495 lines
16 KiB
Python
495 lines
16 KiB
Python
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from operator import gt, lt
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from .libmp.backend import xrange
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from .functions.functions import SpecialFunctions
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from .functions.rszeta import RSCache
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from .calculus.quadrature import QuadratureMethods
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from .calculus.inverselaplace import LaplaceTransformInversionMethods
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from .calculus.calculus import CalculusMethods
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from .calculus.optimization import OptimizationMethods
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from .calculus.odes import ODEMethods
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from .matrices.matrices import MatrixMethods
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from .matrices.calculus import MatrixCalculusMethods
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from .matrices.linalg import LinearAlgebraMethods
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from .matrices.eigen import Eigen
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from .identification import IdentificationMethods
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from .visualization import VisualizationMethods
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from . import libmp
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class Context(object):
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pass
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class StandardBaseContext(Context,
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SpecialFunctions,
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RSCache,
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QuadratureMethods,
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LaplaceTransformInversionMethods,
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CalculusMethods,
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MatrixMethods,
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MatrixCalculusMethods,
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LinearAlgebraMethods,
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Eigen,
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IdentificationMethods,
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OptimizationMethods,
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ODEMethods,
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VisualizationMethods):
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NoConvergence = libmp.NoConvergence
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ComplexResult = libmp.ComplexResult
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def __init__(ctx):
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ctx._aliases = {}
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# Call those that need preinitialization (e.g. for wrappers)
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SpecialFunctions.__init__(ctx)
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RSCache.__init__(ctx)
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QuadratureMethods.__init__(ctx)
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LaplaceTransformInversionMethods.__init__(ctx)
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CalculusMethods.__init__(ctx)
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MatrixMethods.__init__(ctx)
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def _init_aliases(ctx):
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for alias, value in ctx._aliases.items():
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try:
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setattr(ctx, alias, getattr(ctx, value))
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except AttributeError:
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pass
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_fixed_precision = False
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# XXX
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verbose = False
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def warn(ctx, msg):
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print("Warning:", msg)
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def bad_domain(ctx, msg):
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raise ValueError(msg)
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def _re(ctx, x):
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if hasattr(x, "real"):
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return x.real
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return x
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def _im(ctx, x):
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if hasattr(x, "imag"):
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return x.imag
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return ctx.zero
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def _as_points(ctx, x):
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return x
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def fneg(ctx, x, **kwargs):
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return -ctx.convert(x)
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def fadd(ctx, x, y, **kwargs):
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return ctx.convert(x)+ctx.convert(y)
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def fsub(ctx, x, y, **kwargs):
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return ctx.convert(x)-ctx.convert(y)
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def fmul(ctx, x, y, **kwargs):
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return ctx.convert(x)*ctx.convert(y)
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def fdiv(ctx, x, y, **kwargs):
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return ctx.convert(x)/ctx.convert(y)
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def fsum(ctx, args, absolute=False, squared=False):
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if absolute:
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if squared:
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return sum((abs(x)**2 for x in args), ctx.zero)
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return sum((abs(x) for x in args), ctx.zero)
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if squared:
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return sum((x**2 for x in args), ctx.zero)
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return sum(args, ctx.zero)
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def fdot(ctx, xs, ys=None, conjugate=False):
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if ys is not None:
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xs = zip(xs, ys)
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if conjugate:
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cf = ctx.conj
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return sum((x*cf(y) for (x,y) in xs), ctx.zero)
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else:
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return sum((x*y for (x,y) in xs), ctx.zero)
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def fprod(ctx, args):
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prod = ctx.one
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for arg in args:
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prod *= arg
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return prod
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def nprint(ctx, x, n=6, **kwargs):
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"""
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Equivalent to ``print(nstr(x, n))``.
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"""
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print(ctx.nstr(x, n, **kwargs))
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def chop(ctx, x, tol=None):
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"""
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Chops off small real or imaginary parts, or converts
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numbers close to zero to exact zeros. The input can be a
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single number or an iterable::
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>>> from mpmath import *
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>>> mp.dps = 15; mp.pretty = False
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>>> chop(5+1e-10j, tol=1e-9)
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mpf('5.0')
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>>> nprint(chop([1.0, 1e-20, 3+1e-18j, -4, 2]))
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[1.0, 0.0, 3.0, -4.0, 2.0]
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The tolerance defaults to ``100*eps``.
