# csdms-contrib/slepian_alpha

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 function [w,x,N]=gausslegendrecof(l,method,intv) % [w,x,N]=gausslegendrecof(l,method,intv) % % Weights w and abscissas x for Gauss-Legendre integration of a % polynomial function f(x) of degree l. The approximation w(:)'*f(x(:)) % to the integral W(x)f(x), W(x)=1 will be exact on [-1 1] and need only % be rescaled to the desired interval. If intv is specified, % w(:)'*f(x(:)) will do, as we do the scaling for you. % % INPUT: % % l Degree of polynomial in the integrand % method 'jacobi' stable algorithm (Legendre roots only) [default] % 'cofrec' algorithm for l<40 (roots and coefficients) % intv If integration interval specified, returns scaled [w,x] % This may be an Nx2 matrix if you want multiple intervals % % OUTPUT: % % w Weights (vector, or matrix in case of more than one TH) % x Abscissas (roots of Legendre polynomial of degree N) % N Number of points in the integration (length(x)) % % SEE ALSO: GAUSSLEGENDRE % % The abcissa's are the roots of a Legendre polynomials defined on the % same interval. In particular then, this routine can be used to % integrate (products) of Legendre functions themselves, which is useful % in the analysis of spherical harmonics. This returns the N-point % Gauss-Legendre integration points and weights, which is accurate for % polynomials up to degree 2*N-1. Note that these things are symmetric so % for the N-point integration there are only ceil(N/2) unique % weights. The weights are positive, symmetric, and should sum to 2. % As N nodes integrate a polynomial of degree 2N-1 exactly, the number of % nodes returned for a polynomial degree l is ceil((l+1)/2). % % EXAMPLE: (from Numerical Recipes [qgaus, p. 141, Chapter 4.5]) % % [w,x]=gausslegendrecof(19); % % Last modified by fjsimons-at-alum.mit.edu, 03/17/2009 defval('method','jacobi') defval('intv',[]) % Figure out what the N will need to be N=ceil((l+1)/2); switch method case 'jacobi' % Find roots of Legendre polynomial x=legendrecof(N,method); % Find weight values from the derivative of the Legendre polynomial % Calculate Legendre function at N-1; note Pl(1)=1, which is not % the normalization we want in order to make the inner product of the % spherical harmonic be 4 pi! Now, for m=0, the inner product is % 2/(2N+1). These belong to the standard, listed Legendre polynomials. Pl=rindeks(legendre(N-1,x,'sch'),1); Pdiff=-N*Pl(:)./(x(:).^2-1); case 'cofrec' % Find roots of Legendre polynomial of degree N in [-1 1] % These are the abscissa points [x,C1]=legendrecof(N,method); % Find the value of the derivative of the Legendre function at these % points, by involving the lower-degree polynomial as well [y,C2]=legendrecof(N-1,method); x=sort(x); % This is the equation for the derivative of the Legendre polynomial % as found in Numerical Recipes and other texts. Wolfram has this % equation under Legendre-Gauss Quadrature, not under Legendre Polynomials. % Pdiff=N*(x.*polyval(C1,x)-polyval(C2,x))./(x.^2-1); % But we are noting that we are evaluating this at the roots of C1 % so we can save one computation % Note we are using coefficients of Legendre Polynomials that integrate % to 2/(2L+1), i.e. the "classical ones" Pdiff=-N*polyval(C2,x)./(x.^2-1); end % Find weights % Numerical Recipes, Equation (4.5.16) w=2./(1-x.^2)./Pdiff.^2; % Scale to interval if ~isempty(intv) if length(intv(:))==2 a=intv(1); b=intv(2); % Rescale nodes x=a+(x+1)/2*(b-a); % Rescale weights w=w*(b-a)/2; else a=intv(:,1); b=intv(:,2); xg=x;wg=w; x=repmat(NaN,length(xg),length(a)); w=repmat(NaN,length(wg),length(a)); for index=1:length(a) x(:,index)=a(index)+(xg+1)/2*(b(index)-a(index)); w(:,index)=wg*(b(index)-a(index))/2; end end end