Massimo Nocentini

April 19, 2018: splitting from Pascal notebook



Abstract
Theory of matrix functions, with applications to generation matrix $\mathcal{H}$.

In [1]:
from sympy import *
from sympy.abc import n, i, N, x, lamda, phi, z, j, r, k, a, alpha, beta

from commons import *
from matrix_functions import *
from sequences import *
import functions_catalog

init_printing()

In [2]:
d,h = IndexedBase('d'), IndexedBase('h')
alpha, beta, gamma = symbols(r'\alpha \beta \gamma')

Generation matrix $\mathcal{H}$

see https://en.wikipedia.org/wiki/Pascal_matrix


In [3]:
m = 16
H = define(Symbol(r'\mathcal{{H}}_{{ {} }}'.format(m)), 
           Matrix(m, m, lambda n, k: 0 if n==m/2 or n!=k+1 else -h[-(k-(m/S(2))+2)] if n<m/2 else h[k-(m/S(2))]))
H


Out[3]:
$$\mathcal{H}_{ 16 } = \left[\begin{array}{cccccccccccccccc}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\- h_{6} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & - h_{5} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & - h_{4} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & - h_{3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & - h_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & - h_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & - h_{0} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & h_{0} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & h_{1} & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & h_{2} & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & h_{3} & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & h_{4} & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & h_{5} & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & h_{6} & 0\end{array}\right]$$

In [4]:
f = Function('E')
f_expt = define(let=f(z), be=exp(z))
f_expt


Out[4]:
$$E{\left (z \right )} = e^{z}$$

In [5]:
eigendata = spectrum(H)
eigendata


Out[5]:
$$\sigma{\left (\mathcal{H}_{ 16 } \right )} = \left ( \left \{ 1 : \left ( \lambda_{1}, \quad m_{1}\right )\right \}, \quad \left \{ \lambda_{1} : 0\right \}, \quad \left \{ m_{1} : 16\right \}\right )$$

In [6]:
data, eigenvals, multiplicities = eigendata.rhs

In [7]:
Phi_polynomials = component_polynomials(eigendata, early_eigenvals_subs=True)
Phi_polynomials


Out[7]:
$$\left \{ \left ( 1, \quad 1\right ) : \Phi_{ 1, 1 }{\left (z \right )} = 1, \quad \left ( 1, \quad 2\right ) : \Phi_{ 1, 2 }{\left (z \right )} = z, \quad \left ( 1, \quad 3\right ) : \Phi_{ 1, 3 }{\left (z \right )} = \frac{z^{2}}{2}, \quad \left ( 1, \quad 4\right ) : \Phi_{ 1, 4 }{\left (z \right )} = \frac{z^{3}}{6}, \quad \left ( 1, \quad 5\right ) : \Phi_{ 1, 5 }{\left (z \right )} = \frac{z^{4}}{24}, \quad \left ( 1, \quad 6\right ) : \Phi_{ 1, 6 }{\left (z \right )} = \frac{z^{5}}{120}, \quad \left ( 1, \quad 7\right ) : \Phi_{ 1, 7 }{\left (z \right )} = \frac{z^{6}}{720}, \quad \left ( 1, \quad 8\right ) : \Phi_{ 1, 8 }{\left (z \right )} = \frac{z^{7}}{5040}, \quad \left ( 1, \quad 9\right ) : \Phi_{ 1, 9 }{\left (z \right )} = \frac{z^{8}}{40320}, \quad \left ( 1, \quad 10\right ) : \Phi_{ 1, 10 }{\left (z \right )} = \frac{z^{9}}{362880}, \quad \left ( 1, \quad 11\right ) : \Phi_{ 1, 11 }{\left (z \right )} = \frac{z^{10}}{3628800}, \quad \left ( 1, \quad 12\right ) : \Phi_{ 1, 12 }{\left (z \right )} = \frac{z^{11}}{39916800}, \quad \left ( 1, \quad 13\right ) : \Phi_{ 1, 13 }{\left (z \right )} = \frac{z^{12}}{479001600}, \quad \left ( 1, \quad 14\right ) : \Phi_{ 1, 14 }{\left (z \right )} = \frac{z^{13}}{6227020800}, \quad \left ( 1, \quad 15\right ) : \Phi_{ 1, 15 }{\left (z \right )} = \frac{z^{14}}{87178291200}, \quad \left ( 1, \quad 16\right ) : \Phi_{ 1, 16 }{\left (z \right )} = \frac{z^{15}}{1307674368000}\right \}$$

