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Generation of Random Variates

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Generation of Random Variates

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generates random variates from over 870 univariate distributions

mhygfx.m
function mhygfx
%This program is a direct conversion of the corresponding Fortran program in
%S. Zhang & J. Jin "Computation of Special Functions" (Wiley, 1996).
%online: http://iris-lee3.ece.uiuc.edu/~jjin/routines/routines.html
%
%Converted by f2matlab open source project:
%online: https://sourceforge.net/projects/f2matlab/
% written by Ben Barrowes (barrowes@alum.mit.edu)
%

%     ============================================================
%     Purpose: This program computes the hypergeometric function
%     F(a,b,c,x)using subroutine HYGFX
%     Input :  a --- Parameter
%     b --- Parameter
%     c --- Parameter, c <> 0,-1,-2,...
%     x --- Argument(x  1)
%     Output:  HF --- F(a,b,c,x)
%     Example:
%     b = 3.30,  c = 6.70
%     a     F(a,b,c,.25)F(a,b,c,.55)F(a,b,c,.85)
%     ------------------------------------------------------
%     -2.5   .72356129D+00    .46961432D+00   .29106096D+00
%     -0.5   .93610145D+00    .85187390D+00   .75543187D+00
%     0.5   .10689695D+01    .11795358D+01   .13510497D+01
%     2.5   .14051563D+01    .23999063D+01   .57381566D+01
%     a = 3.30,  b = 6.70
%     c     F(a,b,c,.25)F(a,b,c,.55)F(a,b,c,.85)
%     ------------------------------------------------------
%     -5.5   .15090670D+05    .10170778D+11   .58682088D+19
%     -0.5  -.21631479D+04   -.30854772D+07  -.10217370D+13
%     0.5   .26451677D+03    .11967860D+06   .92370648D+10
%     4.5   .41946916D+01    .58092729D+02   .20396914D+05
%     ============================================================
a=[];b=[];c=[];x=[];hf=[];
fprintf(1,'%s \n','please enter a,b,c and x ');
%     READ(*,*)A,B,C,X
a=-2.5;
b=3.3;
c=6.7;
x=.25;
[a,b,c,x,hf]=hygfx(a,b,c,x,hf);
fprintf(1,[repmat(' ',1,1),'a =','%5.2g',',    ','b =','%5.2g',',   ','c =','%5.2g',',    ','x =','%5.2g' ' \n'],a,b,c,x);
fprintf(1,[repmat(' ',1,1),'f(a,b,c,x)=','%16.8g' ' \n'],hf);
%format(1x,',f5.2,',    ',',f5.2,',   ',',f5.2,',    ',',f5.2);
%format(1x,',d16.8);
end
function [a,b,c,x,hf]=hygfx(a,b,c,x,hf,varargin)
%     ====================================================
%     Purpose: Compute hypergeometric function F(a,b,c,x)
%     Input :  a --- Parameter
%     b --- Parameter
%     c --- Parameter, c <> 0,-1,-2,...
%     x --- Argument(x < 1)
%     Output:  HF --- F(a,b,c,x)
%     Routines called:
%(1)GAMMA for computing gamma function
%(2)PSI for computing psi function
%     ====================================================
gc=[];gcab=[];gca=[];gcb=[];g1=[];g2=[];g3=[];ga=[];gb=[];m=[];gam=[];gbm=[];pa=[];pb=[];gabc=[];
  l0=false;
 l1=false;
 l2=false;
 l3=false;
 l4=false;
 l5=false;
hw=0.0;
pi=3.141592653589793d0;
el=.5772156649015329d0;
l0=c == fix(c)&c < 0.0;
l1=1.0d0-x < 1.0d-15&c-a-b <= 0.0;
