dlogm

 Fréchet derivative of the matrix logarithm.

 function [D, fX] = dlogm(X, H)

 Computes the directional derivative (the Fréchet derivative) of logm at X
 along H (square matrices).

 Thus, D = lim_(t -> 0) (logm(X + tH) - logm(X)) / t.

 The second output is fX = logm(X), though it may be less accurate.

 Note: the adjoint of dlogm(X, .) is dlogm(X', .), which is a fact often
 useful to derive gradients of matrix functions involving logm(X).
 (This is wrt the inner product inner = @(A, B) real(trace(A'*B))).
 
 See also: dfunm dexpm dsqrtm

0001 function [D, fX] = dlogm(X, H)
0002 % Fréchet derivative of the matrix logarithm.
0003 %
0004 % function [D, fX] = dlogm(X, H)
0005 %
0006 % Computes the directional derivative (the Fréchet derivative) of logm at X
0007 % along H (square matrices).
0008 %
0009 % Thus, D = lim_(t -> 0) (logm(X + tH) - logm(X)) / t.
0010 %
0011 % The second output is fX = logm(X), though it may be less accurate.
0012 %
0013 % Note: the adjoint of dlogm(X, .) is dlogm(X', .), which is a fact often
0014 % useful to derive gradients of matrix functions involving logm(X).
0015 % (This is wrt the inner product inner = @(A, B) real(trace(A'*B))).
0016 %
0017 % See also: dfunm dexpm dsqrtm
0018 
0019 % This file is part of Manopt: www.manopt.org.
0020 % Original author: Nicolas Boumal, July 3, 2015.
0021 % Contributors:
0022 % Change log:
0023 %
0024 %   June 14, 2019 (NB): now also outputs logm(X) as a by-product.
0025     
0026     [D, fX] = dfunm(@logm, X, H);
0027     
0028 end