"""Coiflet scaling filters $h^{(K)}_k$ -- numberdb.org/T401. For each 1 <= K <= 17 and 0 <= k <= 6K - 1, this generator computes the low-pass reconstruction filter coefficient h_k of the Coiflet wavelet coifK, normalised so that sum_k h_k = sqrt(2), in PyWavelets' rec_lo order. 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 The coefficients are algebraic, but this file records them as real numbers. It starts Newton iteration from the PyWavelets source coefficient rows, solves the Coiflet polynomial equations at two working precisions, and writes only 100 digits after the two solves agree coefficient by coefficient. It checks the normalisation, wavelet moments, scaling moments, even-shift orthogonality, and agreement with the PyWavelets source rows. """ import os import sys from functools import lru_cache import mpmath as mp import numberdb.sage as numberdb TABLE = os.environ.get("NUMBERDB_TABLE", "T401") MAX_K = 17 DIGITS = 100 LOW_WORKING_DIGITS = 180 HIGH_WORKING_DIGITS = 240 NEWTON_STEPS = 3 AGREEMENT_MARGIN = 3 SOURCE_RELATIVE_TOLERANCE = "1e-50" CHECK_RELATIVE_TOLERANCE = "1e-80" PYWAVELETS_SOURCE_ROWS = { 1: ( "-5.142972847076845595317549230122688830344559947132656813651045e-02", "2.389297284707684559531754923012268883034455994713265681365104e-01", "6.028594569415369119063509846024537766068911989426531362730209e-01", "2.721405430584630880936490153975462233931088010573468637269790e-01", "-5.142972847076845595317549230122688830344559947132656813651045e-02", "-1.107027152923154404682450769877311169655440052867343186348954e-02", ), 2: ( "1.158759673871686817889714882853120395708315073355502818875931e-02", "-2.932013798346856448679594524397843054053420947418409889774786e-02", 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_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 _source_row(order): mp.mp.dps = HIGH_WORKING_DIGITS sqrt2 = mp.sqrt(2) return tuple(mp.mpf(value) * sqrt2 for value in PYWAVELETS_SOURCE_ROWS[order]) def _equations_and_jacobian(order, values): length = 6 * order center = 2 * order functions = [] rows = [] row = [mp.mpf(1) for _ in range(length)] functions.append(mp.fsum(values) - mp.sqrt(2)) rows.append(row) for shift in range(1, 3 * order): total = mp.mpf(0) row = [mp.mpf(0) for _ in range(length)] step = 2 * shift for index in range(length - step): total += values[index + step] * values[index] row[index + step] += values[index] row[index] += values[index + step] functions.append(total) rows.append(row) for moment in range(0, 2 * order): row = [((-1) ** index) * (mp.mpf(index) ** moment) for index in range(length)] functions.append(mp.fdot(row, values)) rows.append(row) for moment in range(1, 2 * order): row = [(mp.mpf(index - center) ** moment) for index in range(length)] functions.append(mp.fdot(row, values)) rows.append(row) return mp.matrix(functions), mp.matrix(rows) def _refined_row(order, working_digits): mp.mp.dps = working_digits sqrt2 = mp.sqrt(2) values = mp.matrix([ mp.mpf(value) * sqrt2 for value in PYWAVELETS_SOURCE_ROWS[order] ]) for _step in range(NEWTON_STEPS): functions, jacobian = _equations_and_jacobian(order, values) delta, _residual = mp.qr_solve(jacobian, -functions) values += delta return tuple(values) def _relative_error(one, other): scale = max(abs(other), mp.mpf("1e-1000")) return abs(one - other) / scale def _assert_agreement(order, low, high, digits): mp.mp.dps = HIGH_WORKING_DIGITS tolerance = mp.power(10, -(digits + AGREEMENT_MARGIN)) worst = mp.mpf(0) worst_k = None for k, (a, b) in enumerate(zip(low, high)): error = _relative_error(mp.mpf(a), mp.mpf(b)) if error > worst: worst = error worst_k = k if worst > tolerance: raise ArithmeticError( "K=%d k=%d agrees only to relative error %s" % (order, worst_k, mp.nstr(worst, 12)) ) def _assert_source_agreement(order, row): mp.mp.dps = HIGH_WORKING_DIGITS tolerance = mp.mpf(SOURCE_RELATIVE_TOLERANCE) for k, (got, source) in enumerate(zip(row, _source_row(order))): error = _relative_error(got, source) if error > tolerance: raise ArithmeticError( "K=%d k=%d differs from the PyWavelets source row by %s" % (order, k, mp.nstr(error, 12)) ) def _assert_small(label, value, scale): tolerance = mp.mpf(CHECK_RELATIVE_TOLERANCE) * max(scale, mp.mpf(1)) if abs(value) > tolerance: raise ArithmeticError( "%s is %s, larger than tolerance %s" % (label, mp.nstr(value, 12), mp.nstr(tolerance, 12)) ) def _check_normalisation(order, row): total = mp.fsum(row) _assert_small("K=%d normalisation" % order, total - mp.sqrt(2), mp.sqrt(2)) def _check_orthogonality(order, row): length = 6 * order for shift in range(-(3 * order - 1), 3 * order): terms = [] for k in range(length): other = k - 2 * shift if 0 <= other < length: terms.append(row[k] * row[other]) total = mp.fsum(terms) expected = mp.mpf(1) if shift == 0 else mp.mpf(0) scale = mp.fsum(abs(term) for term in terms) if terms else mp.mpf(1) _assert_small( "K=%d even autocorrelation shift %d" % (order, shift), total - expected, scale, ) def _check_wavelet_moments(order, row): length = 6 * order for moment in range(0, 2 * order): terms = [((-1) ** k) * (mp.mpf(k) ** moment) * row[k] for k in range(length)] _assert_small( "K=%d wavelet moment %d" % (order, moment), mp.fsum(terms), mp.fsum(abs(term) for term in terms), ) def _check_scaling_moments(order, row): length = 6 * order center = 2 * order for moment in range(1, 2 * order): terms = [(mp.mpf(k - center) ** moment) * row[k] for k in range(length)] _assert_small( "K=%d scaling moment %d" % (order, moment), mp.fsum(terms), mp.fsum(abs(term) for term in terms), ) def _check_filter_row(order, row): _check_normalisation(order, row) _check_orthogonality(order, row) _check_wavelet_moments(order, row) _check_scaling_moments(order, row) _assert_source_agreement(order, row) @lru_cache(maxsize=None) def filter_row(order, digits=DIGITS): order = int(order) if not 1 <= order <= MAX_K: raise ValueError("K must satisfy 1 <= K <= %d" % MAX_K) low = _refined_row(order, LOW_WORKING_DIGITS) high = _refined_row(order, HIGH_WORKING_DIGITS) _assert_agreement(order, low, high, digits) mp.mp.dps = HIGH_WORKING_DIGITS row = tuple(mp.mpf(value) for value in high) _check_filter_row(order, row) return row def _format_real(value, digits): mp.mp.dps = HIGH_WORKING_DIGITS return mp.nstr(value, n=digits, strip_zeros=False, min_fixed=-6, max_fixed=digits) def run_integrity_checks(max_k=MAX_K): for order in range(1, max_k + 1): filter_row(order, DIGITS) class CoifletScalingFilters(numberdb.Generator): table = TABLE parameters = ("K", "k") type = "R" digits = DIGITS rigour = "heuristic (agreement-checked)" def enumerate(self, max_k=MAX_K): for order in range(1, max_k + 1): for k in range(6 * order): yield {"K": str(order), "k": str(k)} def value(self, params, digits): order = int(params["K"]) k = int(params["k"]) return _format_real(filter_row(order, digits)[k], digits) def fill_full_draft_once(generator, message): """Write the finished draft document and attach this generator.""" import yaml from numberdb._generate import ( _check_precision, _check_rigour, _producer, _run_name, _source_files, ) from numberdb._write import Entries, attach, to_text 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, generator.table, identity, value) written = to_text(value, wanted, generator.format) _check_precision(generator.table, identity, written, wanted, lowering=False) record = dict(entry) record.pop("digits", None) entries.add(**params, **record, digits=wanted) here = os.path.dirname(os.path.abspath(__file__)) with open(os.path.join(here, "table.yaml"), encoding="utf8") as handle: document = yaml.load(handle, Loader=yaml.BaseLoader) document["Numbers"] = entries.as_list() client = numberdb.Client() headers = { "X-Produced-By": _producer( generator, os.environ.get("NUMBERDB_ASSISTED_BY", "") ), "X-Edit-Message": message, "X-Run-Id": run, "X-Numberdb-Client": "numberdb-python/%s" % numberdb.__version__, } answer = client.submit( "/api/table/%s" % str(generator.table).lstrip("tT"), yaml.dump(document, sort_keys=False, allow_unicode=True), headers, ) attached = [] for name, body in sorted(_source_files(generator).items()): attach( generator.table, name, body, run=run, client=client, message=message, rigour=generator.rigour, ) attached.append(name) answer["attached"] = attached return answer if __name__ == "__main__": _key_from_stdin() generator = CoifletScalingFilters() run_integrity_checks() if os.environ.get("NUMBERDB_PUBLISH") == "1" or "--publish" in sys.argv: print(fill_full_draft_once( generator, message="Coiflet scaling filter coefficients")) else: report = generator.verify(sample=None) print(report) sys.exit(0 if report.ok else 1)