In [8]:
g_expt = Hermite_interpolation_polynomial(f_expt, eigendata, Phi_polynomials)
#g_expt

In [9]:
g_expt = g_expt.subs(eigenvals)
g_expt


Out[9]:
$$\operatorname{E_{ 16 }}{\left (z \right )} = \frac{z^{15}}{1307674368000} + \frac{z^{14}}{87178291200} + \frac{z^{13}}{6227020800} + \frac{z^{12}}{479001600} + \frac{z^{11}}{39916800} + \frac{z^{10}}{3628800} + \frac{z^{9}}{362880} + \frac{z^{8}}{40320} + \frac{z^{7}}{5040} + \frac{z^{6}}{720} + \frac{z^{5}}{120} + \frac{z^{4}}{24} + \frac{z^{3}}{6} + \frac{z^{2}}{2} + z + 1$$

In [10]:
with lift_to_matrix_function(g_expt) as G_expt:
    H_expt = G_expt(H)
H_expt


Out[10]:
$$\operatorname{E_{ 16 }}{\left (\mathcal{H}_{ 16 } \right )} = \left[\begin{array}{cccccccccccccccc}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\- h_{6} & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\frac{h_{5} h_{6}}{2} & - h_{5} & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\- \frac{h_{4} h_{5}}{6} h_{6} & \frac{h_{4} h_{5}}{2} & - h_{4} & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\frac{h_{3} h_{4}}{24} h_{5} h_{6} & - \frac{h_{3} h_{4}}{6} h_{5} & \frac{h_{3} h_{4}}{2} & - h_{3} & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\- \frac{h_{2} h_{3}}{120} h_{4} h_{5} h_{6} & \frac{h_{2} h_{3}}{24} h_{4} h_{5} & - \frac{h_{2} h_{3}}{6} h_{4} & \frac{h_{2} h_{3}}{2} & - h_{2} & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\frac{h_{1} h_{2}}{720} h_{3} h_{4} h_{5} h_{6} & - \frac{h_{1} h_{2}}{120} h_{3} h_{4} h_{5} & \frac{h_{1} h_{2}}{24} h_{3} h_{4} & - \frac{h_{1} h_{2}}{6} h_{3} & \frac{h_{1} h_{2}}{2} & - h_{1} & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\- \frac{h_{0} h_{1}}{5040} h_{2} h_{3} h_{4} h_{5} h_{6} & \frac{h_{0} h_{1}}{720} h_{2} h_{3} h_{4} h_{5} & - \frac{h_{0} h_{1}}{120} h_{2} h_{3} h_{4} & \frac{h_{0} h_{1}}{24} h_{2} h_{3} & - \frac{h_{0} h_{1}}{6} h_{2} & \frac{h_{0} h_{1}}{2} & - h_{0} & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & h_{0} & 1 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{h_{0} h_{1}}{2} & h_{1} & 1 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{h_{0} h_{1}}{6} h_{2} & \frac{h_{1} h_{2}}{2} & h_{2} & 1 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{h_{0} h_{1}}{24} h_{2} h_{3} & \frac{h_{1} h_{2}}{6} h_{3} & \frac{h_{2} h_{3}}{2} & h_{3} & 1 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{h_{0} h_{1}}{120} h_{2} h_{3} h_{4} & \frac{h_{1} h_{2}}{24} h_{3} h_{4} & \frac{h_{2} h_{3}}{6} h_{4} & \frac{h_{3} h_{4}}{2} & h_{4} & 1 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{h_{0} h_{1}}{720} h_{2} h_{3} h_{4} h_{5} & \frac{h_{1} h_{2}}{120} h_{3} h_{4} h_{5} & \frac{h_{2} h_{3}}{24} h_{4} h_{5} & \frac{h_{3} h_{4}}{6} h_{5} & \frac{h_{4} h_{5}}{2} & h_{5} & 1 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{h_{0} h_{1}}{5040} h_{2} h_{3} h_{4} h_{5} h_{6} & \frac{h_{1} h_{2}}{720} h_{3} h_{4} h_{5} h_{6} & \frac{h_{2} h_{3}}{120} h_{4} h_{5} h_{6} & \frac{h_{3} h_{4}}{24} h_{5} h_{6} & \frac{h_{4} h_{5}}{6} h_{6} & \frac{h_{5} h_{6}}{2} & h_{6} & 1\end{array}\right]$$

