Skip to main content
×
Home
    • Aa
    • Aa
  • Get access
    Check if you have access via personal or institutional login
  • Cited by 21
  • Cited by
    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Bader, Philipp Blanes, Sergio Casas, Fernando and Ponsoda, Enrique 2016. Efficient numerical integration of <mml:math altimg="si20.gif" display="inline" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd" xmlns:sa="http://www.elsevier.com/xml/common/struct-aff/dtd"><mml:mi>N</mml:mi></mml:math>th-order non-autonomous linear differential equations. Journal of Computational and Applied Mathematics, Vol. 291, p. 380.


    Botchev, Mikhail A. 2016. Krylov subspace exponential time domain solution of Maxwell’s equations in photonic crystal modeling. Journal of Computational and Applied Mathematics, Vol. 293, p. 20.


    Campos, Carmen and Roman, Jose E. 2016. Parallel iterative refinement in polynomial eigenvalue problems. Numerical Linear Algebra with Applications, Vol. 23, Issue. 4, p. 730.


    Kurbatov, V.G. and Kurbatova, I.V. 2016. Computation of a function of a matrix with close eigenvalues by means of the Newton interpolating polynomial. Linear and Multilinear Algebra, Vol. 64, Issue. 2, p. 111.


    Schild, Axel Agostini, Federica and Gross, E. K. U. 2016. Electronic Flux Density beyond the Born–Oppenheimer Approximation. The Journal of Physical Chemistry A, Vol. 120, Issue. 19, p. 3316.


    Bader, Philipp Blanes, Sergio and Seydaoğlu, Muaz 2015. The Scaling, Splitting, and Squaring Method for the Exponential of Perturbed Matrices. SIAM Journal on Matrix Analysis and Applications, Vol. 36, Issue. 2, p. 594.


    Sastre, J. Ibán͂ez, J. Defez, E. and Ruiz, P. 2015. New Scaling-Squaring Taylor Algorithms for Computing the Matrix Exponential. SIAM Journal on Scientific Computing, Vol. 37, Issue. 1, p. A439.


    Dutra, Dimas Abreu Teixeira, Bruno Otávio Soares and Aguirre, Luis Antonio 2014. Maximum a posteriori state path estimation: Discretization limits and their interpretation. Automatica, Vol. 50, Issue. 5, p. 1360.


    Frommer, Andreas Güttel, Stefan and Schweitzer, Marcel 2014. Efficient and Stable Arnoldi Restarts for Matrix Functions Based on Quadrature. SIAM Journal on Matrix Analysis and Applications, Vol. 35, Issue. 2, p. 661.


    Gratton, Serge and Titley-Peloquin, David 2014. Stochastic Conditioning of Matrix Functions. SIAM/ASA Journal on Uncertainty Quantification, Vol. 2, Issue. 1, p. 763.


    Seddiqi, Hadayat and Humble, Travis S. 2014. Adiabatic quantum optimization for associative memory recall. Frontiers in Physics, Vol. 2,


    Zacur, Ernesto Bossa, Matias and Olmos, Salvador 2014. Left-Invariant Riemannian Geodesics on Spatial Transformation Groups. SIAM Journal on Imaging Sciences, Vol. 7, Issue. 3, p. 1503.


    Al-Mohy, Awad H. Higham, Nicholas J. and Relton, Samuel D. 2013. Computing the Fréchet Derivative of the Matrix Logarithm and Estimating the Condition Number. SIAM Journal on Scientific Computing, Vol. 35, Issue. 4, p. C394.


    Botchev, Mike A. Grimm, Volker and Hochbruck, Marlis 2013. Residual, Restarting, and Richardson Iteration for the Matrix Exponential. SIAM Journal on Scientific Computing, Vol. 35, Issue. 3, p. A1376.


    Higham, Nicholas 2013. Handbook of Linear Algebra, Second Edition.


