/usr/share/openturns/validation/ValidCovarianceAlgorithm.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.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 | > restart:
> assume(lambda>0):
> #F:=1-exp(-lambda*x):
> I1:=Int(1-F(x),x=0..infinity);
> I2:=int(1/2*(1-subs(x=(u-1)/2/(1-(u-1)/2),F))/(1-(u-1)/2)^2,u=0..1);
infinity
/
|
I1 := | 1 - F(x) dx
|
/
0
I2 := 1/3 - F/3
> with(student):
> changevar(x=(u+1)/(u-1),Int(1-F(x),x=0..infinity),u);
> changevar(x=(u+1)/(u-1),Int(F(x),x=-infinity..0),u);
1 u + 1
/ -2 + 2 F(-----)
| u - 1
| --------------- du
| 2
| u - 2 u + 1
/
-1
1 u + 1
/ 2 F(-----)
| u - 1
| ------------ du
| 2
| u - 2 u + 1
/
-1
> plot((u+1)/(u-1),u=-0.99..0.99);
> plot(tanh(x),x=-10..10);
> plot(x/(1+abs(x)),x=-10..10);
> restart:
> with(student):
> assume(lambda>0):
> F:=1-exp(-lambda*x);
> G(u):=solve(F=u,x);
> value(changevar(x=G(u),Int((x-m)^2*diff(F,x),x=0..infinity),u));
F := 1 - exp(-lambda~ x)
ln(1 - u)
G(u) := - ---------
lambda~
2 2
2 + m lambda~ - 2 m lambda~
-----------------------------
2
lambda~
> int((x-m)^2*diff(F,x),x=0..infinity);
2 2
2 + m lambda~ - 2 m lambda~
-----------------------------
2
lambda~
> int((G(u)-m)^2,u=0..1);
2 2
2 + m lambda~ - 2 m lambda~
-----------------------------
2
lambda~
>
|