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"""Values of the even Mathieu functions ce_n(z,q) -- numberdb.org/T437.
This generator computes DLMF-normalised values of the even Mathieu functions
ce_n(z,q), with z measured in radians and stored by the parameter z/pi.
Run it with SageMath:
$ sage -pip install numberdb mpmath scipy
$ sage -python generate.py
$ sage -python generate.py --publish
The computation diagonalises the truncated Fourier matrix for
-d^2/dz^2 + 2q cos(2z) in the orthonormal cosine basis. The values stored in
the table must agree between two working precisions and two truncation lengths.
"""
import math
import os
import sys
from functools import lru_cache
from fractions import Fraction
import numberdb.sage as numberdb
import mpmath as mp
TABLE = os.environ.get("NUMBERDB_TABLE", "T437")
DIGITS = 30
N_VALUES = tuple(range(0, 6))
Q_VALUES = ("1/8", "1/6", "1/4", "1/3", "1/2", "2/3", "3/4", "1", "4/3", "3/2", "2", "3", "4", "6", "8")
Z_DENOMINATOR = 120
Z_STEPS = (0, 10, 12, 15, 20, 24, 30, 36, 40, 45, 48, 50, 60)
# (working decimal digits, largest cosine frequency included)
LOW_PROFILE = (80, 80)
HIGH_PROFILE = (110, 100)
SCIPY_TOLERANCE = mp.mpf("5e-13")
def _progress(message):
print(message, flush=True)
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
numberdb.configure(api_key=token)
def _z_param(step):
if step == 0:
return "0"
denominator = Z_DENOMINATOR
common = math.gcd(step, denominator)
numerator = step // common
denominator //= common
if denominator == 1:
return str(numerator)
return "%d/%d" % (numerator, denominator)
def _z_step(params):
text = str(params["z_over_pi"])
scaled = Fraction(text) * Z_DENOMINATOR
if scaled.denominator != 1 or not 0 <= scaled <= Z_DENOMINATOR // 2:
raise ValueError("z/pi=%s is outside this table" % text)
return scaled.numerator
def _harmonics(parity, max_frequency):
start = int(parity)
return tuple(range(start, int(max_frequency) + 1, 2))
def _matrix_entry_coupling(k, q):
return mp.sqrt(2) * q if k == 0 else q
@lru_cache(maxsize=None)
def _eigensystem(q_text, parity, digits, max_frequency):
mp.mp.dps = int(digits)
coupling = Fraction(q_text)
q = mp.mpf(coupling.numerator) / coupling.denominator
harmonics = _harmonics(parity, max_frequency)
dimension = len(harmonics)
matrix = mp.matrix(dimension)
for i, k in enumerate(harmonics):
diagonal = mp.mpf(k * k)
if k == 1:
diagonal += q
matrix[i, i] = diagonal
for i, k in enumerate(harmonics[:-1]):
coupling = _matrix_entry_coupling(k, q)
matrix[i, i + 1] = coupling
matrix[i + 1, i] = coupling
eigenvalues, eigenvectors = mp.eigsy(matrix)
return harmonics, tuple(eigenvalues), eigenvectors
def _rank_for_n(n):
return int(n) // 2
def _coefficients(n, q_text, profile):
digits, max_frequency = profile
mp.mp.dps = int(digits)
parity = int(n) % 2
harmonics, eigenvalues, eigenvectors = _eigensystem(
str(q_text), parity, int(digits), int(max_frequency)
)
rank = _rank_for_n(n)
coefficients = [mp.mpf(eigenvectors[row, rank]) for row in range(len(harmonics))]
anchor = harmonics.index(int(n))
if coefficients[anchor] < 0:
coefficients = [-c for c in coefficients]
return harmonics, eigenvalues[rank], coefficients
def _angle(step):
return mp.pi * mp.mpf(step) / mp.mpf(Z_DENOMINATOR)
def _basis_value(k, z):
if k == 0:
return 1 / mp.sqrt(2)
return mp.cos(mp.mpf(k) * z)
def _raw_value(n, q_text, step, profile):
harmonics, _eigenvalue, coefficients = _coefficients(n, q_text, profile)
z = _angle(step)
total = mp.mpf("0")
for k, coefficient in zip(harmonics, coefficients):
total += coefficient * _basis_value(k, z)
return total
def _format_decimal(value, digits):
text = mp.nstr(
value,
n=int(digits),
strip_zeros=False,
min_fixed=-100,
max_fixed=100,
)
if text.startswith("-0."):
