Matrix Functions and Nonlinear Matrix Equations

A Multiprecision Derivative-Free Schur-Parlett Algorithm for Computing Matrix Functions (with Xiaobo Liu), SIAM J. Matrix Anal. Appl., 42(3):1401–1422, 2021. Blog post.
A Catalogue of Software for Matrix Functions. Version 3.0 (with Edvin Hopkins), MIMS EPrint 2020.7, Manchester Institute for Mathematical Sciences, The University of Manchester, UK, March 2020. 24 pp.
An Arbitrary Precision Scaling and Squaring Algorithm for the Matrix Exponential (with M. Fasi). SIAM J. Matrix Anal. Appl., 40(4):1233-1256, 2019.
Computing the Wave-Kernel Matrix Functions (with P. Nadukandi). SIAM J. Sci. Comput., 40(6):A4060-A4082, 2018. MATLAB software.
Multiprecision Algorithms for Computing The Matrix Logarithm (with Massimiliano Fasi). SIAM J. Matrix Anal. Appl., 39(1):472-491, 2018.
Computing the Action of Trigonometric and Hyperbolic Matrix Functions (with Peter Kandolf), SIAM J. Sci. Comput., 39(2):A613-A627, 2017.
An Algorithm to Compute the Polar Decomposition of a 3×3 Matrix (with Vanni Noferini) Numer. Algorithms, 73(2):349-369, 2016. MATLAB codes on GitHub.
Matching Exponential-Based and Resolvent-Based Centrality Measures (with Mary Aprahamian and Des Higham), Journal of Complex Networks, 4(2):157-176, 2016.
A Catalogue of Software for Matrix Functions. Version 2.0 (with Edvin Deadman), MIMS EPrint 2016.3, January 2016; updated March 2016.
Matrix Inverse Trigonometric and Inverse Hyperbolic Functions: Theory and Algorithms (with Mary Aprahamian), SIAM J. Matrix Anal. Appl., 37(4):1453-1477, 2016. MATLAB codes on GitHub.
Ranking the Importance of Nuclear Reactions for Activation and Transmutation Events (with Wayne Arter, J. Guy Morgan and Samuel Relton) Nuclear Science and Engineering, 184(4):561-574, 2016.
Testing Matrix Function Algorithms Using Identities (with Edvin Deadman), ACM Trans. Math. Soft., 42(1):4:1-4:15, 2016. MATLAB codes on GitHub.
New Algorithms for Computing the Matrix Sine and Cosine Separately or Simultaneously (with Awad Al-Mohy and Samuel Relton), SIAM J. Sci. Comp., 37(1): A456-A487, 2015. MATLAB codes on Github and MATLAB Central File Exchange.
An Algorithm for the Matrix Lambert W Function (with Massimiliano Fasi and Bruno Iannazzo), SIAM J. Matrix Anal. Appl., 36(2): 669-685, 2015. MATLAB code.
Matrix Functions: Computation In Encyclopedia of Applied and Computational Mathematics, B. Engquist, editor, Springer-Verlag, Berlin, 2015, pages 863-865.
Estimating the Condition Number of the Fréchet Derivative of a Matrix Function (with Samuel Relton), SIAM J. Sci. Comp., 36(6):C617-C634, 2014.
Higher Order Fréchet Derivatives of Matrix Functions and the Level-2 Condition Number (with Samuel Relton), SIAM J. Matrix Anal. Appl., 35(3):1019-1037, 2014.
Functions of Matrices, In Handbook of Linear Algebra, Leslie Hogben, (ed), Second edition, Chapman and Hall/CRC, Boca Raton, FL, USA, 17.1-17.5:, 2014..
The Matrix Unwinding Function, with an Application to Computing the Matrix Exponential (with Mary Aprahamian), SIAM J. Matrix Anal. Appl. 35 (1): 88-109, 2014. MATLAB codes.
