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-rw-r--r--numpy/polynomial/__init__.py2
-rw-r--r--numpy/polynomial/_polybase.py4
-rw-r--r--numpy/polynomial/chebyshev.py127
-rw-r--r--numpy/polynomial/hermite.py179
-rw-r--r--numpy/polynomial/hermite_e.py178
-rw-r--r--numpy/polynomial/laguerre.py119
-rw-r--r--numpy/polynomial/legendre.py121
-rw-r--r--numpy/polynomial/polynomial.py108
-rw-r--r--numpy/polynomial/polytemplate.py927
-rw-r--r--numpy/polynomial/polyutils.py68
-rw-r--r--numpy/polynomial/tests/test_chebyshev.py18
-rw-r--r--numpy/polynomial/tests/test_classes.py28
-rw-r--r--numpy/polynomial/tests/test_hermite.py17
-rw-r--r--numpy/polynomial/tests/test_hermite_e.py21
-rw-r--r--numpy/polynomial/tests/test_laguerre.py17
-rw-r--r--numpy/polynomial/tests/test_legendre.py17
-rw-r--r--numpy/polynomial/tests/test_polynomial.py16
-rw-r--r--numpy/polynomial/tests/test_polyutils.py8
18 files changed, 572 insertions, 1403 deletions
diff --git a/numpy/polynomial/__init__.py b/numpy/polynomial/__init__.py
index e9ca387c3..1200d1c8d 100644
--- a/numpy/polynomial/__init__.py
+++ b/numpy/polynomial/__init__.py
@@ -15,8 +15,6 @@ information can be found in the docstring for the module of interest.
"""
from __future__ import division, absolute_import, print_function
-import warnings
-
from .polynomial import Polynomial
from .chebyshev import Chebyshev
from .legendre import Legendre
diff --git a/numpy/polynomial/_polybase.py b/numpy/polynomial/_polybase.py
index 23608c74a..234b509aa 100644
--- a/numpy/polynomial/_polybase.py
+++ b/numpy/polynomial/_polybase.py
@@ -374,7 +374,7 @@ class ABCPolyBase(object):
return quo, rem
def __pow__(self, other):
- coef = self._pow(self.coef, other, maxpower = self.maxpower)
+ coef = self._pow(self.coef, other, maxpower=self.maxpower)
res = self.__class__(coef, self.domain, self.window)
return res
@@ -721,8 +721,6 @@ class ABCPolyBase(object):
y = self(x)
return x, y
-
-
@classmethod
def fit(cls, x, y, deg, domain=None, rcond=None, full=False, w=None,
window=None):
diff --git a/numpy/polynomial/chebyshev.py b/numpy/polynomial/chebyshev.py
index b4acbbeab..f213ab3fd 100644
--- a/numpy/polynomial/chebyshev.py
+++ b/numpy/polynomial/chebyshev.py
@@ -94,13 +94,14 @@ import numpy.linalg as la
from . import polyutils as pu
from ._polybase import ABCPolyBase
-__all__ = ['chebzero', 'chebone', 'chebx', 'chebdomain', 'chebline',
- 'chebadd', 'chebsub', 'chebmulx', 'chebmul', 'chebdiv', 'chebpow',
- 'chebval', 'chebder', 'chebint', 'cheb2poly', 'poly2cheb',
- 'chebfromroots', 'chebvander', 'chebfit', 'chebtrim', 'chebroots',
- 'chebpts1', 'chebpts2', 'Chebyshev', 'chebval2d', 'chebval3d',
- 'chebgrid2d', 'chebgrid3d', 'chebvander2d', 'chebvander3d',
- 'chebcompanion', 'chebgauss', 'chebweight']
+__all__ = [
+ 'chebzero', 'chebone', 'chebx', 'chebdomain', 'chebline', 'chebadd',
+ 'chebsub', 'chebmulx', 'chebmul', 'chebdiv', 'chebpow', 'chebval',
+ 'chebder', 'chebint', 'cheb2poly', 'poly2cheb', 'chebfromroots',
+ 'chebvander', 'chebfit', 'chebtrim', 'chebroots', 'chebpts1',
+ 'chebpts2', 'Chebyshev', 'chebval2d', 'chebval3d', 'chebgrid2d',
+ 'chebgrid3d', 'chebvander2d', 'chebvander3d', 'chebcompanion',
+ 'chebgauss', 'chebweight']
chebtrim = pu.trimcoef
@@ -109,7 +110,7 @@ chebtrim = pu.trimcoef
# functions and do minimal error checking.
#
-def _cseries_to_zseries(c) :
+def _cseries_to_zseries(c):
"""Covert Chebyshev series to z-series.
Covert a Chebyshev series to the equivalent z-series. The result is
@@ -134,7 +135,7 @@ def _cseries_to_zseries(c) :
return zs + zs[::-1]
-def _zseries_to_cseries(zs) :
+def _zseries_to_cseries(zs):
"""Covert z-series to a Chebyshev series.
Covert a z series to the equivalent Chebyshev series. The result is
@@ -159,7 +160,7 @@ def _zseries_to_cseries(zs) :
return c
-def _zseries_mul(z1, z2) :
+def _zseries_mul(z1, z2):
"""Multiply two z-series.
Multiply two z-series to produce a z-series.
@@ -186,7 +187,7 @@ def _zseries_mul(z1, z2) :
return np.convolve(z1, z2)
-def _zseries_div(z1, z2) :
+def _zseries_div(z1, z2):
"""Divide the first z-series by the second.
Divide `z1` by `z2` and return the quotient and remainder as z-series.