try:
if mp.mpf(text) == 0:
return text[1:]
except ValueError:
pass
return text
def _exact_zero(n, step):
return int(n) % 2 == 1 and int(step) == Z_DENOMINATOR // 2
def ce_value_text(n, q_text, step, digits):
if _exact_zero(n, step):
return 0
low = _raw_value(n, q_text, step, LOW_PROFILE)
high = _raw_value(n, q_text, step, HIGH_PROFILE)
low_text = _format_decimal(low, digits)
high_text = _format_decimal(high, digits)
if low_text != high_text:
raise ArithmeticError(
"ce_%s(q=%s,z=%s*pi) disagrees between profiles: %s vs %s"
% (n, q_text, _z_param(step), low_text, high_text)
)
return high_text
def _q_zero_control(n, step, digits):
mp.mp.dps = digits + 20
z = _angle(step)
if int(n) == 0:
return 1 / mp.sqrt(2)
return mp.cos(mp.mpf(n) * z)
def _norm_squared(coefficients):
total = mp.mpf("0")
for coefficient in coefficients:
total += coefficient * coefficient
return total
def _matrix_residual(n, q_text, profile):
harmonics, eigenvalue, coefficients = _coefficients(n, q_text, profile)
coupling = Fraction(q_text)
q = mp.mpf(coupling.numerator) / coupling.denominator
residual = mp.mpf("0")
for i, k in enumerate(harmonics):
left = mp.mpf(k * k) * coefficients[i]
if k == 1:
left += q * coefficients[i]
if i > 0:
left += _matrix_entry_coupling(harmonics[i - 1], q) * coefficients[i - 1]
if i + 1 < len(harmonics):
left += _matrix_entry_coupling(k, q) * coefficients[i + 1]
residual = max(residual, abs(left - eigenvalue * coefficients[i]))
return residual
def run_integrity_checks():
for n in N_VALUES:
for step in Z_STEPS:
got = _raw_value(n, "0", step, HIGH_PROFILE)
expected = _q_zero_control(n, step, HIGH_PROFILE[0] - 20)
if abs(got - expected) > mp.mpf("1e-95"):
raise ArithmeticError(
"q=0 control failed for n=%d step=%d: %s != %s"
% (n, step, got, expected)
)
for q_text in Q_VALUES:
for n in N_VALUES:
_progress("checking coefficients for n=%d, q=%s" % (n, q_text))
_harmonics, _eigenvalue, coefficients = _coefficients(
n, q_text, HIGH_PROFILE
)
if abs(_norm_squared(coefficients) - 1) > mp.mpf("1e-80"):
raise ArithmeticError("normalization failed for n=%d, q=%s" % (n, q_text))
if _matrix_residual(n, q_text, HIGH_PROFILE) > mp.mpf("1e-80"):
raise ArithmeticError("matrix residual failed for n=%d, q=%s" % (n, q_text))
import scipy.special as scipy_special
for q_text in Q_VALUES:
q_float = float(Fraction(q_text))
for n in N_VALUES:
for step in Z_STEPS:
ours = mp.mpf(ce_value_text(n, q_text, step, DIGITS))
degrees = 180.0 * step / Z_DENOMINATOR
scipy_value = mp.mpf(str(scipy_special.mathieu_cem(n, q_float, degrees)[0]))
if abs(ours - scipy_value) > SCIPY_TOLERANCE:
raise ArithmeticError(
"SciPy check failed for n=%d, q=%s, z=%s degrees: %s vs %s"
% (n, q_text, degrees, ours, scipy_value)
)
class EvenMathieuFunctionValues(numberdb.Generator):
table = TABLE
parameters = ("n", "q", "z_over_pi")
type = "R"
digits = DIGITS
rigour = "heuristic (agreement-checked)"
def enumerate(self):
for n in N_VALUES:
for q_text in Q_VALUES:
for step in Z_STEPS:
yield {
"n": str(n),
"q": q_text,
"z_over_pi": _z_param(step),
}
def value(self, params, digits):
n = int(params["n"])
q_text = str(params["q"])
step = _z_step(params)
if n not in N_VALUES:
raise ValueError("n=%s is outside this table" % n)
if q_text not in Q_VALUES:
raise ValueError("q=%s is outside this table" % q_text)
return ce_value_text(n, q_text, step, digits)
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 = EvenMathieuFunctionValues()
run_integrity_checks()
if "--publish" in sys.argv or os.environ.get("NUMBERDB_PUBLISH") == "1":
print(fill_draft_once(
generator,
message="fill even Mathieu function values by Fourier matrix agreement"))
else:
report = generator.verify(sample=None)
print(report)
sys.exit(0 if report.ok else 1)