    Al-Mohy, Awad H. and Higham, Nicholas J. 2012. Improved Inverse Scaling and Squaring Algorithms for the Matrix Logarithm. SIAM Journal on Scientific Computing, Vol. 34, Issue. 4, p. C153.


    Grimm, Volker 2012. Resolvent Krylov subspace approximation to operator functions. BIT Numerical Mathematics, Vol. 52, Issue. 3, p. 639.


    Ledermann, Daniel and Alexander, Carol 2012. Further properties of random orthogonal matrix simulation. Mathematics and Computers in Simulation, Vol. 83, p. 56.


    Tal-Ezer, Hillel Kosloff, Ronnie and Schaefer, Ido 2012. New, Highly Accurate Propagator for the Linear and Nonlinear Schrödinger Equation. Journal of Scientific Computing, Vol. 53, Issue. 1, p. 211.


    Al-Mohy, Awad H. and Higham, Nicholas J. 2011. Computing the Action of the Matrix Exponential, with an Application to Exponential Integrators. SIAM Journal on Scientific Computing, Vol. 33, Issue. 2, p. 488.


    ×

Computing matrix functions

  • Nicholas J. Higham (a1) and Awad H. Al-Mohy (a1)
  • DOI: http://dx.doi.org/10.1017/S0962492910000036
  • Published online: 01 May 2010
Abstract

The need to evaluate a function f(A) ∈ ℂn×n of a matrix A ∈ ℂn×n arises in a wide and growing number of applications, ranging from the numerical solution of differential equations to measures of the complexity of networks. We give a survey of numerical methods for evaluating matrix functions, along with a brief treatment of the underlying theory and a description of two recent applications. The survey is organized by classes of methods, which are broadly those based on similarity transformations, those employing approximation by polynomial or rational functions, and matrix iterations. Computation of the Fréchet derivative, which is important for condition number estimation, is also treated, along with the problem of computing f(A)b without computing f(A). A summary of available software completes the survey.

Copyright
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

M. Afanasjew , M. Eiermann , O. G. Ernst and S. Güttel (2008), ‘Implementation of a restarted Krylov subspace method for the evaluation of matrix functions’, Linear Algebra Appl. 429, 22932314.

A. H. Al-Mohy and N. J. Higham (2009 a), ‘Computing the Fréchet derivative of the matrix exponential, with an application to condition number estimation’, SIAM J. Matrix Anal. Appl. 30, 16391657.

A. H. Al-Mohy and N. J. Higham (2009 b), ‘A new scaling and squaring algorithm for the matrix exponential’, SIAM J. Matrix Anal. Appl. 31, 970989.

A. H. Al-Mohy and N. J. Higham (2010), ‘The complex step approximation to the Fréchet derivative of a matrix function’, Numer. Algorithms 53, 133148.

A. C. Antoulas (2005), Approximation of Large-Scale Dynamical Systems, SIAM, Philadelphia, PA, USA.

M. Arioli and D. Loghin (2009), ‘Discrete interpolation norms with applications’, SIAM J. Numer. Anal. 47, 29242951.

C. A. Bavely and G. W. Stewart (1979), ‘An algorithm for computing reducing subspaces by block diagonalization’, SIAM J. Numer. Anal. 16, 359367.

H. Berland , B. Skaflestad and W. Wright (2007), ‘EXPINT: A MATLAB package for exponential integrators’, ACM Trans. Math. Software 33, #4.

D. A. Bini , N. J. Higham and B. Meini (2005), ‘Algorithms for the matrix pth root’, Numer. Algorithms 39, 349378.

R. J. Bradford , R. M. Corless , J. H. Davenport , D. J. Jeffrey and S. M. Watt (2002), ‘Reasoning about the elementary functions of complex analysis’, Annals of Mathematics and Artificial Intelligence 36, 303318.

C. Brezinski and J. Van Iseghem (1995), A taste of Padé approximation. In Acta Numerica, Vol. 4, Cambridge University Press, pp. 53103.

