demo_tsscpp_interpolation.sagews

Demonstration of tsscpp_interpolation.sage

Recall the conjectured form of the order-$n$ TSSCPP kernel $K_n$: $$K_n(x,y)_{\alpha, \beta} = \frac{2^{-2(x+y)} \, Q_{x, y, \alpha, \beta}(n)}{\prod_{i=1}^{\lfloor x/2 \rfloor} (4n^2 - (2i-1)^2) \prod_{j=1}^{\lfloor y/2 \rfloor} (4n^2 - (2j-1)^2)} ,$$ where $K_n(x,y)_{\alpha, \beta}$ denotes entry $(\alpha, \beta)$ of $K_n(x,y)$ (note $\alpha, \beta \in \{1,2\}$), and where $Q_{x, y, \alpha, \beta}(n)$ is an even polynomial with integer coefficients and degree at most $2(\lfloor x/2 \rfloor + \lfloor y/2 \rfloor)$). The library tsscpp_interpolation.sage contains tools for interpolating the entries $K_n(x,y)_{\alpha, \beta}$ of $K_n$ based on the conjectured form above. The main functions include:

• make_denom(n, x, y) computes $\prod_{i=1}^{\lfloor x/2 \rfloor} (4n^2 - (2i-1)^2) \prod_{j=1}^{\lfloor y/2 \rfloor} (4n^2 - (2j-1)^2)$

• make_Q_interpolated(x, y, alpha, beta, K_mats) computes the interpolated polynomial $Q_{x, y, \alpha, \beta}(n)$, where K_mats is a list containing the matrices $K_n$

• make_rat_fn(x, y, alpha, beta, K_mats) returns the interpolated rational function for $K_n(x,y)_{\alpha, \beta}$

• make_K_interpolated(K_mats) uses the above functions to interpolate the entries $K_n(x,y)_{\alpha, \beta}$ based on the matrices $K_n$ contained in K_mats

1. Creating the List K_mats

We first create a list object containing the matrices $K_1$ through $K_N$ for $N = 100$, which was computed in the worksheet tsscpp_matrices.sagews. A function make_K_mats included in tsscpp_interpolation.sage performs this task.

import os
load('mat_to_list.sage')
load('tsscpp_interpolation.sage')

N = 100
K_mats = make_K_mats(N)

print('Done')

Done

2. Creating and Saving the Interpolated Kernel $K_n$

With our matrix data contained in the list K_mats, we now interpolate the entries of $K_n$ using the function make_K_interpolated:

K_interpolated = make_K_interpolated(K_mats)
dim = K_interpolated.nrows()
print('The size of the interpolated matrix K_n is {0}-by-{0}\n'.format(dim))

print('Now writing K_interpolated_{0}_by_{0}:'.format(dim))
K_interpolated_file = open('tsscpp_data/tsscpp_K/K_interpolated_{0}_by_{0}.sage'.format(dim), 'w')
K_interpolated_file.write("var('n')" + "\n" + "K_interpolated = Matrix(SR, {0})".format(str(mat_to_list(K_interpolated))))
K_interpolated_file.close()

print('Done')

x=5 completed x=10 completed x=15 completed x=20 completed x=25 completed x=30 completed x=35 completed The size of the interpolated matrix K_n is 76-by-76 Now writing K_interpolated_76_by_76: Done

3. Reading the Interpolated Kernel Back In

Since the computation of the interpolated kernel above is somewhat time-consuming, it is convenient to be able to read the interpolated kernel back in after saving it. Specifically, loading the file K_interpolated_76_by_76.sage, creates the matrix K_interpolated stored in the previous step

N = 100
dim = 2*floor((2*N - 9) / 5)

filename = 'K_interpolated_{0}_by_{0}.sage'.format(dim)
print('Now loading {0}:\n'.format(filename))
# load('tsscpp_data/tsscpp_K/{0}'.format(filename))

print('A sample of the interpolated matrix:')
show(K_interpolated[0:6, 0:6])

print('Done')

Now loading K_interpolated_76_by_76.sage: A sample of the interpolated matrix:

