287 lines
12 KiB
Python
287 lines
12 KiB
Python
import logging
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from sage.all import sage_eval, PolynomialRing, QQ, SR, log, var
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load("sage/util.py")
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# Logger
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logger = logging.getLogger(__name__)
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logging.basicConfig(level=logging.DEBUG)
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class PFOperator:
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"""
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A class representing a Picard-Fuchs operator in a single variable z.
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Independent of the given order, the variables are assumed to be left of
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the derivatives.
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The variables extra_locals are needed for symbolic coefficients used for
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operator Ansätze.
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"""
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def _operator_from_string(self, operator_string: str, extra_locals: dict = None):
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z_names = ['z%d' % i for i in range(self.no_variables)]
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theta_names = ['theta%d' % i for i in range(self.no_variables)]
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base_ring = SR if extra_locals else QQ
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self.ring = PolynomialRing(base_ring, z_names + theta_names)
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self.z_gens = self.ring.gens()[:self.no_variables]
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self.theta_gens = self.ring.gens()[self.no_variables:]
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parse_locals = dict(self.ring.gens_dict())
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if extra_locals:
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parse_locals.update(extra_locals)
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try:
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operator = sage_eval(operator_string, locals=parse_locals)
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return operator
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except Exception as e:
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raise ValueError("Invalid operator string: %s" % e)
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def __init__(self, operator_string: str, no_variables: int = 1, extra_locals: dict = None):
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self.no_variables = no_variables
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self.operator_string = operator_string
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self.operator = self._operator_from_string(operator_string, extra_locals=extra_locals)
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logger.info("Initialised PFOperator: %s", self.operator)
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class PFOperatorAnsatz(PFOperator):
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"""
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A class for Picard-Fuchs operator Ansätze, characterised by number of variables and their z- and theta-multi-degrees.
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"""
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def __init__(self, no_variables: int, theta_degree: int, z_degree: int):
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self.no_variables = no_variables
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self.theta_degree = theta_degree
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self.z_degree = z_degree
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z_indices = _multi_indices(self.no_variables, z_degree)
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theta_indices = _multi_indices(self.no_variables, theta_degree)
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# Every (z_index, theta_index) pair allowed by the degree bounds gets its own
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# fresh unknown, to be solved for once the Ansatz is applied to a period.
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operator_terms = [
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(var("b_" + "_".join(str(x) for x in z_index + theta_index)), (z_index, theta_index))
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for z_index in z_indices
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for theta_index in theta_indices
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]
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def _monomial_factors(index, name):
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return [f"{name}{i}^{exp}" for i, exp in enumerate(index) if exp > 0]
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operator_string = " + ".join(
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"*".join([str(coeff), *_monomial_factors(z_index, "z"), *_monomial_factors(theta_index, "theta")])
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for coeff, (z_index, theta_index) in operator_terms
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)
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extra_locals = {str(coeff): coeff for coeff, _ in operator_terms}
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super().__init__(operator_string=operator_string, no_variables=no_variables, extra_locals=extra_locals)
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class Period:
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"""
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A class representing a period as a formal power series in z-variables and their logs.
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The coefficients are stored in a dictionary of dictionaries, where the first key is the multi-index
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of the logarithmic part and the second key is the multi-index of the z-variables.
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So, for example, the coefficient of
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log(z0)^2 * log(z1) * z0^3 * z1^2 would be stored as coefficients[(2, 1)][(3, 2)]
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"""
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def _initialise_ring(self):
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z_names = ['z%d' % i for i in range(self.no_variables)]
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log_names = ['L%d' % i for i in range(self.no_variables)]
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ring = PolynomialRing(QQ, z_names + log_names)
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z_gens = ring.gens()[:self.no_variables]
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log_gens = ring.gens()[self.no_variables:]
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return ring, z_gens, log_gens # type: (PolynomialRing, list, list)
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def _period_from_string(self, period_string: str) -> dict:
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ring, z_gens, log_gens = self._initialise_ring()
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log_of_z = dict(zip(z_gens, log_gens))
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def log(zi):
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try:
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return log_of_z[zi]
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except (KeyError, TypeError):
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raise ValueError(
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"log(...) may only be applied to one of the z-variables z0, ..., z%d"
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% (self.no_variables - 1)
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)
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parse_locals = dict(ring.gens_dict())
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parse_locals['log'] = log
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try:
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expression = ring(sage_eval(period_string, locals=parse_locals))
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except Exception as e:
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raise ValueError("Invalid period string: %s" % e)
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coefficients = {}
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for coeff, monomial in expression:
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exponents = monomial.exponents()[0]
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z_index = tuple(exponents[:self.no_variables])
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log_index = tuple(exponents[self.no_variables:])
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coefficients.setdefault(log_index, {})[z_index] = coeff
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return coefficients
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def _period_to_string(self) -> str:
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ring, z_gens, log_gens = self._initialise_ring()
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expression = ring(0)
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for log_index, z_dict in self.coefficients.items():
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for z_index, coeff in z_dict.items():
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monomial = coeff
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for i in range(self.no_variables):
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monomial *= (log_gens[i] ** log_index[i]) * (z_gens[i] ** z_index[i])
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expression += monomial
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log_substitutions = {log_gens[i]: log(SR(z_gens[i])) for i in range(self.no_variables)}
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return str(SR(expression).subs(log_substitutions))
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def _max_z_degree(self) -> int:
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# Highest total z-degree (sum of the z-multi-index) among all coefficients.
