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The complex step approximation to the Fréchet derivative of a matrix function. (English) Zbl 1188.65054

The authors show that the Fréchet derivative of a matrix function \(f\) at \(A\) in the direction \(E\), where \(A\) and \(E\) are real matrices, can be approximatecd by \(\text{Im\,}f(A+ ih\,E)/h\) for some suitably small \(h\). This approach is known in the scalar case but has not been applied previously to matrix functions. Its main advantage is that the stepsize \(h\) is allowed to be chosen as small as necessary to obtain an accurate approximation, without cancellation errors contaminating the result in floating point arithmetic. The implementation of the approximation is ease, assuming the availability of complex arithmetic.

MSC:

65F30 Other matrix algorithms (MSC2010)
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
65F35 Numerical computation of matrix norms, conditioning, scaling
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[1] Al-Mohy, A.H., Higham, N.J.: Computing the Fréchet derivative of the matrix exponential, with an application to condition number estimation. SIAM J. Matrix Anal. Appl. 30(4), 1639–1657 (2009) · Zbl 1180.65049 · doi:10.1137/080716426
[2] Al-Mohy, A.H., Higham, N.J.: A new scaling and squaring algorithm for the matrix exponential. MIMS EPrint 2009.9, Manchester Institute for Mathematical Sciences, The University of Manchester, UK. Revised April 2009 (2009) (To appear in SIAM J. Matrix Anal. Appl.) · Zbl 1194.15021
[3] Cox, M.G., Harris, P.M.: Numerical analysis for algorithm design in metrology. Software Support for Metrology Best Practice Guide No. 11, National Physical Laboratory, Teddington (2004)
[4] Davies, P.I., Higham, N.J.: A Schur–Parlett algorithm for computing matrix functions. SIAM J. Matrix Anal. Appl. 25(2), 464–485 (2003) · Zbl 1052.65031 · doi:10.1137/S0895479802410815
[5] Demmel, J.W.: On condition numbers and the distance to the nearest ill-posed problem. Numer. Math. 51, 251–289 (1987) · Zbl 0597.65036 · doi:10.1007/BF01400115
[6] Guo, C.H., Higham, N.J.: A Schur–Newton method for the matrix pth root and its inverse. SIAM J. Matrix Anal. Appl. 28(3), 788–804 (2006) · Zbl 1128.65030 · doi:10.1137/050643374
[7] Higham, N.J.: The Matrix Function Toolbox. http://www.ma.man.ac.uk/\(\sim\)higham/mftoolbox
[8] Higham, N.J.: Exploiting fast matrix multiplication within the level 3 BLAS. ACM Trans. Math. Softw. 16(4), 352–368 (1990) · Zbl 0900.65118 · doi:10.1145/98267.98290
[9] Higham, N.J.: Stability of a method for multiplying complex matrices with three real matrix multiplications. SIAM J. Matrix Anal. Appl. 13(3), 681–687 (1992) · Zbl 0777.65027 · doi:10.1137/0613043
[10] Higham, N.J.: Accuracy and Stability of Numerical Algorithms, 2nd edn. Society for Industrial and Applied Mathematics, Philadelphia (2002) · Zbl 1011.65010
[11] Higham, N.J.: The scaling and squaring method for the matrix exponential revisited. SIAM J. Matrix Anal. Appl. 26(4), 1179–1193 (2005) · Zbl 1081.65037 · doi:10.1137/04061101X
[12] Higham, N.J.: Functions of Matrices: Theory and Computation. Society for Industrial and Applied Mathematics, Philadelphia (2008) · Zbl 1167.15001
[13] Higham, N.J., Tisseur, F.: A block algorithm for matrix 1-norm estimation, with an application to 1-norm pseudospectra. SIAM J. Matrix Anal. Appl. 21(4), 1185–1201 (2000) · Zbl 0959.65061 · doi:10.1137/S0895479899356080
[14] Kågström, B., Ling, P., Van Loan, C.F.: GEMM-based level 3 BLAS: high performance model implementations and performance evaluation benchmark. ACM Trans. Math. Softw. 24(3), 268–302 (1998) · Zbl 0930.65047 · doi:10.1145/292395.292412
[15] Kelley, C.T.: Solving Nonlinear Equations with Newton’s Method. Society for Industrial and Applied Mathematics, Philadelphia (2003) · Zbl 1031.65069
[16] Koikari, S.: An error analysis of the modified scaling and squaring method. Comput. Math. Appl. 53, 1293–1305 (2007) · Zbl 1134.65053 · doi:10.1016/j.camwa.2006.04.032
[17] Koikari, S.: Algorithm 894: On a block Schur–Parlett algorithm for -functions based on the sep-inverse estimate. ACM Trans. Math. Software 36(2), Article 12 (2009) · Zbl 1364.65104
[18] Lai, K.L., Crassidis, J.L.: Extensions of the first and second complex-step derivative approximations. J. Comput. Appl. Math. 219, 276–293 (2008) · Zbl 1145.65016 · doi:10.1016/j.cam.2007.07.026
[19] Lyness, J.N.: Numerical algorithms based on the theory of complex variable. In: Proceedings of the 1967 22nd National Conference, pp. 125–133. ACM, New York (1967)
[20] Lyness, J.N., Moler, C.B.: Numerical differentiation of analytic functions. SIAM J. Numer. Anal. 4(2), 202–210 (1967) · Zbl 0155.48003 · doi:10.1137/0704019
[21] Martins, J.R.R.A., Sturdza, P., Alonso, J.J.: The connection between the complex-step derivative approximation and algorithmic differentiation. AIAA paper AIAA-2001-0921 (2001)
[22] Martins, J.R.R.A., Sturdza, P., Alonso, J.J.: The complex-step derivative approximation. ACM Trans. Math. Softw. 29(3), 245–262 (2003) · Zbl 1072.65027 · doi:10.1145/838250.838251
[23] Najfeld, I., Havel, T.F.: Derivatives of the matrix exponential and their computation. Adv. Appl. Math. 16, 321–375 (1995) · Zbl 0839.15004 · doi:10.1006/aama.1995.1017
[24] Shampine, L.F.: Accurate numerical derivatives in MATLAB. ACM Trans. Math. Softw. 33(4), Article 26, 17 pages (2007) · Zbl 1365.65057
[25] Skaflestad, B., Wright, W.M.: The scaling and modified squaring method for matrix functions related to the exponential. Appl. Numer. Math. 59, 783–799 (2009) · Zbl 1160.65034 · doi:10.1016/j.apnum.2008.03.035
[26] Squire, W., Trapp, G.: Using complex variables to estimate derivatives of real functions. SIAM Rev. 40(1), 110–112 (1998) · Zbl 0913.65014 · doi:10.1137/S003614459631241X
[27] Sun, J.: Perturbation analysis of the matrix sign function. Linear Algebra Appl. 250, 177–206 (1997) · Zbl 0871.15027 · doi:10.1016/0024-3795(95)00522-6
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