@@ -223,19 +224,19 @@ def _zseries_div(z1, z2) :
z2 = z2.copy()
len1 = len(z1)
len2 = len(z2)
- if len2 == 1 :
+ if len2 == 1:
z1 /= z2
return z1, z1[:1]*0
- elif len1 < len2 :
+ elif len1 < len2:
return z1[:1]*0, z1
- else :
+ else:
dlen = len1 - len2
scl = z2[0]
z2 /= scl
quo = np.empty(dlen + 1, dtype=z1.dtype)
i = 0
j = dlen
- while i < j :
+ while i < j:
r = z1[i]
quo[i] = z1[i]
quo[dlen - i] = r
@@ -253,7 +254,7 @@ def _zseries_div(z1, z2) :
return quo, rem
-def _zseries_der(zs) :
+def _zseries_der(zs):
"""Differentiate a z-series.
The derivative is with respect to x, not z. This is achieved using the
@@ -285,7 +286,7 @@ def _zseries_der(zs) :
return d
-def _zseries_int(zs) :
+def _zseries_int(zs):
"""Integrate a z-series.
The integral is with respect to x, not z. This is achieved by a change
@@ -323,7 +324,7 @@ def _zseries_int(zs) :
#
-def poly2cheb(pol) :
+def poly2cheb(pol):
"""
Convert a polynomial to a Chebyshev series.
@@ -368,12 +369,12 @@ def poly2cheb(pol) :
[pol] = pu.as_series([pol])
deg = len(pol) - 1
res = 0
- for i in range(deg, -1, -1) :
+ for i in range(deg, -1, -1):
res = chebadd(chebmulx(res), pol[i])
return res
-def cheb2poly(c) :
+def cheb2poly(c):
"""
Convert a Chebyshev series to a polynomial.
@@ -427,7 +428,7 @@ def cheb2poly(c) :
c0 = c[-2]
c1 = c[-1]
# i is the current degree of c1
- for i in range(n - 1, 1, -1) :
+ for i in range(n - 1, 1, -1):
tmp = c0
c0 = polysub(c[i - 2], c1)
c1 = polyadd(tmp, polymulx(c1)*2)
@@ -452,7 +453,7 @@ chebone = np.array([1])
chebx = np.array([0, 1])
-def chebline(off, scl) :
+def chebline(off, scl):
"""
Chebyshev series whose graph is a straight line.
@@ -482,13 +483,13 @@ def chebline(off, scl) :
-3.0
"""
- if scl != 0 :
+ if scl != 0:
return np.array([off, scl])
- else :
+ else:
return np.array([off])
-def chebfromroots(roots) :
+def chebfromroots(roots):
"""
Generate a Chebyshev series with given roots.
@@ -537,9 +538,9 @@ def chebfromroots(roots) :
array([ 1.5+0.j, 0.0+0.j, 0.5+0.j])
"""
- if len(roots) == 0 :
+ if len(roots) == 0:
return np.ones(1)
- else :
+ else:
[roots] = pu.as_series([roots], trim=False)
roots.sort()
p = [chebline(-r, 1) for r in roots]
@@ -595,10 +596,10 @@ def chebadd(c1, c2):
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
- if len(c1) > len(c2) :
+ if len(c1) > len(c2):
c1[:c2.size] += c2
ret = c1
- else :
+ else:
c2[:c1.size] += c1
ret = c2
return pu.trimseq(ret)
@@ -647,10 +648,10 @@ def chebsub(c1, c2):
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
- if len(c1) > len(c2) :
+ if len(c1) > len(c2):
c1[:c2.size] -= c2
ret = c1
- else :
+ else:
c2 = -c2
c2[:c1.size] += c1
ret = c2
@@ -794,16 +795,16 @@ def chebdiv(c1, c2):
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
- if c2[-1] == 0 :
+ if c2[-1] == 0:
raise ZeroDivisionError()
lc1 = len(c1)
lc2 = len(c2)
- if lc1 < lc2 :
+ if lc1 < lc2:
return c1[:1]*0, c1
- elif lc2 == 1 :
+ elif lc2 == 1:
return c1/c2[-1], c1[:1]*0
- else :
+ else:
z1 = _cseries_to_zseries(c1)
z2 = _cseries_to_zseries(c2)
quo, rem = _zseries_div(z1, z2)
@@ -812,7 +813,7 @@ def chebdiv(c1, c2):
return quo, rem
-def chebpow(c, pow, maxpower=16) :
+def chebpow(c, pow, maxpower=16):
"""Raise a Chebyshev series to a power.
Returns the Chebyshev series `c` raised to the power `pow`. The
@@ -846,25 +847,25 @@ def chebpow(c, pow, maxpower=16) :
# c is a trimmed copy
[c] = pu.as_series([c])
power = int(pow)
- if power != pow or power < 0 :
+ if power != pow or power < 0:
raise ValueError("Power must be a non-negative integer.")
- elif maxpower is not None and power > maxpower :
+ elif maxpower is not None and power > maxpower:
raise ValueError("Power is too large")
- elif power == 0 :
+ elif power == 0:
return np.array([1], dtype=c.dtype)
- elif power == 1 :
+ elif power == 1:
return c
- else :
+ else:
# This can be made more efficient by using powers of two
# in the usual way.
zs = _cseries_to_zseries(c)
prd = zs
- for i in range(2, power + 1) :
+ for i in range(2, power + 1):
prd = np.convolve(prd, zs)
return _zseries_to_cseries(prd)
-def chebder(c, m=1, scl=1, axis=0) :
+def chebder(c, m=1, scl=1, axis=0):
"""
Differentiate a Chebyshev series.
