552 lines
17 KiB
Python
552 lines
17 KiB
Python
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import operator
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from . import libmp
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from .libmp.backend import basestring
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from .libmp import (
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int_types, MPZ_ONE,
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prec_to_dps, dps_to_prec, repr_dps,
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round_floor, round_ceiling,
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fzero, finf, fninf, fnan,
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mpf_le, mpf_neg,
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from_int, from_float, from_str, from_rational,
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mpi_mid, mpi_delta, mpi_str,
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mpi_abs, mpi_pos, mpi_neg, mpi_add, mpi_sub,
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mpi_mul, mpi_div, mpi_pow_int, mpi_pow,
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mpi_from_str,
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mpci_pos, mpci_neg, mpci_add, mpci_sub, mpci_mul, mpci_div, mpci_pow,
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mpci_abs, mpci_pow, mpci_exp, mpci_log,
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ComplexResult,
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mpf_hash, mpc_hash)
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from .matrices.matrices import _matrix
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mpi_zero = (fzero, fzero)
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from .ctx_base import StandardBaseContext
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new = object.__new__
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def convert_mpf_(x, prec, rounding):
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if hasattr(x, "_mpf_"): return x._mpf_
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if isinstance(x, int_types): return from_int(x, prec, rounding)
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if isinstance(x, float): return from_float(x, prec, rounding)
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if isinstance(x, basestring): return from_str(x, prec, rounding)
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raise NotImplementedError
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class ivmpf(object):
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"""
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Interval arithmetic class. Precision is controlled by iv.prec.
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"""
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def __new__(cls, x=0):
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return cls.ctx.convert(x)
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def cast(self, cls, f_convert):
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a, b = self._mpi_
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if a == b:
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return cls(f_convert(a))
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raise ValueError
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def __int__(self):
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return self.cast(int, libmp.to_int)
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def __float__(self):
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return self.cast(float, libmp.to_float)
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def __complex__(self):
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return self.cast(complex, libmp.to_float)
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def __hash__(self):
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a, b = self._mpi_
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if a == b:
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return mpf_hash(a)
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else:
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return hash(self._mpi_)
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@property
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def real(self): return self
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@property
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def imag(self): return self.ctx.zero
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def conjugate(self): return self
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@property
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def a(self):
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a, b = self._mpi_
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return self.ctx.make_mpf((a, a))
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@property
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def b(self):
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a, b = self._mpi_