l2=a == fix(a)&a < 0.0;
l3=b == fix(b)&b < 0.0;
l4=c-a == fix(c-a)&c-a <= 0.0;
l5=c-b == fix(c-b)&c-b <= 0.0;
if(l0|l1);
fprintf(1,'%s \n','the hypergeometric series is divergent');
return;
end;
eps=1.0d-15;
if(x > 0.95)eps=1.0d-8; end;
if(x == 0.0|a == 0.0|b == 0.0);
hf=1.0d0;
return;
elseif(1.0d0-x == eps&c-a-b > 0.0);
[c,gc]=gamma(c,gc);
[dumvar1,gcab]=gamma(c-a-b,gcab);
[dumvar1,gca]=gamma(c-a,gca);
[dumvar1,gcb]=gamma(c-b,gcb);
hf=gc.*gcab./(gca.*gcb);
return;
elseif(1.0d0+x <= eps&abs(c-a+b-1.0)<= eps);
g0=sqrt(pi).*2.0d0.^(-a);
[c,g1]=gamma(c,g1);
[dumvar1,g2]=gamma(1.0d0+a./2.0-b,g2);
[dumvar1,g3]=gamma(0.5d0+0.5.*a,g3);
hf=g0.*g1./(g2.*g3);
return;
elseif(l2|l3);
if(l2)nm=fix(abs(a)); end;
if(l3)nm=fix(abs(b)); end;
hf=1.0d0;
r=1.0d0;
for  k=1:nm;
r=r.*(a+k-1.0d0).*(b+k-1.0d0)./(k.*(c+k-1.0d0)).*x;
hf=hf+r;
end;  k=nm+1;
return;
elseif(l4|l5);
if(l4)nm=fix(abs(c-a)); end;
if(l5)nm=fix(abs(c-b)); end;
hf=1.0d0;
r=1.0d0;
for  k=1:nm;
r=r.*(c-a+k-1.0d0).*(c-b+k-1.0d0)./(k.*(c+k-1.0d0)).*x;
hf=hf+r;
end;  k=nm+1;
hf=(1.0d0-x).^(c-a-b).*hf;
return;
end;
aa=a;
bb=b;
x1=x;
if(x < 0.0d0);
x=x./(x-1.0d0);
if(c > a&b < a&b > 0.0);
a=bb;
b=aa;
end;
b=c-b;
end;
if(x >= 0.75d0);
gm=0.0d0;
if(abs(c-a-b-fix(c-a-b))< 1.0d-15);
m=fix(c-a-b);
[a,ga]=gamma(a,ga);
[b,gb]=gamma(b,gb);
[c,gc]=gamma(c,gc);
[dumvar1,gam]=gamma(a+m,gam);
[dumvar1,gbm]=gamma(b+m,gbm);
[a,pa]=psi(a,pa);
[b,pb]=psi(b,pb);
if(m ~= 0)gm=1.0d0; end;
for  j=1:abs(m)-1;
gm=gm.*j;
end;  j=abs(m)-1+1;
rm=1.0d0;
for  j=1:abs(m);
rm=rm.*j;
end;  j=abs(m)+1;
f0=1.0d0;
r0=1.0d0;
r1=1.0d0;
sp0=0.d0;
sp=0.0d0;
if(m >= 0);
c0=gm.*gc./(gam.*gbm);
c1=-gc.*(x-1.0d0).^m./(ga.*gb.*rm);
for  k=1:m-1;
r0=r0.*(a+k-1.0d0).*(b+k-1.0)./(k.*(k-m)).*(1.0-x);
f0=f0+r0;
end;  k=m-1+1;
for  k=1:m;
sp0=sp0+1.0d0./(a+k-1.0)+1.0./(b+k-1.0)-1.0./k;
end;  k=m+1;
f1=pa+pb+sp0+2.0d0.*el+log(1.0d0-x);
for  k=1:250;
sp=sp+(1.0d0-a)./(k.*(a+k-1.0))+(1.0-b)./(k.*(b+k-1.0));
sm=0.0d0;
for  j=1:m;
sm=sm+(1.0d0-a)./((j+k).*(a+j+k-1.0))+1.0./(b+j+k-1.0);
end;  j=m+1;
rp=pa+pb+2.0d0.*el+sp+sm+log(1.0d0-x);
r1=r1.*(a+m+k-1.0d0).*(b+m+k-1.0)./(k.*(m+k)).*(1.0-x);
f1=f1+r1.*rp;
if(abs(f1-hw)< abs(f1).*eps)break; end;
hw=f1;
end;
hf=f0.*c0+f1.*c1;
elseif(m < 0);
m=-m;
c0=gm.*gc./(ga.*gb.*(1.0d0-x).^m);
c1=-(-1).^m.*gc./(gam.*gbm.*rm);
for  k=1:m-1;
r0=r0.*(a-m+k-1.0d0).*(b-m+k-1.0)./(k.*(k-m)).*(1.0-x);
f0=f0+r0;
end;  k=m-1+1;
for  k=1:m;
sp0=sp0+1.0d0./k;
end;  k=m+1;
f1=pa+pb-sp0+2.0d0.*el+log(1.0d0-x);
for  k=1:250;
sp=sp+(1.0d0-a)./(k.*(a+k-1.0))+(1.0-b)./(k.*(b+k-1.0));
sm=0.0d0;
for  j=1:m;
sm=sm+1.0d0./(j+k);
end;  j=m+1;
rp=pa+pb+2.0d0.*el+sp-sm+log(1.0d0-x);
r1=r1.*(a+k-1.0d0).*(b+k-1.0)./(k.*(m+k)).*(1.0-x);
f1=f1+r1.*rp;
if(abs(f1-hw)< abs(f1).*eps)break; end;
hw=f1;
end;
hf=f0.*c0+f1.*c1;
end;
else;
[a,ga]=gamma(a,ga);
[b,gb]=gamma(b,gb);
[c,gc]=gamma(c,gc);
[dumvar1,gca]=gamma(c-a,gca);
[dumvar1,gcb]=gamma(c-b,gcb);
[dumvar1,gcab]=gamma(c-a-b,gcab);
[dumvar1,gabc]=gamma(a+b-c,gabc);
c0=gc.*gcab./(gca.*gcb);
