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Companion to "Pfaffian Point Processes for Two Classes of Random Plane Partitions"

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Computation of Matrices for the TSSCPP Point Process

Consider the Pfaffian point process Xn\mathscr X_n for order-nn TSSCPP's. The code below computes the matrices MnM_n, Mn−1M_n^{-1}, and KnK_n associated with the Pfaffian point process for order-nn TSSCPP's. Recall that Mn=(mi,j)i,j=δn−1M_n = (m_{i,j})_{i,j = \delta}^{n-1}, where δ\delta is the residue of nn modulo 22 and where mi,j=∑x,y=δ2n−2ϕi(x) ϕj(y) ϵ(x,y) m_{i,j} = \sum_{x,y = \delta}^{2n-2} \phi_i(x) \, \phi_j(y) \, \epsilon(x,y) for ϕk(x)=(kx−k)\phi_k(x) = \binom{k}{x-k} and ϵ(x,y)=sgn⁡(y−x)\epsilon(x,y) = \operatorname{sgn}(y-x). Then the kernel Kn=(Kn(x,y))x,y=12n−2K_n = (K_n(x,y))_{x,y = 1}^{2n-2} is the 4n−4×4n−44n-4 \times 4n-4 matrix with blocks Kn(x,y)=(∑i,j=δn−1ϕi(x) ϕj(y) m~j,i∑i,j=δn−1ϕi(x) (ϵϕj)(y) m~j,i∑i,j=δn−1(ϵϕi)(x) ϕj(y) m~j,i−ϵ(x,y)+∑i,j=δn−1(ϵϕi)(x) (ϵϕj)(y) m~j,i). K_n(x,y) = \begin{pmatrix} \sum_{i,j = \delta}^{n-1} \phi_i(x) \, \phi_j(y) \, \tilde m_{j,i} & \sum_{i,j = \delta}^{n-1} \phi_i(x) \, (\epsilon \phi_j)(y) \, \tilde m_{j,i} \\ \sum_{i,j = \delta}^{n-1} (\epsilon \phi_i)(x) \, \phi_j(y) \, \tilde m_{j,i} & -\epsilon(x,y) + \sum_{i,j = \delta}^{n-1} (\epsilon \phi_i)(x) \, (\epsilon \phi_j)(y) \, \tilde m_{j,i} \end{pmatrix} . Here, m~j,i\tilde m_{j,i} denotes entry (j,i)(j,i) of matrix Mn−1M_n^{-1}, and (ϵϕk)(x)=∑z=δ2n−2sgn⁡(z−x) ϕk(z). (\epsilon \phi_k)(x) = \sum_{z = \delta}^{2n-2} \operatorname{sgn}(z-x) \, \phi_k(z) . Note that for easier computation, we have the following identity for the entries of the matrix MnM_n: mi,j={∑x=2i−j+12j−i(i+jx)if i<j−∑x=2j−i+12i−j(i+jx)if j<i0if i=j. m_{i,j} = \begin{cases} \sum_{x = 2i - j + 1}^{2j - i} \binom{i+j}{x} & \text{if } i < j \\ -\sum_{x = 2j - i + 1}^{2i - j} \binom{i+j}{x} & \text{if } j < i \\ 0 & \text{if } i=j . \end{cases}

1. Computing and Saving the Values ϕk(x)\phi_k(x)

Let Φ=(ϕk(x))k≥0,x≥0\Phi = (\phi_k(x))_{k \geq 0, x \geq 0} be the array containing all values ϕk(x)\phi_k(x), where kk indexes rows and xx indexes columns. Note that this array contains Pascal's triangle, since Φ=(1111211331⋱⋱) \Phi = \begin{pmatrix} 1 & & & & & & & \\ & 1 & 1 & & & & & \\ & & 1 & 2 & 1 & & & \\ & & & 1 & 3 & 3 & 1 \\ & & & & \ddots & & & \ddots \end{pmatrix} Let Φn=(ϕk(x))0≤k≤n−1,0≤x≤2n−2\Phi_n = (\phi_k(x))_{0 \leq k \leq n-1, 0 \leq x \leq 2n-2} denote the truncation of Φ\Phi. Below, we create and save the array ΦN,N=(ϕk(x))0≤k≤N−1,0≤x≤2N−2\Phi_{N,N} = (\phi_k(x))_{0 \leq k \leq N-1, 0 \leq x \leq 2N-2}, where NN is the index of the largest correlation kernel KNK_N we wish to compute.

def phi(x, k):
    return(binomial(k, x-k))

def eps(x, y):
    return(sgn(y-x))

def epsphi(x, k, n):
    delta = n % 2
    return(sum(eps(y, z) * phi(z, k) for z in [delta .. n+m]))

# define functions phi and epsphi

print('Done')
Done 
import os
load('mat_to_list.sage')

N = 100
Phi = Matrix(ZZ, [[phi(x, k) for x in [0 .. 2*N-2]] for k in [0 .. N-1]])

path = 'tsscpp_data/'
file_name = 'Phi_{0}'.format(N)
text_file = open(os.path.join(path, file_name), 'w')
text_file.write(str(mat_to_list(Phi)))
text_file.close()

