Matrix Inverse Trigonometric and Inverse Hyperbolic Functions: : Theory and Algorithms
Pages 1453 - 1477
Abstract
Theoretical and computational aspects of matrix inverse trigonometric and inverse hyperbolic functions are studied. Conditions for existence are given, all possible values are characterized, and the principal values acos, asin, acosh, and asinh are defined and shown to be unique primary matrix functions. Various functional identities are derived, some of which are new even in the scalar case, with care taken to specify precisely the choices of signs and branches. New results include a “round trip” formula that relates $acos(\cos A)$ to $A$ and similar formulas for the other inverse functions. Key tools used in the derivations are the matrix unwinding function and the matrix sign function. A new inverse scaling and squaring type algorithm employing a Schur decomposition and variable-degree Padé approximation is derived for computing acos, and it is shown how it can also be used to compute asin, acosh, and asinh. In numerical experiments the algorithm is found to behave in a forward stable fashion and to be superior to computing these functions via logarithmic formulas.
References
[1]
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Natl. Bureau Standards Appl. Math. Ser. 55, National Bureau of Standards, Washington, D.C., 1964; reprinted by Dover, New York.
[2]
A. H. Al-Mohy and N. J. Higham, A new scaling and squaring algorithm for the matrix exponential, SIAM J. Matrix Anal. Appl., 31 (2009), pp. 970--989, https://doi.org/10.1137/09074721X.
[3]
A. H. Al-Mohy and N. J. Higham, Improved inverse scaling and squaring algorithms for the matrix logarithm, SIAM J. Sci. Comput., 34 (2012), pp. C153--C169, https://doi.org/10.1137/110852553.
[4]
A. H. Al-Mohy, N. J. Higham, and S. D. Relton, Computing the Fréchet derivative of the matrix logarithm and estimating the condition number, SIAM J. Sci. Comput., 35 (2013), pp. C394--C410, https://doi.org/10.1137/120885991.
[5]
A. H. Al-Mohy, N. J. Higham, and S. D. Relton, New algorithms for computing the matrix sine and cosine separately or simultaneously, SIAM J. Sci. Comput., 37 (2015), pp. A456--A487, https://doi.org/10.1137/140973979.
[6]
M. Aprahamian and N. J. Higham, The matrix unwinding function, with an application to computing the matrix exponential, SIAM J. Matrix Anal. Appl., 35 (2014), pp. 88--109, https://doi.org/10.1137/130920137.
[7]
M. Aprahamian and N. J. Higham, Argument Reduction for Computing Periodic Functions, MIMS EPrint, Manchester Institute for Mathematical Sciences, The University of Manchester, UK, 2015, in preparation.
[8]
Å. Björck and S. Hammarling, A Schur method for the square root of a matrix, Linear Algebra Appl., 52/53 (1983), pp. 127--140, https://doi.org/10.1016/0024-3795(83)80010-X.
[9]
R. Bradford, R. M. Corless, J. H. Davenport, D. J. Jeffrey, and S. M. Watt, Reasoning about the elementary functions of complex analysis, Ann. Math. Artif. Intell., 36 (2002), pp. 303--318, https://doi.org/10.1023/A:1016007415899.
[10]
J. R. Cardoso and F. Silva Leite, Computing the inverse matrix hyperbolic sine, in Numerical Analysis and Its Applications, L. Vulkov, J. Waśniewski, and P. Yalamov, eds., Lecture Notes in Comput. Sci. 1988, Springer-Verlag, Berlin, 2001, pp. 160--169, https://doi.org/10.1007/3-540-45262-1_20.
[11]
J. R. Cardoso and F. Silva Leite, Theoretical and numerical considerations about Padé approximants for the matrix logarithm, Linear Algebra Appl., 330 (2001), pp. 31--42, https://doi.org/10.1016/S0024-3795(01)00251-8.
[12]
J. R. Cardoso and F. Silva Leite, The Moser--Veselov equation, Linear Algebra Appl., 360 (2003), pp. 237--248, https://doi.org/10.1016/S0024-3795(02)00450-0.
[13]
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, https://doi.org/10.1137/S0895479899364015.
[14]
R. M. Corless, J. H. Davenport, D. J. Jeffrey, G. Litt, and S. M. Watt, Reasoning about the elementary functions of complex analysis, in Artificial Intelligence and Symbolic Computation, J. A. Campbell and E. Roanes-Lozano, eds., Lecture Notes in Comput. Sci. 1930, Springer-Verlag, Berlin, 2001, pp. 115--126, https://doi.org/10.1007/3-540-44990-6_9.
[15]
R. M. Corless, J. H. Davenport, D. J. Jeffrey, and S. M. Watt, ``According to Abramowitz and Stegun” or arccoth needn't be uncouth, ACM SIGSAM Bull., 34 (2000), pp. 58--65.
[16]
G. W. Cross and P. Lancaster, Square roots of complex matrices, Linear Multilinear Algebra, 1 (1974), pp. 289--293, https://doi.org/10.1080/03081087408817029.