@@ -1057,9 +1058,9 @@ def chebint(c, m=1, k=[], lbnd=0, scl=1, axis=0):
if cnt != m:
raise ValueError("The order of integration must be integer")
- if cnt < 0 :
+ if cnt < 0:
raise ValueError("The order of integration must be non-negative")
- if len(k) > cnt :
+ if len(k) > cnt:
raise ValueError("Too many integration constants")
if iaxis != axis:
raise ValueError("The axis must be integer")
@@ -1073,7 +1074,7 @@ def chebint(c, m=1, k=[], lbnd=0, scl=1, axis=0):
c = np.rollaxis(c, iaxis)
k = list(k) + [0]*(cnt - len(k))
- for i in range(cnt) :
+ for i in range(cnt):
n = len(c)
c *= scl
if n == 1 and np.all(c[0] == 0):
@@ -1162,19 +1163,19 @@ def chebval(x, c, tensor=True):
if isinstance(x, (tuple, list)):
x = np.asarray(x)
if isinstance(x, np.ndarray) and tensor:
- c = c.reshape(c.shape + (1,)*x.ndim)
+ c = c.reshape(c.shape + (1,)*x.ndim)
- if len(c) == 1 :
+ if len(c) == 1:
c0 = c[0]
c1 = 0
- elif len(c) == 2 :
+ elif len(c) == 2:
c0 = c[0]
c1 = c[1]
- else :
+ else:
x2 = 2*x
c0 = c[-2]
c1 = c[-1]
- for i in range(3, len(c) + 1) :
+ for i in range(3, len(c) + 1):
tmp = c0
c0 = c[-i] - c1
c1 = tmp + c1*x2
@@ -1410,7 +1411,7 @@ def chebgrid3d(x, y, z, c):
return c
-def chebvander(x, deg) :
+def chebvander(x, deg):
"""Pseudo-Vandermonde matrix of given degree.
Returns the pseudo-Vandermonde matrix of degree `deg` and sample points
@@ -1457,15 +1458,15 @@ def chebvander(x, deg) :
v = np.empty(dims, dtype=dtyp)
# Use forward recursion to generate the entries.
v[0] = x*0 + 1
- if ideg > 0 :
+ if ideg > 0:
x2 = 2*x
v[1] = x
- for i in range(2, ideg + 1) :
+ for i in range(2, ideg + 1):
v[i] = v[i-1]*x2 - v[i-2]
return np.rollaxis(v, 0, v.ndim)
-def chebvander2d(x, y, deg) :
+def chebvander2d(x, y, deg):
"""Pseudo-Vandermonde matrix of given degrees.
Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
@@ -1528,7 +1529,7 @@ def chebvander2d(x, y, deg) :
return v.reshape(v.shape[:-2] + (-1,))
-def chebvander3d(x, y, z, deg) :
+def chebvander3d(x, y, z, deg):
"""Pseudo-Vandermonde matrix of given degrees.
Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
@@ -1714,13 +1715,13 @@ def chebfit(x, y, deg, rcond=None, full=False, w=None):
y = np.asarray(y) + 0.0
# check arguments.
- if deg < 0 :
+ if deg < 0:
raise ValueError("expected deg >= 0")
if x.ndim != 1:
raise TypeError("expected 1D vector for x")
if x.size == 0:
raise TypeError("expected non-empty vector for x")
- if y.ndim < 1 or y.ndim > 2 :
+ if y.ndim < 1 or y.ndim > 2:
raise TypeError("expected 1D or 2D array for y")
if len(x) != len(y):
raise TypeError("expected x and y to have same length")
@@ -1740,7 +1741,7 @@ def chebfit(x, y, deg, rcond=None, full=False, w=None):
rhs = rhs * w
# set rcond
- if rcond is None :
+ if rcond is None:
rcond = len(x)*np.finfo(x.dtype).eps
# Determine the norms of the design matrix columns.
@@ -1759,9 +1760,9 @@ def chebfit(x, y, deg, rcond=None, full=False, w=None):
msg = "The fit may be poorly conditioned"
warnings.warn(msg, pu.RankWarning)
- if full :
+ if full:
return c, [resids, rank, s, rcond]
- else :
+ else:
return c
@@ -1916,8 +1917,8 @@ def chebweight(x):
The weight function of the Chebyshev polynomials.
The weight function is :math:`1/\sqrt{1 - x^2}` and the interval of
- integration is :math:`[-1, 1]`. The Chebyshev polynomials are orthogonal, but
- not normalized, with respect to this weight function.
+ integration is :math:`[-1, 1]`. The Chebyshev polynomials are
+ orthogonal, but not normalized, with respect to this weight function.
Parameters
----------
diff --git a/numpy/polynomial/hermite.py b/numpy/polynomial/hermite.py
index 43ede58ac..1d3bef390 100644
--- a/numpy/polynomial/hermite.py
+++ b/numpy/polynomial/hermite.py
@@ -66,18 +66,18 @@ import numpy.linalg as la
from . import polyutils as pu
from ._polybase import ABCPolyBase
-__all__ = ['hermzero', 'hermone', 'hermx', 'hermdomain', 'hermline',
- 'hermadd', 'hermsub', 'hermmulx', 'hermmul', 'hermdiv', 'hermpow',
- 'hermval', 'hermder', 'hermint', 'herm2poly', 'poly2herm',
- 'hermfromroots', 'hermvander', 'hermfit', 'hermtrim', 'hermroots',
- 'Hermite', 'hermval2d', 'hermval3d', 'hermgrid2d', 'hermgrid3d',
- 'hermvander2d', 'hermvander3d', 'hermcompanion', 'hermgauss',
- 'hermweight']
+__all__ = [
+ 'hermzero', 'hermone', 'hermx', 'hermdomain', 'hermline', 'hermadd',
+ 'hermsub', 'hermmulx', 'hermmul', 'hermdiv', 'hermpow', 'hermval',
+ 'hermder', 'hermint', 'herm2poly', 'poly2herm', 'hermfromroots',
+ 'hermvander', 'hermfit', 'hermtrim', 'hermroots', 'Hermite',
+ 'hermval2d', 'hermval3d', 'hermgrid2d', 'hermgrid3d', 'hermvander2d',
+ 'hermvander3d', 'hermcompanion', 'hermgauss', 'hermweight']
hermtrim = pu.trimcoef
-def poly2herm(pol) :
+def poly2herm(pol):
"""
poly2herm(pol)
@@ -118,12 +118,12 @@ def poly2herm(pol) :
[pol] = pu.as_series([pol])
deg = len(pol) - 1
res = 0
- for i in range(deg, -1, -1) :
+ for i in range(deg, -1, -1):
res = hermadd(hermmulx(res), pol[i])
return res
-def herm2poly(c) :
+def herm2poly(c):
"""
Convert a Hermite series to a polynomial.