# creates a text file 'Phi_100.txt' in the directory 'tsscpp_data/'

print('Done')
Done

2. Computing and Saving the Values (ϵϕk)(x)(\epsilon \phi_k)(x)

With the array ΦN\Phi_N saved, we now create arrays EΦn=((ϵϕk)(x))δ≤k≤n−1,1≤x≤2n−2E \Phi_n = ((\epsilon \phi_k)(x))_{\delta \leq k \leq n-1, 1 \leq x \leq 2n-2}, which are needed for computation of the matrices KnK_n. Note that (ϵϕk)(x)=∑z=δ2n−2ϵ(x,z) ϕk(z) (\epsilon \phi_k)(x) = \sum_{z = \delta}^{2n-2} \epsilon(x,z) \, \phi_k(z) Depends on the order nn of the TSSCPP point process, and that EΦn=(ϕk(z))δ≤k≤n−1,δ≤z≤2n−2(ϵ(x,z))1≤x≤2n−2,δ≤z≤2n−2T E \Phi_n = (\phi_k(z))_{\delta \leq k \leq n-1, \delta \leq z \leq 2n-2} (\epsilon(x,z))_{1 \leq x \leq 2n-2, \delta \leq z \leq 2n-2}^T Below, we compute all arrays EΦnE \Phi_n for 1≤n≤N1 \leq n \leq N, where N=100N = 100 was chosen previously.

N = 100

Phi_text = open(os.path.join('tsscpp_data/', 'Phi_{0}'.format(N)), 'r')
Phi_data = sage_eval(Phi_text.readline())
Phi = Matrix(ZZ, Phi_data)
Phi_text.close()

# loads phi values stored in 'Phi_100.txt'

path = 'tsscpp_data/tsscpp_Epsphi'
for n in [1 .. N]:
    delta = n % 2
    Eps = Matrix(ZZ, [[eps(x, z) for z in [delta .. 2*n-2]] for x in [1 .. 2*n-2]])
    Epsphi = Phi[delta:n-1+1, delta:2*n-2+1] * Eps.transpose()

    file_name = 'Epsphi_{0}'.format(n)
    text_file = open(os.path.join(path, file_name), 'w')
    text_file.write(str(mat_to_list(Epsphi)))
    text_file.close()
    if n % 5 == 0:
        print('n={0} completed'.format(n))

# creates a series of files 'Epsphi_n.txt' in the directory 'tsscpp_data/tsscpp_Epsphi'

print('Done')
n=5 completed n=10 completed n=15 completed n=20 completed n=25 completed n=30 completed n=35 completed n=40 completed n=45 completed n=50 completed n=55 completed n=60 completed n=65 completed n=70 completed n=75 completed n=80 completed n=85 completed n=90 completed n=95 completed n=100 completed Done

3. Computing and Saving the Matrices MnM_n

Recall that Mn=(mi,j)i,j=δn−1=(∑x=δ2n−2∑y=δ2n−2ϵ(x,y) ϕi(x) ϕj(y))i,j=δn−1=(ϕi(x))i,x(ϵ(x,y))x,y(ϕj(y))y,j M_n = (m_{i,j})_{i,j = \delta}^{n-1} = \left( \sum_{x = \delta}^{2n-2} \sum_{y = \delta}^{2n-2} \epsilon(x,y) \, \phi_i(x) \, \phi_j(y) \right)_{i,j = \delta}^{n-1} = (\phi_i(x))_{i,x} (\epsilon(x,y))_{x,y} (\phi_j(y))_{y,j} where on the right-hand side ii and jj run from δ\delta to n−1n-1, and where xx and yy run from δ\delta to 2n−22n-2. We now compute and save the matrices MnM_n for 1≤n≤N1 \leq n \leq N, where N=100N = 100.

def make_M(n, eps, Phi_arr):
    delta = n % 2
    Eps = Matrix(ZZ, [[eps(x, y) for y in [delta .. 2*n-2]] for x in [delta .. 2*n-2]])
    Phi = Phi_arr[delta:n-1+1, delta:2*n-2+1]
    return(Phi * Eps * Phi.transpose())

# creates M_(m,n) using the function eps and the complete array of phi values Phi_arr

N = 100
path = 'tsscpp_data/'
file_name = 'Phi_{0}'.format(N)

text_file = open(os.path.join(path, file_name), 'r')
Phi_arr_data = sage_eval(text_file.readline())
Phi_arr = Matrix(ZZ, Phi_arr_data)
text_file.close()

# loads phi values from 'Phi_100.txt'

path = 'tsscpp_data/tsscpp_M/'
for n in [1 .. N]:
    M = make_M(n, eps, Phi_arr)
    file_name = 'M_{0}'.format(n)
    text_file = open(os.path.join(path, file_name), 'w')
    text_file.write(str(mat_to_list(M)))
    text_file.close()
    if n % 5 == 0:
        print('n={0} completed'.format(n))

print('Done')
n=5 completed n=10 completed n=15 completed n=20 completed n=25 completed n=30 completed n=35 completed n=40 completed n=45 completed n=50 completed n=55 completed n=60 completed n=65 completed n=70 completed n=75 completed n=80 completed n=85 completed n=90 completed n=95 completed n=100 completed Done

4. Computing and Saving the Matrices Mn−1M_n^{-1}

By simply loading the matrices MnM_n saved in the last step, we now compute the matrices Mn−1M_n^{-1} for 1≤n≤N=1001 \leq n \leq N = 100.

