Wishart

Packages
- n_wishartd.rr
  This is a Risa/Asir package for computing a system of PDEs for the matrix 1F1 over diagonal regions. This package provides two important functions. Note that the version of Risa/Asir must be after 20160401 to run this package.
  - n_wishartd.diagpf(M,[[S1,E1],[S2,E2],...]) computes a system of PDEs for a diagonalized 1F1. M is the number of variables and [Si,Ei] is a diagonal block s.t. S(i+1)=Ei + 1. This function returns a list of operators.
  - n_wishartd.prob_by_hgm(M,N,[P1,...],[S1,...],T|options) computes the probability Pr[l1< T|Sigma], where l1 is the largest eigenvalue of an M-dimensional Wishart matrix with N degrees of freedom and a covariance matrix Sigma=diag(S1,...,S2,...) with Si is repeated Pi times.
  - A sample Risa/Asir session
```
[2278] load("n_wishartd.rr")$
[2450] n_wishartd.diagpf(3,[[1,2],[3,3]]);
[[[[[-1,[]]],(1)*<<0,0,0,2>>],[[[-1/2,[[y1-y3,1]]],[-1/2,[[y3,1]]]],(1)*<<0,1,0,0>>],[[[1,[[y1-y3,1]]],[-c+1,[[y3,1]]],[1,[]]],(1)*<<0,0,0,1>>],[[[a,[[y3,1]]]],(1)*<<0,0,0,0>>]],[[[[-1,[]]],(1)*<<0,3,0,0>>],[[[-3/2,[[y1-y3,1]]],[-3*c+5/2,[[y1,1]]],[3,[]]],(1)*<<0,2,0,0>>],[[[2,[[y1-y3,1]]],[-2,[[y1,1]]]],(1)*<<0,1,0,1>>],[[[-1/2,[[y1-y3,2]]],[-2*c+2,[[y1-y3,1],[y1,1]]],[2,[[y1-y3,1]]],[-2*c^2+4*c-3/2,[[y1,2]]],[4*a+4*c-4,[[y1,1]]],[-2,[]]],(1)*<<0,1,0,0>>],[[[1,[[y1-y3,2]]],[2*c-4,[[y1-y3,1],[y1,1]]],[-2,[[y1-y3,1]]],[-2*c+3,[[y1,2]]],[2,[[y1,1]]]],(1)*<<0,0,0,1>>],[[[2*a,[[y1-y3,1],[y1,1]]],[(4*c-6)*a,[[y1,2]]],[-4*a,[[y1,1]]]],(1)*<<0,0,0,0>>]]]
[2451] n_wishartd.prob_by_hgm(4,10,[2,2],[1/100,1],100);
...
0.999996500887832
         
```
    In this example, after loading the package, a system of PDEs for 1F1(y1,y1,y3) is computed. Then Pr[l1< 100|diag(1/100,1/100,1,1)] is computed when the degrees of freedom=100.
Data
- systems over diagonal regions