Last update: Hnětynková Iveta, doc. RNDr., Ph.D. (28.10.2019)
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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 Last update: Hnětynková Iveta, doc. RNDr., Ph.D. (07.04.2015)
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Lectures and tutorials in a lecture hall. Practicals in computer laboratory (Matlab enviroment). Last update: Hnětynková Iveta, doc. RNDr., Ph.D. (07.09.2020)
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The final exam has written and oral part and covers all material presented in lectures and practicals during the semester. A student, who does not pass the written part is not allowed to continue to the oral part and fails the exam. A student, who does not pass the oral part also fails the exam. In both cases, it is necessary to repeat both parts of the exam. Last update: Hnětynková Iveta, doc. RNDr., Ph.D. (28.10.2019)
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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).
3. Krylov subspaces (the Arnoldi and the Lanczos method for computation of a basis, connections to Jacobi matrices, applications).
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. Overview of other Krylov subspace methods.
7. Matrix functions (definition, evaluation, apllications).
8. Special matrices (definition of selected matrices of special structure and properties, applications). Last update: Hnětynková Iveta, doc. RNDr., Ph.D. (07.09.2020)
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