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M2S.m

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+%{
+Copyright © 2020 Alexey A. Shcherbakov. All rights reserved.
+
+This file is part of GratingFMM.
+
+GratingFMM is free software: you can redistribute it and/or modify
+it under the terms of the GNU General Public License as published by
+the Free Software Foundation, either version 2 of the License, or
+(at your option) any later version.
+
+GratingFMM is distributed in the hope that it will be useful,
+but WITHOUT ANY WARRANTY; without even the implied warranty of
+MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+GNU General Public License for more details.
+
+You should have received a copy of the GNU General Public License
+along with GratingFMM. If not, see <https://www.gnu.org/licenses/>.
+%}
+%% convert a matrix to S-matrix format
+function S = M2S(M)
+	no = size(M,1)/2;
+	S = zeros(no,no,2,2);
+	S(:,:,1,1) = M(1:no,1:no);
+	S(:,:,1,2) = M(1:no,(no+1):(2*no));
+	S(:,:,2,1) = M((no+1):(2*no),1:no);
+	S(:,:,2,2) = M((no+1):(2*no),(no+1):(2*no));
+end

mul_SM.m

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+%{
+Copyright © 2020 Alexey A. Shcherbakov. All rights reserved.
+
+This file is part of GratingFMM.
+
+GratingFMM is free software: you can redistribute it and/or modify
+it under the terms of the GNU General Public License as published by
+the Free Software Foundation, either version 2 of the License, or
+(at your option) any later version.
+
+GratingFMM is distributed in the hope that it will be useful,
+but WITHOUT ANY WARRANTY; without even the implied warranty of
+MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+GNU General Public License for more details.
+
+You should have received a copy of the GNU General Public License
+along with GratingFMM. If not, see <https://www.gnu.org/licenses/>.
+%}
+%% description:
+% multiplication of two S-matrices
+%% input:
+% SM1, SM2: S-matrices of size (n,n,2,2) or (n,2,2)
+% the multiplication order is meaningful!
+%% output:
+% SM: S-matrix of size (n,n,2,2) or (n,2,2)
+%% implementation:
+function [SM] = mul_SM(SM1, SM2)
+	if (numel(size(SM1)) == 4) && (numel(size(SM2)) == 4) % both S-matrices are full
+		if ~isequal(size(SM1),size(SM2))
+			error("mul_SM: different size input");
+		end
+		n = size(SM1,1);
+		SM = zeros(n,n,2,2);
+
+		TM = -SM2(:,:,1,1)*SM1(:,:,2,2);
+		TM(1:n+1:end) = TM(1:n+1:end) + 1;
+		TM = SM1(:,:,1,2)/TM;
+		SM(:,:,1,2) = TM*SM2(:,:,1,2);
+		SM(:,:,1,1) = SM1(:,:,1,1) + TM*SM2(:,:,1,1)*SM1(:,:,2,1);
+
+		TM = -SM1(:,:,2,2)*SM2(:,:,1,1);
+		TM(1:n+1:end) = TM(1:n+1:end) + 1;
+		TM = SM2(:,:,2,1)/TM;
+		SM(:,:,2,1) = TM*SM1(:,:,2,1);
+		SM(:,:,2,2) = SM2(:,:,2,2) + TM*SM1(:,:,2,2)*SM2(:,:,1,2);
+
+	elseif (numel(size(SM1)) == 3) && (numel(size(SM2)) == 4) % first S-matrix is diagonal
+		if size(SM1,1) ~= size(SM2,1)
+			error("mul_SM: different size input");
+		end
+		n = size(SM1,1);
+		SM = zeros(n,n,2,2);
+
+		TM = -SM2(:,:,1,1).*transpose(SM1(:,2,2));
+		TM(1:n+1:end) = TM(1:n+1:end) + 1;
+		TM = diag(SM1(:,1,2))/TM;
+		SM(:,:,1,2) = TM*SM2(:,:,1,2);
+		SM(:,:,1,1) = diag(SM1(:,1,1)) + TM*(SM2(:,:,1,1).*transpose(SM1(:,2,1)));
+
+		TM = -SM1(:,2,2).*SM2(:,:,1,1);
+		TM(1:n+1:end) = TM(1:n+1:end) + 1;
+		TM = SM2(:,:,2,1)/TM;
+		SM(:,:,2,1) = TM.*transpose(SM1(:,2,1));
+		SM(:,:,2,2) = SM2(:,:,2,2) + TM*(SM1(:,2,2).*SM2(:,:,1,2));
+
+	elseif (numel(size(SM1)) == 4) && (numel(size(SM2)) == 3) % second S-matrix is diagonal
+		if size(SM1,2) ~= size(SM2,1)
+			error("mul_SM_SMD: different size input");
+		end
+		n = size(SM1,2);
+		SM = zeros(n,n,2,2);
+
+		TM = -SM2(:,1,1).*SM1(:,:,2,2);
+		TM(1:n+1:end) = TM(1:n+1:end) + 1;
+		TM = SM1(:,:,1,2)/TM;
+		SM(:,:,1,2) = TM.*transpose(SM2(:,1,2));
+		SM(:,:,1,1) = SM1(:,:,1,1) + TM*(SM2(:,1,1).*SM1(:,:,2,1));
+
+		TM = -SM1(:,:,2,2).*transpose(SM2(:,1,1));
+		TM(1:n+1:end) = TM(1:n+1:end) + 1;
+		TM = diag(SM2(:,2,1))/TM;
+		SM(:,:,2,1) = TM*SM1(:,:,2,1);
+		SM(:,:,2,2) = diag(SM2(:,2,2)) + TM*(SM1(:,:,2,2).*transpose(SM2(:,1,2)));
+
+	elseif (numel(size(SM1)) == 3) && (numel(size(SM2)) == 3) % both S-matrices are diagonal
+		if ~isequal(size(SM1),size(SM2))
+			error("mul_SM: different size input");
+		end
+		n = size(SM1,1);
+		SM = zeros(n,2,2);
+
+		TM = SM1(:,1,2)./(1 - SM1(:,2,2).*SM2(:,1,1));
+		SM(:,1,2) = SM2(:,1,2).*TM;
+		SM(:,1,1) = SM1(:,1,1) + TM.*SM2(:,1,1).*SM1(:,2,1);
+
+		TM = SM2(:,2,1)./(1 - SM2(:,1,1).*SM1(:,2,2));
+		SM(:,2,1) = SM1(:,2,1).*TM;
+		SM(:,2,2) = SM2(:,2,2) + SM2(:,1,2).*SM1(:,2,2).*TM;
+
+	else
+		error("mul_SM: incorrect input size");
+
+	end
+end
+%
+% end of mul_SM
+%