A. Cayley (1858), ‘A memoir on the theory of matrices’, Philos. Trans. Roy. Soc. London 148, 1737.

T. Charitos , P. R. de Waal and L. C. van der Gaag (2008), ‘Computing short-interval transition matrices of a discrete-time Markov chain from partially observed data’, Statistics in Medicine 27, 905921.

S. H. Cheng , N. J. Higham , C. S. Kenney and A. J. Laub (2001), ‘Approximating the logarithm of a matrix to specified accuracy’, SIAM J. Matrix Anal. Appl. 22, 11121125.

J. J. Crofts and D. J. Higham (2009), ‘A weighted communicability measure applied to complex brain networks’, J. Roy. Soc. Interface 6, 411414.

P. I. Davies and N. J. Higham (2003), ‘A Schur-Parlett algorithm for computing matrix functions’, SIAM J. Matrix Anal. Appl. 25, 464485.

P. I. Davies and N. J. Higham (2005), Computing f(A)b for matrix functions f. In QCD and Numerical Analysis III (A. Boriçi , A. Frommer , B. Joó , A. Kennedy and B. Pendleton , eds), Vol. 47 of Lecture Notes in Computational Science and Engineering, Springer, Berlin, pp. 1524.

C. Davis (1973), ‘Explicit functional calculus’, Linear Algebra Appl. 6, 193199.

L. Dieci and A. Papini (2000), ‘Padé approximation for the exponential of a block triangular matrix’, Linear Algebra Appl. 308, 183202.

L. Dieci , B. Morini and A. Papini (1996), ‘Computational techniques for real logarithms of matrices’, SIAM J. Matrix Anal. Appl. 17, 570593.

E. Estrada and N. Hatano (2008), ‘Communicability in complex networks’, Phys. Review E 77, 036111.

E. Estrada and J. A. Rodríguez-Velázquez (2005 a), ‘Spectral measures of bipartivity in complex networks’, Phys. Review E 72, 046105.

E. Estrada and J. A. Rodríguez-Velázquez (2005 b), ‘Subgraph centrality in complex networks’, Phys. Review E 71, 056103.

E. Estrada , D. J. Higham and N. Hatano (2009), ‘Communicability betweenness in complex networks’, Physica A 388, 764774.

S. Fiori (2008), ‘Leap-frog-type learning algorithms over the Lie group of unitary matrices’, Neurocomputing 71, 22242244.

R. A. Frazer , W. J. Duncan and A. R. Collar (1938), Elementary Matrices and Some Applications to Dynamics and Differential Equations, Cambridge University Press, Cambridge, UK. 1963 printing.

A. Frommer and V. Simoncini (2008 b), ‘Stopping criteria for rational matrix functions of Hermitian and symmetric matrices’, SIAM J. Sci. Comput. 30, 13871412.

H. H. Goldstine (1977), A History of Numerical Analysis from the 16th through the 19th Century, Springer, New York.

V. Grimm and M. Hochbruck (2008), ‘Rational approximation to trigonometric operator’, BIT 48, 215229.

C.-H. Guo (2009), ‘On Newton's method and Halley's method for the principal pth root of a matrix’, Linear Algebra Appl. 432, 19051922.

C.-H. Guo and N. J. Higham (2006), ‘A Schur-Newton method for the matrix pth root and its inverse’, SIAM J. Matrix Anal. Appl. 28, 788804.

N. Hale , N. J. Higham and L. N. Trefethen (2008), ‘Computing Aα, log(A) and related matrix functions by contour integrals’, SIAM J. Numer. Anal. 46, 25052523.

G. I. Hargreaves and N. J. Higham (2005), ‘Efficient algorithms for the matrix cosine and sine’, Numer. Algorithms 40, 383400.

N. J. Higham (1986 a), ‘Computing the polar decomposition: With applications’, SIAM J. Sci. Statist. Comput. 7, 11601174.

