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function [Q, R] = orthogonalize(M, x, A)
% Orthonormalizes a basis of tangent vectors in the Manopt framework.
% function [orthobasis, R] = orthogonalize(M, x, basis)
% M is a Manopt manifold structure obtained from a factory.
% x is a point on the manifold M.
% basis is a cell containing n linearly independent tangent vectors at x.
% orthobasis is a cell of same size as basis which contains an orthonormal
% basis for the same subspace as that spanned by basis. Orthonormality is
% assessed with respect to the metric on the tangent space to M at x.
% R is upper triangular of size n x n if basis has n vectors, such that:
% basis{k} = sum_j=1^k orthobasis{j} * R(j, k).
% That is: we compute a QR factorization of basis.
% The algorithm is a modified Gram-Schmidt. If elements in the input basis
% are close to being linearly dependent (ill conditioned), then consider
% orthogonalizing twice, or calling orthogonalizetwice directly.
% See also: orthogonalizetwice grammatrix tangentorthobasis
% This file is part of Manopt:
% Original author: Nicolas Boumal, April 28, 2016.
% Contributors:
% Change log:
% Oct. 5, 2017 (NB):
% Changed algorithm to a modified Gram-Schmidt and commented
% about the twice-is-enough trick. Compared to the previous
% version, this algorithm behaves much better if the input basis
% is ill conditioned.
assert(iscell(A), ...
'The input basis must be a cell containing tangent vectors at x');
n = numel(A);
R = zeros(n);
Q = cell(size(A));
for j = 1 : n
v = A{j};
for i = 1 : (j-1)
qi = Q{i};
R(i, j) = M.inner(x, qi, v);
v = M.lincomb(x, 1, v, -R(i, j), qi);
R(j, j) = M.norm(x, v);
Q{j} = M.lincomb(x, 1/R(j, j), v);
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