133
Views
0
CrossRef citations to date
0
Altmetric
Research Article

A technique for improving the computation of functions of triangular matrices

&
Pages 2449-2465 | Received 10 Oct 2021, Accepted 06 Apr 2022, Published online: 29 Apr 2022

References

  • A. Al-Mohy and N. Higham, A new scaling and squaring algorithm for the matrix exponential, SIAM J. Matrix Anal. Appl. 31 (2009), pp. 970–989.
  • A. Al-Mohy and N. Higham, Improved inverse scaling and squaring for the matrix logarithm, SIAM J. Sci. Comput. 34 (2012), pp. C153–C169.
  • A. Al-Mohy, N. Higham, and S. Relton, New algorithms for computing the matrix sine and cosine separately or simultaneously, SIAM J. Sci. Comput. 37 (2015), pp. A456–A487.
  • M. Aprahamian and N. Higham, The matrix unwinding function, with an application to computing the matrix exponential, SIAM J. Matrix Anal. Appl. 35 (2014), pp. 88–109.
  • M. Aprahamian and N. Higham, Matrix inverse trigonometric and inverse hyperbolic functions: Theory and algorithms, SIAM J. Matrix Anal. Appl. 37 (2016), pp. 1453–1477.
  • M. Arioli and M. Benzi, A finite element method for quantum graphs, IMA J. Numer. Anal. 38 (2018), pp. 1119–1163.
  • M. Benzi, E. Estrada, and C. Klymko, Ranking hubs and authorities using matrix functions, Linear Algebra Appl. 438 (2013), pp. 2447–2474.
  • R. Bhatia, Matrix Analysis, Springer-Verlag, New York, 1997
  • A. Björck and S. Hammarling, A Schur method for the square root of a matrix, Linear Algebra Appl. 52–53 (1983), pp. 127–140.
  • A. Bloch and P. Crouch, Optimal control and geodesic flows, Syst. Control Lett. 28 (1996), pp. 65–72.
  • J. Cardoso and A. Sadeghi, Computation of matrix gamma function, BIT Numer. Math. 59 (2019), pp. 343–370.
  • W. Culver, An analytic theory of modeling for a class of minimal-energy control systems (disturbance-free case), SIAM J. Control 2 (1965), pp. 267–294.
  • P. Davies and N. Higham, A Schur–Parlett algorithm for computing matrix functions, SIAM J. Matrix Anal. Appl. 25 (2003), pp. 464–485.
  • M. Fasi, N. Higham, and B. Iannazzo, An algorithm for the matrix Lambert W function, SIAM J. Matrix Anal. Appl. 36 (2015), pp. 669–685.
  • G. Franklin, J. Powell, and M. Workman, Digital Control of Dynamic Systems, 3rd ed., Addison-Wesley, Reading, MA, 1998
  • R. Garrappa and M. Popolizio, Computing the matrix Mittag-Leffler function with applications to fractional calculus, J. Sci. Comput. 77 (2018), pp. 129–153.
  • W. Harris and J. Cardoso, The exponential-mean-log-transference as a possible representation of the optical character of an average eye, Ophthalmic Physiol. Opt. 26 (2006), pp. 380–383.
  • T.F. Havel, I. Najfeld, and J. Yang, Matrix decompositions of two-dimensional nuclear magnetic resonance spectra, Proc. Natl. Acad. Sci. 91 (1994), pp. 7962–7966.
  • N. Higham, The scaling and squaring method for the matrix exponential revisited, SIAM J. Matrix Anal. Appl. 26 (2005), pp. 1179–1193.
  • N. Higham, Functions of Matrices: Theory and Computation, SIAM, Philadelphia, PA, 2008
  • N. Higham and L. Lin, An improved Schur–Padé algorithm for fractional powers of a matrix and their Fréchet derivatives, SIAM J. Matrix Anal. Appl. 34 (2013), pp. 1341–1360.
  • R. Horn and C. Johnson, Topics in Matrix Analysis, Paperback Edition, Cambridge University Press, Cambridge, 1994
  • R. Horn and C. Johnson, Matrix Analysis, 2nd ed., Cambridge University Press, Cambridge , 2013
  • B. Iannazzo and C. Manasse, A Schur logarithmic algorithm for fractional powers of matrices, SIAM J. Matrix Anal. Appl. 34 (2013), pp. 794–813.
  • C. Kenney and A. Laub, Condition estimates for matrix functions, SIAM J. Matrix Anal. Appl. 10 (1989), pp. 191–209.
  • R. Mathias, The spectral norm of a nonnegative matrix, Linear Algebra Appl. 139 (1990), pp. 269–284.
  • C. Moler and C. Van Loan, Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later, SIAM Rev. 45 (2003), pp. 3–49.
  • R. Murray, Z. Li, and S. Sastry, A Mathematical Introduction to Robotic Manipulation, CRC Press, Boca Raton, FL, 1994
  • J. Rossignac and A. Vinacua, Steady affine motions and morphs, ACM Trans. Graph. 30 (2011), pp. 1–16.
  • S. Serbin, Rational approximations of trigonometric matrices with application to second-order systems of differential equations, Appl. Math. Comput. 5 (1979), pp. 75–92.
  • R. Sidje and W. Stewart, A numerical study of large sparse matrix exponentials arising in Markov chains, Comput. Stat. Data Anal. 29 (1999), pp. 345–368.
  • M. Smith, A Schur algorithm for computing matrix pth roots, SIAM J. Matrix Anal. Appl. 24 (2003), pp. 971–989.
  • E. Verriest, The matrix logarithm and the continuization of a discrete process, in Proceedings of the 1991 American Control Conference.June 26–28, Boston, MA, United States, (1991), pp. 184–189 .

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.