N. J. Higham (1986 b), ‘Newton‘s method for the matrix square root’, Math. Comp. 46, 537549.

N. J. Higham (1997), ‘Stable iterations for the matrix square root’, Numer. Algorithms 15, 227242.

N. J. Higham (2001), ‘Evaluating Padé approximants of the matrix logarithm’, SIAM J. Matrix Anal. Appl. 22, 11261135.

N. J. Higham (2002), Accuracy and Stability of Numerical Algorithms, second edn, SIAM, Philadelphia, PA, USA.

N. J. Higham (2005), ‘The scaling and squaring method for the matrix exponential revisited’, SIAM J. Matrix Anal. Appl. 26, 11791193.

N. J. Higham (2008), Functions of Matrices: Theory and Computation, SIAM, Philadelphia, PA, USA.

N. J. Higham (2009), ‘The scaling and squaring method for the matrix exponential revisited’, SIAM Rev. 51, 747764.

N. J. Higham and M. I. Smith (2003), ‘Computing the matrix cosine’, Numer. Algorithms 34, 1326.

N. J. Higham and F. Tisseur (2000), ‘A block algorithm for matrix 1-norm estimation, with an application to 1-norm pseudospectra’, SIAM J. Matrix Anal. Appl. 21, 11851201.

N. J. Higham , D. S. Mackey , N. Mackey and F. Tisseur (2005), ‘Functions preserving matrix groups and iterations for the matrix square root’, SIAM J. Matrix Anal. Appl. 26, 849877.

M. Hochbruck and A. Ostermann (2010), Exponential integrators. In Acta Numerica, Vol. 19, Cambridge University Press, pp. 209286.

R. A. Horn and C. R. Johnson (1991), Topics in Matrix Analysis, Cambridge University Press, Cambridge, UK.

B. Iannazzo (2006), ‘On the Newton method for the matrix Pth root’, SIAM J. Matrix Anal. Appl. 28, 503523.

R. B. Israel , J. S. Rosenthal and J. Z. Wei (2001), ‘Finding generators for Markov chains via empirical transition matrices, with applications to credit ratings’, Mathematical Finance 11, 245265.

R. A. Jarrow , D. Lando and S. M. Turnbull (1997), ‘A Markov model for the term structure of credit risk spreads’, Rev. Financial Stud. 10, 481523.

C. S. Kenney and A. J. Laub (1989 a), ‘Condition estimates for matrix functions’, SIAM J. Matrix Anal. Appl. 10, 191209.

C. S. Kenney and A. J. Laub (1989 b), ‘Padé error estimates for the logarithm of a matrix’, Interna t. J. Control 50, 707730.

C. S. Kenney and A. J. Laub (1991 a), ‘Polar decomposition and matrix sign function condition estimates’, SIAM J. Sci. Statist. Comput. 12, 488504.

C. S. Kenney and A. J. Laub (1991 b), ‘Rational iterative methods for the matrix sign function’, SIAM J. Matrix Anal. Appl. 12, 273291.

C. S. Kenney and A. J. Laub (1998), ‘A Schur-Fréchet algorithm for computing the logarithm and exponential of a matrix’, SIAM J. Matrix Anal. Appl. 19, 640663.

S. Koikari (2009), ‘Algorithm 894: On a block Schur-Parlett algorithm for φ-functions based on the sep-inverse estimate’, ACM Trans. Math. Software 36, #12.

P. Laasonen (1958), ‘On the iterative solution of the matrix equation AX2-I = 0’, Math. Tables Aids Comp. 12, 109116.

B. Laszkiewicz and K. Zietak (2009), ‘A Padé family of iterations for the matrix sector function and the matrix pth root’, Numer. Linear Algebra Appl. 16, 951970.

P.-F. Lavallée , A. Malyshev and M. Sadkane (1997), Spectral portrait of matrices by block diagonalization. In Numerical Analysis and its Applications (L. Vulkov , J. Waśniewski and P. Yalamov , eds), Vol. 1196 of Lecture Notes in Computer Science, Springer, Berlin, pp. 266273.

J. D. Lawson (1967), ‘Generalized Runge-Kutta processes for stable systems with large Lipschitz constants’, SIAM J. Numer. Anal. 4, 372380.

