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fftSO3_prod.m
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fftSO3_prod.m
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function [ FG, fg ] = fftSO3_prod( func1, func2, B, isreal )
if ~exist('isreal','var') || isempty(isreal)
isreal = false;
end
% grid over SO(3)
alpha = reshape(pi/B*(0:(2*B-1)),1,1,[]);
beta = reshape(pi/(4*B)*(2*(0:(2*B-1))+1),1,1,[]);
gamma = reshape(pi/B*(0:(2*B-1)),1,1,[]);
ca = cos(alpha);
sa = sin(alpha);
cb = cos(beta);
sb = sin(beta);
cg = cos(gamma);
sg = sin(gamma);
Ra = [ca,-sa,zeros(1,1,2*B);sa,ca,zeros(1,1,2*B);zeros(1,1,2*B),zeros(1,1,2*B),ones(1,1,2*B)];
Rb = [cb,zeros(1,1,2*B),sb;zeros(1,1,2*B),ones(1,1,2*B),zeros(1,1,2*B);-sb,zeros(1,1,2*B),cb];
Rg = [cg,-sg,zeros(1,1,2*B);sg,cg,zeros(1,1,2*B);zeros(1,1,2*B),zeros(1,1,2*B),ones(1,1,2*B)];
R = zeros(3,3,2*B,2*B,2*B);
for i = 1:2*B
for j = 1:2*B
for k = 1:2*B
R(:,:,i,j,k) = Ra(:,:,i)*Rb(:,:,j)*Rg(:,:,k);
end
end
end
% weights
w = zeros(1,2*B);
for j = 1:2*B
w(j) = 1/(4*B^3)*sin(beta(j))*sum(1./(2*(0:B-1)+1).*sin((2*(0:B-1)+1)*beta(j)));
end
% function values
f = zeros(2*B,2*B,2*B);
g = zeros(2*B,2*B,2*B);
for i = 1:2*B
for j = 1:2*B
for k = 1:2*B
f(i,j,k) = func1(R(:,:,i,j,k));
g(i,j,k) = func2(R(:,:,i,j,k));
end
end
end
fg = f.*g;
% Wigner_d
lmax = B-1;
d = zeros(2*lmax+1,2*lmax+1,lmax+1,2*B);
for j = 1:2*B
d(:,:,:,j) = Wigner_d(beta(j),lmax);
end
% CG coefficient
CG = cell(B,B);
for l1 = 0:lmax
for l2 = 0:lmax
CG{l1+1,l2+1} = clebsch_gordan(l1,l2);
end
end
% fft
F = zeros(2*lmax+1,2*lmax+1,lmax+1);
G = zeros(2*lmax+1,2*lmax+1,lmax+1);
FG = zeros(2*lmax+1,2*lmax+1,lmax+1);
if isreal
% real analysis
% Psi
Psi = zeros(2*lmax+1,2*lmax+1,lmax+1,2*B);
for k = 1:2*B
for l = 0:lmax
for m = -l:l
for n = -l:0
if m==0 && n==0
Psi(m+lmax+1,n+lmax+1,l+1,k) = d(lmax+1,lmax+1,l+1,k);
elseif m==0 || n==0
Psi(m+lmax+1,n+lmax+1,l+1,k) = (-1)^(m-n)*sqrt(2)...
*d(abs(m)+lmax+1,abs(n)+lmax+1,l+1,k);
else
Psi(m+lmax+1,n+lmax+1,l+1,k) =...
(-1)^(m-n)*d(abs(m)+lmax+1,abs(n)+lmax+1,l+1,k)+...
(-1)^m*sign(m)*d(abs(m)+lmax+1,-abs(n)+lmax+1,l+1,k);
end
end
end
if l>0
Psi(-l+lmax+1:l+lmax+1,1+lmax+1:l+lmax+1,l+1,k) =...
flip(Psi(-l+lmax+1:l+lmax+1,-l+lmax+1:-1+lmax+1,l+1,k),2);
end
end
end
% cg matrix
T = cell(2*lmax+1,1);
for l = 0:2*lmax
T{l+1} = zeros(2*l+1,2*l+1);
for m = -l:l
if m<0
T{l+1}(m+l+1,m+l+1) = 1i;
T{l+1}(m+l+1,-m+l+1) = -1i*(-1)^(-m);
elseif m==0
T{l+1}(l+1,l+1) = sqrt(2);
else
T{l+1}(m+l+1,m+l+1) = (-1)^(m);
T{l+1}(m+l+1,-m+l+1) = 1;
end
end
T{l+1} = T{l+1}/sqrt(2);
end
cg = cell(B,B);
for l1 = 0:lmax
for l2 = 0:lmax
cg{l1+1,l2+1} = kron(conj(T{l1+1}),conj(T{l2+1}))...
*CG{l1+1,l2+1}*blkdiag(T{abs(l1-l2)+1:l1+l2+1}).';
end
end
% fft for f
S12 = zeros(2*B,2*B,2*B);
for kk = 1:2*B
S12(kk,:,:) = fftshift(fft2(reshape(f(:,kk,:),2*B,2*B)));
end
for l = 0:lmax
for m = -l:l
indm = m+lmax+2;
for n = -l:l
indn = n+lmax+2;
ind_n = -n+lmax+2;
sin_mang = -imag(S12(:,indm,indn));
sin_ma_ng = -imag(S12(:,indm,ind_n));
cos_mang = real(S12(:,indm,indn));
cos_ma_ng = real(S12(:,indm,ind_n));
if (m>=0 && n>=0) || (m<0 && n<0)
F(m+lmax+1,n+lmax+1,l+1) = sum(w.'.*(...