@@ -174,7 +174,7 @@ def herm2poly(c) :
c0 = c[-2]
c1 = c[-1]
# i is the current degree of c1
- for i in range(n - 1, 1, -1) :
+ for i in range(n - 1, 1, -1):
tmp = c0
c0 = polysub(c[i - 2], c1*(2*(i - 1)))
c1 = polyadd(tmp, polymulx(c1)*2)
@@ -198,7 +198,7 @@ hermone = np.array([1])
hermx = np.array([0, 1/2])
-def hermline(off, scl) :
+def hermline(off, scl):
"""
Hermite series whose graph is a straight line.
@@ -228,13 +228,13 @@ def hermline(off, scl) :
5.0
"""
- if scl != 0 :
+ if scl != 0:
return np.array([off, scl/2])
- else :
+ else:
return np.array([off])
-def hermfromroots(roots) :
+def hermfromroots(roots):
"""
Generate a Hermite series with given roots.
@@ -284,9 +284,9 @@ def hermfromroots(roots) :
array([ 0.+0.j, 0.+0.j])
"""
- if len(roots) == 0 :
+ if len(roots) == 0:
return np.ones(1)
- else :
+ else:
[roots] = pu.as_series([roots], trim=False)
roots.sort()
p = [hermline(-r, 1) for r in roots]
@@ -340,10 +340,10 @@ def hermadd(c1, c2):
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
- if len(c1) > len(c2) :
+ if len(c1) > len(c2):
c1[:c2.size] += c2
ret = c1
- else :
+ else:
c2[:c1.size] += c1
ret = c2
return pu.trimseq(ret)
@@ -388,10 +388,10 @@ def hermsub(c1, c2):
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
- if len(c1) > len(c2) :
+ if len(c1) > len(c2):
c1[:c2.size] -= c2
ret = c1
- else :
+ else:
c2 = -c2
c2[:c1.size] += c1
ret = c2
@@ -501,13 +501,13 @@ def hermmul(c1, c2):
elif len(c) == 2:
c0 = c[0]*xs
c1 = c[1]*xs
- else :
+ else:
nd = len(c)
c0 = c[-2]*xs
c1 = c[-1]*xs
- for i in range(3, len(c) + 1) :
+ for i in range(3, len(c) + 1):
tmp = c0
- nd = nd - 1
+ nd = nd - 1
c0 = hermsub(c[-i]*xs, c1*(2*(nd - 1)))
c1 = hermadd(tmp, hermmulx(c1)*2)
return hermadd(c0, hermmulx(c1)*2)
@@ -560,16 +560,16 @@ def hermdiv(c1, c2):
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
- if c2[-1] == 0 :
+ if c2[-1] == 0:
raise ZeroDivisionError()
lc1 = len(c1)
lc2 = len(c2)
- if lc1 < lc2 :
+ if lc1 < lc2:
return c1[:1]*0, c1
- elif lc2 == 1 :
+ elif lc2 == 1:
return c1/c2[-1], c1[:1]*0
- else :
+ else:
quo = np.empty(lc1 - lc2 + 1, dtype=c1.dtype)
rem = c1
for i in range(lc1 - lc2, - 1, -1):
@@ -580,7 +580,7 @@ def hermdiv(c1, c2):
return quo, pu.trimseq(rem)
-def hermpow(c, pow, maxpower=16) :
+def hermpow(c, pow, maxpower=16):
"""Raise a Hermite series to a power.
Returns the Hermite series `c` raised to the power `pow`. The
@@ -617,24 +617,24 @@ def hermpow(c, pow, maxpower=16) :
# c is a trimmed copy
[c] = pu.as_series([c])
power = int(pow)
- if power != pow or power < 0 :
+ if power != pow or power < 0:
raise ValueError("Power must be a non-negative integer.")
- elif maxpower is not None and power > maxpower :
+ elif maxpower is not None and power > maxpower:
raise ValueError("Power is too large")
- elif power == 0 :
+ elif power == 0:
return np.array([1], dtype=c.dtype)
- elif power == 1 :
+ elif power == 1:
return c
- else :
+ else:
# This can be made more efficient by using powers of two
# in the usual way.
prd = c
- for i in range(2, power + 1) :
+ for i in range(2, power + 1):
prd = hermmul(prd, c)
return prd
-def hermder(c, m=1, scl=1, axis=0) :
+def hermder(c, m=1, scl=1, axis=0):
"""
Differentiate a Hermite series.
