csdms-contrib/slepian_alpha

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 function [G,l]=gaunt(l1,l2,l3,m1,m2,m3,meth,L,C0,S0,C3,S3) % [G,l]=GAUNT(l1,l2,l3,m1,m2,m3,meth,L,C0,S0,C3,S3) % % Gaunt coefficients: the l1m1th expansion coefficient of the product % Yl2m2 x Yl3m3 where Ylm is a Condon-Shortley complex spherical % harmonic. % % INPUT: % % l1,l2,l3 Top row of the Wigner 3j symbol [may be vector for 'table'] % m1,m2,m3 Bottom row of the Wigner 3j symbol [may be vector for 'table'] % meth 'recursive' Recursive method and Wigner symbols (WIGNER3JM) % 'gl' Gauss-Legendre integration % 'table' Table look-up prestored from recursive (THREEJ, ZEROJ) % 'guseinov' Binomial method of Guseinov (no sign) % L Bandwidth of the table vectors (only if they are supplied next) % C0,S0 The column/element vectors of the table for ZEROJ if available % C3,S3 The column/element vectors of the table for THREEJ if available % % OUTPUT: % % G The scalar Gaunt coefficient % (for scalar input: for 'guseinov', 'gl' and 'table') % A vector of Gaunt coefficients % (for scalar input: for 'recursive'; for vector inputs) % l The first degrees from 0 to the maximum allowed value % % EXAMPLE: % % gaunt('demo1') % Generate Table 1 [Sebilleau, 1996]; compare approaches % gaunt('demo2') % Generate Table 2 [Guseinov, 1995]; compare approaches % gaunt('demo3') % Compare the recursive method with the prestored tables % gaunt('demo4') % Compare the Gauss-Legendre method with the prestored tables % % Last modified by fjsimons-at-alum.mit.edu, 01/18/2007 % Evaluates the integral of the product of three complex fully % orthonormalized spherical harmonics, with the Condon-Shortley phase... % l1,m1 belong to the harmonic whose complex conjugate appears in the % integral... Thus this is the l1m1th expansion coefficient of the % product Yl2m2 x Yl3m3 in the complex spherical harmonic basis Yl1m1. % % Note that an alternative formulation contains Wigner 3j-symbols with a % bottom row, which must sum to one, of -m1 m2 m3 and that thus, % m3=m1-m2. Flipping all the order signs leaves the overall sign intact. % m1 regulates the sign by the factor (-1)^m1. % SHOULD BUILD IN RULES BEFORE YOU EVEN START % MUST CHECK! THREEJ and GAUNT DO NOT BEHAVE THE WAY I WANT THEM TO % REGARDING REPEATED VALUES OF A DEGREE BUT NOT THE CORRESPONDING ORDERS! % SHOULD REWRITE THIS TO BE NOTATIONALLY MORE UNIFORM if ~isstr(l1) defval('meth','table') defval('L',[]) defval('C0',[]) defval('S0',[]) defval('C3',[]) defval('S3',[]) %disp(sprintf('Using method %s',meth)) switch meth case 'guseinov' if length(l1)>1||length(l2)>1||length(l3)>1; error('Need scalar input'); end G=abs(guseinov(l1,l2,l3,m1,m2,m3)); Gp=abs(guseinov(l1,l2,l3,-m1,-m2,-m3)); if abs(G-Gp)>1e-10 warning('No stable result. Choose (n)either value.'); G=[G Gp]; end % Returns single value l=l1; case 'recursive' if length(l1)>1||length(l2)>1||length(l3)>1; error('Need scalar input'); end [wm,l]=wigner3jm(l1,l2,l3,-m1,m2,m3); % Note that WIGNER0J uses a different algorithm [w0,ll]=wigner3jm(l1,l2,l3,0,0,0); % Returns an array of all allowable values difer(l-ll) G=(-1)^m1*sqrt((2*l+1)*(2*l2+1)*(2*l3+1)/4/pi).*wm.*w0; case 'gl' if length(l1)>1||length(l2)>1||length(l3)>1; error('Need scalar input'); end % Construct function names first... remember the complex conjugate integrand=inline(sprintf(... ['(-1)^%i*xlm(%i,%i,acos(x)).*'... 'xlm(%i,%i,acos(x)).*'... 'xlm(%i,%i,acos(x))'],m1,... l1,-m1,l2,m2,l3,m3)); % And the complex exponential with zero sum of orders evaluates to % 2*pi, but to zero if the selection rules aren't applied... like so G=gausslegendre([-1 1],integrand,l1+l2+l3)*2*pi*~[-m1+m2+m3]; l=l1; case 'table' % Used to be THREEJ, now have ZEROJ w0=zeroj(l1,l2,l3,L,[],C0,S0); % disp('Done zero-bottom row') % The result may be a whole vector % Could probably load them all at the same time to save time % Put in condition that if they're all zero you don't have to anymore if all(m1==0) && all(m2==0) && all(m3==0) wm=w0; else wm=threej(l1,l2,l3,-m1,m2,m3,L,[],C3,S3); % disp('Done general-bottom row') end G=(-1).