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function [Q, R] = orthogonalizetwice(M, x, A)
% Orthonormalizes a basis of tangent vectors twice for increased accuracy.
% function [orthobasis, R] = orthogonalizetwice(M, x, basis)
% See help for orthogonalize. This function calls that algorithm twice.
% This is useful if elements in the input basis are close to being linearly
% dependent (ill conditioned). See in code for details.
% See also: orthogonalize grammatrix tangentorthobasis
% This file is part of Manopt:
% Original author: Nicolas Boumal, Oct. 5, 2017.
% Contributors:
% Change log:
[Q1, R1] = orthogonalize(M, x, A);
[Q , R2] = orthogonalize(M, x, Q1);
R = R2*R1; % This is upper triangular since R1 and R2 are.