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Tytuł artykułu

The polar decomposition : properties, applications and algorithms

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Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
In the paper we review the numerical methods for computing the polar decomposition of a matrix. Numerical tests comparing these methods are included. Moreover, the applications of the polar decomposition and the most important its properties are mentioned.
Słowa kluczowe
Rocznik
Tom
Strony
23--49
Opis fizyczny
Bibliogr. 43 poz.
Twórcy
  • Department of Mathematics Higher Engineering School, ul. Luboszycka 3, Opole, Poland
  • Institute of Computer Science, University of Wrocław, ul. Przesmyckiego 20, Wrocław, Poland
Bibliografia
  • [1] L. Autonne, Sur les groupes linéaires, réels et orthogonaux, Bull. Soc. Math. France 30 (1902), 121-134.
  • [2] I. Y. Bar-Itzhack and K. A. Fegley, Orthogonalization techniques of a direction cosine matrix, IEEE Trans. Aeorospace Electron. Systems 5 (1969), 798-804.
  • [3] A. Barrlund, Perturbation bounds on the polar decomposition, BIT 30 (1989), 101-113.
  • [4] R. Bhatia and F. Kittaneh, Approximation by positive operators, Linear Algebra Appl. 161 (1992), 1-9.
  • [5] Å. Björck and C. Bowie, An iterative algorithm for computing the best estimate of an orthogonal matrix, SIAM J. Numer. Anal. 8 (1971), 358-364.
  • [6] R. Byers, Solving the algebraic Riccati equation with the matrix sign function, Linear Algebra Appl. 85 (1987), 267-279.
  • [7] J. Demmel and A. McKenney, A test matrix generation suite, LAPACK Working Note 9, Courant Institute, New York 1989.
  • [8] J. J. Dongarra, J. R. Bunch, C. B. Moler and G. W. Stewart, LINPACK Users’s Guide, SIAM, Philadelphia 1979.
  • [9] K. Fan and A. J. Hoffman, Some matrix inequalities in the space of matrices, Proc. Amer. Math. Soc. 6 (1955), 111-116.
  • [10] W. Gander, On Halley’s iteration method, Amer. Math. Monthly 92 (1985), 131-134.
  • [11] W. Gander, Algorithms for polar decomposition, SIAM J. Sci. Statist. Comput. 11 (1990), 1102-1115.
  • [12] F. R. Gantmacher, Theory of Matrices, vol. 1, Chelsea, New York 1959.
  • [13] P. E. Gill, W. Murray and M. H. Wright, Practical Optimization, Academic Press, London 1981.
  • [14] G. H. Golub and Ch. F. Van Loan, Matrix Computation, John Hopkins University Press, Maryland 1989.
  • [15] P. R. Halmos, Positive approximants of operators, Indiana Univ. Math. I. 21 (1972), 951-960.
  • [16] G. Hardt-Olejniczak and K. Ziętak, Moduł AL-Algebra liniowa, in: Elementy informatyki. Pakiet oprogramowania edukacyjnego, M. M. Sysło, Red., Inst. Inf. Uniwersytetu Wrocławskiego i OFEK-Poznań 1992.
  • [17] N. J. Higham, Computing the polar decomposition - with applications, SIAM J. Sci. Stat. Comput. 7 (1986), 1160-1173.
  • [18] N. J. Higham, Newton’s method for the matrix square root, Math. Comput. 46 (1986), 537-549.
  • [19] N. J. Higham, Computing real square roots of a real matrix, Linear Algebra Appl. 88/89 (1987), 405-430.
  • [20] N. J. Higham, Fortran codes for estimating the one-norm of real or complex matrix, with applications to condition estimation, ACM Trans. Math. Software 14 (1988), 381-396.
  • [21] N. J. Higham, Computing a nearest symmetric positive semidefinite matrix, Linear Algebra Appl. 103 (1988), 103-118.