In [11]:
production_matrix(H_expt.rhs[7:,7:])


Out[11]:
$$\left[\begin{matrix}0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\0 & h_{0} & 1 & 0 & 0 & 0 & 0 & 0\\0 & \frac{h_{0}}{2} \left(- 2 h_{0} + h_{1}\right) & - h_{0} + h_{1} & 1 & 0 & 0 & 0 & 0\\0 & \frac{h_{0} h_{1}}{6} \left(3 h_{0} - 3 h_{1} + h_{2}\right) & \frac{h_{1}}{2} \left(h_{0} - 2 h_{1} + h_{2}\right) & - h_{1} + h_{2} & 1 & 0 & 0 & 0\\0 & \frac{h_{0} h_{1}}{24} \left(- 4 h_{0} + 6 h_{1} - 4 h_{2} + h_{3}\right) h_{2} & \frac{h_{1} h_{2}}{6} \left(- h_{0} + 3 h_{1} - 3 h_{2} + h_{3}\right) & \frac{h_{2}}{2} \left(h_{1} - 2 h_{2} + h_{3}\right) & - h_{2} + h_{3} & 1 & 0 & 0\\0 & \frac{h_{0} h_{1}}{120} \left(5 h_{0} - 10 h_{1} + 10 h_{2} - 5 h_{3} + h_{4}\right) h_{2} h_{3} & \frac{h_{1} h_{2}}{24} \left(h_{0} - 4 h_{1} + 6 h_{2} - 4 h_{3} + h_{4}\right) h_{3} & \frac{h_{2} h_{3}}{6} \left(- h_{1} + 3 h_{2} - 3 h_{3} + h_{4}\right) & \frac{h_{3}}{2} \left(h_{2} - 2 h_{3} + h_{4}\right) & - h_{3} + h_{4} & 1 & 0\\0 & \frac{h_{0} h_{1}}{720} \left(- 6 h_{0} + 15 h_{1} - 20 h_{2} + 15 h_{3} - 6 h_{4} + h_{5}\right) h_{2} h_{3} h_{4} & \frac{h_{1} h_{2}}{120} \left(- h_{0} + 5 h_{1} - 10 h_{2} + 10 h_{3} - 5 h_{4} + h_{5}\right) h_{3} h_{4} & \frac{h_{2} h_{3}}{24} \left(h_{1} - 4 h_{2} + 6 h_{3} - 4 h_{4} + h_{5}\right) h_{4} & \frac{h_{3} h_{4}}{6} \left(- h_{2} + 3 h_{3} - 3 h_{4} + h_{5}\right) & \frac{h_{4}}{2} \left(h_{3} - 2 h_{4} + h_{5}\right) & - h_{4} + h_{5} & 1\\0 & \frac{h_{0} h_{1}}{5040} \left(7 h_{0} - 21 h_{1} + 35 h_{2} - 35 h_{3} + 21 h_{4} - 7 h_{5} + h_{6}\right) h_{2} h_{3} h_{4} h_{5} & \frac{h_{1} h_{2}}{720} \left(h_{0} - 6 h_{1} + 15 h_{2} - 20 h_{3} + 15 h_{4} - 6 h_{5} + h_{6}\right) h_{3} h_{4} h_{5} & \frac{h_{2} h_{3}}{120} \left(- h_{1} + 5 h_{2} - 10 h_{3} + 10 h_{4} - 5 h_{5} + h_{6}\right) h_{4} h_{5} & \frac{h_{3} h_{4}}{24} \left(h_{2} - 4 h_{3} + 6 h_{4} - 4 h_{5} + h_{6}\right) h_{5} & \frac{h_{4} h_{5}}{6} \left(- h_{3} + 3 h_{4} - 3 h_{5} + h_{6}\right) & \frac{h_{5}}{2} \left(h_{4} - 2 h_{5} + h_{6}\right) & - h_{5} + h_{6}\end{matrix}\right]$$