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return self.ctx.make_mpf((b, b))
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@property
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def mid(self):
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ctx = self.ctx
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v = mpi_mid(self._mpi_, ctx.prec)
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return ctx.make_mpf((v, v))
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@property
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def delta(self):
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ctx = self.ctx
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v = mpi_delta(self._mpi_, ctx.prec)
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return ctx.make_mpf((v,v))
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@property
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def _mpci_(self):
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return self._mpi_, mpi_zero
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def _compare(*args):
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raise TypeError("no ordering relation is defined for intervals")
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__gt__ = _compare
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__le__ = _compare
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__gt__ = _compare
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__ge__ = _compare
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def __contains__(self, t):
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t = self.ctx.mpf(t)
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return (self.a <= t.a) and (t.b <= self.b)
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def __str__(self):
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return mpi_str(self._mpi_, self.ctx.prec)
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def __repr__(self):
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if self.ctx.pretty:
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return str(self)
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a, b = self._mpi_
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n = repr_dps(self.ctx.prec)
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a = libmp.to_str(a, n)
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b = libmp.to_str(b, n)
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return "mpi(%r, %r)" % (a, b)
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def _compare(s, t, cmpfun):
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if not hasattr(t, "_mpi_"):
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try:
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t = s.ctx.convert(t)
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except:
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return NotImplemented
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return cmpfun(s._mpi_, t._mpi_)
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def __eq__(s, t): return s._compare(t, libmp.mpi_eq)
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def __ne__(s, t): return s._compare(t, libmp.mpi_ne)
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def __lt__(s, t): return s._compare(t, libmp.mpi_lt)
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def __le__(s, t): return s._compare(t, libmp.mpi_le)
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def __gt__(s, t): return s._compare(t, libmp.mpi_gt)
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def __ge__(s, t): return s._compare(t, libmp.mpi_ge)
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def __abs__(self):
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return self.ctx.make_mpf(mpi_abs(self._mpi_, self.ctx.prec))
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def __pos__(self):
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return self.ctx.make_mpf(mpi_pos(self._mpi_, self.ctx.prec))
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def __neg__(self):
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return self.ctx.make_mpf(mpi_neg(self._mpi_, self.ctx.prec))
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def ae(s, t, rel_eps=None, abs_eps=None):
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return s.ctx.almosteq(s, t, rel_eps, abs_eps)
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class ivmpc(object):
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def __new__(cls, re=0, im=0):
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re = cls.ctx.convert(re)
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im = cls.ctx.convert(im)
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y = new(cls)
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y._mpci_ = re._mpi_, im._mpi_
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return y
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def __hash__(self):