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"""
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if tol is None:
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tol = 100*ctx.eps
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try:
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x = ctx.convert(x)
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absx = abs(x)
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if abs(x) < tol:
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return ctx.zero
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if ctx._is_complex_type(x):
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#part_tol = min(tol, absx*tol)
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part_tol = max(tol, absx*tol)
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if abs(x.imag) < part_tol:
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return x.real
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if abs(x.real) < part_tol:
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return ctx.mpc(0, x.imag)
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except TypeError:
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if isinstance(x, ctx.matrix):
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return x.apply(lambda a: ctx.chop(a, tol))
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if hasattr(x, "__iter__"):
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return [ctx.chop(a, tol) for a in x]
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return x
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def almosteq(ctx, s, t, rel_eps=None, abs_eps=None):
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r"""
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Determine whether the difference between `s` and `t` is smaller
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than a given epsilon, either relatively or absolutely.
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Both a maximum relative difference and a maximum difference
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('epsilons') may be specified. The absolute difference is
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defined as `|s-t|` and the relative difference is defined
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as `|s-t|/\max(|s|, |t|)`.
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If only one epsilon is given, both are set to the same value.
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If none is given, both epsilons are set to `2^{-p+m}` where
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`p` is the current working precision and `m` is a small
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integer. The default setting typically allows :func:`~mpmath.almosteq`
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to be used to check for mathematical equality
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in the presence of small rounding errors.
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**Examples**
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>>> from mpmath import *
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>>> mp.dps = 15
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>>> almosteq(3.141592653589793, 3.141592653589790)
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True
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>>> almosteq(3.141592653589793, 3.141592653589700)
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False
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>>> almosteq(3.141592653589793, 3.141592653589700, 1e-10)
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True
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>>> almosteq(1e-20, 2e-20)
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True
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>>> almosteq(1e-20, 2e-20, rel_eps=0, abs_eps=0)
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False
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"""
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t = ctx.convert(t)
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if abs_eps is None and rel_eps is None:
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rel_eps = abs_eps = ctx.ldexp(1, -ctx.prec+4)
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if abs_eps is None:
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abs_eps = rel_eps
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elif rel_eps is None:
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rel_eps = abs_eps
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diff = abs(s-t)
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if diff <= abs_eps:
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return True
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abss = abs(s)
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abst = abs(t)
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if abss < abst:
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err = diff/abst
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else:
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err = diff/abss
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return err <= rel_eps
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def arange(ctx, *args):
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r"""
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This is a generalized version of Python's :func:`~mpmath.range` function
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that accepts fractional endpoints and step sizes and
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returns a list of ``mpf`` instances. Like :func:`~mpmath.range`,
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:func:`~mpmath.arange` can be called with 1, 2 or 3 arguments:
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``arange(b)``
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`[0, 1, 2, \ldots, x]`
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``arange(a, b)``
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`[a, a+1, a+2, \ldots, x]`
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``arange(a, b, h)``
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`[a, a+h, a+h, \ldots, x]`
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where `b-1 \le x < b` (in the third case, `b-h \le x < b`).
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Like Python's :func:`~mpmath.range`, the endpoint is not included. To
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produce ranges where the endpoint is included, :func:`~mpmath.linspace`
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is more convenient.
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**Examples**
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>>> from mpmath import *
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>>> mp.dps = 15; mp.pretty = False
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>>> arange(4)
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[mpf('0.0'), mpf('1.0'), mpf('2.0'), mpf('3.0')]
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>>> arange(1, 2, 0.25)
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[mpf('1.0'), mpf('1.25'), mpf('1.5'), mpf('1.75')]
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>>> arange(1, -1, -0.75)
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[mpf('1.0'), mpf('0.25'), mpf('-0.5')]
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"""
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if not len(args) <= 3:
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raise TypeError('arange expected at most 3 arguments, got %i'
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% len(args))
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if not len(args) >= 1:
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raise TypeError('arange expected at least 1 argument, got %i'
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% len(args))
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# set default
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a = 0
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dt = 1
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# interpret arguments
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if len(args) == 1:
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b = args[0]
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elif len(args) >= 2:
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a = args[0]
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b = args[1]
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if len(args) == 3:
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dt = args[2]
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a, b, dt = ctx.mpf(a), ctx.mpf(b), ctx.mpf(dt)
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assert a + dt != a, 'dt is too small and would cause an infinite loop'
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# adapt code for sign of dt
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if a > b:
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if dt > 0:
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return []
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op = gt
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else:
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if dt < 0:
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return []
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op = lt
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# create list
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result = []
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i = 0
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t = a
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while 1:
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t = a + dt*i
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i += 1
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if op(t, b):
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result.append(t)
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else:
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break
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return result
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def linspace(ctx, *args, **kwargs):
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"""
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``linspace(a, b, n)`` returns a list of `n` evenly spaced
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samples from `a` to `b`. The syntax ``linspace(mpi(a,b), n)``
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is also valid.