In [12]:
H_expt.rhs.subs({h[0]:1,h[1]:1,h[2]:1, h[3]:1,h[4]:1,h[5]:1,h[6]:1})


Out[12]:
$$\left[\begin{array}{cccccccccccccccc}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\-1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\frac{1}{2} & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\- \frac{1}{6} & \frac{1}{2} & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\frac{1}{24} & - \frac{1}{6} & \frac{1}{2} & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\- \frac{1}{120} & \frac{1}{24} & - \frac{1}{6} & \frac{1}{2} & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\frac{1}{720} & - \frac{1}{120} & \frac{1}{24} & - \frac{1}{6} & \frac{1}{2} & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\- \frac{1}{5040} & \frac{1}{720} & - \frac{1}{120} & \frac{1}{24} & - \frac{1}{6} & \frac{1}{2} & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{2} & 1 & 1 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{6} & \frac{1}{2} & 1 & 1 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{24} & \frac{1}{6} & \frac{1}{2} & 1 & 1 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{120} & \frac{1}{24} & \frac{1}{6} & \frac{1}{2} & 1 & 1 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{720} & \frac{1}{120} & \frac{1}{24} & \frac{1}{6} & \frac{1}{2} & 1 & 1 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{5040} & \frac{1}{720} & \frac{1}{120} & \frac{1}{24} & \frac{1}{6} & \frac{1}{2} & 1 & 1\end{array}\right]$$

In [13]:
H_expt.rhs.subs({h[0]:1,h[1]:2,h[2]:3, h[3]:4,h[4]:5,h[5]:6,h[6]:7})


Out[13]:
$$\left[\begin{array}{cccccccccccccccc}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\-7 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\21 & -6 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\-35 & 15 & -5 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\35 & -20 & 10 & -4 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\-21 & 15 & -10 & 6 & -3 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\7 & -6 & 5 & -4 & 3 & -2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\-1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 & 1 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 3 & 3 & 1 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 4 & 6 & 4 & 1 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 5 & 10 & 10 & 5 & 1 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 6 & 15 & 20 & 15 & 6 & 1 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 7 & 21 & 35 & 35 & 21 & 7 & 1\end{array}\right]$$

In [14]:
H_expt.rhs.subs({h[0]:1,h[1]:1,h[2]:2, h[3]:3,h[4]:5,h[5]:8,h[6]:13})


Out[14]:
$$\left[\begin{array}{cccccccccccccccc}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\-13 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\52 & -8 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\- \frac{260}{3} & 20 & -5 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\65 & -20 & \frac{15}{2} & -3 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\-26 & 10 & -5 & 3 & -2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\frac{13}{3} & -2 & \frac{5}{4} & -1 & 1 & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\- \frac{13}{21} & \frac{1}{3} & - \frac{1}{4} & \frac{1}{4} & - \frac{1}{3} & \frac{1}{2} & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{2} & 1 & 1 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{3} & 1 & 2 & 1 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{4} & 1 & 3 & 3 & 1 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{4} & \frac{5}{4} & 5 & \frac{15}{2} & 5 & 1 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{3} & 2 & 10 & 20 & 20 & 8 & 1 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{13}{21} & \frac{13}{3} & 26 & 65 & \frac{260}{3} & 52 & 13 & 1\end{array}\right]$$

In [15]:
H_expt.rhs.subs({h[0]:1,h[1]:1,h[2]:2, h[3]:5,h[4]:14,h[5]:42})


Out[15]:
$$\left[\begin{array}{cccccccccccccccc}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\- h_{6} & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\21 h_{6} & -42 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\- 98 h_{6} & 294 & -14 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\frac{245 h_{6}}{2} & -490 & 35 & -5 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\- 49 h_{6} & 245 & - \frac{70}{3} & 5 & -2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\frac{49 h_{6}}{6} & -49 & \frac{35}{6} & - \frac{5}{3} & 1 & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\- \frac{7 h_{6}}{6} & \frac{49}{6} & - \frac{7}{6} & \frac{5}{12} & - \frac{1}{3} & \frac{1}{2} & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{2} & 1 & 1 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{3} & 1 & 2 & 1 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{5}{12} & \frac{5}{3} & 5 & 5 & 1 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{7}{6} & \frac{35}{6} & \frac{70}{3} & 35 & 14 & 1 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{49}{6} & 49 & 245 & 490 & 294 & 42 & 1 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{7 h_{6}}{6} & \frac{49 h_{6}}{6} & 49 h_{6} & \frac{245 h_{6}}{2} & 98 h_{6} & 21 h_{6} & h_{6} & 1\end{array}\right]$$

