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