Verified computation of square roots of a matrix

SIAM Journal on Matrix Analysis and Applications

Volumn 31, Issue 3, 2009, Pages 1279-1302

  Frommer, Andreas ^a   Hashemi, Behnam ^b  

^a UNIVERSITY OF WUPPERTAL   (Germany)

^b AMIRKABIR UNIVERSITY OF TECHNOLOGY   (Iran)

Author keywords

Brouwer's fixed point theorem; Circular arithmetic; Interval analysis; Krawczyk's method; Kronecker structures; Matrix square root

Indexed keywords

BLOCK DIAGONAL; BROUWER'S FIXED POINT THEOREM; CIRCULAR ARITHMETIC; COMPUTATIONALLY EFFICIENT; EIGENVECTOR MATRICES; FLOATING-POINT ALGORITHMS; ILL-CONDITIONED; INTERVAL ANALYSIS; INTERVAL ARITHMETIC; INTERVAL MATRIX; MATRIX; MATRIX SQUARE ROOTS; N-MATRIX; NUMERICAL EXPERIMENTS; SIMILARITY TRANSFORMATION; SPECTRAL DECOMPOSITION; SQUARE MATRICES; SQUARE ROOTS; SYMMETRIC MATRICES; VERIFIED COMPUTATIONS;

APPROXIMATION ALGORITHMS; COMPUTATIONAL COMPLEXITY; EIGENVALUES AND EIGENFUNCTIONS; NUMERICAL METHODS;

COMPUTATIONAL EFFICIENCY;

EID: 72449198921     PISSN: 08954798     EISSN: 10957162     Source Type: Journal    
DOI: 10.1137/090757058     Document Type: Article

References (36)

1. G. ALEFELD AND J. HERZBERGER, Introduction to Interval Computations, Computer Science and Applied Mathematics, Academic Press, New York, 1983.
2. G. ALEFELD AND N. SCHNEIDER, On square roots of M-matrices, Linear Algebra Appl., 42 (1982), pp. 119-132.
3. R. BARTELS AND G. STEWART, Solution of the matrix equation AX +XB = C, Comm. ACM, 15 (1972), pp. 820-826.
4. C. A. BAVELY AND G. W. STEWART, An algorithm for computing reducing subspaces by block diagonalization, SIAM J. Numer. Anal., 16 (1979), pp. 359-367.
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. L. BROUWER, Über Abbildungen von Mannigfaltigkeiten, Math. Ann., 71 (1912), pp. 97-115.
7. G. CROSS AND P. LANCASTER, Square roots of complex matrices, Linear Multilinear Algebra, 1 (1974), pp. 289-293.
8. E. D. DENMAN, Roots of real matrices, Linear Algebra Appl., 36 (1981), pp. 133-139.
9. E. D. DENMAN AND A. N. BEAVERS, The matrix sign function and computations in systems, Appl. Math. Comput., 2 (1976), pp. 63-94.
10. I. DUFF, R. GRIMES, AND J. LEWIS, Users' Guide for Harwell-Boeing Sparse Matrix Test Problems Collection, Tech. report RAL-92-086, Computing and Information Systems Department, Rutherford Appleton Laboratory, Didcot, UK, 1992.
11. N. J. HIGHAM, The Matrix Computation Toolbox, http://www.ma.man.ac.uk/ ~higham/mctoolbox.
12. N. J. HIGHAM, Newton's method for the matrix square root, Math. Comput., 46 (1986), pp. 537-549.
13. N. J. HIGHAM, Computing real square roots of a real matrix, Linear Algebra Appl., 88/89 (1987), pp. 405-430.
14. N. J. HIGHAM, Stable iterations for the matrix square root, Numer. Algorithms, 15 (1997), pp. 227-242.
15. N. J. HIGHAM, The Matrix Computation Toolbox for MATLAB (Version 1.0), Tech. report, Numerical Analysis Report 410, University of Manchester, Manchester, UK, 2002, http://www.ma.man.ac.uk/nareports.
16. N. J. HIGHAM, Functions of Matrices: Theory and Computation, SIAM, Philadelphia, 2008.
17. R. A. HORN AND C. R. JOHNSON, Topics in Matrix Analysis, Cambridge University Press, Cambridge, UK, 1994.
18. W. HOSKINS AND D. WALTON, A faster method of computing the square root of a matrix, IEEE Trans. Automat. Control, 23 (1978), pp. 494-495.
19. R. B. KEARFOTT, Rigorous Global Search: Continuous Problems, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996.
20. R. B. KEARFOTT, M. NAKAO, A. NEUMAIER, S. RUMP, S. SHARY, AND P. VAN HENTENRYCK, Standardized Notation in Interval Analysis, 2005, http://www.mat.univie.ac.at/~neum/ms/notation.pdf.
21. R. KLATTE, U. W. KULISCH, A. WIETHOFF, C. LAWO, AND M. RAUCH, C-XSC. A C++ Class Library for Extended Scientific Computing, Springer-Verlag, Berlin, 1993.
22. R. KRAWCZYK, Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken, Computing, 4 (1969), pp. 187-201.
23. W. LUTHER AND W. OTTEN, The complex arithmetic-geometric mean and multiple-precision matrix functions, in Scientific Computing and Validated Numerics, Math. Res. 90, G. Alefeld, A. Frommer, and B. Lang, eds., Akademie Verlag, Berlin, 1996, pp. 52-58.
24. 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.
25. R. E. MOORE, A test for existence of solutions to nonlinear systems, SIAM J. Numer. Anal., 14 (1977), pp. 611-615.
26. R. E. MOORE, R. B. KEARFOTT, AND M. J. CLOUD, Introduction to Interval Analysis, SIAM, Philadelphia, 2009.
27. A. NEUMAIER, Interval Methods for Systems of Equations, Encyclopedia Math. Appl. 37, Cambridge University Press, Cambridge, UK, 1990.
28. N. I. of STANDARDS AND TECHNOLOGY, Matrix Market: A Web Resource for Test Matrix Collections, http://math.nist.gov/Matrix-Market/.
29. T. OGITA, S. M. RUMP, AND S. OISHI, Accurate sum and dot product, SIAM J. Sci. Comput., 26 (2005), pp. 1955-1988.
30. J. ROHN, VERSOFT: Verification Software in MATLAB/INTLAB, http://uivtx.cs.cas.cz/rohn/matlab.
31. S. M. RUMP, Solving algebraic problems with high accuracy, in A New Approach to Scientific Computation, Comput. Sci. Appl. Math. 7, W. Miranker and E. Kaucher, eds., Academic Press, New York, 1983, pp. 51-120.
32. S. M. RUMP, Verification methods for dense and sparse systems of equations, in Topics in Validated Computations, Stud. Comput. Math. 5, J. Herzberger, ed., Elsevier, Amsterdam, 1994, pp. 63-135.
33. S. M. RUMP, Expansion and estimation of the range of nonlinear functions, Math. Comput., 65 (1996), pp. 1503-1512.
34. S. M. RUMP, INTLAB - INTerval LABoratory, in Developments in Reliable Computing, T. Csendes, ed., Kluwer Academic Publishers, Dordrecht, The Netherlands, 1999, pp. 77-104.
35. M. SMITH, Numerical Computation of Matrix Functions, Ph.D. thesis, University of Manchester, Manchester, UK, 2002.
36. R. S. VARGA, Matrix Iterative Analysis, 2nd ed., Springer Ser. Comput. Math. 27, Springer, Berlin, 2000.