@@ -712,7 +712,7 @@ def hermder(c, m=1, scl=1, axis=0) :
n = len(c)
if cnt >= n:
c = c[:1]*0
- else :
+ else:
for i in range(cnt):
n = n - 1
c *= scl
@@ -816,9 +816,9 @@ def hermint(c, m=1, k=[], lbnd=0, scl=1, axis=0):
if cnt != m:
raise ValueError("The order of integration must be integer")
- if cnt < 0 :
+ if cnt < 0:
raise ValueError("The order of integration must be non-negative")
- if len(k) > cnt :
+ if len(k) > cnt:
raise ValueError("Too many integration constants")
if iaxis != axis:
raise ValueError("The axis must be integer")
@@ -832,7 +832,7 @@ def hermint(c, m=1, k=[], lbnd=0, scl=1, axis=0):
c = np.rollaxis(c, iaxis)
k = list(k) + [0]*(cnt - len(k))
- for i in range(cnt) :
+ for i in range(cnt):
n = len(c)
c *= scl
if n == 1 and np.all(c[0] == 0):
@@ -924,22 +924,22 @@ def hermval(x, c, tensor=True):
if isinstance(x, (tuple, list)):
x = np.asarray(x)
if isinstance(x, np.ndarray) and tensor:
- c = c.reshape(c.shape + (1,)*x.ndim)
+ c = c.reshape(c.shape + (1,)*x.ndim)
x2 = x*2
- if len(c) == 1 :
+ if len(c) == 1:
c0 = c[0]
c1 = 0
- elif len(c) == 2 :
+ elif len(c) == 2:
c0 = c[0]
c1 = c[1]
- else :
+ else:
nd = len(c)
c0 = c[-2]
c1 = c[-1]
- for i in range(3, len(c) + 1) :
+ for i in range(3, len(c) + 1):
tmp = c0
- nd = nd - 1
+ nd = nd - 1
c0 = c[-i] - c1*(2*(nd - 1))
c1 = tmp + c1*x2
return c0 + c1*x2
@@ -1174,7 +1174,7 @@ def hermgrid3d(x, y, z, c):
return c
-def hermvander(x, deg) :
+def hermvander(x, deg):
"""Pseudo-Vandermonde matrix of given degree.
Returns the pseudo-Vandermonde matrix of degree `deg` and sample points
@@ -1229,15 +1229,15 @@ def hermvander(x, deg) :
dtyp = x.dtype
v = np.empty(dims, dtype=dtyp)
v[0] = x*0 + 1
- if ideg > 0 :
+ if ideg > 0:
x2 = x*2
v[1] = x2
- for i in range(2, ideg + 1) :
+ for i in range(2, ideg + 1):
v[i] = (v[i-1]*x2 - v[i-2]*(2*(i - 1)))
return np.rollaxis(v, 0, v.ndim)
-def hermvander2d(x, y, deg) :
+def hermvander2d(x, y, deg):
"""Pseudo-Vandermonde matrix of given degrees.
Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
@@ -1300,7 +1300,7 @@ def hermvander2d(x, y, deg) :
return v.reshape(v.shape[:-2] + (-1,))
-def hermvander3d(x, y, z, deg) :
+def hermvander3d(x, y, z, deg):
"""Pseudo-Vandermonde matrix of given degrees.
Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
@@ -1491,13 +1491,13 @@ def hermfit(x, y, deg, rcond=None, full=False, w=None):
y = np.asarray(y) + 0.0
# check arguments.
- if deg < 0 :
+ if deg < 0:
raise ValueError("expected deg >= 0")
if x.ndim != 1:
raise TypeError("expected 1D vector for x")
if x.size == 0:
raise TypeError("expected non-empty vector for x")
- if y.ndim < 1 or y.ndim > 2 :
+ if y.ndim < 1 or y.ndim > 2:
raise TypeError("expected 1D or 2D array for y")
if len(x) != len(y):
raise TypeError("expected x and y to have same length")
@@ -1517,7 +1517,7 @@ def hermfit(x, y, deg, rcond=None, full=False, w=None):
rhs = rhs * w
# set rcond
- if rcond is None :
+ if rcond is None:
rcond = len(x)*np.finfo(x.dtype).eps
# Determine the norms of the design matrix columns.
@@ -1536,9 +1536,9 @@ def hermfit(x, y, deg, rcond=None, full=False, w=None):
msg = "The fit may be poorly conditioned"
warnings.warn(msg, pu.RankWarning)
- if full :
+ if full:
return c, [resids, rank, s, rcond]
- else :
+ else:
return c
@@ -1568,7 +1568,6 @@ def hermcompanion(c):
.. versionadded::1.7.0
"""
- accprod = np.multiply.accumulate
# c is a trimmed copy
[c] = pu.as_series([c])
if len(c) < 2:
@@ -1578,13 +1577,13 @@ def hermcompanion(c):
n = len(c) - 1
mat = np.zeros((n, n), dtype=c.dtype)
- scl = np.hstack((1., np.sqrt(2.*np.arange(1, n))))
- scl = np.multiply.accumulate(scl)
+ scl = np.hstack((1., 1./np.sqrt(2.*np.arange(n - 1, 0, -1))))
+ scl = np.multiply.accumulate(scl)[::-1]
top = mat.reshape(-1)[1::n+1]
bot = mat.reshape(-1)[n::n+1]
top[...] = np.sqrt(.5*np.arange(1, n))
bot[...] = top
- mat[:, -1] -= (c[:-1]/c[-1])*(scl/scl[-1])*.5
+ mat[:, -1] -= scl*c[:-1]/(2.0*c[-1])
return mat
@@ -1636,9 +1635,9 @@ def hermroots(c):
"""
# c is a trimmed copy
[c] = pu.as_series([c])
- if len(c) <= 1 :
+ if len(c) <= 1:
return np.array([], dtype=c.dtype)
- if len(c) == 2 :
+ if len(c) == 2:
return np.array([-.5*c[0]/c[1]])
m = hermcompanion(c)
@@ -1647,6 +1646,49 @@ def hermroots(c):
return r
+def _normed_hermite_n(x, n):
+ """
+ Evaluate a normalized Hermite polynomial.