[17]
L. Dieci, B. Morini, and A. Papini, Computational techniques for real logarithms of matrices, SIAM J. Matrix Anal. Appl., 17 (1996), pp. 570--593, https://doi.org/10.1137/S0895479894273614.
[18]
GNU Octave, http://www.octave.org.
[19]
N. J. Higham, The Matrix Computation Toolbox, http://www.maths.manchester.ac.uk/ higham/mctoolbox.
[20]
N. J. Higham, The Matrix Function Toolbox, http://www.maths.manchester.ac.uk/ higham/mftoolbox.
[21]
N. J. Higham, Functions of Matrices: Theory and Computation, SIAM, Philadelphia, 2008, https://doi.org/10.1137/1.9780898717778.
[22]
N. J. Higham, The scaling and squaring method for the matrix exponential revisited, SIAM Rev., 51 (2009), pp. 747--764, https://doi.org/10.1137/090768539.
[23]
N. J. Higham and E. Deadman, A Catalogue of Software for Matrix Functions. Version 2.0, MIMS EPrint 2016.3, Manchester Institute for Mathematical Sciences, The University of Manchester, UK, Jan. 2016, http://eprints.ma.man.ac.uk/2431; updated March 2016.
[24]
N. J. Higham and F. Tisseur, A block algorithm for matrix $1$-norm estimation, with an application to 1-norm pseudospectra, SIAM J. Matrix Anal. Appl., 21 (2000), pp. 1185--1201, https://doi.org/10.1137/S0895479899356080.
[25]
R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, UK, 1991.
[26]
B. Iannazzo, Numerical Solution of Certain Nonlinear Matrix Equations, Ph.D. thesis, Università degli Studi di Pisa, Pisa, Italy, 2007.
[27]
W. Kahan, Branch cuts for complex elementary functions or much ado about nothing's sign bit, in The State of the Art in Numerical Analysis, A. Iserles and M. J. D. Powell, eds., Oxford University Press, 1987, pp. 165--211.
[28]
C. S. Kenney and A. J. Laub, Condition estimates for matrix functions, SIAM J. Matrix Anal. Appl., 10 (1989), pp. 191--209, https://doi.org/10.1137/0610014.
[29]
R. Mathias, A chain rule for matrix functions and applications, SIAM J. Matrix Anal. Appl., 17 (1996), pp. 610--620, https://doi.org/10.1137/S0895479895283409.
[30]
Multiprecision Computing Toolbox, Advanpix, Tokyo, http://www.advanpix.com.
[31]
Y. Nakatsukasa and R. W. Freund, Computing fundamental matrix decompositions accurately via the matrix sign function in two iterations: The power of Zolotarev's functions, SIAM Rev., 58 (2016), pp. 461--493, https://doi.org/10.1137/140990334.
[32]
F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, eds., NIST Handbook of Mathematical Functions, Cambridge University Press, Cambridge, UK, 2010, http://dlmf.nist.gov.
[33]
M. S. Paterson and L. J. Stockmeyer, On the number of nonscalar multiplications necessary to evaluate polynomials, SIAM J. Comput., 2 (1973), pp. 60--66, https://doi.org/10.1137/0202007.
[34]
P. Penfield, Jr., Principal values and branch cuts in complex APL, SIGAPL APL Quote Quad, 12 (1981), pp. 248--256, https://doi.org/10.1145/390007.805368.
[35]
G. Pólya and G. Szegö, Problems and Theorems in Analysis II. Theory of Functions. Zeros. Polynomials. Determinants. Number Theory. Geometry, Springer-Verlag, New York, 1998, https://doi.org/10.1007/978-3-642-61905-2; reprint of the 1976 edition.
[36]
K. Ruedenberg, Free-electron network model for conjugated systems. V. Energies and electron distributions in the FE MO model and in the LCAO MO model, J. Chem. Phys., 22 (1954), pp. 1878--1894, https://doi.org/10.1063/1.1739935.
[37]
S. M. Serbin, Rational approximations of trigonometric matrices with application to second-order systems of differential equations, Appl. Math. Comput., 5 (1979), pp. 75--92, https://doi.org/10.1016/0096-3003(79)90011-0.
[38]
V. Stotland, O. Schwartz, and S. Toledo, High-Performance Algorithms for Computing the Sign Function of Triangular Matrices, arXiv preprint, http://arxiv.org/abs/1607.06303 arXiv:1607.06303v1 [cs.NA], 2016.
[39]
T. G. Wright, Eigtool, 2002, http://www.comlab.ox.ac.uk/pseudospectra/eigtool.
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© 2016, Society for Industrial and Applied Mathematics.
This work is licensed under a Creative Commons Attribution 4.0 International License
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Society for Industrial and Applied Mathematics
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Publication History
Published: 01 January 2016
Author Tags
- matrix function
- inverse trigonometric functions
- inverse hyperbolic functions
- matrix inverse sine
- matrix inverse cosine
- matrix inverse hyperbolic sine
- matrix inverse hyperbolic cosine
- matrix exponential
- matrix logarithm
- matrix sign function
- rational approximation
- Padé approximation
- MATLAB
- GNU Octave
- Fréchet derivative
- condition number
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