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# dexpm

## PURPOSE Fréchet derivative of the matrix exponential.

## SYNOPSIS function [D, fX] = dexpm(X, H)

## DESCRIPTION ``` Fréchet derivative of the matrix exponential.

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

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

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

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

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

## CROSS-REFERENCE INFORMATION This function calls:
• dfunm Fréchet derivative of matrix functions.
This function is called by:

## SOURCE CODE ```0001 function [D, fX] = dexpm(X, H)
0002 % Fréchet derivative of the matrix exponential.
0003 %
0004 % function [D, fX] = dexpm(X, H)
0005 %
0006 % Computes the directional derivative (the Fréchet derivative) of expm at X
0007 % along H (square matrices).
0008 %
0009 % Thus, D = lim_(t -> 0) (expm(X + tH) - expm(X)) / t.
0010 %
0011 % Note: the adjoint of dexpm(X, .) is dexpm(X', .), which is a fact often
0012 % useful to derive gradients of matrix functions involving expm(X).
0013 % (This is wrt the inner product inner = @(A, B) real(trace(A'*B))).
0014 %
0015 % The second output is fX = expm(X), though it may be less accurate.
0016 %