+
+ Compute the value of the normalized Hermite polynomial of degree ``n``
+ at the points ``x``.
+
+
+ Parameters
+ ----------
+ x : ndarray of double.
+ Points at which to evaluate the function
+ n : int
+ Degree of the normalized Hermite function to be evaluated.
+
+ Returns
+ -------
+ values : ndarray
+ The shape of the return value is described above.
+
+ Notes
+ -----
+ .. versionadded:: 1.10.0
+
+ This function is needed for finding the Gauss points and integration
+ weights for high degrees. The values of the standard Hermite functions
+ overflow when n >= 207.
+
+ """
+ if n == 0:
+ return np.ones(x.shape)/np.sqrt(np.sqrt(np.pi))
+
+ c0 = 0.
+ c1 = 1./np.sqrt(np.sqrt(np.pi))
+ nd = float(n)
+ for i in range(n - 1):
+ tmp = c0
+ c0 = -c1*np.sqrt((nd - 1.)/nd)
+ c1 = tmp + c1*x*np.sqrt(2./nd)
+ nd = nd - 1.0
+ return c0 + c1*x*np.sqrt(2)
+
+
def hermgauss(deg):
"""
Gauss-Hermite quadrature.
@@ -1689,22 +1731,21 @@ def hermgauss(deg):
# first approximation of roots. We use the fact that the companion
# matrix is symmetric in this case in order to obtain better zeros.
- c = np.array([0]*deg + [1])
+ c = np.array([0]*deg + [1], dtype=np.float64)
m = hermcompanion(c)
- x = la.eigvals(m)
+ x = la.eigvalsh(m)
x.sort()
# improve roots by one application of Newton
- dy = hermval(x, c)
- df = hermval(x, hermder(c))
+ dy = _normed_hermite_n(x, ideg)
+ df = _normed_hermite_n(x, ideg - 1) * np.sqrt(2*ideg)
x -= dy/df
# compute the weights. We scale the factor to avoid possible numerical
# overflow.
- fm = hermval(x, c[1:])
+ fm = _normed_hermite_n(x, ideg - 1)
fm /= np.abs(fm).max()
- df /= np.abs(df).max()
- w = 1/(fm * df)
+ w = 1/(fm * fm)
# for Hermite we can also symmetrize
w = (w + w[::-1])/2
diff --git a/numpy/polynomial/hermite_e.py b/numpy/polynomial/hermite_e.py
index 874b42470..fce13a84e 100644
--- a/numpy/polynomial/hermite_e.py
+++ b/numpy/polynomial/hermite_e.py
@@ -66,18 +66,19 @@ import numpy.linalg as la
from . import polyutils as pu
from ._polybase import ABCPolyBase
-__all__ = ['hermezero', 'hermeone', 'hermex', 'hermedomain', 'hermeline',
- 'hermeadd', 'hermesub', 'hermemulx', 'hermemul', 'hermediv', 'hermpow',
- 'hermeval',
- 'hermeder', 'hermeint', 'herme2poly', 'poly2herme', 'hermefromroots',
- 'hermevander', 'hermefit', 'hermetrim', 'hermeroots', 'HermiteE',
- 'hermeval2d', 'hermeval3d', 'hermegrid2d', 'hermegrid3d', 'hermevander2d',
- 'hermevander3d', 'hermecompanion', 'hermegauss', 'hermeweight']
+__all__ = [
+ 'hermezero', 'hermeone', 'hermex', 'hermedomain', 'hermeline',
+ 'hermeadd', 'hermesub', 'hermemulx', 'hermemul', 'hermediv',
+ 'hermepow', 'hermeval', 'hermeder', 'hermeint', 'herme2poly',
+ 'poly2herme', 'hermefromroots', 'hermevander', 'hermefit', 'hermetrim',
+ 'hermeroots', 'HermiteE', 'hermeval2d', 'hermeval3d', 'hermegrid2d',
+ 'hermegrid3d', 'hermevander2d', 'hermevander3d', 'hermecompanion',
+ 'hermegauss', 'hermeweight']
hermetrim = pu.trimcoef
-def poly2herme(pol) :
+def poly2herme(pol):
"""
poly2herme(pol)
@@ -118,12 +119,12 @@ def poly2herme(pol) :
[pol] = pu.as_series([pol])
deg = len(pol) - 1
res = 0
- for i in range(deg, -1, -1) :
+ for i in range(deg, -1, -1):
res = hermeadd(hermemulx(res), pol[i])
return res
-def herme2poly(c) :
+def herme2poly(c):
"""
Convert a Hermite series to a polynomial.
@@ -173,7 +174,7 @@ def herme2poly(c) :
c0 = c[-2]
c1 = c[-1]
# i is the current degree of c1
- for i in range(n - 1, 1, -1) :
+ for i in range(n - 1, 1, -1):
tmp = c0
c0 = polysub(c[i - 2], c1*(i - 1))
c1 = polyadd(tmp, polymulx(c1))
@@ -197,7 +198,7 @@ hermeone = np.array([1])
hermex = np.array([0, 1])
-def hermeline(off, scl) :
+def hermeline(off, scl):
"""
Hermite series whose graph is a straight line.
@@ -228,13 +229,13 @@ def hermeline(off, scl) :
5.0
"""
- if scl != 0 :
+ if scl != 0:
return np.array([off, scl])
- else :
+ else:
return np.array([off])
-def hermefromroots(roots) :
+def hermefromroots(roots):
"""
Generate a HermiteE series with given roots.