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if not self.coefficients:
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return 0
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return max(
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sum(z_index)
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for z_dict in self.coefficients.values()
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for z_index in z_dict
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)
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def _truncate_coefficients(self, order: int) -> dict:
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# Drop every (log_index, z_index) entry whose total z-degree exceeds order,
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# removing it from the dictionary rather than merely zeroing it out.
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truncated = {}
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for log_index, z_dict in self.coefficients.items():
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kept = {z_index: coeff for z_index, coeff in z_dict.items() if sum(z_index) <= order}
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if kept:
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truncated[log_index] = kept
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return truncated
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def _apply_theta(self, coefficients: dict, index: int) -> dict:
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"""
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Applies the logarithmic derivative theta_i = z_i * d/dz_i once to a coefficients dict of the same
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shape as self.coefficients. It uses the product rule
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theta_i(z^a log(z)^k) = a_i * z^a log(z)^k + k_i * z^a log(z)^(k - e_i).
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"""
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result = {}
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for log_index, z_dict in coefficients.items():
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for z_index, coeff in z_dict.items():
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a_i = z_index[index]
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if a_i != 0:
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inner = result.setdefault(log_index, {})
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inner[z_index] = inner.get(z_index, 0) + a_i * coeff
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k_i = log_index[index]
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if k_i != 0:
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lowered_log_index = log_index[:index] + (k_i - 1,) + log_index[index + 1:]
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inner = result.setdefault(lowered_log_index, {})
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inner[z_index] = inner.get(z_index, 0) + k_i * coeff
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return result
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def apply_operator(self, pf_operator: PFOperator) -> "Period":
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"""
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Applies a PFOperator to this period and returns the result as a new Period,
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truncated to self.order. Each monomial of the operator is normalised as
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coeff * z^p * theta^q (see PFOperator's docstring), so theta^q is applied
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to the period first and the result is then multiplied by coeff * z^p.
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"""
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if pf_operator.no_variables != self.no_variables:
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raise ValueError(
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"Variable count mismatch: period has %d variable(s), operator has %d."
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% (self.no_variables, pf_operator.no_variables)
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)
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result_coefficients = {}
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for monomial_coeff, monomial in pf_operator.operator:
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exponents = monomial.exponents()[0]
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z_exponents = exponents[:self.no_variables]
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theta_exponents = exponents[self.no_variables:]
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term = self.coefficients
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for i, power in enumerate(theta_exponents):
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for _ in range(power):
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term = self._apply_theta(term, i)
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for log_index, z_dict in term.items():
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inner = result_coefficients.setdefault(log_index, {})
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for z_index, coeff in z_dict.items():
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shifted_z_index = tuple(z_index[i] + z_exponents[i] for i in range(self.no_variables))
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inner[shifted_z_index] = inner.get(shifted_z_index, 0) + monomial_coeff * coeff
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result_coefficients = {
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log_index: {z_index: c for z_index, c in z_dict.items() if c != 0}
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for log_index, z_dict in result_coefficients.items()
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}
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result_coefficients = {log_index: z_dict for log_index, z_dict in result_coefficients.items() if z_dict}
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return Period(
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no_variables=self.no_variables,
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coefficients=result_coefficients,
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order=self.order,
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)
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def __init__(self, no_variables: int = 1, coefficients: dict = None, period_string: str = None, order: int = None):
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self.no_variables = no_variables
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# Use coefficients or period_string to initialize the period
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if period_string is not None:
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if coefficients is not None:
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raise ValueError("Provide either coefficients or period_string, not both.")
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self.coefficients = self._period_from_string(period_string)
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self.period_string = period_string
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logger.debug("Using string for initialisation.")
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elif coefficients is None:
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self.coefficients = {}
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else:
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self.coefficients = coefficients
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self.period_string = self._period_to_string()
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logger.debug("Using coefficients for initialisation.")
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if order is None:
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self.order = self._max_z_degree()
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logger.debug("Order not provided, using maximum z-degree: %d", self.order)
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else:
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self.coefficients = self._truncate_coefficients(order)
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self.order = order
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self.period_string = self._period_to_string()
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logger.debug("Truncated coefficients to order %d.", self.order)
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logger.info("Initialised Period in %d variables at order %d.", self.no_variables, self.order)
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class PeriodAnsatz(Period):
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"""
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A class for period Ansätze, characterised by number of variables and their z- and log-multi-degrees.
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"""
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def __init__(self, no_variables: int, z_degree: int, log_degree: int):
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self.no_variables = no_variables
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self.z_degree = z_degree
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self.log_degree = log_degree
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log_indices = _multi_indices(self.no_variables, log_degree)
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z_indices = _multi_indices(self.no_variables, z_degree)
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# Every (log_index, z_index) pair allowed by the degree bounds gets its own
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# fresh unknown, to be solved for once a PFOperator is applied to the Ansatz.
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coefficients = {
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log_index: {
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z_index: var("a_" + "_".join(str(x) for x in log_index + z_index))
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for z_index in z_indices
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}
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for log_index in log_indices
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}
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super().__init__(no_variables=no_variables, coefficients=coefficients, order=z_degree)
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self.expansion_coefficients = self.coefficients
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logger.info(
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"Initialised PeriodAnsatz with %d unknown coefficient(s).",
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sum(len(z_dict) for z_dict in self.expansion_coefficients.values()),
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)
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"""
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To do:
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- add method to Period that finds operators that annihilate it (using operator ansatz class)
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- add class PFideal containing a list of PFOperators
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""" |