J. R. R. A. Martins , P. Sturdza and J. J. Alonso (2003), ‘The complex-step derivative approximation’, ACM Trans. Math. Software 29, 245262.

R. Mathias (1996), ‘A chain rule for matrix functions and applications’, SIAM J. Matrix Anal. Appl. 17, 610620.

C. B. Moler and C. F. Van Loan (1978), ‘Nineteen dubious ways to compute the exponential of a matrix’, SIAM Rev. 20, 801836.

C. B. Moler and C. F. Van Loan (2003), ‘Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later’, SIAM Rev. 45, 349.

B. N. Parlett (1976), ‘A recurrence among the elements of functions of triangular matrices’, Linear Algebra Appl. 14, 117121.

M. S. Paterson and L. J. Stockmeyer (1973), ‘On the number of nonscalar multiplications necessary to evaluate polynomials’, SIAM J. Comput. 2, 6066.

H.-O. Peitgen , H. Jürgens and D. Saupe (1992), Fractals for the Classroom, Part Two: Complex Systems and Mandelbrot Set, Springer, New York.

G. M. Phillips (2000), Two Millennia of Mathematics: From Archimedes to Gauss, Springer, New York.

M. Popolizio and V. Simoncini (2008), ‘Acceleration techniques for approximating the matrix exponential operator’, SIAM J. Matrix Anal. Appl. 30, 657683.

P. J. Psarrakos (2002), ‘On the mth roots of a complex matrix’, Electron. J. Linear Algebra 9, 3241.

P. Pulay (1966), ‘An iterative method for the determination of the square root of a positive definite matrix’, Z. Angew. Math. Mech. 46, 151.

R. F. Rinehart (1955), ‘The equivalence of definitions of a matric function’, Amer. Math. Monthly 62, 395414.

J. D. Roberts (1980), ‘Linear model reduction and solution of the algebraic Riccati equation by use of the sign function’, Internat. J. Control 32, 677687. First issued as report CUED/B-Control/TR13, Department of Engineering, University of Cambridge, 1971.

Y. Saad (1992), ‘Analysis of some Krylov subspace approximations to the matrix exponential operator’, SIAM J. Numer. Anal. 29, 209228.

Y. Saad (2003), Iterative Methods for Sparse Linear Systems, second edn, SIAM, Philadelphia, PA, USA.

S. M. Serbin and S. A. Blalock (1980), ‘An algorithm for computing the matrix cosine’, SIAM J. Sci. Statist. Comput. 1, 198204.

R. B. Sidje (1998), ‘Expokit: A software package for computing matrix exponentials’, ACM Trans. Math. Software 24, 130156.

M. I. Smith (2003), ‘A Schur algorithm for computing matrix pth roots’, SIAM J. Matrix Anal. Appl. 24, 971989.

W. Squire and G. Trapp (1998), ‘Using complex variables to estimate derivatives of real functions’, SIAM Rev. 40, 110112.

C. F. Van Loan (1978), ‘Computing integrals involving the matrix exponential’, IEEE Trans. Automat. Control AC-23, 395404.

C. F. Van Loan (1979), ‘A note on the evaluation of matrix polynomials’, IEEE Trans. Automat. Control AC-24, 320321.

R. S. Varga (2000), Matrix Iterative Analysis, second edn, Springer, Berlin.

R. C. Ward (1977), ‘Numerical computation of the matrix exponential with accuracy estimate’, SIAM J. Numer. Anal. 14, 600610.

F. V. Waugh and M. E. Abel (1967), ‘On fractional powers of a matrix’, J. Amer. Statist. Assoc. 62, 10181021.

D. Yuan and W. Kernan (2007), ‘Explicit solutions for exit-only radioactive decay chains’, J. Appl. Phys. 101, 094907 112.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Acta Numerica
  • ISSN: 0962-4929
  • EISSN: 1474-0508
  • URL: /core/journals/acta-numerica
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×