A schur-newton method for the matrix pth root and its inverse

SIAM Journal on Matrix Analysis and Applications

Volumn 28, Issue 3, 2006, Pages 788-804

  Guo, Chun Hua ^a   Higham, Nicholas J ^b  

^a UNIVERSITY OF REGINA   (Canada)

^b UNIVERSITY OF MANCHESTER   (United Kingdom)

Author keywords

Convergence; Inverse; Markov model; Matrix logarithm; Matrix pth root; Newton's method; Numerical stability; Preprocessing; Principal pth root; Schur decomposition; Transition matrix

Indexed keywords

ALGORITHMS; CONVERGENCE OF NUMERICAL METHODS; EIGENVALUES AND EIGENFUNCTIONS; MARKOV PROCESSES; MATHEMATICAL MODELS;

INVERSE MATRIX; MARKOV MODELS; MATRIX PTH ROOTS; NEWTON ITERATION; TRANSITION MATRIX; TRIANGULAR FACTORS;

NEWTON-RAPHSON METHOD;

EID: 34547217286     PISSN: 08954798     EISSN: 10957162     Source Type: Journal    
DOI: 10.1137/050643374     Document Type: Article

References (35)

1. D. H. BAILEY, MPFUN: A Portable High Performance Multiprecision Package, Technical Report RNR-90-022, NASA Ames Research Center, Moffett Field, CA, 1990.
2. D. H. BAILEY, A Fortran 90-based multiprecision system, ACM Trans. Math. Software, 21 (1995), pp. 379-387.
3. D. H. BAILEY, Y. HIDA, X. S. LI, AND B. THOMPSON, ARPREC: An Arbitrary Precision Computation Package, Technical Report LBNL-53651, Lawrence Berkeley National Laboratory, Berkeley, CA, 2002.
4. D. A. BINI, N. J. HIGHAM, AND B. MEINI, Algorithms for the matrix pth root, Numer. Algorithms, 39 (2005), pp. 349-378.
5. A. BJÖRCK AND S. HAMMARLING, A Schur method for the square root of a matrix, Linear Algebra Appl., 52/53 (1983), pp. 127-140.
6. S. H. CHENG, N. J. HIGHAM, C. S. KENNEY, AND A. J. LAUB, Approximating the logarithm of a matrix to specified accuracy, SIAM J. Matrix Anal. Appl., 22 (2001), pp. 1112-1125.
7. M. CORNEA-HASEGAN AND B. NORIN, IA-64 floating-point operations and the IEEE standard for binary floating-point arithmetic, Intel Technology Journal, 3 (1999), ftp://download.intel.com/ technology/itj/q41999/pdf/ia64fpbf.pdf.
8. J. R. CUTHBERT, On uniqueness of the logarithm for Markov semi-groups, J. London Math. Soc., 4 (1972), pp. 623-630.
9. J.-C. EVARD AND F. JAFARI, A complex Rolle's theorem, Amer. Math. Monthly, 99 (1992), pp. 858-861.
10. G. H. GOLUB AND C. F. VAN LOAN, Matrix Computations, 3rd ed., Johns Hopkins University Press, Baltimore, MD, 1996.
11. N. J. HIGHAM, Functions of a Matrix: Theory and Computation, book in preparation.
12. N. J. HIGHAM, Computing the polar decomposition - with applications, SIAM J. Sci. Statist. Comput., 7 (1986), pp. 1160-1174.
13. N. J. HIGHAM, Newton's method for the matrix square root, Math. Comp., 46 (1986), pp. 537-549.
14. N. J. HIGHAM, Computing real square roots of a real matrix, Linear Algebra Appl., 88/89 (1987), pp. 405-430.
15. N. J. HIGHAM, Stable iterations for the matrix square root, Numer. Algorithms, 15 (1997), pp. 227-242.
16. N. J. HIGHAM, Accuracy and Stability of Numerical Algorithms, 2nd ed., SIAM, Philadelphia, PA, 2002.
17. R. A. HORN AND C. R. JOHNSON, Topics in Matrix Analysis, Cambridge University Press, Cambridge, UK, 1991.
18. B. IANNAZZO, A note on computing the matrix square root, Calcolo, 40 (2003), pp. 273-283.
19. B. IANNAZZO, On the Newton method for the matrix pth root, SIAM J. Matrix Anal. Appl., 28 (2006), pp. 503-523.
20. R. B. ISRAEL, J. S. ROSENTHAL, AND J. Z. WEI, Finding generators for Markov chains via empirical transition matrices, with applications to credit ratings, Math. Finance, 11 (2001), pp. 245-265.
21. A. H. KARP AND P. MARKSTEIN, High-precision division and square root, ACM Trans. Math. Software, 23 (1997), pp. 561-589.
22. C S. KENNEY AND A. J. LAUB, Condition estimates for matrix functions, SIAM J. Matrix Anal. Appl., 10 (1989), pp. 191-209.
23. Ç. K. KOCCEDIL; AND B. BAKKALOǦLU, Halley's method for the matrix sector function, IEEE Trans. Automat. Control, 40 (1995), pp. 944-949.
24. A. KREININ AND M. SIDELNIKOVA, Regularization algorithms for transition matrices, Algo Research Quarterly, 4 (2001), pp. 23-40.
25. 2 - I = 0, Math. Tables Aids Comput., 12 (1958), pp. 109-116.
26. S. LAKIĆ, On the computation of the matrix k-th root, Z. Angew. Math. Mech., 78 (1998), pp. 167-172.
27. B. MEINI, The matrix square root from a new functional perspective: Theoretical results and computational issues, SIAM J. Matrix Anal. Appl., 26 (2004), pp. 362-376.
28. H.-O. PEITGEN, H. JÜRGENS, AND D. SAUPE, Fractals for the Classroom. Part Two: Complex Systems and Mandelbrot Set, Springer-Verlag, New York, 1992.
29. B. PHILIPPE, An algorithm to improve nearly orthonormal sets of vectors on a vector processor, SIAM J. AIg. Discrete Methods, 8 (1987), pp. 396-403.
30. M. SCHROEDER, Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise, W. H. Freeman, New York, 1991.
31. L. S. SHIEH, Y. T. TSAY, AND C. T. WANG, Matrix sector functions and their applications to system theory, IEE Proc., 131 (1984), pp. 171-181.
32. B. SINGER AND S. SPILERMAN, The representation of social processes by Markov models, Amer. J. Sociology, 82 (1976), pp. 1-54.
33. M. I. SMITH, A Schur algorithm for computing matrix pth roots, SIAM J. Matrix Anal. Appl., 24 (2003), pp. 971-989.
34. R. A. SMITH, Infinite product expansions for matrix n-th roots, J. Austral. Math. Soc., 8 (1968), pp. 242-249.
35. F. V. WAUGH AND M. E. ABEL, On fractional powers of a matrix, J. Amer. Statist. Assoc., 62 (1967), pp. 1018-1021.