188 lines
6.3 KiB
Python
188 lines
6.3 KiB
Python
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import math
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from sympy.sets.sets import Interval
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from sympy.calculus.singularities import is_increasing, is_decreasing
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from sympy.codegen.rewriting import Optimization
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from sympy.core.function import UndefinedFunction
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"""
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This module collects classes useful for approimate rewriting of expressions.
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This can be beneficial when generating numeric code for which performance is
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of greater importance than precision (e.g. for preconditioners used in iterative
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methods).
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"""
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class SumApprox(Optimization):
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"""
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Approximates sum by neglecting small terms.
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Explanation
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===========
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If terms are expressions which can be determined to be monotonic, then
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bounds for those expressions are added.
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Parameters
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==========
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bounds : dict
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Mapping expressions to length 2 tuple of bounds (low, high).
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reltol : number
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Threshold for when to ignore a term. Taken relative to the largest
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lower bound among bounds.
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Examples
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========
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>>> from sympy import exp
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>>> from sympy.abc import x, y, z
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>>> from sympy.codegen.rewriting import optimize
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>>> from sympy.codegen.approximations import SumApprox
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>>> bounds = {x: (-1, 1), y: (1000, 2000), z: (-10, 3)}
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>>> sum_approx3 = SumApprox(bounds, reltol=1e-3)
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>>> sum_approx2 = SumApprox(bounds, reltol=1e-2)
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>>> sum_approx1 = SumApprox(bounds, reltol=1e-1)
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>>> expr = 3*(x + y + exp(z))
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>>> optimize(expr, [sum_approx3])
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3*(x + y + exp(z))
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>>> optimize(expr, [sum_approx2])
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3*y + 3*exp(z)
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>>> optimize(expr, [sum_approx1])
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3*y
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"""
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def __init__(self, bounds, reltol, **kwargs):
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super().__init__(**kwargs)
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self.bounds = bounds
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self.reltol = reltol
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def __call__(self, expr):
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return expr.factor().replace(self.query, lambda arg: self.value(arg))
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def query(self, expr):
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return expr.is_Add
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def value(self, add):
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for term in add.args:
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if term.is_number or term in self.bounds or len(term.free_symbols) != 1:
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continue
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fs, = term.free_symbols
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if fs not in self.bounds:
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continue
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intrvl = Interval(*self.bounds[fs])
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if is_increasing(term, intrvl, fs):
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self.bounds[term] = (
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term.subs({fs: self.bounds[fs][0]}),
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term.subs({fs: self.bounds[fs][1]})
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)
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elif is_decreasing(term, intrvl, fs):
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self.bounds[term] = (
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term.subs({fs: self.bounds[fs][1]}),
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term.subs({fs: self.bounds[fs][0]})
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)
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else:
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return add
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if all(term.is_number or term in self.bounds for term in add.args):
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bounds = [(term, term) if term.is_number else self.bounds[term] for term in add.args]
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largest_abs_guarantee = 0
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for lo, hi in bounds:
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if lo <= 0 <= hi:
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continue
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largest_abs_guarantee = max(largest_abs_guarantee,
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min(abs(lo), abs(hi)))
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new_terms = []
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for term, (lo, hi) in zip(add.args, bounds):
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if max(abs(lo), abs(hi)) >= largest_abs_guarantee*self.reltol:
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new_terms.append(term)
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return add.func(*new_terms)
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else:
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return add
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class SeriesApprox(Optimization):
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""" Approximates functions by expanding them as a series.
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Parameters
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==========
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bounds : dict
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Mapping expressions to length 2 tuple of bounds (low, high).
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reltol : number
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Threshold for when to ignore a term. Taken relative to the largest
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lower bound among bounds.
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max_order : int
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Largest order to include in series expansion
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n_point_checks : int (even)
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The validity of an expansion (with respect to reltol) is checked at
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discrete points (linearly spaced over the bounds of the variable). The
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number of points used in this numerical check is given by this number.
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Examples
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========
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>>> from sympy import sin, pi
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>>> from sympy.abc import x, y
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>>> from sympy.codegen.rewriting import optimize
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>>> from sympy.codegen.approximations import SeriesApprox
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>>> bounds = {x: (-.1, .1), y: (pi-1, pi+1)}
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>>> series_approx2 = SeriesApprox(bounds, reltol=1e-2)
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>>> series_approx3 = SeriesApprox(bounds, reltol=1e-3)
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>>> series_approx8 = SeriesApprox(bounds, reltol=1e-8)
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>>> expr = sin(x)*sin(y)
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>>> optimize(expr, [series_approx2])
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x*(-y + (y - pi)**3/6 + pi)
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>>> optimize(expr, [series_approx3])
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(-x**3/6 + x)*sin(y)
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>>> optimize(expr, [series_approx8])
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sin(x)*sin(y)
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"""
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def __init__(self, bounds, reltol, max_order=4, n_point_checks=4, **kwargs):
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super().__init__(**kwargs)
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self.bounds = bounds
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self.reltol = reltol
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self.max_order = max_order
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if n_point_checks % 2 == 1:
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raise ValueError("Checking the solution at expansion point is not helpful")
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self.n_point_checks = n_point_checks
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self._prec = math.ceil(-math.log10(self.reltol))
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def __call__(self, expr):
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return expr.factor().replace(self.query, lambda arg: self.value(arg))
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def query(self, expr):
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return (expr.is_Function and not isinstance(expr, UndefinedFunction)
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and len(expr.args) == 1)
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def value(self, fexpr):
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free_symbols = fexpr.free_symbols
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if len(free_symbols) != 1:
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return fexpr
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symb, = free_symbols
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if symb not in self.bounds:
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return fexpr
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lo, hi = self.bounds[symb]
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x0 = (lo + hi)/2
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cheapest = None
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for n in range(self.max_order+1, 0, -1):
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fseri = fexpr.series(symb, x0=x0, n=n).removeO()
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n_ok = True
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for idx in range(self.n_point_checks):
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x = lo + idx*(hi - lo)/(self.n_point_checks - 1)
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val = fseri.xreplace({symb: x})
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ref = fexpr.xreplace({symb: x})
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if abs((1 - val/ref).evalf(self._prec)) > self.reltol:
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n_ok = False
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break
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if n_ok:
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cheapest = fseri
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else:
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break
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if cheapest is None:
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return fexpr
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else:
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return cheapest
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