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(a, b), (c,d) = self._mpci_
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if a == b and c == d:
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return mpc_hash((a, c))
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else:
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return hash(self._mpci_)
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def __repr__(s):
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if s.ctx.pretty:
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return str(s)
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return "iv.mpc(%s, %s)" % (repr(s.real), repr(s.imag))
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def __str__(s):
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return "(%s + %s*j)" % (str(s.real), str(s.imag))
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@property
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def a(self):
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(a, b), (c,d) = self._mpci_
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return self.ctx.make_mpf((a, a))
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@property
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def b(self):
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(a, b), (c,d) = self._mpci_
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return self.ctx.make_mpf((b, b))
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@property
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def c(self):
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(a, b), (c,d) = self._mpci_
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return self.ctx.make_mpf((c, c))
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@property
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def d(self):
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(a, b), (c,d) = self._mpci_
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return self.ctx.make_mpf((d, d))
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@property
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def real(s):
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return s.ctx.make_mpf(s._mpci_[0])
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@property
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def imag(s):
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return s.ctx.make_mpf(s._mpci_[1])
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def conjugate(s):
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a, b = s._mpci_
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return s.ctx.make_mpc((a, mpf_neg(b)))
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def overlap(s, t):
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t = s.ctx.convert(t)
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real_overlap = (s.a <= t.a <= s.b) or (s.a <= t.b <= s.b) or (t.a <= s.a <= t.b) or (t.a <= s.b <= t.b)
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imag_overlap = (s.c <= t.c <= s.d) or (s.c <= t.d <= s.d) or (t.c <= s.c <= t.d) or (t.c <= s.d <= t.d)
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return real_overlap and imag_overlap
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def __contains__(s, t):
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t = s.ctx.convert(t)
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return t.real in s.real and t.imag in s.imag
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def _compare(s, t, ne=False):
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if not isinstance(t, s.ctx._types):
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try:
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t = s.ctx.convert(t)
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except:
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return NotImplemented
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if hasattr(t, '_mpi_'):
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tval = t._mpi_, mpi_zero
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elif hasattr(t, '_mpci_'):
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tval = t._mpci_
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if ne:
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return s._mpci_ != tval
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return s._mpci_ == tval
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def __eq__(s, t): return s._compare(t)
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def __ne__(s, t): return s._compare(t, True)
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def __lt__(s, t): raise TypeError("complex intervals cannot be ordered")
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__le__ = __gt__ = __ge__ = __lt__
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def __neg__(s): return s.ctx.make_mpc(mpci_neg(s._mpci_, s.ctx.prec))
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def __pos__(s): return s.ctx.make_mpc(mpci_pos(s._mpci_, s.ctx.prec))