c1=gc.*gabc./(ga.*gb).*(1.0d0-x).^(c-a-b);
hf=0.0d0;
r0=c0;
r1=c1;
for  k=1:250;
r0=r0.*(a+k-1.0d0).*(b+k-1.0)./(k.*(a+b-c+k)).*(1.0-x);
r1=r1.*(c-a+k-1.0d0).*(c-b+k-1.0)./(k.*(c-a-b+k)).*(1.0-x);
hf=hf+r0+r1;
if(abs(hf-hw)< abs(hf).*eps)break; end;
hw=hf;
end;
hf=hf+c0+c1;
end;
else;
a0=1.0d0;
if(c > a&c < 2.0d0.*a& c > b&c < 2.0d0.*b);
a0=(1.0d0-x).^(c-a-b);
a=c-a;
b=c-b;
end;
hf=1.0d0;
r=1.0d0;
for  k=1:250;
r=r.*(a+k-1.0d0).*(b+k-1.0d0)./(k.*(c+k-1.0d0)).*x;
hf=hf+r;
if(abs(hf-hw)<= abs(hf).*eps)break; end;
hw=hf;
end;
hf=a0.*hf;
end;
if(x1 < 0.0d0);
x=x1;
c0=1.0d0./(1.0d0-x).^aa;
hf=c0.*hf;
end;
a=aa;
b=bb;
if(k > 120)fprintf(1,[repmat(' ',1,1),'warning% you should check the accuracy' ' \n']); end;
%format(1x,'warning% you should check the accuracy');
return;
end
function [x,ga]=gamma(x,ga,varargin);
%     ==================================================
%     Purpose: Compute gamma function (x)
%     Input :  x  --- Argument of (x)
%(x is not equal to 0,-1,-2,)
%     Output:  GA --- (x)
%     ==================================================
 g=zeros(1,26);
pi=3.141592653589793d0;
if(x == fix(x));
if(x > 0.0d0);
ga=1.0d0;
m1=x-1;
for  k=2:m1;
ga=ga.*k;
end;  k=m1+1;
else;
ga=1.0d+300;
end;
else;
if(abs(x)> 1.0d0);
z=abs(x);
m=fix(z);
r=1.0d0;
for  k=1:m;
r=r.*(z-k);
end;  k=m+1;
z=z-m;
else;
z=x;
end;
g(:)=[1.0d0,0.5772156649015329d0,-0.6558780715202538d0,-0.420026350340952d-1,0.1665386113822915d0,-.421977345555443d-1,-.96219715278770d-2,.72189432466630d-2,-.11651675918591d-2,-.2152416741149d-3,.1280502823882d-3,-.201348547807d-4,-.12504934821d-5,.11330272320d-5,-.2056338417d-6,.61160950d-8,.50020075d-8,-.11812746d-8,.1043427d-9,.77823d-11,-.36968d-11,.51d-12,-.206d-13,-.54d-14,.14d-14,.1d-15];
gr=g(26);
for  k=25:-1:1;
gr=gr.*z+g(k);
end;  k=1-1;
ga=1.0d0./(gr.*z);
if(abs(x)> 1.0d0);
ga=ga.*r;
if(x < 0.0d0)ga=-pi./(x.*ga.*sin(pi.*x)); end;
end;
end;
return;
end
function [x,ps]=psi(x,ps,varargin);
%     ======================================
%     Purpose: Compute Psi function
%     Input :  x  --- Argument of psi(x)
%     Output:  PS --- psi(x)
%     ======================================
xa=abs(x);
pi=3.141592653589793d0;
el=.5772156649015329d0;
s=0.0d0;
if(x == fix(x)&x <= 0.0);
ps=1.0d+300;
return;
elseif(xa == fix(xa));
n=xa;
for  k=1 :n-1;
s=s+1.0d0./k;
end;  k=n-1+1;
ps=-el+s;
elseif(xa+.5 == fix(xa+.5));
n=xa-.5;
for  k=1:n;
s=s+1.0./(2.0d0.*k-1.0d0);
end;  k=n+1;
ps=-el+2.0d0.*s-1.386294361119891d0;
else;
if(xa < 10.0);
n=10-fix(xa);
for  k=0:n-1;
s=s+1.0d0./(xa+k);
end;  k=n-1+1;
xa=xa+n;
end;
x2=1.0d0./(xa.*xa);
a1=-.8333333333333d-01;
a2=.83333333333333333d-02;
a3=-.39682539682539683d-02;
a4=.41666666666666667d-02;
a5=-.75757575757575758d-02;
a6=.21092796092796093d-01;
a7=-.83333333333333333d-01;
a8=.4432598039215686d0;
ps=log(xa)-.5d0./xa+x2.*(((((((a8.*x2+a7).*x2+a6).*x2+a5).*x2+a4).*x2+a3).*x2+a2).*x2+a1);
ps=ps-s;
end;
if(x < 0.0)ps=ps-pi.*cos(pi.*x)./sin(pi.*x)-1.0d0./x; end;
return;
end

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