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Inversion of selected structures of block matrices of chosen mechatronic systems

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Języki publikacji
EN
Abstrakty
EN
This paper describes how to calculate the number of algebraic operations necessary to implement block matrix inversion that occurs, among others, in mathematical models of modern positioning systems of mass storage devices. The inversion method of block matrices is presented as well. The presented form of general formulas describing the calculation complexity of inverted form of block matrix were prepared for three different cases of division into internal blocks. The obtained results are compared with a standard Gaussian method and the “inv” method used in Matlab. The proposed method for matrix inversion is much more effective in comparison in standard Matlab matrix inversion “inv” function (almost two times faster) and is much less numerically complex than standard Gauss method.
Rocznik
Strony
853--863
Opis fizyczny
Bibliogr. 24 poz., rys., tab., wykr.
Twórcy
  • Department of Mechatronics, Silesian University of Technology, 10A Akademicka St., 44-100 Gliwice, Poland
autor
  • Department of Mechatronics, Silesian University of Technology, 10A Akademicka St., 44-100 Gliwice, Poland
autor
  • Department of Mechatronics, Silesian University of Technology, 10A Akademicka St., 44-100 Gliwice, Poland
autor
  • Institute of Mathematics, Silesian University of Technology, 23 Kaszubska St., 44-100 Gliwice, Poland
Bibliografia
  • [1] J. de Jesus Rubio, C. Torres, C. Aguilar, “Optimal control based in a mathematical model applied to robotic arms”, International Journal of Innovative Computing, Information and Control 7(8), 5045‒5062, (2011).
  • [2] T. Trawiński, K. Kluszczyński, W. Kolton, “Lumped parameter model of double armature VCM motor for head positioning system of mass storage devices”, Przegląd Elektrotechniczny 87(12b), 184‒187, [in Polish] (2011).
  • [3] D. Słota, T. Trawiński, R. Wituła, “Inversion of dynamic matrices of HDD head positioning system”, Appl. Math. Model. 35(3), 1497‒1505, (2011).
  • [4] T. Trawiński, “Kinematic chains of branched head positioning system of hard disk drives”, Przegląd Elektrotechniczny 87(3), 204‒207,(2011).
  • [5] T. Trawiński, “Block form of inverse inductance matrix for poliharmonic model of an induction machine”, XXVIII International Conference on Fundamentals of Electrotechnics and Circuit Theory, IC-SPETO’2005, II, p.371, (2005).
  • [6] T. Trawiński, Modeling of Driving Lay-out of Mass Storage Head Positioning System, Wydawnictwa Politechniki Śląskiej, Gliwice, [in Polish] (2010).
  • [7] N. Jakovcevic Stor, I. Slapnicar, J. L. Barlow, “Accurate eigenvalue decomposition of arrowhead matrices and applications”, Linear Algebra and its Applications, 464, 62–89, (2015).
  • [8] F. Diele, N. Mastronardi, M. Van Barel, E. Van Camp, “On computing the spectral decomposition of symmetric arrowhead matrices”, Computational Science and Its Applications – ICCSA 2004 Lecture Notes in Computer Science, 3044, 932–941, (2004).
  • [9] L. Shen, B. W. Suter, “Bounds for eigenvalues of arrowhead matrices and their applications to hub matrices and wireless communications”, EURASIP Journal on Advances in Signal Processing, 2009, (2009).
  • [10] G. A. Gravvanis, K. M. Giannoutakis, “Parallel exact and approximate arrow-type inverses on symmetric multiprocessor systems”, Computational Science – ICCS 2006 Lecture Notes in Computer Science, 3991, 506‒513, (2006).
  • [11] G. A. Gravvanis, “Solving symmetric arrowhead and special tridiagonal linear systems by fast approximate inverse preconditioning”, Journal of Mathematical Modelling and Algorithms 1(4), 269‒282, (2002).
  • [12] J. Tanton, “A dozen questions about triangular numbers”, Math Horizons 13(2), 6‒7, 21‒23 (2005).
  • [13] P. A. Piza, “On the squares of some triangular numbers”, Mathematics magazine 23(1), 15‒16, (1949).
  • [14] J. A. Ewell, “A trio of triangular number theorems”, American Mathematical Monthly 105(9), 848‒849, (1998).
  • [15] “Lozanic triangle”, The On-Line Encyclopedia of Integer Sequences, (2014).
  • [16] K. A. Atkinson, An Introduction to Numerical Analysis, John Wiley and Sons (2nd ed.), New York, 1989.
  • [17] W. Hołubowski, D. Kurzyk, T. Trawiński, “A fast method for computing the inverse of symmetric block arrowhead matrices”, Appl. Math. Inf. Sci. 9(2L), 1‒5, (2015).
  • [18] T. Trawiński, “Inversion method of matrices with chosen structure with help of block matrices”, Przegląd Elektrotechniczny 85(6), 98‒101, [in Polish] (2009).
  • [19] H. T. Kung, B. Wilsey Suter, “A hub matrix theory and applications to wireless communications”. EURASIP Journal on Advances in Signal Processing 2007(1), 1‒8, (2007).
  • [20] E. Mizutani, J. W. Demmel, “On structure-exploiting trustregion regularized nonlinear least squares algorithms for neuralnetwork learning”, Neural Networks 16(5), 745‒753, (2003).
  • [21] M. Bixon, J. Jortner, “Intramolecular radiationless transitions”, The Journal of Chemical Physics 48(2), 715‒726, (1968).
  • [22] J. W. Gadzuk, “Localized vibrational modes in Fermi liquids. General theory”, Physical Review B 24(4), 1651, (1981).
  • [23] A. Ratajczak, “Trajectory reproduction and trajectory tracking problem for the nonholonomic systems”, Bull. Pol. Ac.: Tech. 64(1), 63‒70 (2016).
  • [24] A. Mazur, M. Cholewiński, “Implementation of factitious force method for control of 5R manipulator with skid-steering platform REX”, Bull. Pol. Ac.: Tech. 64(1), 71‒80 (2016).
Uwagi
PL
Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-9e7a5cd7-45a5-417b-84a8-c916ae5b2237
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