@@ -284,9 +285,9 @@ def hermefromroots(roots) :
array([ 0.+0.j, 0.+0.j])
"""
- if len(roots) == 0 :
+ if len(roots) == 0:
return np.ones(1)
- else :
+ else:
[roots] = pu.as_series([roots], trim=False)
roots.sort()
p = [hermeline(-r, 1) for r in roots]
@@ -340,10 +341,10 @@ def hermeadd(c1, c2):
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
- if len(c1) > len(c2) :
+ if len(c1) > len(c2):
c1[:c2.size] += c2
ret = c1
- else :
+ else:
c2[:c1.size] += c1
ret = c2
return pu.trimseq(ret)
@@ -388,10 +389,10 @@ def hermesub(c1, c2):
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
- if len(c1) > len(c2) :
+ if len(c1) > len(c2):
c1[:c2.size] -= c2
ret = c1
- else :
+ else:
c2 = -c2
c2[:c1.size] += c1
ret = c2
@@ -501,13 +502,13 @@ def hermemul(c1, c2):
elif len(c) == 2:
c0 = c[0]*xs
c1 = c[1]*xs
- else :
+ else:
nd = len(c)
c0 = c[-2]*xs
c1 = c[-1]*xs
- for i in range(3, len(c) + 1) :
+ for i in range(3, len(c) + 1):
tmp = c0
- nd = nd - 1
+ nd = nd - 1
c0 = hermesub(c[-i]*xs, c1*(nd - 1))
c1 = hermeadd(tmp, hermemulx(c1))
return hermeadd(c0, hermemulx(c1))
@@ -558,16 +559,16 @@ def hermediv(c1, c2):
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
- if c2[-1] == 0 :
+ if c2[-1] == 0:
raise ZeroDivisionError()
lc1 = len(c1)
lc2 = len(c2)
- if lc1 < lc2 :
+ if lc1 < lc2:
return c1[:1]*0, c1
- elif lc2 == 1 :
+ elif lc2 == 1:
return c1/c2[-1], c1[:1]*0
- else :
+ else:
quo = np.empty(lc1 - lc2 + 1, dtype=c1.dtype)
rem = c1
for i in range(lc1 - lc2, - 1, -1):
@@ -578,7 +579,7 @@ def hermediv(c1, c2):
return quo, pu.trimseq(rem)
-def hermepow(c, pow, maxpower=16) :
+def hermepow(c, pow, maxpower=16):
"""Raise a Hermite series to a power.
Returns the Hermite series `c` raised to the power `pow`. The
@@ -615,24 +616,24 @@ def hermepow(c, pow, maxpower=16) :
# c is a trimmed copy
[c] = pu.as_series([c])
power = int(pow)
- if power != pow or power < 0 :
+ if power != pow or power < 0:
raise ValueError("Power must be a non-negative integer.")
- elif maxpower is not None and power > maxpower :
+ elif maxpower is not None and power > maxpower:
raise ValueError("Power is too large")
- elif power == 0 :
+ elif power == 0:
return np.array([1], dtype=c.dtype)
- elif power == 1 :
+ elif power == 1:
return c
- else :
+ else:
# This can be made more efficient by using powers of two
# in the usual way.
prd = c
- for i in range(2, power + 1) :
+ for i in range(2, power + 1):
prd = hermemul(prd, c)
return prd
-def hermeder(c, m=1, scl=1, axis=0) :
+def hermeder(c, m=1, scl=1, axis=0):
"""
Differentiate a Hermite_e series.
@@ -710,7 +711,7 @@ def hermeder(c, m=1, scl=1, axis=0) :
n = len(c)
if cnt >= n:
return c[:1]*0
- else :
+ else:
for i in range(cnt):
n = n - 1
c *= scl
@@ -814,9 +815,9 @@ def hermeint(c, m=1, k=[], lbnd=0, scl=1, axis=0):
if cnt != m:
raise ValueError("The order of integration must be integer")
- if cnt < 0 :
+ if cnt < 0:
raise ValueError("The order of integration must be non-negative")
- if len(k) > cnt :
+ if len(k) > cnt:
raise ValueError("Too many integration constants")
if iaxis != axis:
raise ValueError("The axis must be integer")
@@ -830,7 +831,7 @@ def hermeint(c, m=1, k=[], lbnd=0, scl=1, axis=0):
c = np.rollaxis(c, iaxis)
k = list(k) + [0]*(cnt - len(k))
- for i in range(cnt) :
+ for i in range(cnt):
n = len(c)
c *= scl
if n == 1 and np.all(c[0] == 0):
@@ -922,21 +923,21 @@ def hermeval(x, c, tensor=True):
if isinstance(x, (tuple, list)):
x = np.asarray(x)
if isinstance(x, np.ndarray) and tensor:
- c = c.reshape(c.shape + (1,)*x.ndim)
+ c = c.reshape(c.shape + (1,)*x.ndim)
- if len(c) == 1 :
+ if len(c) == 1:
c0 = c[0]
c1 = 0
- elif len(c) == 2 :
+ elif len(c) == 2:
c0 = c[0]
c1 = c[1]
- else :
+ else:
nd = len(c)
c0 = c[-2]
c1 = c[-1]
- for i in range(3, len(c) + 1) :
+ for i in range(3, len(c) + 1):
tmp = c0
- nd = nd - 1
+ nd = nd - 1
c0 = c[-i] - c1*(nd - 1)
c1 = tmp + c1*x
return c0 + c1*x
@@ -1171,7 +1172,7 @@ def hermegrid3d(x, y, z, c):
return c
-def hermevander(x, deg) :
+def hermevander(x, deg):
"""Pseudo-Vandermonde matrix of given degree.
