python3 <<'PY'
import math
def F(M,N=200,verbose=False):
    m11,m12=M[0]; m21,m22=M[1]
    alpha=m11+m12+m21+m22
    mu1=m11+m12; nu1=m11+m21
    best=-1.0; ba=None
    for iq in range(1,N):
        q=alpha*iq/N; q2=alpha-q
        lo=max(0.0,q-(alpha-mu1)); hi=min(q,mu1)
        for ir in range(N+1):
            r=lo+(hi-lo)*ir/N
            den=r*(alpha-q)/q-(mu1-r)
            if abs(den)<1e-13: continue
            c=(m11*(alpha-q)-(mu1-r)*nu1)/den
            if c<0 or c>nu1 or c>q or nu1-c>q2: continue
            r1=r/q; s1=c/q; r2=(mu1-r)/q2; s2=(nu1-c)/q2
            val=q*(math.sqrt(m11*r1*s1)+math.sqrt(m12*r1*(1-s1))+math.sqrt(m21*(1-r1)*s1)+math.sqrt(m22*(1-r1)*(1-s1)))+q2*(math.sqrt(m11*r2*s2)+math.sqrt(m12*r2*(1-s2))+math.sqrt(m21*(1-r2)*s2)+math.sqrt(m22*(1-r2)*(1-s2)))
            if val>best: best=val; ba=(q,r,c)
    if verbose: print("arg",ba)
    return best
M=[[0.25,0.0],[1/12,0.05]]
print(F(M,200,True))
PY