  • [22] N. J. Higham, Matrix nearness problem and applications, in: Applications of Matrix Theory, M. J. C. Cover and S. Barnett, Eds, Oxford Univ. Press, Oxford 1989, pp. 1-27.
  • [23] N. J. Higham, Analysis of the Cholesky decomposition of a semi-definite matrix, in: Reliable Numerical Computation, M. G. Cox and S. J. Hammarling, Eds, Oxford University Press, Oxford 1990, pp. 161-185.
  • [24] N. J. Higham, The matrix sign decomposition and its relation to the polar decomposition, Numer. Anal. Report 225, Dept. of Math. Univ. of Manchester, Manchester 1993, to appear in Linear Algebra. Appl. 212/213.
  • [25] N. J. Higham and P. Papadimitriou, A new parallel algorithm for computing the singular value decomposition, in: Proc. The Fifth SIAM Conf. on Appl. Lin. Algebra, J. G. Lewis, Ed., SIAM, Philadelphia 1994, pp. 80-84.
  • [26] N. J. Higham and P. Papadimitriou, A parallel algorithm for computing the polar decomposition, Parallel Computing 20 (1994), 1161-1173.
  • [27] N. J. Higham and R. S. Schreiber, Fast polar decomposition of an arbitrary matrix SIAM J. Sci. Stat. Comput. 11 (1990), 648-655.
  • [28] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge 1985.
  • [29] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge 1991.
  • [30] Ch. Kenney and A. J. Laub, On scaling Newton’s method for polar decomposition and the matrix sign function, SIAM J. Matrix Anal. Appl. 13 (1992), 688-706.
  • [31] A. Kiełbasiński, A note on rouding-error analysis of Cholesky factorization, Linear Algebra Appl. 88/89 (1987), 487-494.
  • [32] A. Kiełbasiński and II. Schwetlick, Numerische lineare Algebra, VEB Deutscher Verlag der Wissenschaften, Berlin 1988.
  • [33] Ch. L. Lawson and R. J. Hanson, Solving Least Squares Problems, Prentice Hall, New York 1974.
  • [34] R. Mathias, Perturbation bounds for the polar decomposition, SIAM J. Matrix Anal. Appl. 14 (1993), 588-597.
  • [35] C. Moler, J. Little and S. Bangert, PRO-MATLAB User’s Guide, The Math Works Inc., Sherborn 1987.
  • [36] P. Pandey, Ch. Kenney and A. J. Laub, A parallel algorithm for the matrix sign function, Int. J. High Speed Computing 2 (1990), 181-191.
  • [37] C. R. Rao, Matrix approximation and reduction of dimensionality in multivariate statistical analysis, in: Multivariate Analysis V, Proc. of the Fifth Intern. Symp. on Multiv. Anal., P. R. Krishnaiah, Ed., North - Holland, Amsterdam 1980.
  • [38] J. Roberts, Linear model reduction and solution of the Riccati equation by use the sign function, Internat. J. Control 32 (1980), 677-687.
  • [39] R. S. Schreiber and B. N. Parlett, Block reflectors: Theory and computation, SIAM I. Numer. Anal. 25 (1988), 189-205.
  • [40] G. W. Stewart, On efficient generation of random orthogonal matrices with an application to condition estimators, SIAM J. Numer. Anal. 17 (1980), 403-409.
  • [41] G. A. Watson, Solving generalizations of orthogonal Procrustes problem, in: World Scientific Series in Applicable Analysis 2, Contributions in Numerical Mathematics, R. P. Agarwal, Ed., World Scientific, Singapore 1993, pp. 413-426.
  • [42] J. H. Wilkinson and C. Reinsch, Handbook for Automatic Computation. Linear Algebra, Springer, Berlin 1971.
  • [43] P. Zieliński, Rozkład polarny macierzy, Inst. Inf. Uniwersytetu Wrocławskiego, Wrocław 1993 (Master thesis).
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-90a277a8-bc3d-4056-bcb9-68b6e4baf489
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