### Expected Euler Characteristic

#### Papers

1. N.Takayama, L.Jiu, S.Kuriki, Y.Zhang, Computations of the Expected Euler Characteristic for the Largest Eigenvalue of a Real Wishart Matrix, arxiv:1903.10099
2. References for the HGM and the HGD

#### Programs

1. R program to make a simulation: matnorm.r . In order to run the program, install the package mnormt by
install.packages("mnormt");

2. The computer algebra Risa/Asir programs for Example 5 and other examples: float-exyi5b.rr , ode-yi3.rr , tk_series , ann3.txt ,
3. Mathematica program to evalute the approximate formula of Theorem 3 in the Appendix: ec-selb.m .

#### Example 5

$m=n=2$ case with $M = \pmatrix{1 & 0 \cr 2 & 3 \cr}$ and $S={\rm diag}(10^3,10^2)$ where $\Sigma = S^{-1}$.
Simulation by R.
source("matnorm.r");
ans<-seq(0,10);
k<-1;
for (i in seq(3.8,3.81,by=0.001)) {
ans[k]<-yiex5(x=i,try=100000);
print(i); print(ans[k]);
k<-k+1;
}
print(ans)


Approximation by our Theorem 1 and ODE solution (HGM) on the computer algebra system Risa/Asir
load("float-exyi5b.rr");


#### Example: Appendix, Theorem 3

Evalute $P(\lambda_1 > 20)$ for $10 \times 10$ matrix ($m=n=10$) with $M=0$ and $\Sigma=I_{10}/s$, $s=0.1$.
Simulation by R with 100000 random matrices.
source("matnorm.r");
selb_sim(m=10,s=0.1,x=20,try=100000)


Evaluation by Mathematica and our approximate formula by the Euler characteristic heuristic.
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Last update: April 6, 2020