/usr/share/openturns/validation/ValidFrankCopula.txt is in openturns-validation 1.5-7build2.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
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> CDF_frank:=-log(1+(exp(-theta*u)-1)*(exp(-theta*v)-1)/(exp(-theta)-1))
> /theta;
(exp(-theta u) - 1) (exp(-theta v) - 1)
ln(1 + ---------------------------------------)
exp(-theta) - 1
CDF_frank := - -----------------------------------------------
theta
> PDF_frank:=factor(diff(diff(CDF_frank,u),v));
PDF_frank := - exp(-theta u) theta exp(-theta v) (exp(-theta) - 1)/
(exp(-theta) + exp(-theta u) exp(-theta v) - exp(-theta u)
2
- exp(-theta v))
> algsubs(exp(-theta*v)=V,algsubs(exp(-theta*u)=U,PDF_frank));
theta (exp(-theta) - 1) U V
- ----------------------------
2
(exp(-theta) - V + U V - U)
> du:=factor(diff(PDF_frank,u));
> dv:=factor(diff(PDF_frank,v));
> with(codegen,optimize,makeproc,cost,prep2trans):
> optimize(CDF_frank);
2
du := - theta exp(-theta u) exp(-theta v) (exp(-theta) - 1) (
-exp(-theta) + exp(-theta u) exp(-theta v) - exp(-theta u)
/
+ exp(-theta v)) / (exp(-theta)
/
+ exp(-theta u) exp(-theta v) - exp(-theta u)
3
- exp(-theta v))
2
dv := - theta exp(-theta u) exp(-theta v) (exp(-theta) - 1) (
-exp(-theta) + exp(-theta u) exp(-theta v) + exp(-theta u)
/
- exp(-theta v)) / (exp(-theta)
/
+ exp(-theta u) exp(-theta v) - exp(-theta u)
3
- exp(-theta v))
t2 = exp(-theta u), t5 = exp(-theta v), t8 = exp(-theta),
(t2 - 1) (t5 - 1) t13
t13 = ln(1 + -----------------), t16 = - -----
t8 - 1 theta
> optimize(du);
> optimize(dv);
2
t1 = theta , t3 = exp(-theta u), t6 = exp(-theta v),
t8 = exp(-theta), t10 = t3 t6, t13 = t8 + t10 - t3 - t6,
2 t1 t3 t6 (t8 - 1) (-t8 + t10 - t3 + t6)
t14 = t13 , t19 = - ---------------------------------------
t13 t14
2
t1 = theta , t3 = exp(-theta u), t6 = exp(-theta v),
t8 = exp(-theta), t10 = t3 t6, t13 = t8 + t10 - t3 - t6,
2 t1 t3 t6 (t8 - 1) (-t8 + t10 + t3 - t6)
t14 = t13 , t19 = - ---------------------------------------
t13 t14
> factor(du/dv);
(-exp(-theta) + exp(-theta u) exp(-theta v) - exp(-theta u)
+ exp(-theta v))/(-exp(-theta) + exp(-theta u) exp(-theta v)
+ exp(-theta u) - exp(-theta v))
> theta:=2.5;evalf(Int(Int((u-0.5)*(v-0.5)*PDF_frank,u=0..1),v=0..1));
>
theta := 2.5
0.03213516003
> theta:='theta';CDF_frank;
theta := theta
(exp(-theta u) - 1) (exp(-theta v) - 1)
ln(1 + ---------------------------------------)
exp(-theta) - 1
- -----------------------------------------------
theta
> limit(CDF_frank,theta=0);
u v
> factor((subs(v=u,(expand(1+(exp(-theta*u)-1)*(exp(-theta*v)-1)/(exp(-t
> heta)-1)))))-1);
2
exp(theta) (exp(theta u) - 1)
- -------------------------------
2
(-1 + exp(theta)) exp(theta u)
> f:=(exp(-u*theta))/(exp(-theta)-1);
exp(-theta u)
f := ---------------
exp(-theta) - 1
> g:=solve(f=y,u);
-ln(-y (-1 + exp(theta))) + theta
g := ---------------------------------
theta
> map(expand,subs(y=f+subs(u=v,f),g));
> expand(CDF_frank);
/ / exp(-theta u) exp(-theta v) \
|-ln(|- --------------- - ---------------| (-1 + exp(theta)))
\ \ exp(-theta) - 1 exp(-theta) - 1/
\
+ theta|/theta
/
(exp(-theta u) - 1) (exp(-theta v) - 1)
ln(1 + ---------------------------------------)
exp(-theta) - 1
- -----------------------------------------------
theta
> factor((1+(exp(-theta*u)-1)*(exp(-theta*v)-1)/(exp(-theta)-1)));
(exp(-theta) + exp(-theta u) exp(-theta v) - exp(-theta u)
- exp(-theta v))/(exp(-theta) - 1)
> subs(v=0,CDF_frank);
(exp(-theta u) - 1) (exp(0) - 1)
ln(1 + --------------------------------)
exp(-theta) - 1
- ----------------------------------------
theta
> restart:
> phi:=-log((exp(-theta*t)-1)/(exp(-theta)-1));
> psi:=solve(phi=y,t);
> #assume(p>0, p < 1, theta>0):
> q:=simplify(solve(subs(y=2*phi,psi)=p,t));
exp(-theta t) - 1
phi := -ln(-----------------)
exp(-theta) - 1
-ln(1 - exp(theta) + exp(y + theta)) + y + theta
psi := ------------------------------------------------
theta
/ /-1 + exp(p theta)\1/2 (p - 1) theta
q := |-ln(|-----------------| exp(- -------------)
\ \ -1 + exp(theta) / 2
/-1 + exp(p theta)\1/2 theta (-3 + p)
- |-----------------| exp(- --------------) + exp(theta))
\ -1 + exp(theta) / 2
\
+ theta|/theta
/
> diff(simplify(q-(
> 1-log(
> (sqrt(
> (exp(theta*p)-1)/
> (exp(theta)-1)*exp(theta)/exp(theta*p)
> )*(1-exp(theta))+exp(theta)
> )/theta)),symbolic),t);
0
>
|