SPECIAL TOPICS OF NUMERICAL ANALYSIS

Description.

Norms of vectors and matrices. Characteristic and minimum polynomial of a matrix. Properties of characteristic quantities of special matrices (symmetric, Hermitian, normal, orthogonal, etc.). Matrix factorizations (triangulation -Schur, diagonalization of a matrix, Jordan, SVD, LU, QR and Cholesky form). Applications of factorizations in solving a linear system, stability check of a system, root of square matrix, etc. Perron-Frobenius theory. Applications to spectral radius localization and graph theory. Arithmetic field matrix and its characteristic properties. Polynomial matrix theory and applications to differential equations. Interpolation and numerical diffusion. Methods of interpolation (Linear, Cubic, Lagrange, Spline, Hermite). Construction of interpolation kernels. Applications in 2 dimensions, in medical images. Construction of discrete derivative filters (finite and infinite impulse response, FIR and IIR). Numerical solution of Partial Differential Equations (PDE). Applications of PDE in image processing and modelling of biological systems, (de-noising, segmentation, active contours). Numerical integration methods.

Bibliography:
1. James E. Gentle, Matrix Algebra, Ηλεκτρονικό Βιβλίο, εκδότης HEAL-Link Springer ebooks.
2. Thomas S. Shores, Applied Linear Algebra and Matrix Analysis, Ηλεκτρονικό Βιβλίο, εκδότης HEAL-Link Springer ebooks.
3. Alan J. Laub, Ανάλυση Μητρώων για Επιστήμονες και Μηχανικούς, εκδόσεις ΚΛΕΙΔΑΡΙΘΜΟΣ ΕΠΕ, 1η έκδοση 2010.
4. R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, second edition, 2013.
5. R.A. Horn and C.R. Johnson, Τopics in Matrix Analysis, Cambridge University Press, Cambridge, fourth edition, 1999.
6. L.N. Trefethen and D. Bau III, Numerical Linear Algebra, SIAM, 1997.
7. G.W. Stewart and Ji-g Sun, Matrix perturbation theory, Academic Press, 1990.
8. Γεώργιος Δονάτος και Μαρία Αδάμ, Γραμμική Άλγεβρα, εκδόσεις Γ. ΔΑΡΔΑΝΟΣ – Κ. ΔΑΡΔΑΝΟΣ Ο.Ε., 2008, Κωδ. Ευδόξου [31174].
9. Μιχαήλ Βραχάτης, Αριθμητική Ανάλυση: Εισαγωγή εκδόσεις ΚΛΕΙΔΑΡΙΘΜΟΣ ΕΠΕ, 1η έκδοση 2011.
10. Νικόλαος Μισυρλής, Αριθμητική Ανάλυση, ΕΤΑΙΡΕΙΑ ΑΞΙΟΠΟΙΗΣΕΩΣ ΚΑΙ ΔΙΑΧΕΙΡΙΣΕΩΣ ΤΗΣ ΠΕΡΙΟΥΣΙΑΣ ΤΟΥ ΠΑΝΕΠΙΣΤΗΜΙΟΥ ΑΘΗΝΩΝ, ΑΘΗΝΑ 2η έκδοση 2017.
11. Γ. Σοφιανός και Ε.Θ. Τυχόπουλος, Αριθμητική ανάλυση, εκδόσεις ΣΤΑΜΟΥΛΗ Α.Ε., 2005.
12. Σαρρής και Θ. Καρακασίδης, Αριθμητικές Μέθοδοι και Εφαρμογές για Μηχανικούς, ΕΚΔΟΣΕΙΣ Α. ΤΖΙΟΛΑ & ΥΙΟΙ Α.Ε., 2η έκδοση 2017.
13. Carlo Vercellis, Business Intelligence: Data Mining and Optimization for Decision Making, John Wiley & Sons, 2009.
14. Heppner, F. and Grenander, U.: A stochastic nonlinear model for coordinated bird flocks. In S.Krasner, Ed., The Ubiquity of Chaos. AAAS Publications, Washington, DC, 1990.
15. Kennedy, J., Eberhart, R. C., and Shi, Y., Swarm intelligence, San Francisco: Morgan Kaufmann Publishers, 2001.
16. KE Parsopoulos, MN Vrahatis, Particle swarm optimization and intelligence: advances and applications, Information Science Reference, 2010.
17. T. M. Lehmann, Gonner, C., and Spitzer, K. Survey: Interpolation methods in medical image processing, IEEE Transactions on Medical Imaging, 18(11), (1999), 1049-1075.
18. Infinite impulse response (IIR) prefiltering algorithm described in the work of Unser et al [IEEE T. SP, 41, 1993]
19. T. Blu, P. Thevenaz, and . Unser, Complete Parameterization of P iecewise-Polynomial Interpolation Kernels, IEEE TRANSACTIONS on Image Processing, 12(11), 2003.
20. K. Delibasis, A. Kechriniotis and Il. Maglogiannis, On Centered and Compact Signal and Image Derivatives for Feature Extraction, 9th Artificial Intelligence Applications and Innovations (AIAI), Sep 2013, Paphos, Greece. pp.318-327, ff10.1007/978-3-642-41142-7_33ff.
21. https://hal.inria.fr/hal-01459628