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"""Krawtchouk polynomials K_n(x; p, N) in the Askey normalisation -- numberdb.org/T270.
This generator fills the table of KLS/DLMF Krawtchouk polynomials
K_n(x; p, N) = _2F_1(-n, -x; -N; 1/p).
The table stores the Askey-scheme normalisation with K_n(0; p, N) = 1,
not the Hamming-scheme rescaling used in coding theory.
Run it with SageMath:
$ sage -pip install numberdb # once
$ sage -python generate.py # check the table against this code
$ sage -python generate.py --publish # fill the draft, with NUMBERDB_API_KEY set
"""
import json
import os
import sys
import urllib.request
import numberdb.sage as numberdb
from sage.arith.misc import binomial
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
from sage.rings.rational_field import QQ
TABLE = os.environ.get("NUMBERDB_TABLE", "T270")
# Measured before filling the draft: these five probabilities and N <= 16 give
# 760 entries. The longest written value has 549 characters, and the entries
# block is 112.4 KB in the dry-run measurement.
P_VALUES = (
QQ(1) / QQ(4),
QQ(1) / QQ(3),
QQ(1) / QQ(2),
QQ(2) / QQ(3),
QQ(3) / QQ(4),
)
MAX_N = 16
R = PolynomialRing(QQ, "x")
x = R.gen()
def _key_from_stdin():
if os.environ.get("NUMBERDB_KEY_FROM_STDIN") != "1":
return
token = sys.stdin.read().strip()
if "=" in token and token.split("=", 1)[0].isupper():
token = token.split("=", 1)[1].strip().strip("'\"")
if token:
os.environ["NUMBERDB_API_KEY"] = token
def _rising_scalar(argument, count):
"""The scalar Pochhammer symbol (argument)_count over QQ."""
value = QQ(1)
argument = QQ(argument)
for offset in range(count):
value *= argument + QQ(offset)
return value
def _falling_binomial(argument, count):
"""The polynomial binomial(argument, count), with exact divisions."""
value = R.one()
argument = R(argument)
for offset in range(count):
value *= argument - QQ(offset)
value *= QQ(1) / QQ(offset + 1)
return R(value)
def krawtchouk_askey(p, N, n):
"""The KLS/DLMF Krawtchouk polynomial K_n(x; p, N)."""
p = QQ(p)
N, n = int(N), int(n)
total = R.one()
term = R.one()
inverse_p = QQ(1) / p
for j in range(n):
term *= QQ(j - n) * (-x + QQ(j)) * inverse_p
term *= QQ(1) / ((QQ(j) - QQ(N)) * QQ(j + 1))
total += term
return R(total)
class AskeyKrawtchoukPolynomials(numberdb.Generator):
"""Generator for T270, the Askey-normalised Krawtchouk polynomials."""
table = TABLE
parameters = ("p", "N", "n")
type = "Q[]"
rigour = "exact"
def enumerate(self, max_N=MAX_N):
for p in P_VALUES:
for N in range(1, max_N + 1):
for n in range(N + 1):
yield {"p": str(p), "N": str(N), "n": str(n)}
def value(self, params, digits):
return krawtchouk_askey(
QQ(params["p"]),
int(params["N"]),
int(params["n"]),
)
def _computed_values():
return {
(p, N, n): krawtchouk_askey(p, N, n)
for p in P_VALUES
for N in range(1, MAX_N + 1)
for n in range(N + 1)
}
def _hamming_krawtchouk(q, N, n):
total = R.zero()
for j in range(n + 1):
total += (
QQ((-1) ** j)
* (QQ(q) - QQ(1)) ** (n - j)
* _falling_binomial(x, j)
* _falling_binomial(QQ(N) - x, n - j)
)
return R(total)
def _check_hamming_scaling(values):
for q in (2, 3, 4, 5):
p = QQ(q - 1) / QQ(q)
for N in range(1, MAX_N + 1):
for n in range(N + 1):
factor = QQ(binomial(N, n)) * QQ(q - 1) ** n
expected = factor * krawtchouk_askey(p, N, n)
if _hamming_krawtchouk(q, N, n) != expected:
raise ArithmeticError(
"Hamming scaling failed at q=%d, N=%d, n=%d" % (q, N, n)
)
if p in P_VALUES and values[(p, N, n)] != krawtchouk_askey(p, N, n):
raise ArithmeticError(
"stored Krawtchouk value changed at p=%s, N=%d, n=%d"
% (p, N, n)
)
def _check_generating_function(values):
Z = PolynomialRing(QQ, "z")
z = Z.gen()
for p in P_VALUES:
a = (QQ(1) - p) / p
for N in range(1, MAX_N + 1):
polynomials = [values[(p, N, n)] for n in range(N + 1)]
for point in range(N + 1):
rhs = (QQ(1) - a * z) ** point * (QQ(1) + z) ** (N - point)
for n, polynomial in enumerate(polynomials):
expected = QQ(binomial(N, n)) * polynomial(point)
if rhs[n] != expected:
raise ArithmeticError(
"generating function failed at p=%s, N=%d, x=%d, n=%d"
% (p, N, point, n)
)
def _check_orthogonality(values):
for p in P_VALUES:
for N in range(1, MAX_N + 1):
