Home > manopt > tools > tangent2vec.m

tangent2vec

PURPOSE ^

Expands a tangent vector into an orthogonal basis in the Manopt framework

SYNOPSIS ^

function vec = tangent2vec(M, x, basis, u)

DESCRIPTION ^

 Expands a tangent vector into an orthogonal basis in the Manopt framework

 vec = tangent2vec(M, x, basis, u)

 The inverse operation is lincomb (see below).

 M is a Manopt manifold structure obtained from a factory.
 x is a point on the manifold M.
 basis is a cell containing n orthonormal tangent vectors at x, forming an
       orthonormal basis of the tangent space at x.
 u is a tangent vector at x

 vec is a column vector of length n which contains the coefficients of the
     expansion of u into the basis. Thus:

    vec(k) = <basis{k}, u>_x          <- vec = tangent2vec(M, x, basis, u)

    u = sum_{k=1}^n  vec(k)*basis{k}    <- u = lincomb(M, x, basis, vec)

 Note that tangent2vec is an isometry, that is, up to numerical round-off
 errors, with u and v two tangent vectors at x:

    M.inner(x, u, v)  ==  uu'*vv,

 where uu = tangent2vec(M, x, basis, u), vv = tangent2vec(M, x, basis, v).

 See also: lincomb tangentorthobasis orthogonalize grammatrix hessianmatrix

CROSS-REFERENCE INFORMATION ^

This function calls: This function is called by:

SOURCE CODE ^

0001 function vec = tangent2vec(M, x, basis, u)
0002 % Expands a tangent vector into an orthogonal basis in the Manopt framework
0003 %
0004 % vec = tangent2vec(M, x, basis, u)
0005 %
0006 % The inverse operation is lincomb (see below).
0007 %
0008 % M is a Manopt manifold structure obtained from a factory.
0009 % x is a point on the manifold M.
0010 % basis is a cell containing n orthonormal tangent vectors at x, forming an
0011 %       orthonormal basis of the tangent space at x.
0012 % u is a tangent vector at x
0013 %
0014 % vec is a column vector of length n which contains the coefficients of the
0015 %     expansion of u into the basis. Thus:
0016 %
0017 %    vec(k) = <basis{k}, u>_x          <- vec = tangent2vec(M, x, basis, u)
0018 %
0019 %    u = sum_{k=1}^n  vec(k)*basis{k}    <- u = lincomb(M, x, basis, vec)
0020 %
0021 % Note that tangent2vec is an isometry, that is, up to numerical round-off
0022 % errors, with u and v two tangent vectors at x:
0023 %
0024 %    M.inner(x, u, v)  ==  uu'*vv,
0025 %
0026 % where uu = tangent2vec(M, x, basis, u), vv = tangent2vec(M, x, basis, v).
0027 %
0028 % See also: lincomb tangentorthobasis orthogonalize grammatrix hessianmatrix
0029 
0030 % This file is part of Manopt: www.manopt.org.
0031 % Original author: Nicolas Boumal, Feb. 3, 2017.
0032 % Contributors:
0033 % Change log:
0034 
0035 
0036     n = numel(basis);
0037     
0038     vec = zeros(n, 1);
0039     
0040     for k = 1 : n
0041         
0042         vec(k) = M.inner(x, basis{k}, u);
0043         
0044     end
0045 
0046 end

Generated on Fri 08-Sep-2017 12:43:19 by m2html © 2005