orthogonalize

 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

0001 function [Q, R] = orthogonalize(M, x, A)
0002 % Orthonormalizes a basis of tangent vectors in the Manopt framework.
0003 %
0004 % function [orthobasis, R] = orthogonalize(M, x, basis)
0005 %
0006 % M is a Manopt manifold structure obtained from a factory.
0007 % x is a point on the manifold M.
0008 % basis is a cell containing n linearly independent tangent vectors at x.
0009 %
0010 % orthobasis is a cell of same size as basis which contains an orthonormal
0011 % basis for the same subspace as that spanned by basis. Orthonormality is
0012 % assessed with respect to the metric on the tangent space to M at x.
0013 % R is upper triangular of size n x n if basis has n vectors, such that:
0014 %
0015 %   basis{k} = sum_j=1^k orthobasis{j} * R(j, k).
0016 %
0017 % That is: we compute a QR factorization of basis.
0018 %
0019 % The algorithm is a modified Gram-Schmidt. If elements in the input basis
0020 % are close to being linearly dependent (ill conditioned), then consider
0021 % orthogonalizing twice, or calling orthogonalizetwice directly.
0022 %
0023 % See also: orthogonalizetwice grammatrix tangentorthobasis
0024 
0025 % This file is part of Manopt: www.manopt.org.
0026 % Original author: Nicolas Boumal, April 28, 2016.
0027 % Contributors:
0028 % Change log:
0029 %
0030 %       Oct. 5, 2017 (NB):
0031 %           Changed algorithm to a modified Gram-Schmidt and commented
0032 %           about the twice-is-enough trick. Compared to the previous
0033 %           version, this algorithm behaves much better if the input basis
0034 %           is ill conditioned.
0035 
0036     assert(iscell(A), ...
0037          'The input basis must be a cell containing tangent vectors at x');
0038 
0039     n = numel(A);
0040     R = zeros(n);
0041     Q = cell(size(A));
0042     
0043     for j = 1 : n
0044         
0045         v = A{j};
0046         
0047         for i = 1 : (j-1)
0048            
0049             qi = Q{i};
0050             
0051             R(i, j) = M.inner(x, qi, v);
0052             
0053             v = M.lincomb(x, 1, v, -R(i, j), qi);
0054             
0055         end
0056         
0057         R(j, j) = M.norm(x, v);
0058         
0059         Q{j} = M.lincomb(x, 1/R(j, j), v);
0060         
0061     end
0062 
0063 end