N = 100

for n in [1 .. N]:
    path = 'tsscpp_data/tsscpp_M/'
    file_name = 'M_{0}'.format(n)
    text_file = open(os.path.join(path, file_name), 'r')
    M_data = sage_eval(text_file.readline())
    M = Matrix(ZZ, M_data)
    text_file.close()

    path = 'tsscpp_data/tsscpp_M_inv'
    file_name = 'M_inv_{0}'.format(n)
    text_file = open(os.path.join(path, file_name), 'w')
    text_file.write(str(mat_to_list(M.inverse())))
    text_file.close()
    if n % 5 == 0:
        print('n={0} completed'.format(n))

print('Done')
n=5 completed n=10 completed n=15 completed n=20 completed n=25 completed n=30 completed n=35 completed n=40 completed n=45 completed n=50 completed n=55 completed n=60 completed n=65 completed n=70 completed n=75 completed n=80 completed n=85 completed n=90 completed n=95 completed n=100 completed Done

5. Computing and Saving the Kernels KnK_n

Our final step is to compute the kernels KnK_n for 1≤n≤N=1001 \leq n \leq N = 100. Recalling that Kn=(Kn(x,y))x,y=12n−2K_n = (K_n(x,y))_{x,y = 1}^{2n-2}, we construct four separate arrays (Kn(x,y)α,β)x,y=12n−2(K_n(x,y)_{\alpha, \beta})_{x,y = 1}^{2n-2}, where 1≤α,β≤21 \leq \alpha, \beta \leq 2. Notice that (Kn(x,y)1,1)x,y=12n−2=(∑i=δn−1∑j=δn−1ϕi(x) ϕj(y) m~j,i)x,y=12n−2=(ϕi(x))i,xTMn−T(ϕj(y))j,y (K_n(x,y)_{1,1})_{x,y = 1}^{2n-2} = \left( \sum_{i=\delta}^{n-1} \sum_{j=\delta}^{n-1} \phi_i(x) \, \phi_j(y) \, \tilde m_{j,i} \right)_{x,y = 1}^{2n-2} = (\phi_i(x))_{i,x}^T M_n^{-T} (\phi_j(y))_{j,y} where Mn−TM_n^{-T} denotes the transpose of Mn−1M_n^{-1}. On the right-hand side, ii and jj run from δ\delta to n−1n-1, and xx and yy run from 11 to 2n−22n-2. Similarly, we have (Kn(x,y)1,2)x,y=12n−2=(ϕi(x))i,xTMn−T((ϵϕj)(y))j,y(Kn(x,y)2,1)x,y=12n−2=((ϵϕi)(x))i,xTMn−T(ϕj(y))j,y(Kn(x,y)2,2)x,y=12n−2=−(ϵ(x,y))x,y=12n−2+((ϵϕi)(x))i,xTMn−T((ϵϕj)(y))j,y\begin{align*} (K_n(x,y)_{1,2})_{x,y = 1}^{2n-2} &= (\phi_i(x))_{i,x}^T M_n^{-T} ((\epsilon \phi_j)(y))_{j,y} \\ (K_n(x,y)_{2,1})_{x,y = 1}^{2n-2} &= ((\epsilon \phi_i)(x))_{i,x}^T M_n^{-T} (\phi_j(y))_{j,y} \\ (K_n(x,y)_{2,2})_{x,y = 1}^{2n-2} &= -(\epsilon(x,y))_{x,y = 1}^{2n-2} + ((\epsilon \phi_i)(x))_{i,x}^T M_n^{-T} ((\epsilon \phi_j)(y))_{j,y} \end{align*} These four arrays are then combined to form KnK_n.

def make_K(n, N):
    delta = n % 2

    Phi_text = open(os.path.join('tsscpp_data/', 'Phi_{0}'.format(N)), 'r')
    Phi_data = sage_eval(Phi_text.readline())
    Phi_arr = Matrix(ZZ, Phi_data)
    Phi = Phi_arr[delta:n-1+1, 1:2*n-2+1]
    Phi_text.close()

    Epsphi_text = open(os.path.join('tsscpp_data/tsscpp_Epsphi/', 'Epsphi_{0}'.format(n)), 'r')
    Epsphi_data = sage_eval(Epsphi_text.readline())
    Epsphi = Matrix(ZZ, Epsphi_data)
    Epsphi_text.close()

    Eps = Matrix(ZZ, [[eps(x, y) for y in [1 .. 2*n-2]] for x in [1 .. 2*n-2]])

    M_inv_text = open(os.path.join('tsscpp_data/tsscpp_M_inv/', 'M_inv_{0}'.format(n)))
    M_inv_data = sage_eval(M_inv_text.readline())
    M_inv = Matrix(QQ, M_inv_data)
    M_inv_text.close()

    # loads values of phi, epsphi, eps, and the inverse of matrix M_n

    K_11 = Phi.transpose() * M_inv.transpose() * Phi
    K_12 = Phi.transpose() * M_inv.transpose() * Epsphi
    K_21 = Epsphi.transpose() * M_inv.transpose() * Phi
    K_22 = -Eps + Epsphi.transpose() * M_inv.transpose() * Epsphi

