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Lecturer(s)
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Černá Dana, doc. RNDr. Ph.D.
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Course content
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Lectures: 1. Basic concepts. Numerical model. Sources of errors. Algorithm complexity. Parallelization of numerical problems. 2. Solution of systems of linear algebraic equations. Vector and matrix standards. Matrix norms consistent and cocompatible with the vector norm. 3. Theorems on the calculation of basic matrix norms. 4. Oldenburger theorem. Theorem on system conditioning. 5. Direct methods. Gaussian elimination - algorithm and its complexity, stability, pivotation, Gaussian elimination for three-diagonal matrix, matrix filling problems. 6. LU decomposition - algorithm, pivoting. Cholesky decomposition - derivation of an algorithm. 7. Iterative methods. Matrix iteration methods, necessary and sufficient convergence conditions, error estimation. 8. Jacobi method, Gauss-Seidel method, successive overrelaxation method. 9. Conjugate gradient method - derivation of the algorithm, construction of A-orthogonal vectors, orthogonality of residues. 10. Solution of systems with singular matrices. Pseudoinverse matrix - existence and uniqueness, theorem on solving systems using pseudoinverse matrix. Least-squares method. 11. Singular decomposition - existence, uniqueness, algorithm, use in linear algebra. 12. Interpolation. Lagrange's interpolation polynomial - existence and uniqueness, error estimation, Lagrange's and Newton's form, Runge's phenomenon. 13. Hermite's interpolation polynomial - existence and uniqueness, error estimation, Lagrange's and Newton's form. 14. Reserve. Exercises: The exercise is focused on the implementation of the discussed numerical methods in Matlab.
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Learning activities and teaching methods
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Monological explanation (lecture, presentation,briefing)
- Class attendance
- 56 hours per semester
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Learning outcomes
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The subject is concerned with the analysis and implementation of numerical methods for solving linear algebraic problems. It is focused on the direct and iteration methods for solving systems of linear algebraic equations with regular matrices and methods for the numerical solution of systems with singular matrices. Another topic is Lagrange and Hermite interpolation.
Construction of the mathematical and numerical model. Basic approximate and numerical methods: Methods of linear algebra.
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Prerequisites
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Condition of registration: Calculus 1, Calculus 2.
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Assessment methods and criteria
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Combined examination
Credit: active participation in exercises and tests Exam: combined, consists of theoretical and numerical parts
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Recommended literature
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Ch. Woodford, C. Philips. Numerical Methods with Worked Examples: Matlab Edition. Dordrecht, 2012. ISBN 978-94-007-1365-9.
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PRESS, W.H.:. Numerical Recipes 3rd Edition, The Art of Scientific Computing. 2007. ISBN 9780521880688.
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Quarteroni, A. - Saleri, F.:. Scientific Computing with MATLAB.. Berlin, Springer, 2000.
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Stoer J., Bulirsch R.:. Introduction to Numerical Analysis. Springer. ISBN 0-387-95452-X.
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