-0.5*(cos_ma_ng-cos_mang).*reshape(Psi(-m+lmax+1,n+lmax+1,l+1,:),2*B,1)...
+0.5*(cos_ma_ng+cos_mang).*reshape(Psi(m+lmax+1,n+lmax+1,l+1,:),2*B,1)));
else
F(m+lmax+1,n+lmax+1,l+1) = sum(w.'.*(...
-0.5*(sin_mang+sin_ma_ng).*reshape(Psi(-m+lmax+1,n+lmax+1,l+1,:),2*B,1)...
+0.5*(sin_mang-sin_ma_ng).*reshape(Psi(m+lmax+1,n+lmax+1,l+1,:),2*B,1)));
end
end
end
end
% fft for g
S12 = zeros(2*B,2*B,2*B);
for kk = 1:2*B
S12(kk,:,:) = fftshift(fft2(reshape(g(:,kk,:),2*B,2*B)));
end
for l = 0:lmax
for m = -l:l
indm = m+lmax+2;
for n = -l:l
indn = n+lmax+2;
ind_n = -n+lmax+2;
sin_mang = -imag(S12(:,indm,indn));
sin_ma_ng = -imag(S12(:,indm,ind_n));
cos_mang = real(S12(:,indm,indn));
cos_ma_ng = real(S12(:,indm,ind_n));
if (m>=0 && n>=0) || (m<0 && n<0)
G(m+lmax+1,n+lmax+1,l+1) = sum(w.'.*(...
-0.5*(cos_ma_ng-cos_mang).*reshape(Psi(-m+lmax+1,n+lmax+1,l+1,:),2*B,1)...
+0.5*(cos_ma_ng+cos_mang).*reshape(Psi(m+lmax+1,n+lmax+1,l+1,:),2*B,1)));
else
G(m+lmax+1,n+lmax+1,l+1) = sum(w.'.*(...
-0.5*(sin_mang+sin_ma_ng).*reshape(Psi(-m+lmax+1,n+lmax+1,l+1,:),2*B,1)...
+0.5*(sin_mang-sin_ma_ng).*reshape(Psi(m+lmax+1,n+lmax+1,l+1,:),2*B,1)));
end
end
end
end
% fft for product
for l1 = 0:lmax
for l2 = 0:lmax
ind1 = -l1+lmax+1:l1+lmax+1;
ind2 = -l2+lmax+1:l2+lmax+1;
A = cg{l1+1,l2+1}.'*kron(F(ind1,ind1,l1+1),G(ind2,ind2,l2+1))*conj(cg{l1+1,l2+1});
A = A*(2*l1+1)*(2*l2+1);
for l = 0:lmax
if l>=abs(l1-l2) && l<=l1+l2
indA = l^2-(l1-l2)^2+1 : l^2-(l1-l2)^2+2*l+1;
indFG = -l+lmax+1:l+lmax+1;
FG(indFG,indFG,l+1) = FG(indFG,indFG,l+1)+A(indA,indA)/(2*l+1);
end
end
end
end
FG = real(FG);
else
% complex analysis
% fft for f
F1 = zeros(2*B,2*B,2*B);
for k = 1:2*B
F1(:,k,:) = fftn(f(:,k,:));
end
F1 = fftshift(fftshift(F1,1),3);
F1 = flip(flip(F1,1),3);
F = zeros(2*lmax+1,2*lmax+1,lmax+1);
for l = 0:lmax
for m = -l:l
for n = -l:l
F(m+lmax+1,n+lmax+1,l+1) = sum(w.*F1(m+lmax+1,:,n+lmax+1).*...
permute(d(m+lmax+1,n+lmax+1,l+1,:),[1,4,3,2]));
end
end
end
% fft for g
G1 = zeros(2*B,2*B,2*B);
for k = 1:2*B
G1(:,k,:) = fftn(g(:,k,:));
end
G1 = fftshift(fftshift(G1,1),3);
G1 = flip(flip(G1,1),3);
G = zeros(2*lmax+1,2*lmax+1,lmax+1);
for l = 0:lmax
for m = -l:l
for n = -l:l
G(m+lmax+1,n+lmax+1,l+1) = sum(w.*G1(m+lmax+1,:,n+lmax+1).*...
permute(d(m+lmax+1,n+lmax+1,l+1,:),[1,4,3,2]));
end
end
end
% fft for product
for l1 = 0:lmax
for l2 = 0:lmax
ind1 = -l1+lmax+1:l1+lmax+1;
ind2 = -l2+lmax+1:l2+lmax+1;
A = CG{l1+1,l2+1}.'*kron(F(ind1,ind1,l1+1),G(ind2,ind2,l2+1))*CG{l1+1,l2+1};
A = A*(2*l1+1)*(2*l2+1);
for l = 0:lmax
if l>=abs(l1-l2) && l<=l1+l2
indA = l^2-(l1-l2)^2+1 : l^2-(l1-l2)^2+2*l+1;
indFG = -l+lmax+1:l+lmax+1;
FG(indFG,indFG,l+1) = FG(indFG,indFG,l+1)+A(indA,indA)/(2*l+1);
end
end
end
end
end
end