Solve Sylvester equation without input checks. function X = sylvester_nochecks(A, B, C) Solves the Sylvester equation A*X + X*B = C, where A is an m-by-m matrix, B is an n-by-n matrix, and X and C are two m-by-n matrices. This is a stripped-down version of Matlab's own sylvester function that bypasses any input checks. This is significantly faster for small m and n, which is often useful in Manopt. See also: sylvester lyapunov_symmetric

- fixedrankfactory_2factors Manifold of m-by-n matrices of rank k with balanced quotient geometry.
- rotationsfactory Returns a manifold structure to optimize over rotation matrices.
- stiefelfactory Returns a manifold structure to optimize over orthonormal matrices.

0001 function X = sylvester_nochecks(A, B, C) 0002 % Solve Sylvester equation without input checks. 0003 % 0004 % function X = sylvester_nochecks(A, B, C) 0005 % 0006 % Solves the Sylvester equation A*X + X*B = C, where A is an m-by-m matrix, 0007 % B is an n-by-n matrix, and X and C are two m-by-n matrices. 0008 % 0009 % This is a stripped-down version of Matlab's own sylvester function that 0010 % bypasses any input checks. This is significantly faster for small m and 0011 % n, which is often useful in Manopt. 0012 % 0013 % See also: sylvester lyapunov_symmetric 0014 0015 % This file is part of Manopt: www.manopt.org. 0016 % Original author: Nicolas Boumal, July 19, 2018 0017 % Contributors: This is a modification of Matlab's built-in sylvester. 0018 % Change log: 0019 % July 30, 2020 (NB): 0020 % Changed call from builtin('_sylvester_tri', ...) to 0021 % matlab.internal.math.sylvester_tri(...), which seems necessary for 0022 % more recent versions of Matlab. 0023 % Also had to remove the second output: the 'unique' flag. 0024 0025 flag = 'real'; 0026 if ~isreal(A) || ~isreal(B) || ~isreal(C) 0027 flag = 'complex'; 0028 end 0029 0030 [QA, TA] = schur(A, flag); 0031 [QB, TB] = schur(B, flag); 0032 0033 % Solve Sylvester Equation TA*Y + Y*TB = QA'*C*QB. 0034 Y = matlab.internal.math.sylvester_tri(TA, TB, QA'*C*QB, ... 0035 'I', 'I', 'notransp'); 0036 % Use this call instead for older versions of Matlab: 0037 % [Y, info] = builtin('_sylvester_tri', TA, TB, QA'*C*QB); 0038 0039 X = QA*Y*QB'; 0040 0041 end

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