function [pdftwappx, cdftwappx] = tracywidom_appx(x, i) 
% 
%
% [pdftwappx, cdftwappx]=tracywidom_appx(x, i)
%
% SHIFTED GAMMA APPROXIMATION FOR THE TRACY-WIDOM LAWS, by M. Chiani, 2012
%
% TW ~ Gamma[k,theta]-alpha   
%
% [pdf,cdf]=tracywidom_appx(x,2) gives the approximated pdf(x) and CDF(x) of the TW2   
%
% [pdf,cdf]=tracywidom_appx(x,i) for i=1,2,4 gives TW1, TW2, TW4     
%
% see paper M.Chiani, "Distribution of the largest eigenvalue for real 
% Wishart and Gaussian random matrices and a simple approximation for the 
% Tracy-Widom distribution", submitted 2012, ArXiv
%
kappx = [46.44604884387787, 79.6594870666346, 0, 146.0206131050228];   %  K, THETA, ALPHA 
thetaappx = [0.18605402228279347, 0.10103655775856243, 0, 0.05954454047933292];  
alphaappx = [9.848007781128567, 9.819607173436484, 0, 11.00161520109004]; 
cdftwappx = cdfgamma(x+alphaappx(i), thetaappx(i), kappx(i));   % Chiani's appx: Tracy-Widom as a shifted Gamma
pdftwappx = pdfgamma(x+alphaappx(i), thetaappx(i), kappx(i));   % Chiani's appx: Tracy-Widom as a shifted Gamma 
end
function [pdfgamma]=pdfgamma(x, ta, ka) % Utility: PDF of a Gamma 
 if(x > 0)
     pdfgamma=1/(gamma(ka)*ta^ka) * x.^(ka - 1) .* exp(-x/ta);
 else
     pdfgamma=0 ;
 end
end
function[cdfgamma]=cdfgamma(x, ta, ka)  % Utility: CDF of a Gamma
 if(x > 0) 
     cdfgamma=gammainc(x/ta,ka); 
 else
     cdfgamma=0; 
 end
 
end