An Improved Schur-Padé Algorithm for Fractional Powers of a Matrix and their Fréchet Derivatives (with Lijing Lin), SIAM J. Matrix Anal. Appl. 34 (3): 1341-1360, 2013. MATLAB codes at MATLAB Central File Exchange;
Computing the Fréchet Derivative of the Matrix Logarithm and Estimating the Condition Number (with Awad Al-Mohy and Samuel Relton), SIAM J. Sci. Comp. 35 (4), C394-C410, 2013. MATLAB codes at MATLAB Central File Exchange;
Blocked Schur Algorithms for Computing the Matrix Square Root (with Edvin Deadman and Rui Ralha), In P. Manninen and P. Ãster, eds, Applied Parallel and Scientific Computing: 11th International Conference, PARA 2012, Helsinki, Finland, Lecture Notes in Computer Science, Vol 7782, pp, 171-182, Springer, 2013.
Backward Stability of Iterations for Computing the Polar Decomposition (with Yuji Nakatsukasa), SIAM J. Matrix Anal. Appl. 33(2):460-479, 2012.
Improved Inverse Scaling and Squaring Algorithms for the Matrix Logarithm (with Awad Al-Mohy), SIAM J. Sci. Comp. 34 (4): C152-C169, 2012. MATLAB codes. MATLAB codes at MATLAB Central File Exchange.
A Schur-Padé Algorithm for Fractional Powers of a Matrix (with Lijing Lin), SIAM J. Matrix Anal. Appl. 32(3):1056-1078, 2011. MATLAB codes. MATLAB codes at MATLAB Central File Exchange.
Computing the Action of the Matrix Exponential, with an Application to Exponential Integrators (with Awad Al-Mohy), SIAM J. Sci. Comp. 33 (2): 488-511, 2011. MATLAB codes. MATLAB codes at MATLAB Central File Exchange.
On pth Roots of Stochastic Matrices (with Lijing Lin), Linear Algebra Appl. 435(3):448-463, 2011. Published version.
Computing Matrix Functions (with Awad Al-Mohy), Acta Numerica 19: 159-208, 2010.
The Canonical Generalized Polar Decomposition (with Christian Mehl and Françoise Tisseur), SIAM J. Matrix Anal. Appl. 31(4): 2163-2180, 2010.
The Complex Step Approximation to the Fréchet Derivative of a Matrix Function (with Awad Al-Mohy), Numer. Alg. 53(1): 133-148, 2010. Published version.
The Scaling and Squaring Method for the Matrix Exponential Revisited, SIAM Review 51(4): 747-764, 2009.
A New Scaling and Squaring Algorithm for the Matrix Exponential (with Awad Al-Mohy), SIAM J. Matrix Anal. Appl. 31(3): 970-989, 2009. MATLAB codes.
Computing the Fréchet Derivative of the Matrix Exponential, with an Application to Condition Number Estimation (with Awad Al-Mohy), SIAM J. Matrix Anal. Appl. 30(4): 1639-1657, 2009.
Computing $A^\alpha$ , $\log(A)$ and Related Matrix Functions by Contour Integrals (with Nicholas Hale and Lloyd N. Trefethen), SIAM J. Numer. Anal. 46 (5):2505-2523, 2008.
Iterative Solution of a Nonsymmetric Algebraic Riccati Equation (with Chun-Hua Guo), SIAM J. Matrix Anal. Appl., 29(2):396-412, 2007.
The solution of S exp(S) = A is not always the Lambert W function of A (with R. M. Corless, H. Ding, and D. J. Jeffrey), ISSAC ’07: Proceedings of the 2007 International Symposium on Symbolic and Algebraic Computation, pages 116-121. ACM Press, New York, 2007.
Functions of Matrices, in Handbook of Linear Algebra, edited by L. Hogben, Chapman and Hall/CRC, Boca Raton, FL, USA, pages 11.1-11.13, 2006.