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This function is often more convenient than :func:`~mpmath.arange`
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for partitioning an interval into subintervals, since
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the endpoint is included::
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>>> from mpmath import *
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>>> mp.dps = 15; mp.pretty = False
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>>> linspace(1, 4, 4)
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[mpf('1.0'), mpf('2.0'), mpf('3.0'), mpf('4.0')]
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You may also provide the keyword argument ``endpoint=False``::
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>>> linspace(1, 4, 4, endpoint=False)
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[mpf('1.0'), mpf('1.75'), mpf('2.5'), mpf('3.25')]
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"""
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if len(args) == 3:
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a = ctx.mpf(args[0])
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b = ctx.mpf(args[1])
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n = int(args[2])
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elif len(args) == 2:
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assert hasattr(args[0], '_mpi_')
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a = args[0].a
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b = args[0].b
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n = int(args[1])
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else:
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raise TypeError('linspace expected 2 or 3 arguments, got %i' \
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% len(args))
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if n < 1:
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raise ValueError('n must be greater than 0')
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if not 'endpoint' in kwargs or kwargs['endpoint']:
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if n == 1:
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return [ctx.mpf(a)]
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step = (b - a) / ctx.mpf(n - 1)
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y = [i*step + a for i in xrange(n)]
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y[-1] = b
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else:
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step = (b - a) / ctx.mpf(n)
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y = [i*step + a for i in xrange(n)]
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return y
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def cos_sin(ctx, z, **kwargs):
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return ctx.cos(z, **kwargs), ctx.sin(z, **kwargs)
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def cospi_sinpi(ctx, z, **kwargs):
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return ctx.cospi(z, **kwargs), ctx.sinpi(z, **kwargs)
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def _default_hyper_maxprec(ctx, p):
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return int(1000 * p**0.25 + 4*p)
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_gcd = staticmethod(libmp.gcd)
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list_primes = staticmethod(libmp.list_primes)
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isprime = staticmethod(libmp.isprime)
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bernfrac = staticmethod(libmp.bernfrac)
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moebius = staticmethod(libmp.moebius)
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_ifac = staticmethod(libmp.ifac)
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_eulernum = staticmethod(libmp.eulernum)
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_stirling1 = staticmethod(libmp.stirling1)
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_stirling2 = staticmethod(libmp.stirling2)
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def sum_accurately(ctx, terms, check_step=1):
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prec = ctx.prec
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try:
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extraprec = 10
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while 1:
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ctx.prec = prec + extraprec + 5
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max_mag = ctx.ninf
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s = ctx.zero
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k = 0