$\displaystyle \left(\begin{array}{rrrrrr} 0 & \frac{5 \, {\left(n^{2} - 1\right)}}{2 \, {\left(4 \, n^{2} - 1\right)}} & \frac{n^{2} - 4}{4 \, n^{2} - 1} & \frac{3 \, {\left(n^{2} + 1\right)}}{2 \, {\left(4 \, n^{2} - 1\right)}} & \frac{7 \, {\left(37 \, n^{6} + 7 \, n^{4} - 692 \, n^{2} + 288\right)}}{16 \, {\left(64 \, n^{6} - 176 \, n^{4} + 76 \, n^{2} - 9\right)}} & -\frac{131 \, n^{6} - 1039 \, n^{4} - 1396 \, n^{2} + 1224}{16 \, {\left(64 \, n^{6} - 176 \, n^{4} + 76 \, n^{2} - 9\right)}} \\ -\frac{5 \, {\left(n^{2} - 1\right)}}{2 \, {\left(4 \, n^{2} - 1\right)}} & 0 & -\frac{3 \, {\left(n^{2} + 1\right)}}{2 \, {\left(4 \, n^{2} - 1\right)}} & 0 & \frac{5 \, {\left(65 \, n^{6} - 175 \, n^{4} - 322 \, n^{2} - 72\right)}}{16 \, {\left(64 \, n^{6} - 176 \, n^{4} + 76 \, n^{2} - 9\right)}} & \frac{59 \, n^{6} + 299 \, n^{4} + 866 \, n^{2} + 576}{16 \, {\left(64 \, n^{6} - 176 \, n^{4} + 76 \, n^{2} - 9\right)}} \\ -\frac{n^{2} - 4}{4 \, n^{2} - 1} & \frac{3 \, {\left(n^{2} + 1\right)}}{2 \, {\left(4 \, n^{2} - 1\right)}} & 0 & \frac{5 \, {\left(n^{2} - 1\right)}}{2 \, {\left(4 \, n^{2} - 1\right)}} & \frac{391 \, n^{6} - 1799 \, n^{4} + 1624 \, n^{2} - 2736}{16 \, {\left(64 \, n^{6} - 176 \, n^{4} + 76 \, n^{2} - 9\right)}} & \frac{3 \, {\left(83 \, n^{6} - 147 \, n^{4} + 112 \, n^{2} + 792\right)}}{16 \, {\left(64 \, n^{6} - 176 \, n^{4} + 76 \, n^{2} - 9\right)}} \\ -\frac{3 \, {\left(n^{2} + 1\right)}}{2 \, {\left(4 \, n^{2} - 1\right)}} & 0 & -\frac{5 \, {\left(n^{2} - 1\right)}}{2 \, {\left(4 \, n^{2} - 1\right)}} & 0 & -\frac{5 \, {\left(65 \, n^{6} - 175 \, n^{4} - 322 \, n^{2} - 72\right)}}{16 \, {\left(64 \, n^{6} - 176 \, n^{4} + 76 \, n^{2} - 9\right)}} & -\frac{59 \, n^{6} + 299 \, n^{4} + 866 \, n^{2} + 576}{16 \, {\left(64 \, n^{6} - 176 \, n^{4} + 76 \, n^{2} - 9\right)}} \\ -\frac{7 \, {\left(37 \, n^{6} + 7 \, n^{4} - 692 \, n^{2} + 288\right)}}{16 \, {\left(64 \, n^{6} - 176 \, n^{4} + 76 \, n^{2} - 9\right)}} & -\frac{5 \, {\left(65 \, n^{6} - 175 \, n^{4} - 322 \, n^{2} - 72\right)}}{16 \, {\left(64 \, n^{6} - 176 \, n^{4} + 76 \, n^{2} - 9\right)}} & -\frac{391 \, n^{6} - 1799 \, n^{4} + 1624 \, n^{2} - 2736}{16 \, {\left(64 \, n^{6} - 176 \, n^{4} + 76 \, n^{2} - 9\right)}} & \frac{5 \, {\left(65 \, n^{6} - 175 \, n^{4} - 322 \, n^{2} - 72\right)}}{16 \, {\left(64 \, n^{6} - 176 \, n^{4} + 76 \, n^{2} - 9\right)}} & 0 & \frac{358 \, n^{6} - 1337 \, n^{4} + 7 \, n^{2} - 1548}{8 \, {\left(64 \, n^{6} - 176 \, n^{4} + 76 \, n^{2} - 9\right)}} \\ \frac{131 \, n^{6} - 1039 \, n^{4} - 1396 \, n^{2} + 1224}{16 \, {\left(64 \, n^{6} - 176 \, n^{4} + 76 \, n^{2} - 9\right)}} & -\frac{59 \, n^{6} + 299 \, n^{4} + 866 \, n^{2} + 576}{16 \, {\left(64 \, n^{6} - 176 \, n^{4} + 76 \, n^{2} - 9\right)}} & -\frac{3 \, {\left(83 \, n^{6} - 147 \, n^{4} + 112 \, n^{2} + 792\right)}}{16 \, {\left(64 \, n^{6} - 176 \, n^{4} + 76 \, n^{2} - 9\right)}} & \frac{59 \, n^{6} + 299 \, n^{4} + 866 \, n^{2} + 576}{16 \, {\left(64 \, n^{6} - 176 \, n^{4} + 76 \, n^{2} - 9\right)}} & -\frac{358 \, n^{6} - 1337 \, n^{4} + 7 \, n^{2} - 1548}{8 \, {\left(64 \, n^{6} - 176 \, n^{4} + 76 \, n^{2} - 9\right)}} & 0 \end{array}\right)$