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def __abs__(s): return s.ctx.make_mpf(mpci_abs(s._mpci_, s.ctx.prec))
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def ae(s, t, rel_eps=None, abs_eps=None):
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return s.ctx.almosteq(s, t, rel_eps, abs_eps)
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def _binary_op(f_real, f_complex):
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def g_complex(ctx, sval, tval):
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return ctx.make_mpc(f_complex(sval, tval, ctx.prec))
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def g_real(ctx, sval, tval):
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try:
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return ctx.make_mpf(f_real(sval, tval, ctx.prec))
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except ComplexResult:
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sval = (sval, mpi_zero)
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tval = (tval, mpi_zero)
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return g_complex(ctx, sval, tval)
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def lop_real(s, t):
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if isinstance(t, _matrix): return NotImplemented
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ctx = s.ctx
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if not isinstance(t, ctx._types): t = ctx.convert(t)
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if hasattr(t, "_mpi_"): return g_real(ctx, s._mpi_, t._mpi_)
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if hasattr(t, "_mpci_"): return g_complex(ctx, (s._mpi_, mpi_zero), t._mpci_)
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return NotImplemented
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def rop_real(s, t):
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ctx = s.ctx
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if not isinstance(t, ctx._types): t = ctx.convert(t)
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if hasattr(t, "_mpi_"): return g_real(ctx, t._mpi_, s._mpi_)
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if hasattr(t, "_mpci_"): return g_complex(ctx, t._mpci_, (s._mpi_, mpi_zero))
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return NotImplemented
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def lop_complex(s, t):
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if isinstance(t, _matrix): return NotImplemented
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ctx = s.ctx
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if not isinstance(t, s.ctx._types):
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try:
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t = s.ctx.convert(t)
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except (ValueError, TypeError):
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return NotImplemented
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return g_complex(ctx, s._mpci_, t._mpci_)
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def rop_complex(s, t):
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ctx = s.ctx
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if not isinstance(t, s.ctx._types):
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t = s.ctx.convert(t)
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return g_complex(ctx, t._mpci_, s._mpci_)
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return lop_real, rop_real, lop_complex, rop_complex
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ivmpf.__add__, ivmpf.__radd__, ivmpc.__add__, ivmpc.__radd__ = _binary_op(mpi_add, mpci_add)
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ivmpf.__sub__, ivmpf.__rsub__, ivmpc.__sub__, ivmpc.__rsub__ = _binary_op(mpi_sub, mpci_sub)
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ivmpf.__mul__, ivmpf.__rmul__, ivmpc.__mul__, ivmpc.__rmul__ = _binary_op(mpi_mul, mpci_mul)
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ivmpf.__div__, ivmpf.__rdiv__, ivmpc.__div__, ivmpc.__rdiv__ = _binary_op(mpi_div, mpci_div)
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ivmpf.__pow__, ivmpf.__rpow__, ivmpc.__pow__, ivmpc.__rpow__ = _binary_op(mpi_pow, mpci_pow)
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ivmpf.__truediv__ = ivmpf.__div__; ivmpf.__rtruediv__ = ivmpf.__rdiv__
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ivmpc.__truediv__ = ivmpc.__div__; ivmpc.__rtruediv__ = ivmpc.__rdiv__
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class ivmpf_constant(ivmpf):
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def __new__(cls, f):
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self = new(cls)
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self._f = f
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return self
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def _get_mpi_(self):
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prec = self.ctx._prec[0]