    # computes the four arrays of entries of K_n

    L1 = [0 .. 2*(2*n-2-1), step=2]
    L2 = [1 .. 2*(2*n-2-1) + 1, step=2]
    K = Matrix(QQ, 2*(2*n-2))

    K[L1,L1] = K_11
    K[L1,L2] = K_12
    K[L2,L1] = K_21
    K[L2,L2] = K_22

    # combines the four arrays into K_n

    return(K)

print('Done')
Done 
N = 100

for n in [1 .. N]:
    K = make_K(n, N)
    K_file = open(os.path.join('tsscpp_data/tsscpp_K/', 'K_{0}').format(n), 'w')
    K_file.write(str(mat_to_list(K)))
    K_file.close()
    if n % 5 == 0:
        print('n={0} completed'.format(n))

print('Done')
n=5 completed n=10 completed n=15 completed n=20 completed n=25 completed n=30 completed n=35 completed n=40 completed n=45 completed n=50 completed n=55 completed n=60 completed n=65 completed n=70 completed n=75 completed n=80 completed n=85 completed n=90 completed n=95 completed n=100 completed Done

Loading and Viewing Matrix Data

Below, we access and print a few of the matrices MnM_n, Mn−1M_n^{-1}, and KnK_n:

def retrieve_mat(name, n):
    if name == 'M' or name == 'M_inv' or name == 'K':
        mat_file = open('tsscpp_data/tsscpp_{0}/{0}_{1}'.format(name, n), 'r')
        mat_data = sage_eval(mat_file.readline())
        mat = Matrix(QQ, mat_data)
        mat_file.close()
        return(mat)
    else:
        print("Error: first argument must be 'M', 'M_inv', or 'K'")
        return

# reads in the matrix M_(n,m), M_inv_(n,m), or K_(n,m)

print('Done')
Done 
a = 2
b = 8

print('Matrices M_{0} through M_{1} are as follows:\n'.format(a, b))

for n in [a .. b]:
    M = retrieve_mat('M', n)
    print('M_{0} ='.format(n))
    show(M)
Matrices M_2 through M_8 are as follows: M_2 =
(02−20)\displaystyle \left(\begin{array}{rr} 0 & 2 \\ -2 & 0 \end{array}\right)
M_3 =
(07−70)\displaystyle \left(\begin{array}{rr} 0 & 7 \\ -7 & 0 \end{array}\right)
M_4 =
(0248−20716−4−7025−8−16−250)\displaystyle \left(\begin{array}{rrrr} 0 & 2 & 4 & 8 \\ -2 & 0 & 7 & 16 \\ -4 & -7 & 0 & 25 \\ -8 & -16 & -25 & 0 \end{array}\right)
M_5 =
(071632−702563−16−25091−32−63−910)\displaystyle \left(\begin{array}{rrrr} 0 & 7 & 16 & 32 \\ -7 & 0 & 25 & 63 \\ -16 & -25 & 0 & 91 \\ -32 & -63 & -91 & 0 \end{array}\right)
M_6 =
(02481632−207163264−4−702563128−8−16−25091246−16−32−63−910336−32−64−128−246−3360)\displaystyle \left(\begin{array}{rrrrrr} 0 & 2 & 4 & 8 & 16 & 32 \\ -2 & 0 & 7 & 16 & 32 & 64 \\ -4 & -7 & 0 & 25 & 63 & 128 \\ -8 & -16 & -25 & 0 & 91 & 246 \\ -16 & -32 & -63 & -91 & 0 & 336 \\ -32 & -64 & -128 & -246 & -336 & 0 \end{array}\right)
M_7 =
(07163264128−702563128256−16−25091246511−32−63−910336957−64−128−246−33601254−128−256−511−957−12540)\displaystyle \left(\begin{array}{rrrrrr} 0 & 7 & 16 & 32 & 64 & 128 \\ -7 & 0 & 25 & 63 & 128 & 256 \\ -16 & -25 & 0 & 91 & 246 & 511 \\ -32 & -63 & -91 & 0 & 336 & 957 \\ -64 & -128 & -246 & -336 & 0 & 1254 \\ -128 & -256 & -511 & -957 & -1254 & 0 \end{array}\right)
M_8 =
(0248163264128−207163264128256−4−702563128256512−8−16−250912465111024−16−32−63−9103369572035−32−64−128−246−336012543718−64−128−256−511−957−125404719−128−256−512−1024−2035−3718−47190)\displaystyle \left(\begin{array}{rrrrrrrr} 0 & 2 & 4 & 8 & 16 & 32 & 64 & 128 \\ -2 & 0 & 7 & 16 & 32 & 64 & 128 & 256 \\ -4 & -7 & 0 & 25 & 63 & 128 & 256 & 512 \\ -8 & -16 & -25 & 0 & 91 & 246 & 511 & 1024 \\ -16 & -32 & -63 & -91 & 0 & 336 & 957 & 2035 \\ -32 & -64 & -128 & -246 & -336 & 0 & 1254 & 3718 \\ -64 & -128 & -256 & -511 & -957 & -1254 & 0 & 4719 \\ -128 & -256 & -512 & -1024 & -2035 & -3718 & -4719 & 0 \end{array}\right)
a = 2
b = 8

print('Matrices M_inv_{0} through M_inv_{1} are as follows:\n'.format(a, b))

