python3 <<'PY'
import math
def F(M,N=400):
    m11,m12=M[0]; m21,m22=M[1]
    alpha=m11+m12+m21+m22
    mu1=m11+m12; nu1=m11+m21
    best=-1; arg=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)
        if hi<lo: continue
        for ir in range(1,N):
            r=lo+(hi-lo)*ir/N
            den=r*(alpha-q)/q - (mu1-r)
            if abs(den)<1e-15: continue
            c=(m11*(alpha-q)-(mu1-r)*nu1)/den
            if c<-1e-9 or c>nu1+1e-9 or c>q+1e-9 or (nu1-c)>q2+1e-9: continue
            c=min(max(c,0.0),nu1)
            comps=[]
            comps.append((q,[r/q,1-r/q],[c/q,1-c/q]))
            comps.append((q2,[(mu1-r)/q2,1-(mu1-r)/q2],[(nu1-c)/q2,1-(nu1-c)/q2]))
            val=0
            for w,rr,ss in comps:
                for i in range(2):
                    for j in range(2):
                        val+=w*math.sqrt(M[i][j]*rr[i]*ss[j])
            if val>best: best=val; arg=(q,r,c)
    return best,arg
M=[[0.375,0.125],[0.125,0.375]]
print(F(M,300))
PY