^m1(:)'.*sqrt((2*l1(:)'+1).*(2*l2(:)'+1).*(2*l3(:)'+1)/4/pi)... .*wm.*w0; l=l1; otherwise error('Specify valid method') end elseif strcmp(l1,'demo1') % For Table I in Sebilleau (1995) % Compare Guseinovs' approach with the Wigner 3j approach (no signs) difer(indeks(abs(gaunt(10,10,12,-9,3,-12,'recursive')),'end')-... indeks(gaunt(10,10,12,-9,3,-12,'guseinov'),'end')); difer(indeks(abs(gaunt(12,15,5,-2,3,-5,'recursive')),'end')-... indeks(gaunt(12,15,5,-2,3,-5,'guseinov'),'end')); difer(indeks(abs(gaunt(20,20,40,-1,-1,0,'recursive')),'end')-... indeks(gaunt(20,20,40,-1,-1,0,'guseinov'),'end')); difer(indeks(abs(gaunt(29,29,34,-10,-5,-5,'recursive')),'end')-... indeks(gaunt(29,29,34,-10,-5,-5,'guseinov'),'end')); elseif strcmp(l1,'demo2') % For Table I in Guseinov (1995) L1=[20 20 20 20 25 25 25 40 40 40 40 60 60 38 2 80 80 80 80]; L2=[15 15 17 9 35 35 35 37 37 21 5 58 58 58 58 77 77 77 77]; L3=[35 31 15 15 60 48 38 75 59 37 37 118 58 60 60 155 131 83 5]; M1=[-3 3 -3 -3 12 12 12 2 2 -2 -2 3 3 1 1 1 1 1 1]; M2=[2 -2 -5 -5 -17 -17 -17 -1 -1 -3 -3 2 1 -2 -2 -3 -3 -3 -3]; M3=M1-M2; clear comp for ind=1:length(L1) l1=L1(ind); l2=L2(ind); l3=L3(ind); m1=M1(ind); m2=M2(ind); m3=M3(ind); % In Condon-Shortley convention; recursively; with sign t0=cputime; G1=indeks(gaunt(l1,l2,l3,m1,m2,m3,'recursive'),'end'); tg1=cputime-t0; t0=cputime; % In Condon-Shortley convention; Gauss-Legendre; with sign G2=indeks(gaunt(l1,l2,l3,m1,m2,m3,'gl'),'end'); tg2=cputime-t0; t0=cputime; % In Guseinov's stupid phase convention; no sign G3=min(abs(gaunt(l1,l2,l3,m1,m2,m3,'guseinov'))); tg3=cputime-t0; if G3>1e3; G3=NaN; end comp(ind,:)=[G1 G2 G3]; cpu(ind,:)=[tg1 tg2 tg3]; end % Take the recursive calculation to be the reference... based on the % table it indeed appears to be the most accurate; % The first comparison should be down to the sign incro=max([L1 ; L2 ; L3]); [k,l]=sort(incro); figure(1); clf semilogy(k,abs(comp(l,1)-comp(l,2)),'-o') hold on % The second comparison is regardless of the sign semilogy(k,abs(abs(comp(l,1))-abs(comp(l,3))),'r-+') % And so is the third comparison semilogy(k,abs(abs(comp(l,2))-abs(comp(l,3))),'g-x') hold off ll=legend('Recursion vs Gauss-Legendre',... 'Recursion vs Guseinov',... 'Gauss-Legendre vs Guseinov'); tt=title('Accuracy of Gaunt coefficient determination'); grid on xl=xlabel('maximum degree'); yl=ylabel('difference'); incro=sum([L1 ; L2 ; L3]); [k,l]=sort(incro); figdisp([],1) figure(2); clf plot(k,cpu(l,1),'-o') hold on plot(k,cpu(l,2),'r-+') plot(k,cpu(l,3),'g-x') hold off ll=legend('Recursion','Gauss-Legendre','Guseinov'); tt=title('CPU times of Gaunt coefficient determination'); grid on xl=xlabel('maximum degree'); yl=ylabel('time'); figdisp([],2) elseif strcmp(l1,'demo3') difer(indeks(gaunt(10,10,12,-9,3,-12,'recursive'),'end')-... gaunt(10,10,12,-9,3,-12,'table')); difer(indeks(gaunt(12,15,5,-2,3,-5,'recursive'),'end')-... gaunt(12,15,5,-2,3,-5,'table')); L1=[20 20 20 20]; L2=[15 15 17 9]; L3=[13 31 15 15]; M1=[-3 3 -3 -3]; M2=[2 -2 -5 -5]; M3=M1-M2; for ind=1:length(L1) l1=L1(ind); l2=L2(ind); l3=L3(ind); m1=M1(ind); m2=M2(ind); m3=M3(ind); difer(indeks(gaunt(l1,l2,l2,m1,m2,m3,'recursive'),'end')-... gaunt(l1,l2,l2,m1,m2,m3,'table')); difer(gaunt(l1,l2,l2,m1,m2,m3,'gl')-... gaunt(l1,l2,l2,m1,m2,m3,'table')); end elseif strcmp(l1,'demo4') L1=[20 20 20 20]; L2=[15 15 17 9]; L3=[13 19 15 15]; M1=[-3 3 -3 -3]; M2=[2 -2 -5 -5]; M3=M1-M2; Gt=gaunt(L1,L2,L3,M1,M2,M3,'table'); for index=1:length(Gt) Gg(index)=gaunt(L1(index),L2(index),L3(index),... M1(index),M2(index),M3(index),'gl'); end difer(Gt-Gg) else error('Specify valid option') end