A Schur-Newton Method for the Matrix p’th Root and its Inverse (with Chun-Hua Guo), SIAM J. Matrix Anal. Appl., 28(3):788-804, 2006.
Efficient Algorithms for the Matrix Cosine and Sine (with Gareth Hargreaves), Numerical Algorithms, 40(4):383-400, 2005.
Functions Preserving Matrix Groups and Iterations for the Matrix Square Root (with D. Steven Mackey, Niloufer Mackey and Françoise Tisseur), SIAM J. Matrix Anal. Appl., 26(3):849-877, 2005.
The Scaling and Squaring Method for the Matrix Exponential Revisited, SIAM J. Matrix Anal. Appl., 26(4):1179-1193, 2005.
Algorithms for the Matrix p’th Root (with Dario A. Bini and Beatrice Meini), Numerical Algorithms, 39(4):349-378, 2005.
Computing f(A)b for Matrix Functions f (with Philip Davies). In A. Borici, A. Frommer, B. Joo, A. Kennedy, and B. Pendleton, editors, QCD and Numerical Analysis III, volume 47 of Lecture Notes in Computational Science and Engineering, pages 15-24. Springer-Verlag, Berlin, 2005.
Computing the Polar Decomposition and the Matrix Sign Decomposition in Matrix Groups (with Steven Mackey, Niloufer Mackey and Françoise Tisseur), SIAM J. Matrix Anal. Appl., 25(4):1178-1192, 2004.
J-Orthogonal Matrices: Properties and Generation, SIAM Review, 45(3), 504-519, 2003.
Computing the Matrix Cosine (with Matthew I. Smith), Numerical Algorithms, 34:13-26, 2003.
A Schur-Parlett Algorithm for Computing Matrix Functions (with Philip Davies), SIAM J. Matrix Anal. Appl., 25(2):464-485, 2003.
Solving a Quadratic Matrix Equation by Newton’s Method with Exact Line Searches (with Hyun-Min Kim), SIAM J. Matrix Anal. Appl., 23(2): 303-316, 2001.
Evaluating Padé Approximants of the Matrix Logarithm, SIAM J. Matrix Anal. Appl. 22(4): 1126-1135, 2001.
Approximating the Logarithm of a Matrix to Specified Accuracy with Sheung Hun Cheng, Charles S. Kenney and Alan J. Laub), SIAM J. Matrix Anal. Appl. 22(4): 1112-1125, 2001.
Numerical Analysis of a Quadratic Matrix Equation (with Hyun-Min Kim), IMA J. Numer. Anal., 20(4): 499-519, 2000.
Return to the Middle Ages: A Half-Angle Iteration for the Logarithm of a Unitary Matrix, (with Sheung Hun Cheng, Charles S. Kenney, Alan J. Laub), Proceedings of the Fourteenth International Symposium of Mathematical Theory of Networks and Systems, Perpignan, France, 2000.
A New sqrtm for MATLAB, NA Report 336, January 1999.
Stable Iterations for the Matrix Square Root, Numerical Algorithms, 15(2): 227-242, 1997.
A Parallel Algorithm for Computing the Polar Decomposition (with Pythagoras Papadimitriou), Parallel Computing, 20(8):1161-1173, Aug. 1994.
The Matrix Sign Decomposition and its Relation to the Polar Decomposition, Linear Algebra and Appl., 212/213:3-20, 1994.
Fast polar decomposition of an arbitrary matrix (with R. S. Schreiber), SIAM J. Sci. Stat. Comput., 11(4):648-655, July 1990.
Newton’s Method for the Matrix Square Root, Math. Comp., 46(174):537-549, 1986.
Computing Real Square Roots of a Real Matrix, Linear Algebra and Appl., 88/89:405-430, 1987.
Computing the Polar Decomposition—with Applications, SIAM J. Sci. Stat. Comput., 7(4):1160-1174, 1986.

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