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for term in terms():
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s += term
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if (not k % check_step) and term:
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term_mag = ctx.mag(term)
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max_mag = max(max_mag, term_mag)
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sum_mag = ctx.mag(s)
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if sum_mag - term_mag > ctx.prec:
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break
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k += 1
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cancellation = max_mag - sum_mag
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if cancellation != cancellation:
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break
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if cancellation < extraprec or ctx._fixed_precision:
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||
|
|
break
|
||
|
|
extraprec += min(ctx.prec, cancellation)
|
||
|
|
return s
|
||
|
|
finally:
|
||
|
|
ctx.prec = prec
|
||
|
|
|
||
|
|
def mul_accurately(ctx, factors, check_step=1):
|
||
|
|
prec = ctx.prec
|
||
|
|
try:
|
||
|
|
extraprec = 10
|
||
|
|
while 1:
|
||
|
|
ctx.prec = prec + extraprec + 5
|
||
|
|
max_mag = ctx.ninf
|
||
|
|
one = ctx.one
|
||
|
|
s = one
|
||
|
|
k = 0
|
||
|
|
for factor in factors():
|
||
|
|
s *= factor
|
||
|
|
term = factor - one
|
||
|
|
if (not k % check_step):
|
||
|
|
term_mag = ctx.mag(term)
|
||
|
|
max_mag = max(max_mag, term_mag)
|
||
|
|
sum_mag = ctx.mag(s-one)
|
||
|
|
#if sum_mag - term_mag > ctx.prec:
|
||
|
|
# break
|
||
|
|
if -term_mag > ctx.prec:
|
||
|
|
break
|
||
|
|
k += 1
|
||
|
|
cancellation = max_mag - sum_mag
|
||
|
|
if cancellation != cancellation:
|
||
|
|
break
|
||
|
|
if cancellation < extraprec or ctx._fixed_precision:
|
||
|
|
break
|
||
|
|
extraprec += min(ctx.prec, cancellation)
|
||
|
|
return s
|
||
|
|
finally:
|
||
|
|
ctx.prec = prec
|
||
|
|
|
||
|
|
def power(ctx, x, y):
|
||
|
|
r"""Converts `x` and `y` to mpmath numbers and evaluates
|
||
|
|
`x^y = \exp(y \log(x))`::
|
||
|
|
|
||
|
|
>>> from mpmath import *
|
||
|
|
>>> mp.dps = 30; mp.pretty = True
|
||
|
|
>>> power(2, 0.5)
|
||
|
|
1.41421356237309504880168872421
|
||
|
|
|
||
|
|
This shows the leading few digits of a large Mersenne prime
|
||
|
|
(performing the exact calculation ``2**43112609-1`` and
|
||
|
|
displaying the result in Python would be very slow)::
|
||
|
|
|
||
|
|
>>> power(2, 43112609)-1
|
||
|
|
3.16470269330255923143453723949e+12978188
|
||
|
|
"""
|
||
|
|
return ctx.convert(x) ** ctx.convert(y)
|
||
|
|
|
||
|
|
def _zeta_int(ctx, n):
|
||
|
|
return ctx.zeta(n)
|
||
|
|
|
||
|
|
def maxcalls(ctx, f, N):
|
||
|
|
"""
|
||
|
|
Return a wrapped copy of *f* that raises ``NoConvergence`` when *f*
|
||
|
|
has been called more than *N* times::
|
||
|
|
|
||
|
|
>>> from mpmath import *
|
||
|
|
>>> mp.dps = 15
|
||
|
|
>>> f = maxcalls(sin, 10)
|
||
|
|
>>> print(sum(f(n) for n in range(10)))
|
||
|
|
1.95520948210738
|
||
|
|
>>> f(10) # doctest: +IGNORE_EXCEPTION_DETAIL
|
||
|
|
Traceback (most recent call last):
|
||
|
|
...
|
||
|
|
NoConvergence: maxcalls: function evaluated 10 times
|
||
|
|
|
||
|
|
"""
|
||
|
|
counter = [0]
|
||
|
|
def f_maxcalls_wrapped(*args, **kwargs):
|
||
|
|
counter[0] += 1
|
||
|
|
if counter[0] > N:
|
||
|
|
raise ctx.NoConvergence("maxcalls: function evaluated %i times" % N)
|
||
|
|
return f(*args, **kwargs)
|
||
|
|
return f_maxcalls_wrapped
|
||
|
|
|
||
|
|
def memoize(ctx, f):
|
||
|
|
"""
|
||
|
|
Return a wrapped copy of *f* that caches computed values, i.e.
|
||
|
|
a memoized copy of *f*. Values are only reused if the cached precision
|
||
|
|
is equal to or higher than the working precision::
|
||
|
|
|
||
|
|
>>> from mpmath import *
|
||
|
|
>>> mp.dps = 15; mp.pretty = True
|
||
|
|
>>> f = memoize(maxcalls(sin, 1))
|
||
|
|
>>> f(2)
|
||
|
|
0.909297426825682
|
||
|
|
>>> f(2)
|
||
|
|
0.909297426825682
|
||
|
|
>>> mp.dps = 25
|
||
|
|
>>> f(2) # doctest: +IGNORE_EXCEPTION_DETAIL
|
||
|
|
Traceback (most recent call last):
|
||
|
|
...
|
||
|
|
NoConvergence: maxcalls: function evaluated 1 times
|
||
|
|
|
||
|
|
"""
|
||
|
|
f_cache = {}
|
||
|
|
def f_cached(*args, **kwargs):
|
||
|
|
if kwargs:
|
||
|
|
key = args, tuple(kwargs.items())
|
||
|
|
else:
|
||
|
|
key = args
|
||
|
|
prec = ctx.prec
|
||
|
|
if key in f_cache:
|
||
|
|
cprec, cvalue = f_cache[key]
|
||
|
|
if cprec >= prec:
|
||
|
|
return +cvalue
|
||
|
|
value = f(*args, **kwargs)
|
||
|
|
f_cache[key] = (prec, value)
|
||
|
|
return value
|
||
|
|
f_cached.__name__ = f.__name__
|
||
|
|
f_cached.__doc__ = f.__doc__
|
||
|
|
return f_cached
|