Done

4. Creating and Saving the Limiting Kernel $K_\infty$

Besides its tools for interpolating the entries of $K_n$, tsscpp_interpolation.sage also includes funtions for computing the limiting kernel $K_\infty$ from the interpolated kernel $K_n$. Below, we use rat_limit to create and save this limiting kernel.

i = 0
j = 4

print('Entry ({0},{1}) of the interpolated kernel is'.format(i+1, j+1))
show(K_interpolated[i,j])

print('and its limit as n goes to infinity is')
show(rat_limit(K_interpolated[i,j]))

Entry (1,5) of the interpolated kernel is

$\displaystyle \frac{7 \, {\left(37 \, n^{6} + 7 \, n^{4} - 692 \, n^{2} + 288\right)}}{16 \, {\left(64 \, n^{6} - 176 \, n^{4} + 76 \, n^{2} - 9\right)}}$

and its limit as n goes to infinity is

$\displaystyle \frac{259}{1024}$

N = 100
dim = 2*floor((2*N - 9) / 5)

K_infinity = Matrix(QQ, dim)
for i in [0 .. dim-1]:
    for j in [0 .. dim-1]:
        K_infinity[i,j] = rat_limit(K_interpolated[i,j])
    if i % 5 == 0:
        print('i={0} complete'.format(i))

K_infinity_file = open('tsscpp_data/tsscpp_K/K_infinity_{0}_by_{0}.sage'.format(dim), 'w')
K_infinity_file.write("K_infinity = Matrix(QQ, {0})".format(str(mat_to_list(K_infinity))))
K_infinity_file.close()

print('Done')

i=0 complete i=5 complete i=10 complete i=15 complete i=20 complete i=25 complete i=30 complete i=35 complete i=40 complete i=45 complete i=50 complete i=55 complete i=60 complete i=65 complete i=70 complete i=75 complete Done

5. Reading the Limiting Kernel Back In

As before, we read the limiting kernel back in by loading K_infinity_76_by_76.sage, which stores this matrix as K_infinity.

N = 100
dim = 2*floor((2*N - 9) / 5)

load('tsscpp_data/tsscpp_K/K_infinity_{0}_by_{0}.sage'.format(dim))

print('A sample of the limiting kernel:')
show(K_infinity[0:6, 0:6])

print('Done')

A sample of the limiting kernel:

$\displaystyle \left(\begin{array}{rrrrrr} 0 & \frac{5}{8} & \frac{1}{4} & \frac{3}{8} & \frac{259}{1024} & -\frac{131}{1024} \\ -\frac{5}{8} & 0 & -\frac{3}{8} & 0 & \frac{325}{1024} & \frac{59}{1024} \\ -\frac{1}{4} & \frac{3}{8} & 0 & \frac{5}{8} & \frac{391}{1024} & \frac{249}{1024} \\ -\frac{3}{8} & 0 & -\frac{5}{8} & 0 & -\frac{325}{1024} & -\frac{59}{1024} \\ -\frac{259}{1024} & -\frac{325}{1024} & -\frac{391}{1024} & \frac{325}{1024} & 0 & \frac{179}{256} \\ \frac{131}{1024} & -\frac{59}{1024} & -\frac{249}{1024} & \frac{59}{1024} & -\frac{179}{256} & 0 \end{array}\right)$

Done