analc_endsversion/AnalyticLC/LaplaceCoeffs.m

function Output = LaplaceCoeffs(varargin)
% LaplaceCoeff: get a Laplace coefficient (definition: Solar System
% Dynamics, eq. 6.67)

%Inputs: 'init' - calculate a table of the laplace coefficients and save it
%           in ~/Matlab/Data/LaplaceCoeffs.mat
%           'load' = return all the table and the row/col values in a
%           structure
%           alph,s,j = return the laplace coefficient for the given alph,s,j.
%           alph can be a vector
%           alph,s,j,D = return the D order derivative of the laplace
%           coefficient for the given alph, s, j.

% Yair Judkovsky, 6.4.2020

persistent jVec sVec DVec alphaVec bTable Nj Ns ND


%return a specific laplace coeff
if isnumeric(varargin{1})
    
    alph = varargin{1};
    s = varargin{2};
    j = abs(varargin{3}); %b_s_j = b_s__-j, Solar System Dynamics eq. 6.69
    
    %a recursive path of the function for the case there's a 4th input -
    %the laplace coeff derivative.
    if length(varargin)>3
        D = varargin{4}; %derivative order
    else
        D = 0;
    end          
              
    jInd = j+1;
    sInd = s+0.5;
    DInd = D+1;
    
    %if the requested derivative is too high, calculate by recursive method
    if DInd>ND
        %2nd and higher derivatives of the coeff, Solar System Dynamics eq. 6.71
        Output = s*(LaplaceCoeffs(alph,s+1,j-1,D-1)-2*alph.*LaplaceCoeffs(alph,s+1,j,D-1)+LaplaceCoeffs(alph,s+1,j+1,D-1)-2*(D-1)*LaplaceCoeffs(alph,s+1,j,D-2));
        return
    end
    
    row = sub2ind([Nj,Ns,ND],jInd,sInd,DInd);
    
    %find the column corresponding to the alpha given, currently by finding the
    %closest value in the table
    F = (length(alphaVec));
    col = round(alph*F);
    if col==0
        col = 1;
    end
    
    %return the value from the table
    Output = bTable(row,col);
    return
end

%perform the calculation. Table will be made such that each row
%corresponds to some j,s combination, and each column corresponds to an
%alpha value.
if isequal(varargin{1},'init')
    
    %define the integrand function
    laplace_integrand=@(j1,s1,alph1) (@(phi) (cos(j1.*phi)./(1-2.*alph1.*cos(phi)+alph1.^2).^s1));
    
    warning off
    
    %set range of j, s, D, alpha. Can also be read from varargin if
    %exists, where the 2,3,4,5 arguments will be Ns, Nj,ND and dalpha
    Ns =3;
    Nj = 5;
    ND = 4;
    dalpha = 1e-2;
    %set integration tolerance
    AbsTol = 1e-5;
    Path = '~/Matlab/Data/LaplaceCoeffs.mat';
    
    %read inputs from varargin, if they exist
    if length(varargin)==5
        Ns = varargin{2};
        Nj = varargin{3};
        ND = varargin{4};
        dalpha = varargin{5};
    end
    
    if length(varargin)>5
        AbsTol = varargin{6};
    end
    
    if length(varargin)>6
        Path = varargin{7};
    end
    
    %set a vector of s and j values
    sVec = (1:Ns)-0.5;
    jVec = (1:Nj)-1;
    DVec = (1:ND)-1;
      
    %define the alpha vector from dalpha
    alphaVec = dalpha:dalpha:1;
      
    %pre-allocate
    bTable = zeros(Nj*Ns*ND,length(alphaVec));
    
    for jInd = 1:length(jVec)
        for sInd = 1:length(sVec)
            for DInd = 1:length(DVec)
                for alphaInd = 1:length(alphaVec)
                    
                    row = sub2ind([Nj,Ns,ND],jInd,sInd,DInd);
                    
                    val = CalculateSingleCoeff(jVec(jInd),sVec(sInd),alphaVec(alphaInd),DVec(DInd),AbsTol,laplace_integrand);
                    
                    bTable(row,alphaInd)=val;
                    
                end
            end
        end
    end
    save(Path,'jVec','sVec','alphaVec','DVec','bTable','Nj','Ns','ND');
    disp('calculated laplace coeffs');
    Output = [];
    return
end

%load the table including the persistent variables for future use. Also
%return the persistent variables in a structure as an output.
if isequal(varargin{1},'load')
    Path = '~/Matlab/Data/LaplaceCoeffs.mat';
    if nargin>1
        Path = varargin{2};
    end
    load(Path);
    Output = load(Path);
    return
end


%         if D==0
%             Output = LaplaceCoeffs(alph,s,j);
%         elseif D==1 %first derivative of the coeff, Solar System Dynamics eq. 6.70
%             Output = s*(LaplaceCoeffs(alph,s+1,j-1)-2*alph.*LaplaceCoeffs(alph,s+1,j)+LaplaceCoeffs(alph,s+1,j+1));
%         else %2nd and higher derivatives of the coeff, Solar System Dynamics eq. 6.71
%             Output = s*(LaplaceCoeffs(alph,s+1,j-1,D-1)-2*alph.*LaplaceCoeffs(alph,s+1,j,D-1)+LaplaceCoeffs(alph,s+1,j+1,D-1)-2*(D-1)*LaplaceCoeffs(alph,s+1,j,D-2));
%         end
%         return
        
function val = CalculateSingleCoeff(j,s,alph,D,AbsTol,laplace_integrand)

% %define the integrand function
% laplace_integrand=@(j1,s1,alph1) (@(phi) (cos(j1.*phi)./(1-2.*alph1.*cos(phi)+alph1.^2).^s1));

if D==0
    val = 1/pi*integral(laplace_integrand(j,s,alph),0,2*pi,'AbsTol',AbsTol);
elseif D==1 %first derivative of the coeff, Solar System Dynamics eq. 6.70
    val = s*(CalculateSingleCoeff(j-1,s+1,alph,D-1,AbsTol,laplace_integrand)...
            -2*alph.*CalculateSingleCoeff(j,s+1,alph,D-1,AbsTol,laplace_integrand)...
            +CalculateSingleCoeff(j+1,s+1,alph,D-1,AbsTol,laplace_integrand));
else %2nd and higher derivatives of the coeff, Solar System Dynamics eq. 6.71
    val = s*(CalculateSingleCoeff(j-1,s+1,alph,D-1,AbsTol,laplace_integrand)...
            -2*alph.*CalculateSingleCoeff(j,s+1,alph,D-1,AbsTol,laplace_integrand)...
            +CalculateSingleCoeff(j+1,s+1,alph,D-1,AbsTol,laplace_integrand)...
            -2*(D-1)*CalculateSingleCoeff(j,s+1,alph,D-2,AbsTol,laplace_integrand));
end