In [16]:
H_expt.rhs.subs({h[0]:factorial(0),h[1]:factorial(1),h[2]:factorial(2), 
                 h[3]:factorial(3),h[4]:factorial(4), h[5]:factorial(5), 
                 h[6]:factorial(6)})


Out[16]:
$$\left[\begin{array}{cccccccccccccccc}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\-720 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\43200 & -120 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\-345600 & 1440 & -24 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\518400 & -2880 & 72 & -6 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\-207360 & 1440 & -48 & 6 & -2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\34560 & -288 & 12 & -2 & 1 & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\- \frac{34560}{7} & 48 & - \frac{12}{5} & \frac{1}{2} & - \frac{1}{3} & \frac{1}{2} & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{2} & 1 & 1 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{3} & 1 & 2 & 1 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{2} & 2 & 6 & 6 & 1 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{12}{5} & 12 & 48 & 72 & 24 & 1 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 48 & 288 & 1440 & 2880 & 1440 & 120 & 1 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{34560}{7} & 34560 & 207360 & 518400 & 345600 & 43200 & 720 & 1\end{array}\right]$$

In [17]:
H_expt.rhs.subs({h[0]:2**0,h[1]:2**1,h[2]:2**2, h[3]:2**3,h[4]:2**4,h[5]:2**5,h[6]:2**6})


Out[17]:
$$\left[\begin{array}{cccccccccccccccc}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\-64 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\1024 & -32 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\- \frac{16384}{3} & 256 & -16 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\frac{32768}{3} & - \frac{2048}{3} & 64 & -8 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\- \frac{131072}{15} & \frac{2048}{3} & - \frac{256}{3} & 16 & -4 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\frac{131072}{45} & - \frac{4096}{15} & \frac{128}{3} & - \frac{32}{3} & 4 & -2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\- \frac{131072}{315} & \frac{2048}{45} & - \frac{128}{15} & \frac{8}{3} & - \frac{4}{3} & 1 & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 & 1 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{4}{3} & 4 & 4 & 1 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{8}{3} & \frac{32}{3} & 16 & 8 & 1 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{128}{15} & \frac{128}{3} & \frac{256}{3} & 64 & 16 & 1 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{2048}{45} & \frac{4096}{15} & \frac{2048}{3} & \frac{2048}{3} & 256 & 32 & 1 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{131072}{315} & \frac{131072}{45} & \frac{131072}{15} & \frac{32768}{3} & \frac{16384}{3} & 1024 & 64 & 1\end{array}\right]$$

In [18]:
H_expt.rhs.subs({h[0]:1,h[1]:3,h[2]:6, h[3]:10,h[4]:15,h[5]:21,h[6]:28})


Out[18]:
$$\left[\begin{array}{cccccccccccccccc}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\-28 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\294 & -21 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\-1470 & \frac{315}{2} & -15 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\3675 & -525 & 75 & -10 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\-4410 & \frac{1575}{2} & -150 & 30 & -6 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\2205 & - \frac{945}{2} & \frac{225}{2} & -30 & 9 & -3 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\-315 & \frac{315}{4} & - \frac{45}{2} & \frac{15}{2} & -3 & \frac{3}{2} & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{3}{2} & 3 & 1 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 9 & 6 & 1 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{15}{2} & 30 & 30 & 10 & 1 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{45}{2} & \frac{225}{2} & 150 & 75 & 15 & 1 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{315}{4} & \frac{945}{2} & \frac{1575}{2} & 525 & \frac{315}{2} & 21 & 1 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 315 & 2205 & 4410 & 3675 & 1470 & 294 & 28 & 1\end{array}\right]$$

Areated generation matrix $\mathcal{H}$


In [19]:
m = 16
H = define(Symbol(r'\mathcal{{H}}_{{ {} }}'.format(m)), 
           Matrix(m, m, lambda n, k: 0 if n==m/2 or n!=k+2 else -h[-(k-(m/S(2))+3)] if n<m/2 else h[k-(m/S(2))+1]))
H


