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dsqrtm

PURPOSE ^

Fréchet derivative of the matrix square root.

SYNOPSIS ^

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

DESCRIPTION ^

 Fréchet derivative of the matrix square root.

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

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

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

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

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

CROSS-REFERENCE INFORMATION ^

This function calls: This function is called by:

SOURCE CODE ^

0001 function [D, fX] = dsqrtm(X, H)
0002 % Fréchet derivative of the matrix square root.
0003 %
0004 % function [D, fX] = dsqrtm(X, H)
0005 %
0006 % Computes the directional derivative (the Fréchet derivative) of sqrtm at
0007 % X along H (square matrices).
0008 %
0009 % Thus, D = lim_(t -> 0) (sqrtm(X + tH) - sqrtm(X)) / t.
0010 %
0011 % The second output is fX = sqrtm(X), though it may be less accurate.
0012 %
0013 % Note: the adjoint of dsqrtm(X, .) is dsqrtm(X', .), which is a fact often
0014 % useful to derive gradients of matrix functions involving sqrtm(X).
0015 % (This is wrt the inner product inner = @(A, B) real(trace(A'*B))).
0016 %
0017 % See also: dfunm dlogm dexpm
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 sqrtm(X) as a by-product.
0025     
0026     % Note: following Higham, 'Functions of Matrices', 2008, page 58: this
0027     % could also be computed as fX = sqrtm(X), then solving the Sylvester
0028     % equation fX*D + D*fX = H, e.g. via , D = sylvester(fX, fX, H).
0029     % If X has special structure (e.g., if it is symmetric or Hermitian),
0030     % then this may be faster and more accurate. This should be tested
0031     % before considering a replacement.
0032 
0033     [D, fX] = dfunm(@sqrtm, X, H);
0034     
0035 end

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