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a = self._f(prec, round_floor)
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b = self._f(prec, round_ceiling)
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return a, b
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_mpi_ = property(_get_mpi_)
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class MPIntervalContext(StandardBaseContext):
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def __init__(ctx):
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ctx.mpf = type('ivmpf', (ivmpf,), {})
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ctx.mpc = type('ivmpc', (ivmpc,), {})
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ctx._types = (ctx.mpf, ctx.mpc)
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ctx._constant = type('ivmpf_constant', (ivmpf_constant,), {})
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ctx._prec = [53]
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ctx._set_prec(53)
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ctx._constant._ctxdata = ctx.mpf._ctxdata = ctx.mpc._ctxdata = [ctx.mpf, new, ctx._prec]
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ctx._constant.ctx = ctx.mpf.ctx = ctx.mpc.ctx = ctx
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ctx.pretty = False
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StandardBaseContext.__init__(ctx)
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ctx._init_builtins()
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def _mpi(ctx, a, b=None):
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if b is None:
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return ctx.mpf(a)
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return ctx.mpf((a,b))
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def _init_builtins(ctx):
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ctx.one = ctx.mpf(1)
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ctx.zero = ctx.mpf(0)
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ctx.inf = ctx.mpf('inf')
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ctx.ninf = -ctx.inf
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ctx.nan = ctx.mpf('nan')
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ctx.j = ctx.mpc(0,1)
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ctx.exp = ctx._wrap_mpi_function(libmp.mpi_exp, libmp.mpci_exp)
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ctx.sqrt = ctx._wrap_mpi_function(libmp.mpi_sqrt)
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ctx.ln = ctx._wrap_mpi_function(libmp.mpi_log, libmp.mpci_log)
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ctx.cos = ctx._wrap_mpi_function(libmp.mpi_cos, libmp.mpci_cos)
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ctx.sin = ctx._wrap_mpi_function(libmp.mpi_sin, libmp.mpci_sin)
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ctx.tan = ctx._wrap_mpi_function(libmp.mpi_tan)
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ctx.gamma = ctx._wrap_mpi_function(libmp.mpi_gamma, libmp.mpci_gamma)
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ctx.loggamma = ctx._wrap_mpi_function(libmp.mpi_loggamma, libmp.mpci_loggamma)
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ctx.rgamma = ctx._wrap_mpi_function(libmp.mpi_rgamma, libmp.mpci_rgamma)
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ctx.factorial = ctx._wrap_mpi_function(libmp.mpi_factorial, libmp.mpci_factorial)
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|
ctx.fac = ctx.factorial
|
||
|
|
|
||
|
|
ctx.eps = ctx._constant(lambda prec, rnd: (0, MPZ_ONE, 1-prec, 1))
|
||
|
|
ctx.pi = ctx._constant(libmp.mpf_pi)
|
||
|
|
ctx.e = ctx._constant(libmp.mpf_e)
|
||
|
|
ctx.ln2 = ctx._constant(libmp.mpf_ln2)
|
||
|
|
ctx.ln10 = ctx._constant(libmp.mpf_ln10)
|
||
|
|
ctx.phi = ctx._constant(libmp.mpf_phi)
|
||
|
|
ctx.euler = ctx._constant(libmp.mpf_euler)
|
||
|
|
ctx.catalan = ctx._constant(libmp.mpf_catalan)
|
||
|
|
ctx.glaisher = ctx._constant(libmp.mpf_glaisher)
|
||
|
|
ctx.khinchin = ctx._constant(libmp.mpf_khinchin)
|
||
|
|
ctx.twinprime = ctx._constant(libmp.mpf_twinprime)
|
||
|
|
|
||
|
|
def _wrap_mpi_function(ctx, f_real, f_complex=None):
|
||
|
|
def g(x, **kwargs):
|
||
|
|
if kwargs:
|
||
|
|
prec = kwargs.get('prec', ctx._prec[0])
|
||
|
|
else:
|
||
|
|
prec = ctx._prec[0]
|
||
|
|
x = ctx.convert(x)
|
||
|
|
if hasattr(x, "_mpi_"):
|
||
|
|
return ctx.make_mpf(f_real(x._mpi_, prec))
|
||
|
|
if hasattr(x, "_mpci_"):
|
||
|
|
return ctx.make_mpc(f_complex(x._mpci_, prec))
|
||
|
|
raise ValueError
|
||
|
|
return g
|
||
|
|
|
||
|
|
@classmethod
|
||
|
|
def _wrap_specfun(cls, name, f, wrap):
|
||
|
|
if wrap:
|
||
|
|
def f_wrapped(ctx, *args, **kwargs):
|
||
|
|
convert = ctx.convert
|
||
|
|
args = [convert(a) for a in args]
|
||
|
|
prec = ctx.prec
|
||
|
|
try:
|
||
|
|
ctx.prec += 10
|
||
|
|
retval = f(ctx, *args, **kwargs)
|
||
|
|
finally:
|
||
|
|
ctx.prec = prec
|
||
|
|
return +retval
|
||
|
|
else:
|
||
|
|
f_wrapped = f