Out[19]:
$$\mathcal{H}_{ 16 } = \left[\begin{array}{cccccccccccccccc}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\- h_{5} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & - h_{4} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & - h_{3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & - h_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & - h_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & - h_{0} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & h_{0} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & h_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & h_{2} & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & h_{3} & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & h_{4} & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & h_{5} & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & h_{6} & 0 & 0\end{array}\right]$$

In [20]:
f = Function('E')
f_expt = define(let=f(z), be=exp(z))
f_expt


Out[20]:
$$E{\left (z \right )} = e^{z}$$

In [21]:
eigendata = spectrum(H)
eigendata


Out[21]:
$$\sigma{\left (\mathcal{H}_{ 16 } \right )} = \left ( \left \{ 1 : \left ( \lambda_{1}, \quad m_{1}\right )\right \}, \quad \left \{ \lambda_{1} : 0\right \}, \quad \left \{ m_{1} : 16\right \}\right )$$

In [22]:
data, eigenvals, multiplicities = eigendata.rhs

In [23]:
Phi_polynomials = component_polynomials(eigendata, early_eigenvals_subs=True)
Phi_polynomials


Out[23]:
$$\left \{ \left ( 1, \quad 1\right ) : \Phi_{ 1, 1 }{\left (z \right )} = 1, \quad \left ( 1, \quad 2\right ) : \Phi_{ 1, 2 }{\left (z \right )} = z, \quad \left ( 1, \quad 3\right ) : \Phi_{ 1, 3 }{\left (z \right )} = \frac{z^{2}}{2}, \quad \left ( 1, \quad 4\right ) : \Phi_{ 1, 4 }{\left (z \right )} = \frac{z^{3}}{6}, \quad \left ( 1, \quad 5\right ) : \Phi_{ 1, 5 }{\left (z \right )} = \frac{z^{4}}{24}, \quad \left ( 1, \quad 6\right ) : \Phi_{ 1, 6 }{\left (z \right )} = \frac{z^{5}}{120}, \quad \left ( 1, \quad 7\right ) : \Phi_{ 1, 7 }{\left (z \right )} = \frac{z^{6}}{720}, \quad \left ( 1, \quad 8\right ) : \Phi_{ 1, 8 }{\left (z \right )} = \frac{z^{7}}{5040}, \quad \left ( 1, \quad 9\right ) : \Phi_{ 1, 9 }{\left (z \right )} = \frac{z^{8}}{40320}, \quad \left ( 1, \quad 10\right ) : \Phi_{ 1, 10 }{\left (z \right )} = \frac{z^{9}}{362880}, \quad \left ( 1, \quad 11\right ) : \Phi_{ 1, 11 }{\left (z \right )} = \frac{z^{10}}{3628800}, \quad \left ( 1, \quad 12\right ) : \Phi_{ 1, 12 }{\left (z \right )} = \frac{z^{11}}{39916800}, \quad \left ( 1, \quad 13\right ) : \Phi_{ 1, 13 }{\left (z \right )} = \frac{z^{12}}{479001600}, \quad \left ( 1, \quad 14\right ) : \Phi_{ 1, 14 }{\left (z \right )} = \frac{z^{13}}{6227020800}, \quad \left ( 1, \quad 15\right ) : \Phi_{ 1, 15 }{\left (z \right )} = \frac{z^{14}}{87178291200}, \quad \left ( 1, \quad 16\right ) : \Phi_{ 1, 16 }{\left (z \right )} = \frac{z^{15}}{1307674368000}\right \}$$

In [24]:
g_expt = Hermite_interpolation_polynomial(f_expt, eigendata, Phi_polynomials)
#g_expt

In [25]:
g_expt = g_expt.subs(eigenvals)
g_expt


Out[25]:
$$\operatorname{E_{ 16 }}{\left (z \right )} = \frac{z^{15}}{1307674368000} + \frac{z^{14}}{87178291200} + \frac{z^{13}}{6227020800} + \frac{z^{12}}{479001600} + \frac{z^{11}}{39916800} + \frac{z^{10}}{3628800} + \frac{z^{9}}{362880} + \frac{z^{8}}{40320} + \frac{z^{7}}{5040} + \frac{z^{6}}{720} + \frac{z^{5}}{120} + \frac{z^{4}}{24} + \frac{z^{3}}{6} + \frac{z^{2}}{2} + z + 1$$