for n in [a .. b]:
    M_inv = retrieve_mat('M_inv', n)
    print('M_inv_{0} ='.format(n))
    show(M_inv)
Matrices M_inv_2 through M_inv_8 are as follows: M_inv_2 =
(0−12120)\displaystyle \left(\begin{array}{rr} 0 & -\frac{1}{2} \\ \frac{1}{2} & 0 \end{array}\right)
M_inv_3 =
(0−17170)\displaystyle \left(\begin{array}{rr} 0 & -\frac{1}{7} \\ \frac{1}{7} & 0 \end{array}\right)
M_inv_4 =
(0−2542821−1625420−421221−8214210−12116−2211210)\displaystyle \left(\begin{array}{rrrr} 0 & -\frac{25}{42} & \frac{8}{21} & -\frac{1}{6} \\ \frac{25}{42} & 0 & -\frac{4}{21} & \frac{2}{21} \\ -\frac{8}{21} & \frac{4}{21} & 0 & -\frac{1}{21} \\ \frac{1}{6} & -\frac{2}{21} & \frac{1}{21} & 0 \end{array}\right)
M_inv_5 =
(0−73321143−254297330−3242916429−21143324290−742925429−1642974290)\displaystyle \left(\begin{array}{rrrr} 0 & -\frac{7}{33} & \frac{21}{143} & -\frac{25}{429} \\ \frac{7}{33} & 0 & -\frac{32}{429} & \frac{16}{429} \\ -\frac{21}{143} & \frac{32}{429} & 0 & -\frac{7}{429} \\ \frac{25}{429} & -\frac{16}{429} & \frac{7}{429} & 0 \end{array}\right)
M_inv_6 =
(0−17528664143−41349286−3521752860−321433281859−19018595143−64143321430−16818591231859−7286413−328185916818590−5618595338−492861901859−12318595618590−213718352−51437286−53382137180)\displaystyle \left(\begin{array}{rrrrrr} 0 & -\frac{175}{286} & \frac{64}{143} & -\frac{4}{13} & \frac{49}{286} & -\frac{3}{52} \\ \frac{175}{286} & 0 & -\frac{32}{143} & \frac{328}{1859} & -\frac{190}{1859} & \frac{5}{143} \\ -\frac{64}{143} & \frac{32}{143} & 0 & -\frac{168}{1859} & \frac{123}{1859} & -\frac{7}{286} \\ \frac{4}{13} & -\frac{328}{1859} & \frac{168}{1859} & 0 & -\frac{56}{1859} & \frac{5}{338} \\ -\frac{49}{286} & \frac{190}{1859} & -\frac{123}{1859} & \frac{56}{1859} & 0 & -\frac{21}{3718} \\ \frac{3}{52} & -\frac{5}{143} & \frac{7}{286} & -\frac{5}{338} & \frac{21}{3718} & 0 \end{array}\right)
M_inv_7 =
(0−31311195746−761574617247−17583983130−28828732482873−12247644199−1119574628828730−23157466571218348−4441997615746−248287323157460−2681218348498398−1724712247−657121834826812183480−33167961758398−644199444199−49839833167960)\displaystyle \left(\begin{array}{rrrrrr} 0 & -\frac{3}{13} & \frac{1119}{5746} & -\frac{761}{5746} & \frac{17}{247} & -\frac{175}{8398} \\ \frac{3}{13} & 0 & -\frac{288}{2873} & \frac{248}{2873} & -\frac{12}{247} & \frac{64}{4199} \\ -\frac{1119}{5746} & \frac{288}{2873} & 0 & -\frac{231}{5746} & \frac{6571}{218348} & -\frac{44}{4199} \\ \frac{761}{5746} & -\frac{248}{2873} & \frac{231}{5746} & 0 & -\frac{2681}{218348} & \frac{49}{8398} \\ -\frac{17}{247} & \frac{12}{247} & -\frac{6571}{218348} & \frac{2681}{218348} & 0 & -\frac{33}{16796} \\ \frac{175}{8398} & -\frac{64}{4199} & \frac{44}{4199} & -\frac{49}{8398} & \frac{33}{16796} & 0 \end{array}\right)
M_inv_8 =
(0−2134817−3025839810784199−31924323−1364621340−4171476271383−109567138331323−25054914323−8174170−762371383717271383−223231845491−332330258398−14762713837623713830−33887138313673319124−6354927125547510982−107841991095671383−7172713833388713830−5803319124367992712554−4910982319−3132322323−1367331912458033191240−12324548146137−243232505491−1845491635492712554−367992712554123245480−14320865813646−43233323−75109824910982−1461371432086580)\displaystyle \left(\begin{array}{rrrrrrrr} 0 & -\frac{21}{34} & \frac{8}{17} & -\frac{3025}{8398} & \frac{1078}{4199} & -\frac{3}{19} & \frac{24}{323} & -\frac{13}{646} \\ \frac{21}{34} & 0 & -\frac{4}{17} & \frac{14762}{71383} & -\frac{10956}{71383} & \frac{31}{323} & -\frac{250}{5491} & \frac{4}{323} \\ -\frac{8}{17} & \frac{4}{17} & 0 & -\frac{7623}{71383} & \frac{7172}{71383} & -\frac{22}{323} & \frac{184}{5491} & -\frac{3}{323} \\ \frac{3025}{8398} & -\frac{14762}{71383} & \frac{7623}{71383} & 0 & -\frac{3388}{71383} & \frac{13673}{319124} & -\frac{63549}{2712554} & \frac{75}{10982} \\ -\frac{1078}{4199} & \frac{10956}{71383} & -\frac{7172}{71383} & \frac{3388}{71383} & 0 & -\frac{5803}{319124} & \frac{36799}{2712554} & -\frac{49}{10982} \\ \frac{3}{19} & -\frac{31}{323} & \frac{22}{323} & -\frac{13673}{319124} & \frac{5803}{319124} & 0 & -\frac{123}{24548} & \frac{14}{6137} \\ -\frac{24}{323} & \frac{250}{5491} & -\frac{184}{5491} & \frac{63549}{2712554} & -\frac{36799}{2712554} & \frac{123}{24548} & 0 & -\frac{143}{208658} \\ \frac{13}{646} & -\frac{4}{323} & \frac{3}{323} & -\frac{75}{10982} & \frac{49}{10982} & -\frac{14}{6137} & \frac{143}{208658} & 0 \end{array}\right)
a = 2
b = 5

