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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.

See also: dfunm dlogm dsqrtm```

## 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 %
0017 % See also: dfunm dlogm 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 expm(X) as a by-product.
0025
0026     [D, fX] = dfunm(@expm, X, H);
0027
0028 end```

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