In [26]:
with lift_to_matrix_function(g_expt) as G_expt:
    H_expt = G_expt(H)
H_expt


Out[26]:
$$\operatorname{E_{ 16 }}{\left (\mathcal{H}_{ 16 } \right )} = \left[\begin{array}{cccccccccccccccc}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\- h_{5} & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & - h_{4} & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\frac{h_{3} h_{5}}{2} & 0 & - h_{3} & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & \frac{h_{2} h_{4}}{2} & 0 & - h_{2} & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\- \frac{h_{1} h_{3}}{6} h_{5} & 0 & \frac{h_{1} h_{3}}{2} & 0 & - h_{1} & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & - \frac{h_{0} h_{2}}{6} h_{4} & 0 & \frac{h_{0} h_{2}}{2} & 0 & - h_{0} & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & - \frac{h_{0}^{2} h_{2}}{24} h_{4} & 0 & \frac{h_{0}^{2} h_{2}}{6} & 0 & - \frac{h_{0}^{2}}{2} & 0 & h_{0} & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & h_{1} & 0 & 1 & 0 & 0 & 0 & 0 & 0\\0 & - \frac{h_{0}^{2} h_{2}^{2}}{120} h_{4} & 0 & \frac{h_{0}^{2} h_{2}^{2}}{24} & 0 & - \frac{h_{0}^{2} h_{2}}{6} & 0 & \frac{h_{0} h_{2}}{2} & 0 & h_{2} & 0 & 1 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{h_{1} h_{3}}{2} & 0 & h_{3} & 0 & 1 & 0 & 0 & 0\\0 & - \frac{h_{0}^{2} h_{2}^{2}}{720} h_{4}^{2} & 0 & \frac{h_{0}^{2} h_{2}^{2}}{120} h_{4} & 0 & - \frac{h_{0}^{2} h_{2}}{24} h_{4} & 0 & \frac{h_{0} h_{2}}{6} h_{4} & 0 & \frac{h_{2} h_{4}}{2} & 0 & h_{4} & 0 & 1 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{h_{1} h_{3}}{6} h_{5} & 0 & \frac{h_{3} h_{5}}{2} & 0 & h_{5} & 0 & 1 & 0\\0 & - \frac{h_{0}^{2} h_{2}^{2}}{5040} h_{4}^{2} h_{6} & 0 & \frac{h_{0}^{2} h_{2}^{2}}{720} h_{4} h_{6} & 0 & - \frac{h_{0}^{2} h_{2}}{120} h_{4} h_{6} & 0 & \frac{h_{0} h_{2}}{24} h_{4} h_{6} & 0 & \frac{h_{2} h_{4}}{6} h_{6} & 0 & \frac{h_{4} h_{6}}{2} & 0 & h_{6} & 0 & 1\end{array}\right]$$

In [27]:
H_expt.rhs.subs({h[0]:1,h[1]:2,h[2]:3, h[3]:4,h[4]:5,h[5]:6,h[6]:7})


Out[27]:
$$\left[\begin{array}{cccccccccccccccc}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\-6 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & -5 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\12 & 0 & -4 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & \frac{15}{2} & 0 & -3 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\-8 & 0 & 4 & 0 & -2 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & - \frac{5}{2} & 0 & \frac{3}{2} & 0 & -1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & - \frac{5}{8} & 0 & \frac{1}{2} & 0 & - \frac{1}{2} & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\0 & - \frac{3}{8} & 0 & \frac{3}{8} & 0 & - \frac{1}{2} & 0 & \frac{3}{2} & 0 & 3 & 0 & 1 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 0 & 4 & 0 & 1 & 0 & 0 & 0\\0 & - \frac{5}{16} & 0 & \frac{3}{8} & 0 & - \frac{5}{8} & 0 & \frac{5}{2} & 0 & \frac{15}{2} & 0 & 5 & 0 & 1 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 12 & 0 & 6 & 0 & 1 & 0\\0 & - \frac{5}{16} & 0 & \frac{7}{16} & 0 & - \frac{7}{8} & 0 & \frac{35}{8} & 0 & \frac{35}{2} & 0 & \frac{35}{2} & 0 & 7 & 0 & 1\end{array}\right]$$