Computational Mathematics II

winter semester 2013-14

Syllabus

1. Iterative methods for solving
   • linear systems of equations: sparse formats of matrices, stationary iterative methods (incl. Jacobi, Gauss-Seidel, SOR, Richardson), projection methods (incl. steepest descent method, minimal residual method), Krylov subspace methods (incl. CG and GMRES), preconditioning (ILU, spectral equivalence of matrices, PCG, preconditioned GMRES, termination conditions), structural methods (reduction to Schur matrix), multigrid methods (only 2-grid was discussed) Basic references: posiiton vi. (Saad - Iterative... ) and 4. (Kelley - Iterative....)
   • nonlinear systems of equations - Banach iterations, multidimensional Newton methods, the chord method, Newton method with Jacobian approximated by finite differences, inexact Newton methods. Basic reference: position 4. (Kelley - Iterative....)
2. Eigenproblems - methods for finding (approximations) of eigenvalues and eigenvectors of a matrix - similar matrices, transformation of a matrix into a similar matrix in the Hessenberg form by using Householder reflections, the power and inverse iterations, the Rayleigh quotient iteration, pure QR and shifted QR methods, "divide and conquer" method, the Hyman method. The basic reference: position 8. (Trefethen, Bau, NLA) and 7. (Stoer at al, Intro. to Numer. Anal.- in particular Hyman met.)
3. Multidimension integration - nothing was done - the dimensionality curse, the Monte Carlo (MC) and quasi MC methods should have been discussed

Lecture notes

References

Text books

1. James W. Demmel, Applied Numerical Linear Algebra. Society for Industrial and Applied Mathematics (SIAM), Philadelphia 1997.
2. Peter Deuflhard, Andreas Hohmann. Numerical analysis in modern scientific computing, vol. 43 in Texts in Applied Mathematics. Springer-Verlag, New York, 2nd edition, 2003. An introduction.
3. J.M. Jankowscy, M. Dryja. Przegląd metod i algorytmów numerycznych, tom I i II. Biblioteka inżynierii oprogramowania. Wydawnictwo Naukowo-Techniczne, Warszawa, 1995. (In Polish)
4. C. T. Kelley. Iterative methods for linear and nonlinear equations, vol. 16 of Frontiers in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1995 - covers iterative methods for linear equations (CG, PCG and GMRES), and solving nonlinear equations
5. A. Kiełbasiński, H. Schwetlick. Numeryczna algebra liniowa. Wydawnictwa Naukowo-Techniczne, 1992. (In Polish, there is an German edition, too)
6. D.Kincaid, W.Cheney, Numerical Analysis, 2nd editione, Brooks/Cole, 1996. A general textbook for numerical methods.
7. J. Stoer, R. Bulirsch. Introduction to numerical analysis. Translated from the German by R. Bartels, W. Gautschi and C. Witzgall. Third edition. Texts in Applied Mathematics, 12. Springer-Verlag, New York, 2002
8. Lloyd N. Trefethen, David Bau, III, Numerical linear algebra. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1997.

Monographs or advanced text books

1. John E. Dennis Jr., Robert B. Schnabel. Numerical methods for unconstrained optimization and nonlinear equations. Prentice-Hall Series in Computational Mathematics. Prentice-Hall Inc., En- glewood Cliffs, N.J., 1983.
2. Peter Deuflhard. Newton methods for Nonlinear Problems. Affine Invariance and Adaptive Algori- thms. Springer International, 2002.
3. Eugene G. Dyakonov. Optimization in solving elliptic problems. CRC Press, Boca Raton, FL, 1996. Translated from the 1989 Russian original, Translation edited and with a preface by Steve McCormick.
4. Gene H. Golub, Charles F. Van Loan. Matrix computations. Johns Hopkins Studies in the Mathe- matical Sciences. Johns Hopkins University Press, Baltimore, MD, 3rd ed., 1996.
5. J. M. Ortega, W. C. Rheinboldt. Iterative solutions of nonlinear equations in several variables. Academic Press, New York, 1970.
6. Yousef Saad. Iterative methods for sparse linear systems. Society for Industrial and Applied Ma- thematics, Philadelphia, PA, 2nd ed., 2003. On-line
7. Yousef Saad, Numerical Methods for Large Eigenvalue Problems. Society for Industrial and Applied Ma- thematics, Philadelphia, PA, 2nd ed., 2011. On-line
8. A. A. Samarski, J. S. Nikołajew. Metody rozwiązywania równań siatkowych. PWN 1988. (In Polish)
9. Barry F. Smith, Petter E. Bjorstad, William D. Gropp. Domain decomposition. Cambridge Univer- sity Press, Cambridge, 1996. Parallel multilevel methods for elliptic partial differential equations.
10. Andrea Toselli, Olof Widlund. Domain decomposition methods - algorithms and theory, volume 34 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, 2005.
11. J. F. Traub. Iterative Methods for the Solution of Equations. Englewood Cliffs, New York, 1964.