|
||
|
|
setattr(cls, name, f_wrapped)
|
||
|
|
|
||
|
|
def _set_prec(ctx, n):
|
||
|
|
ctx._prec[0] = max(1, int(n))
|
||
|
|
ctx._dps = prec_to_dps(n)
|
||
|
|
|
||
|
|
def _set_dps(ctx, n):
|
||
|
|
ctx._prec[0] = dps_to_prec(n)
|
||
|
|
ctx._dps = max(1, int(n))
|
||
|
|
|
||
|
|
prec = property(lambda ctx: ctx._prec[0], _set_prec)
|
||
|
|
dps = property(lambda ctx: ctx._dps, _set_dps)
|
||
|
|
|
||
|
|
def make_mpf(ctx, v):
|
||
|
|
a = new(ctx.mpf)
|
||
|
|
a._mpi_ = v
|
||
|
|
return a
|
||
|
|
|
||
|
|
def make_mpc(ctx, v):
|
||
|
|
a = new(ctx.mpc)
|
||
|
|
a._mpci_ = v
|
||
|
|
return a
|
||
|
|
|
||
|
|
def _mpq(ctx, pq):
|
||
|
|
p, q = pq
|
||
|
|
a = libmp.from_rational(p, q, ctx.prec, round_floor)
|
||
|
|
b = libmp.from_rational(p, q, ctx.prec, round_ceiling)
|
||
|
|
return ctx.make_mpf((a, b))
|
||
|
|
|
||
|
|
def convert(ctx, x):
|
||
|
|
if isinstance(x, (ctx.mpf, ctx.mpc)):
|
||
|
|
return x
|
||
|
|
if isinstance(x, ctx._constant):
|
||
|
|
return +x
|
||
|
|
if isinstance(x, complex) or hasattr(x, "_mpc_"):
|
||
|
|
re = ctx.convert(x.real)
|
||
|
|
im = ctx.convert(x.imag)
|
||
|
|
return ctx.mpc(re,im)
|
||
|
|
if isinstance(x, basestring):
|
||
|
|
v = mpi_from_str(x, ctx.prec)
|
||
|
|
return ctx.make_mpf(v)
|
||
|
|
if hasattr(x, "_mpi_"):
|
||
|
|
a, b = x._mpi_
|
||
|
|
else:
|
||
|
|
try:
|
||
|
|
a, b = x
|
||
|
|
except (TypeError, ValueError):
|
||
|
|
a = b = x
|
||
|
|
if hasattr(a, "_mpi_"):
|
||
|
|
a = a._mpi_[0]
|
||
|
|
else:
|
||
|
|
a = convert_mpf_(a, ctx.prec, round_floor)
|
||
|
|
if hasattr(b, "_mpi_"):
|
||
|
|
b = b._mpi_[1]
|
||
|
|
else:
|
||
|
|
b = convert_mpf_(b, ctx.prec, round_ceiling)
|
||
|
|
if a == fnan or b == fnan:
|
||
|
|
a = fninf
|
||
|
|
b = finf
|
||
|
|
assert mpf_le(a, b), "endpoints must be properly ordered"
|
||
|
|
return ctx.make_mpf((a, b))
|
||
|
|
|
||
|
|
def nstr(ctx, x, n=5, **kwargs):
|
||
|
|
x = ctx.convert(x)
|
||
|
|
if hasattr(x, "_mpi_"):
|
||
|
|
return libmp.mpi_to_str(x._mpi_, n, **kwargs)
|
||
|
|
if hasattr(x, "_mpci_"):
|
||
|
|
re = libmp.mpi_to_str(x._mpci_[0], n, **kwargs)
|
||
|
|
im = libmp.mpi_to_str(x._mpci_[1], n, **kwargs)
|
||
|
|
return "(%s + %s*j)" % (re, im)
|
||
|
|
|
||
|
|
def mag(ctx, x):
|
||
|
|
x = ctx.convert(x)
|
||
|
|
if isinstance(x, ctx.mpc):
|
||
|
|
return max(ctx.mag(x.real), ctx.mag(x.imag)) + 1
|
||
|
|
a, b = libmp.mpi_abs(x._mpi_)
|
||
|
|
sign, man, exp, bc = b
|
||
|
|
if man:
|
||
|
|
return exp+bc
|
||
|
|
if b == fzero:
|
||
|
|
return ctx.ninf
|
||
|
|
if b == fnan:
|
||
|
|
return ctx.nan
|
||
|
|
return ctx.inf
|
||
|
|
|
||
|
|
def isnan(ctx, x):
|
||
|
|
return False
|
||
|
|
|
||
|
|
def isinf(ctx, x):
|
||
|
|
return x == ctx.inf
|
||
|
|
|
||
|
|
def isint(ctx, x):
|
||
|
|
x = ctx.convert(x)
|
||
|
|
a, b = x._mpi_
|
||
|
|
if a == b:
|
||
|
|
sign, man, exp, bc = a
|
||
|
|
if man:
|
||
|
|
return exp >= 0
|
||
|
|
return a == fzero
|
||
|
|
return None
|
||
|
|
|
||
|
|
def ldexp(ctx, x, n):
|
||
|
|
a, b = ctx.convert(x)._mpi_
|
||
|
|
a = libmp.mpf_shift(a, n)
|
||
|
|
b = libmp.mpf_shift(b, n)
|
||
|
|
return ctx.make_mpf((a,b))
|
||
|
|
|
||
|
|
def absmin(ctx, x):
|
||
|
|
return abs(ctx.convert(x)).a
|
||
|
|
|
||
|
|
def absmax(ctx, x):
|
||
|
|
return abs(ctx.convert(x)).b
|
||
|
|
|
||
|
|
def atan2(ctx, y, x):
|
||
|
|
y = ctx.convert(y)._mpi_
|
||
|
|
x = ctx.convert(x)._mpi_
|
||
|
|
return ctx.make_mpf(libmp.mpi_atan2(y,x,ctx.prec))
|
||
|
|
|
||
|
|
def _convert_param(ctx, x):
|
||
|
|
if isinstance(x, libmp.int_types):
|
||
|
|
return x, 'Z'
|
||
|
|
if isinstance(x, tuple):
|
||
|
|
p, q = x
|
||
|
|
return (ctx.mpf(p) / ctx.mpf(q), 'R')
|
||
|
|
x = ctx.convert(x)
|
||
|
|
if isinstance(x, ctx.mpf):
|
||
|
|
return x, 'R'
|
||
|
|
if isinstance(x, ctx.mpc):
|
||
|
|
return x, 'C'
|
||
|
|
raise ValueError
|
||
|
|
|
||
|
|
def _is_real_type(ctx, z):
|
||
|
|
return isinstance(z, ctx.mpf) or isinstance(z, int_types)
|
||
|
|
|
||
|
|
def _is_complex_type(ctx, z):
|
||
|
|
return isinstance(z, ctx.mpc)
|
||
|
|
|
||
|
|
def hypsum(ctx, p, q, types, coeffs, z, maxterms=6000, **kwargs):
|
||
|
|
coeffs = list(coeffs)
|
||
|
|
num = range(p)
|
||
|
|
den = range(p,p+q)
|
||
|
|
#tol = ctx.eps
|
||
|
|
s = t = ctx.one
|
||
|
|
k = 0
|
||
|
|
while 1:
|
||
|
|
for i in num: t *= (coeffs[i]+k)
|
||
|
|
for i in den: t /= (coeffs[i]+k)
|
||
|
|
k += 1; t /= k; t *= z; s += t
|
||
|
|
if t == 0:
|
||
|
|
return s
|
||
|
|
#if abs(t) < tol:
|
||
|
|
# return s
|
||
|
|
if k > maxterms:
|
||
|
|
raise ctx.NoConvergence
|
||
|
|
|
||
|
|
|
||
|
|
# Register with "numbers" ABC
|
||
|
|
# We do not subclass, hence we do not use the @abstractmethod checks. While
|
||
|
|
# this is less invasive it may turn out that we do not actually support
|
||
|
|
# parts of the expected interfaces. See
|
||
|
|
# http://docs.python.org/2/library/numbers.html for list of abstract
|
||
|
|
# methods.
|
||
|
|
try:
|
||
|
|
import numbers
|
||
|
|
numbers.Complex.register(ivmpc)
|
||
|
|
numbers.Real.register(ivmpf)
|
||
|
|
except ImportError:
|
||
|
|
pass
|