polynomials = [values[(p, N, n)] for n in range(N + 1)]
for m, left in enumerate(polynomials):
for n, right in enumerate(polynomials):
total = QQ(0)
for point in range(N + 1):
total += (
QQ(binomial(N, point))
* p ** point
* (QQ(1) - p) ** (N - point)
* left(point)
* right(point)
)
expected = QQ(0)
if m == n:
expected = ((QQ(1) - p) / p) ** n / QQ(binomial(N, n))
if total != expected:
raise ArithmeticError(
"orthogonality failed at p=%s, N=%d, m=%d, n=%d"
% (p, N, m, n)
)
def _check_special_value(values):
for (p, N, n), polynomial in values.items():
if polynomial(0) != 1:
raise ArithmeticError("K_n(0) failed at p=%s, N=%d, n=%d" % (p, N, n))
def _check_self_duality(values):
for p in P_VALUES:
for N in range(1, MAX_N + 1):
for n in range(N + 1):
for point in range(N + 1):
if values[(p, N, n)](point) != values[(p, N, point)](n):
raise ArithmeticError(
"self-duality failed at p=%s, N=%d, n=%d, x=%d"
% (p, N, n, point)
)
def _check_leading_coefficient(values):
for (p, N, n), polynomial in values.items():
expected = p ** (-n) / _rising_scalar(-N, n)
if polynomial.monomial_coefficient(x ** n) != expected:
raise ArithmeticError(
"leading coefficient failed at p=%s, N=%d, n=%d" % (p, N, n)
)
def run_integrity_checks(values=None):
if values is None:
values = _computed_values()
_check_hamming_scaling(values)
_check_generating_function(values)
_check_orthogonality(values)
_check_special_value(values)
_check_self_duality(values)
_check_leading_coefficient(values)
def stored_values():
"""Read the table from the API and parse its stored polynomials."""
key = os.environ.get("NUMBERDB_API_KEY")
if not key:
raise RuntimeError("NUMBERDB_API_KEY is not set")
request = urllib.request.Request(
"https://numberdb.org/api/table?id=%s" % TABLE,
headers={"Authorization": "Bearer " + key},
)
with urllib.request.urlopen(request, timeout=60) as response:
tree = json.load(response)
if "error" in tree:
raise RuntimeError(tree["error"])
found = {}
for p_text, by_N in tree.get("Numbers", {}).items():
for N_text, by_n in by_N.items():
for n_text, polynomial_text in by_n.items():
found[(QQ(p_text), int(N_text), int(n_text))] = R(polynomial_text)
expected_keys = set(_computed_values())
if set(found) != expected_keys:
missing = sorted(expected_keys - set(found))[:5]
extra = sorted(set(found) - expected_keys)[:5]
raise ArithmeticError("stored key set disagrees, missing=%s extra=%s" % (missing, extra))
return found
def fill_draft_once(generator, message):
"""Fill a fresh draft without the client's empty upsert probe."""
from numberdb._generate import (
_check_precision,
_check_rigour,
_producer,
_run_name,
_source_files,
)
from numberdb._write import Entries, attach, submit_entries, to_text
table = generator.table
run = _run_name(generator)
entries = Entries(*generator.parameters)
for params in generator.enumerate():
params = dict(params)
wanted = generator.digits_for(params)
entry = generator._entry(params, wanted)
value = entry["number"]
identity = ",".join(str(params[name]) for name in generator.parameters)
_check_rigour(generator, table, identity, value)
written = to_text(value, wanted, generator.format)
_check_precision(table, identity, written, wanted, lowering=False)
record = dict(entry)
record.pop("digits", None)
entries.add(**params, **record, digits=wanted)
answer = submit_entries(
table,
entries,
message=message,
produced_by=_producer(generator, os.environ.get("NUMBERDB_ASSISTED_BY", "")),
upsert=False,
run=run,
rigour=generator.rigour,
)
for name, body in sorted(_source_files(generator).items()):
attach(table, name, body, run=run, message=message, rigour=generator.rigour)
return answer
if __name__ == "__main__":
_key_from_stdin()
generator = AskeyKrawtchoukPolynomials()
run_integrity_checks()
if os.environ.get("NUMBERDB_PUBLISH") == "1" or "--publish" in sys.argv:
print(fill_draft_once(
generator,
message="exact Krawtchouk polynomials in the Askey normalisation"))
elif os.environ.get("NUMBERDB_API_KEY"):
report = generator.verify(sample=None)
print(report)
if not report.ok:
sys.exit(1)
run_integrity_checks(stored_values())
print("stored identity checks passed")
else:
print("identity checks passed; NUMBERDB_API_KEY is not set, so verify() was skipped")