# csdms-contrib/slepian_alpha

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 function [K11,K21,K22]=cdlkernel(x1,x2,L,m,spd) % [K11,K21,K22]=CDLKERNEL(x1,x2,L,m,spd) % % Calculates a Christoffel-Darboux kernel with Legendre functions % whose harmonics are normalized to 4pi (option 'fnr' elsewhere). % Kernels are usually calculated at Gauss-Legendre integration points, at % a mixture of GL points and a complete set, and at the complete set. % % INPUT: % % x1 Vector with first set of points % x2 Vector with second set of points % L Maximum degree of the expansion % m Single angular order (set to zero) % spd 1 Fast method using the Christoffel-Darboux identity % 2 Slow method involving all L+1 summations % % OUTPUT: % % K11 The first kernel at point set x1 (GL points) % \suml_{l=0}^L P_{l}(x1)P_{l}(x1) % K21 The interpolating kernel mixing x1 and x2 % \suml_{l=0}^L P_{l}(x2)P_{l}(x1) % K22 The full kernel at point set x2 (complete set) % \suml_{l=0}^L P_{l}(x2)P_{l}(x2) % % EXAMPLE: % % cdlkernel('demo1') % Compares the two methods % % Last modified by fjsimons-at-alum.mit.edu, June 3rd, 2004 if ~isstr(x1) defval('m',0') defval('spd',1) if m~=0 spd=2; end switch spd case 1 % Using the Christoffel-Darboux identity % First, get P'(L+1), P(L) and P(L+1) % Since we're dividing polynomials don't worry about sqrt(2l+1) [Pdp1,P1,Pp1]=legendrediff(L+1,x1,'sch'); [Pdp2,P2,Pp2]=legendrediff(L+1,x2,'sch'); % Then, also get P'(L) Pd1=legendrediff(L,x1,'sch'); Pd2=legendrediff(L,x2,'sch'); X1=x1(:)*ones(1,length(x1)); X2=x2(:)*ones(1,length(x2)); X21=x2(:)*ones(1,length(x1)); X12=x1*ones(1,length(x2)); % For the non-diagonal elements warning off K11=(L+1)*(Pp1(:)*P1(:)'-P1(:)*Pp1(:)')./(X1-X1'); K22=(L+1)*(Pp2(:)*P2(:)'-P2(:)*Pp2(:)')./(X2-X2'); K21=(L+1)*(Pp2(:)*P1(:)'-P2(:)*Pp1(:)')./(X21-X12'); warning on % For the diagonal elements K11(ondiag(K11))=(L+1)*(Pdp1.*P1-Pd1.*Pp1); K22(ondiag(K22))=(L+1)*(Pdp2.*P2-Pd2.*Pp2); if ~~sum(sum((X21-X12')==0)) error('Diagonal elements not properly accounted for'); end case 2 % Straightforward brute-force calculation % Initialize kernels K11=zeros(length(x1),length(x1)); K21=zeros(length(x2),length(x1)); K22=zeros(length(x2),length(x2)); for l=m:L % Make sure Plm(1)=sqrt(2l+1) so Ylm normalized to 4\pi Plm=(rindeks(legendre(l,x1(:)','sch')*sqrt(2*l+1),m+1)); Plmint=(rindeks(legendre(l,x2(:)','sch')*sqrt(2*l+1),m+1)); % Kernel at Gauss-Legendre points K11=K11+Plm(:)*Plm(:)'; % Kernel to go from Gauss-Legendre to full resolution K21=K21+Plmint(:)*Plm(:)'; % Kernel at full resolution K22=K22+Plmint(:)*Plmint(:)'; end end elseif strcmp(x1,'demo1') defval('th0',40) defval('m',0) defval('SN',5) defval('nth',720) nth0=ceil(th0/180*nth); TH=linspace(0,th0/180*pi,nth0); xint=cos(TH); th0=th0*pi/180; index=0; els=[0:50]; for L=els index=index+1; [w,xGL]=gausslegendrecof(2*L,[],[cos(th0) 1]); tic [KGL1,Kint1,K1]=cdlkernel(xGL,xint,L,m,1); tic1(index)=toc; tic [KGL2,Kint2,K2]=cdlkernel(xGL,xint,L,m,2); tic2(index)=toc; erro(index)=mean(abs(K1(:)-K2(:))); end ah(1)=subplot(121); p{1}=plot(2*els,erro,'+'); grid on; title('Accuracy') xlabel('Product degree'); ylabel('Difference') ah(2)=subplot(122); p{2}=plot(2*els,tic1,'bo',2*els,tic2,'kv'); title('Time Cost'); xlabel('Product degree') ; ylabel('Seconds') yll=ylim; ylim([0 yll(2)]) set(p{1},'MarkerF','r','MarkerE','r') set(p{2}(1),'MarkerF','b','MarkerE','b') set(p{2}(2),'MarkerF','r','MarkerE','r') set(ah(2),'YScale','Log') l=legend('Christoffel-Darboux formula','Complete summation',4); grid on longticks(ah) figdisp else error('Specify valid option') end