# csdms-contrib/slepian_alpha

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 function varargout=legendreint01(l,m,x0) % val=legendreint01(l,m,x0) % % Evaluates integrals of a Schmidt semi-normalized real Legendre % polynomial P_lm(x)dx between x0 to 1 for l=0 or 1, analytically. % % INPUT: % % l Angular degree of the polynomial, l>=0 % m Angular order of the polynomial, 0<=m<=l % x0 Lower bound(s) for the integral(s) % % OUTPUT: % % val Integral(s) of LEGENDRE(N,x,'sch') or LIBBRECHT(N,x,'sch') % % SEE ALSO: % % LIBBRECHT, PAUL, LEGENREPRODINT % % EXAMPLE: % % legendreint01('demo1') Compare analytic to numerical Legendre functions % legendreint01('demo2') Compare Gauss-Legendre integration to analytic % legendreint01('demo3') Evaluates Gauss-Legendre integration errors % % Last modified by plattner-at-princeton.edu, 05/17/2011 % Last modified by fjsimons-at-alum.mit.edu, 06/01/2011 % Calculates the integral of a single Legendre function between x0 and 1 if ~isstr(l) if l<0 || m<0 || m>l error('Bad choice of l and m'); end if m>1 error('Use PAUL or LEGENDREPRODINT instead.') end if l==0 % l=0 m=0, the integral of 1 val=(1-x0); elseif l==1 if m==0 % l=1 m=0, the integral of x val=(1-x0.^2)/2; else % l=1 m=1, the integral of sqrt(1-x.^2) % See Research Book 9 page 79 for these alternative expressions %val=( +acos(x0)/2-(x0.*sqrt(1-x0.^2))/2); %val= coscos([],1,1,[asin(x0(:)) repmat(pi/2,length(x0),1)])'; %val=-sinsin([],1,1,[acos(x0(:)) repmat(0,length(x0),1)])'; val=(pi/4-asin(x0)/2-(x0.*sqrt(1-x0.^2))/2); end end % Optional output varns={val}; varargout=varns(1:nargout); elseif strcmp(l,'demo1') x=linspace(-1,1,100); clf subplot(311) plot(x,ones(size(x)),'k') hold on plot(x,legendre(0,x,'sch'),'o') plot(x,libbrecht(0,x,'sch'),'+') subplot(312) plot(x,x,'k') hold on plot(x,rindeks(legendre(1,x,'sch'),1),'o') plot(x,rindeks(libbrecht(1,x,'sch'),1),'+') subplot(313) plot(x,sqrt(1-x.^2),'k') hold on plot(x,rindeks(legendre(1,x,'sch'),2),'o') plot(x,rindeks(libbrecht(1,x,'sch'),2),'+') elseif strcmp(l,'demo2') x=linspace(-1,1,100); ngl=5; integrand1=inline('ones(size(x))'); integrand2=inline('x'); integrand3=inline('sqrt(1-x.^2)'); [gl1,gl2,gl3]=deal(zeros(size(x))); for i=1:length(x) % Here by GL integration gl1(i)=gausslegendre([x(i) 1],integrand1,ngl); gl2(i)=gausslegendre([x(i) 1],integrand2,ngl); gl3(i)=gausslegendre([x(i) 1],integrand3,ngl); end % Now the code itself val1=legendreint01(0,0,x); val2=legendreint01(1,0,x); val3=legendreint01(1,1,x); % And another piece of code at a random point randi=ceil(rand*length(x)); opti={'automatic','dumb','gl','paul'}; % Note that the choice may be overridden inside LEGENDREPRODINT rando=ceil(rand*length(opti)); % Use various algorithms explicitly at random vol1=legendreprodint(0,0,0,0,x(randi),opti{rando}); vol2=legendreprodint(1,0,0,0,x(randi),opti{rando}); vol3=legendreprodint(1,1,0,0,x(randi),opti{rando}); clf subplot(311) plot(x,val1,'k') hold on plot(x,gl1,'b+') plot(x(randi),vol1,'rv','MarkerF','r','markers',6) subplot(312) plot(x,val2,'k') hold on plot(x,gl2,'b+') plot(x(randi),vol2,'rv','MarkerF','r','markers',6) subplot(313) plot(x,val3,'k') hold on plot(x,gl3,'b+') plot(x(randi),vol3,'rv','MarkerF','r','markers',6) elseif strcmp(l,'demo3') x=-1; ngl=linspace(1,1000,25); integrand1=inline('ones(size(x))'); integrand2=inline('x'); integrand3=inline('sqrt(1-x.^2)'); [gl1,gl2,gl3]=deal(zeros(size(x))); for i=1:length(ngl) % Here by GL integration gl1(i)=gausslegendre([x 1],integrand1,ngl(i)); gl2(i)=gausslegendre([x 1],integrand2,ngl(i)); gl3(i)=gausslegendre([x 1],integrand3,ngl(i)); end % Now the code itself val1=legendreint01(0,0,x); val2=legendreint01(1,0,x); val3=legendreint01(1,1,x); clf subplot(311) semilogy(ngl,abs(gl1-val1),'k') subplot(312) semilogy(ngl,abs(gl2-val2),'k') subplot(313) semilogy(ngl,abs(gl3-val3),'k') end