function M = euclideanfactory(m, n)
% Returns a manifold struct to optimize over real matrices.
%
% function M = euclideanfactory(m)
% function M = euclideanfactory(m, n)
% function M = euclideanfactory([n1, n2, ...])
%
% Returns M, a structure describing the Euclidean space of real matrices,
% equipped with the standard Frobenius distance and associated trace inner
% product, as a manifold for Manopt.
%
% m and n in general can be vectors to handle multidimensional arrays.
% If either of m or n is a vector, they are concatenated as [m, n].
%
% Using this simple linear manifold, Manopt can be used to solve standard
% unconstrained optimization problems, for example in replacement of
% Matlab's fminunc.
%
% See also: euclideancomplexfactory

% This file is part of Manopt: www.manopt.org.
% Original author: Nicolas Boumal, Dec. 30, 2012.
% Contributors: Bamdev Mishra, May 4, 2015.
% Change log: 
%
%   July 5, 2013 (NB):
%       Added egred2rgrad, ehess2rhess, mat, vec, tangent.
%   May 4, 2015 (BM):
%       Added functionality to handle multidimensional arrays.
%   July 2, 2024 (NB):
%       Improved the name string. Made retr2 = exp available.


    % The size can be defined using both m and n, or simply with m.
    % If m is a scalar, then n is implicitly 1.
    if ~exist('n', 'var') || isempty(n)
        if isscalar(m)
            n = 1;
        else
            n = [];
        end
    end
    
    dimensions_vec = [m(:)', n(:)']; % We have a row vector.
    
    M.size = @() dimensions_vec;
    
    dimensions_string = [sprintf('%d', dimensions_vec(1)), ...
                         sprintf(' x %d', dimensions_vec(2:end))];
    M.name = @() sprintf('Euclidean space R^(%s)', dimensions_string);
    
    M.dim = @() prod(dimensions_vec);
    
    M.inner = @(x, d1, d2) d1(:).'*d2(:);
    
    M.norm = @(x, d) norm(d(:), 'fro');
    
    M.dist = @(x, y) norm(x(:) - y(:), 'fro');
    
    M.typicaldist = @() sqrt(prod(dimensions_vec));
    
    M.proj = @(x, d) d;
    
    M.egrad2rgrad = @(x, g) g;
    
    M.ehess2rhess = @(x, eg, eh, d) eh;
    
    M.tangent = M.proj;
    
    M.exp = @exp;
    function y = exp(x, d, t)
        if nargin == 3
            y = x + t*d;
        else
            y = x + d;
        end
    end
    
    M.retr = M.exp;
    M.retr2 = M.exp;
    
    M.log = @(x, y) y-x;

    M.hash = @(x) ['z' hashmd5(x(:))];
    
    M.rand = @() randn(dimensions_vec);
    
    M.randvec = @randvec;
    function u = randvec(x) %#ok<INUSD>
        u = randn(dimensions_vec);
        u = u / norm(u(:), 'fro');
    end
    
    M.lincomb = @matrixlincomb;
    
    M.zerovec = @(x) zeros(dimensions_vec);
    
    M.transp = @(x1, x2, d) d;
    M.isotransp = M.transp; % the transport is isometric
    
    M.pairmean = @(x1, x2) .5*(x1+x2);
    
    M.vec = @(x, u_mat) u_mat(:);
    M.mat = @(x, u_vec) reshape(u_vec, dimensions_vec);
    M.vecmatareisometries = @() true;
    M.lie_identity = @() zeros(dimensions_vec);

end