toeplitz2.m

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+%{
+Copyright © 2020 Alexey A. Shcherbakov. All rights reserved.
+
+This file is part of GratingFMM.
+
+GratingFMM is free software: you can redistribute it and/or modify
+it under the terms of the GNU General Public License as published by
+the Free Software Foundation, either version 2 of the License, or
+(at your option) any later version.
+
+GratingFMM is distributed in the hope that it will be useful,
+but WITHOUT ANY WARRANTY; without even the implied warranty of
+MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+GNU General Public License for more details.
+
+You should have received a copy of the GNU General Public License
+along with GratingFMM. If not, see <https://www.gnu.org/licenses/>.
+%}
+%% description:
+% return a 2D Toeplitz matrix (2D analog of the matlab function 'toeplitz')
+%% input:
+% M: vector of size (2*ny-1)*(2*nx-1)
+% nx, ny: block dimensions of Toeplitz matrix
+%% output:
+% T: 2D block-Toeplitz matrix of size nx*ny
+%% implementation:
+function [T] = toeplitz2(M, nx, ny)
+	CT = cell(1,2*nx-1);
+	ind = toeplitz(linspace(nx,2*nx-1,nx),flip(linspace(1,nx,nx)));		
+	for i = 1:2*nx-1
+		CT{1,i} = toeplitz(M(ny:2*ny-1,i),M(ny:-1:1,i));
+	end
+	T = cell2mat(CT(ind));
+end
+%
+% end of function toeplitz2
+%