print('Kernels K_{0} through K_{1} are as follows:\n'.format(a, b))

for n in [a .. b]:
    K = retrieve_mat('K', n)
    print('K_{0} ='.format(n))
    show(K)
Kernels K_2 through K_5 are as follows: K_2 =
(012012−120−120012012−120−120)\displaystyle \left(\begin{array}{rrrr} 0 & \frac{1}{2} & 0 & \frac{1}{2} \\ -\frac{1}{2} & 0 & -\frac{1}{2} & 0 \\ 0 & \frac{1}{2} & 0 & \frac{1}{2} \\ -\frac{1}{2} & 0 & -\frac{1}{2} & 0 \end{array}\right)
K_3 =
(047173727017−37−470−370271717−27−1737047272717−17−370−470−27−17−1727−27−27−27270470470−17−2717−470−27−17−17−17−1717027027372717−27−4717−270)\displaystyle \left(\begin{array}{rrrrrrrr} 0 & \frac{4}{7} & \frac{1}{7} & \frac{3}{7} & \frac{2}{7} & 0 & \frac{1}{7} & -\frac{3}{7} \\ -\frac{4}{7} & 0 & -\frac{3}{7} & 0 & \frac{2}{7} & \frac{1}{7} & \frac{1}{7} & -\frac{2}{7} \\ -\frac{1}{7} & \frac{3}{7} & 0 & \frac{4}{7} & \frac{2}{7} & \frac{2}{7} & \frac{1}{7} & -\frac{1}{7} \\ -\frac{3}{7} & 0 & -\frac{4}{7} & 0 & -\frac{2}{7} & -\frac{1}{7} & -\frac{1}{7} & \frac{2}{7} \\ -\frac{2}{7} & -\frac{2}{7} & -\frac{2}{7} & \frac{2}{7} & 0 & \frac{4}{7} & 0 & \frac{4}{7} \\ 0 & -\frac{1}{7} & -\frac{2}{7} & \frac{1}{7} & -\frac{4}{7} & 0 & -\frac{2}{7} & -\frac{1}{7} \\ -\frac{1}{7} & -\frac{1}{7} & -\frac{1}{7} & \frac{1}{7} & 0 & \frac{2}{7} & 0 & \frac{2}{7} \\ \frac{3}{7} & \frac{2}{7} & \frac{1}{7} & -\frac{2}{7} & -\frac{4}{7} & \frac{1}{7} & -\frac{2}{7} & 0 \end{array}\right)
K_4 =
(02542421174227−114−221−1142−27542−22112−25420−174201342221−142−421−314121−11413−421174202542131142121−542−17−142−12116−17420−25420−1342−221142421314−121114−13−27−1342−1313420914521174227−542221−12114−221−1142221−9140−1142−2211142211420221142−121−142−521114201217514121161142421542−421−1742221−120−514−17−542132731417−314−27−114−17514012012−542−121142121542−221−51417−120−16−13221114121−114−221−142−121542016016−12−13−1613120−16−13−1213−160)\displaystyle \left(\begin{array}{rrrrrrrrrrrr} 0 & \frac{25}{42} & \frac{4}{21} & \frac{17}{42} & \frac{2}{7} & -\frac{1}{14} & -\frac{2}{21} & -\frac{11}{42} & -\frac{2}{7} & \frac{5}{42} & -\frac{2}{21} & \frac{1}{2} \\ -\frac{25}{42} & 0 & -\frac{17}{42} & 0 & \frac{13}{42} & \frac{2}{21} & -\frac{1}{42} & -\frac{4}{21} & -\frac{3}{14} & \frac{1}{21} & -\frac{1}{14} & \frac{1}{3} \\ -\frac{4}{21} & \frac{17}{42} & 0 & \frac{25}{42} & \frac{1}{3} & \frac{11}{42} & \frac{1}{21} & -\frac{5}{42} & -\frac{1}{7} & -\frac{1}{42} & -\frac{1}{21} & \frac{1}{6} \\ -\frac{17}{42} & 0 & -\frac{25}{42} & 0 & -\frac{13}{42} & -\frac{2}{21} & \frac{1}{42} & \frac{4}{21} & \frac{3}{14} & -\frac{1}{21} & \frac{1}{14} & -\frac{1}{3} \\ -\frac{2}{7} & -\frac{13}{42} & -\frac{1}{3} & \frac{13}{42} & 0 & \frac{9}{14} & \frac{5}{21} & \frac{17}{42} & \frac{2}{7} & -\frac{5}{42} & \frac{2}{21} & -\frac{1}{2} \\ \frac{1}{14} & -\frac{2}{21} & -\frac{11}{42} & \frac{2}{21} & -\frac{9}{14} & 0 & -\frac{11}{42} & -\frac{2}{21} & \frac{1}{14} & \frac{2}{21} & \frac{1}{42} & 0 \\ \frac{2}{21} & \frac{1}{42} & -\frac{1}{21} & -\frac{1}{42} & -\frac{5}{21} & \frac{11}{42} & 0 & \frac{1}{2} & \frac{1}{7} & \frac{5}{14} & \frac{1}{21} & \frac{1}{6} \\ \frac{11}{42} & \frac{4}{21} & \frac{5}{42} & -\frac{4}{21} & -\frac{17}{42} & \frac{2}{21} & -\frac{1}{2} & 0 & -\frac{5}{14} & -\frac{1}{7} & -\frac{5}{42} & \frac{1}{3} \\ \frac{2}{7} & \frac{3}{14} & \frac{1}{7} & -\frac{3}{14} & -\frac{2}{7} & -\frac{1}{14} & -\frac{1}{7} & \frac{5}{14} & 0 & \frac{1}{2} & 0 & \frac{1}{2} \\ -\frac{5}{42} & -\frac{1}{21} & \frac{1}{42} & \frac{1}{21} & \frac{5}{42} & -\frac{2}{21} & -\frac{5}{14} & \frac{1}{7} & -\frac{1}{2} & 0 & -\frac{1}{6} & -\frac{1}{3} \\ \frac{2}{21} & \frac{1}{14} & \frac{1}{21} & -\frac{1}{14} & -\frac{2}{21} & -\frac{1}{42} & -\frac{1}{21} & \frac{5}{42} & 0 & \frac{1}{6} & 0 & \frac{1}{6} \\ -\frac{1}{2} & -\frac{1}{3} & -\frac{1}{6} & \frac{1}{3} & \frac{1}{2} & 0 & -\frac{1}{6} & -\frac{1}{3} & -\frac{1}{2} & \frac{1}{3} & -\frac{1}{6} & 0 \end{array}\right)
K_5 =
(020337331333119429−41429−73429−29143−894292514329143739100429−103925429−235429−20330−133304514334429−113−68429−73429384295395642968429−6742917429−152429−7331333020331514291094297429−49429−191431429234293542912143−81433143−23143−13330−20330−45143−344291136842973429−38429−539−56429−6842967429−17429152429−119429−45143−151429451430234514315142992429−76429−86429−82429−100429833−2542922942941429−34429−10942934429−230−109429−3442938429374298429−3143−4429−133−1429−842973429113−7429−113−45143109429024442995429149429539−142920429−764295429−101429291436842949429−68429−15142934429−2444290−149429−121431117442976429−4142919429−136429894297342919143−73429−92429−38429−9542914942902444299842914642928143−121437143−47143−25143−38429−14293842976429−37429−14942912143−2444290−38143−71429−2842971429−7429106429−29143−539−2342953986429−8429−539−111−98429381430212429284291844297429149429−739−56429−35429564298242931431429−74429−14642971429−2124290−140429−739−3542998429−100429−68429−12143684291004294429−20429−76429−2814328429−284291404290561430561431039674298143−67429−833133764294142912143−71429−184429739−561430−14143−73143−25429−17429−314317429254291429−5429−19429−71437429−74293542901414301414323542915242923143−152429−229429842910142913642947143−106429−149429−98429−5614373143−141430)\displaystyle \left(\begin{array}{rrrrrrrrrrrrrrrr} 0 & \frac{20}{33} & \frac{7}{33} & \frac{13}{33} & \frac{119}{429} & -\frac{41}{429} & -\frac{73}{429} & -\frac{29}{143} & -\frac{89}{429} & \frac{25}{143} & \frac{29}{143} & \frac{7}{39} & \frac{100}{429} & -\frac{10}{39} & \frac{25}{429} & -\frac{235}{429} \\ -\frac{20}{33} & 0 & -\frac{13}{33} & 0 & \frac{45}{143} & \frac{34}{429} & -\frac{1}{13} & -\frac{68}{429} & -\frac{73}{429} & \frac{38}{429} & \frac{5}{39} & \frac{56}{429} & \frac{68}{429} & -\frac{67}{429} & \frac{17}{429} & -\frac{152}{429} \\ -\frac{7}{33} & \frac{13}{33} & 0 & \frac{20}{33} & \frac{151}{429} & \frac{109}{429} & \frac{7}{429} & -\frac{49}{429} & -\frac{19}{143} & \frac{1}{429} & \frac{23}{429} & \frac{35}{429} & \frac{12}{143} & -\frac{8}{143} & \frac{3}{143} & -\frac{23}{143} \\ -\frac{13}{33} & 0 & -\frac{20}{33} & 0 & -\frac{45}{143} & -\frac{34}{429} & \frac{1}{13} & \frac{68}{429} & \frac{73}{429} & -\frac{38}{429} & -\frac{5}{39} & -\frac{56}{429} & -\frac{68}{429} & \frac{67}{429} & -\frac{17}{429} & \frac{152}{429} \\ -\frac{119}{429} & -\frac{45}{143} & -\frac{151}{429} & \frac{45}{143} & 0 & \frac{2}{3} & \frac{45}{143} & \frac{151}{429} & \frac{92}{429} & -\frac{76}{429} & -\frac{86}{429} & -\frac{82}{429} & -\frac{100}{429} & \frac{8}{33} & -\frac{25}{429} & \frac{229}{429} \\ \frac{41}{429} & -\frac{34}{429} & -\frac{109}{429} & \frac{34}{429} & -\frac{2}{3} & 0 & -\frac{109}{429} & -\frac{34}{429} & \frac{38}{429} & \frac{37}{429} & \frac{8}{429} & -\frac{3}{143} & -\frac{4}{429} & -\frac{1}{33} & -\frac{1}{429} & -\frac{8}{429} \\ \frac{73}{429} & \frac{1}{13} & -\frac{7}{429} & -\frac{1}{13} & -\frac{45}{143} & \frac{109}{429} & 0 & \frac{244}{429} & \frac{95}{429} & \frac{149}{429} & \frac{5}{39} & -\frac{1}{429} & \frac{20}{429} & -\frac{76}{429} & \frac{5}{429} & -\frac{101}{429} \\ \frac{29}{143} & \frac{68}{429} & \frac{49}{429} & -\frac{68}{429} & -\frac{151}{429} & \frac{34}{429} & -\frac{244}{429} & 0 & -\frac{149}{429} & -\frac{12}{143} & \frac{1}{11} & \frac{74}{429} & \frac{76}{429} & -\frac{41}{429} & \frac{19}{429} & -\frac{136}{429} \\ \frac{89}{429} & \frac{73}{429} & \frac{19}{143} & -\frac{73}{429} & -\frac{92}{429} & -\frac{38}{429} & -\frac{95}{429} & \frac{149}{429} & 0 & \frac{244}{429} & \frac{98}{429} & \frac{146}{429} & \frac{28}{143} & -\frac{12}{143} & \frac{7}{143} & -\frac{47}{143} \\ -\frac{25}{143} & -\frac{38}{429} & -\frac{1}{429} & \frac{38}{429} & \frac{76}{429} & -\frac{37}{429} & -\frac{149}{429} & \frac{12}{143} & -\frac{244}{429} & 0 & -\frac{38}{143} & -\frac{71}{429} & -\frac{28}{429} & \frac{71}{429} & -\frac{7}{429} & \frac{106}{429} \\ -\frac{29}{143} & -\frac{5}{39} & -\frac{23}{429} & \frac{5}{39} & \frac{86}{429} & -\frac{8}{429} & -\frac{5}{39} & -\frac{1}{11} & -\frac{98}{429} & \frac{38}{143} & 0 & \frac{212}{429} & \frac{28}{429} & \frac{184}{429} & \frac{7}{429} & \frac{149}{429} \\ -\frac{7}{39} & -\frac{56}{429} & -\frac{35}{429} & \frac{56}{429} & \frac{82}{429} & \frac{3}{143} & \frac{1}{429} & -\frac{74}{429} & -\frac{146}{429} & \frac{71}{429} & -\frac{212}{429} & 0 & -\frac{140}{429} & -\frac{7}{39} & -\frac{35}{429} & \frac{98}{429} \\ -\frac{100}{429} & -\frac{68}{429} & -\frac{12}{143} & \frac{68}{429} & \frac{100}{429} & \frac{4}{429} & -\frac{20}{429} & -\frac{76}{429} & -\frac{28}{143} & \frac{28}{429} & -\frac{28}{429} & \frac{140}{429} & 0 & \frac{56}{143} & 0 & \frac{56}{143} \\ \frac{10}{39} & \frac{67}{429} & \frac{8}{143} & -\frac{67}{429} & -\frac{8}{33} & \frac{1}{33} & \frac{76}{429} & \frac{41}{429} & \frac{12}{143} & -\frac{71}{429} & -\frac{184}{429} & \frac{7}{39} & -\frac{56}{143} & 0 & -\frac{14}{143} & -\frac{73}{143} \\ -\frac{25}{429} & -\frac{17}{429} & -\frac{3}{143} & \frac{17}{429} & \frac{25}{429} & \frac{1}{429} & -\frac{5}{429} & -\frac{19}{429} & -\frac{7}{143} & \frac{7}{429} & -\frac{7}{429} & \frac{35}{429} & 0 & \frac{14}{143} & 0 & \frac{14}{143} \\ \frac{235}{429} & \frac{152}{429} & \frac{23}{143} & -\frac{152}{429} & -\frac{229}{429} & \frac{8}{429} & \frac{101}{429} & \frac{136}{429} & \frac{47}{143} & -\frac{106}{429} & -\frac{149}{429} & -\frac{98}{429} & -\frac{56}{143} & \frac{73}{143} & -\frac{14}{143} & 0 \end{array}\right)