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This course should extend and strenghten theoretical foundations of methods of numerical linear algebra. Its main goals are:
to present an introduction into perturbation theory and numerical stability;
to extend knowledge about several modern methods;
to emphasize analysis of the methods and algorithms, including effects of rounding errors;
to use the studied topics for demonstration of connections between various mathematical tools and disciplines.
Prerequisites: NNUM006, NNUM042.
Last update: T_KNM (18.05.2008)
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Theoretical foundations of methods of numerical linear algebra. Last update: T_KNM (18.05.2008)
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[1] G.H. Golub and C.F. Van Loan, Matrix computations (Third edition), Johns Hopkins University Press, Baltimore, MD, 1996 [2] N.J. Higham, Accuracy and stability of numerical algorithms (second edition), SIAM, Philadelphia, PA, 2002 [3] J. Liesen and Z. Strakoš, On numerical stability in large scale numerical computations, ZAMM, 85, 2005, pp. 307-325, [4] G. Meurant and Z. Strakoš, The Lanczos and conjugate gradient algorithms in finite precision arithmetic, Acta Numerica, 15, pp. 471-542, 2006 [5] S.G. Nash (Ed.), A history of scientific computing (Papers from the Conference on the History of Scientific and Numeric Computation held at Princeton University, Princeton, ACM Press, New York, 1990 [6] B.N. Parlett, The symmetric eigenvalue problem, SIAM, Philadelphia, 1998 [7] D.P. O'Leary, Z. Strakoš and P. Tichý, On sensitivity of Gauss-Christoffel quadrature, Numerische Mathematik, accepted for publication, 2007 [8] I. Hnětynková and Z. Strakoš, Lanczos tridiagonalization and core problems, Linear Algebra and its Applications, 421, pp. 243-251, 2007 [9] Z. Strakoš, D.P. O'Leary, C.C. Paige and P. Tichý, On unexpected consequences of numerical stability analysis of Krylov subspace methods, zvaná plenární přednáška, 22nd Biennial Conference on NA, Dundee, June 2007 [10] C. Brezinski and L. Wuytack (Eds.), Numerical Analysis: Historical Developments in the 20th Century. Elsevier, Amsterdam, 2001. [11] L. Eldén, Matrix methods in Data Mining and Pattern Recognition, SIAM, Philadelphia, 2007 Last update: T_KNM (18.05.2008)
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Lectures and discussions in a lecture hall. Last update: T_KNM (18.05.2008)
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Oral exam reflecting the content of the course. Last update: STRAKOS/MFF.CUNI.CZ (30.04.2008)
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1. Introduction to perturbation theory and to numerical stability. 2. The Lanczos tridiagonalization, the method of conjugate gradients and Gauss-Christoffel quadrature, their relationship. 3. Gauss quadrature from the view of analysis, algebra and numerical metods. 4. Golub-Kahan bidiagonalization and core problem in linear algebraic models. Last update: T_KNM (18.05.2008)
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The course assumes standard knowledge of linear algebra, calculus, elements of complex analysis, basic knowledge of numerical methods, including methods of numerical linear algebra. It is offered for students of various specializations starting from the seventh semester. Students are expected to have attended the courses NNUM006 and NNUM042. Last update: T_KNM (18.05.2008)
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