Returns the pseudo-Vandermonde matrix of degree `deg` and sample points
@@ -1226,14 +1227,14 @@ def hermevander(x, deg) :
dtyp = x.dtype
v = np.empty(dims, dtype=dtyp)
v[0] = x*0 + 1
- if ideg > 0 :
+ if ideg > 0:
v[1] = x
- for i in range(2, ideg + 1) :
+ for i in range(2, ideg + 1):
v[i] = (v[i-1]*x - v[i-2]*(i - 1))
return np.rollaxis(v, 0, v.ndim)
-def hermevander2d(x, y, deg) :
+def hermevander2d(x, y, deg):
"""Pseudo-Vandermonde matrix of given degrees.
Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
@@ -1296,7 +1297,7 @@ def hermevander2d(x, y, deg) :
return v.reshape(v.shape[:-2] + (-1,))
-def hermevander3d(x, y, z, deg) :
+def hermevander3d(x, y, z, deg):
"""Pseudo-Vandermonde matrix of given degrees.
Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
@@ -1487,13 +1488,13 @@ def hermefit(x, y, deg, rcond=None, full=False, w=None):
y = np.asarray(y) + 0.0
# check arguments.
- if deg < 0 :
+ if deg < 0:
raise ValueError("expected deg >= 0")
if x.ndim != 1:
raise TypeError("expected 1D vector for x")
if x.size == 0:
raise TypeError("expected non-empty vector for x")
- if y.ndim < 1 or y.ndim > 2 :
+ if y.ndim < 1 or y.ndim > 2:
raise TypeError("expected 1D or 2D array for y")
if len(x) != len(y):
raise TypeError("expected x and y to have same length")
@@ -1513,7 +1514,7 @@ def hermefit(x, y, deg, rcond=None, full=False, w=None):
rhs = rhs * w
# set rcond
- if rcond is None :
+ if rcond is None:
rcond = len(x)*np.finfo(x.dtype).eps
# Determine the norms of the design matrix columns.
@@ -1532,9 +1533,9 @@ def hermefit(x, y, deg, rcond=None, full=False, w=None):
msg = "The fit may be poorly conditioned"
warnings.warn(msg, pu.RankWarning)
- if full :
+ if full:
return c, [resids, rank, s, rcond]
- else :
+ else:
return c
@@ -1565,7 +1566,6 @@ def hermecompanion(c):
.. versionadded::1.7.0
"""
- accprod = np.multiply.accumulate
# c is a trimmed copy
[c] = pu.as_series([c])
if len(c) < 2:
@@ -1575,13 +1575,13 @@ def hermecompanion(c):
n = len(c) - 1
mat = np.zeros((n, n), dtype=c.dtype)
- scl = np.hstack((1., np.sqrt(np.arange(1, n))))
- scl = np.multiply.accumulate(scl)
+ scl = np.hstack((1., 1./np.sqrt(np.arange(n - 1, 0, -1))))
+ scl = np.multiply.accumulate(scl)[::-1]
top = mat.reshape(-1)[1::n+1]
bot = mat.reshape(-1)[n::n+1]
top[...] = np.sqrt(np.arange(1, n))
bot[...] = top
- mat[:, -1] -= (c[:-1]/c[-1])*(scl/scl[-1])
+ mat[:, -1] -= scl*c[:-1]/c[-1]
return mat
@@ -1633,9 +1633,9 @@ def hermeroots(c):
"""
# c is a trimmed copy
[c] = pu.as_series([c])
- if len(c) <= 1 :
+ if len(c) <= 1:
return np.array([], dtype=c.dtype)
- if len(c) == 2 :
+ if len(c) == 2:
return np.array([-c[0]/c[1]])
m = hermecompanion(c)
@@ -1644,6 +1644,49 @@ def hermeroots(c):
return r
+def _normed_hermite_e_n(x, n):
+ """
+ Evaluate a normalized HermiteE polynomial.
+
+ Compute the value of the normalized HermiteE polynomial of degree ``n``
+ at the points ``x``.
+
+
+ Parameters
+ ----------
+ x : ndarray of double.
+ Points at which to evaluate the function
+ n : int
+ Degree of the normalized HermiteE function to be evaluated.
+
+ Returns
+ -------
+ values : ndarray
+ The shape of the return value is described above.
+
+ Notes
+ -----
+ .. versionadded:: 1.10.0
+
+ This function is needed for finding the Gauss points and integration
+ weights for high degrees. The values of the standard HermiteE functions
+ overflow when n >= 207.
+
+ """
+ if n == 0:
+ return np.ones(x.shape)/np.sqrt(np.sqrt(2*np.pi))
+
+ c0 = 0.
+ c1 = 1./np.sqrt(np.sqrt(2*np.pi))
+ nd = float(n)
+ for i in range(n - 1):
+ tmp = c0
+ c0 = -c1*np.sqrt((nd - 1.)/nd)
+ c1 = tmp + c1*x*np.sqrt(1./nd)
+ nd = nd - 1.0
+ return c0 + c1*x
+
+
def hermegauss(deg):
"""
Gauss-HermiteE quadrature.
@@ -1688,20 +1731,19 @@ def hermegauss(deg):
# matrix is symmetric in this case in order to obtain better zeros.
c = np.array([0]*deg + [1])
m = hermecompanion(c)
- x = la.eigvals(m)
+ x = la.eigvalsh(m)
x.sort()
# improve roots by one application of Newton
- dy = hermeval(x, c)
- df = hermeval(x, hermeder(c))
+ dy = _normed_hermite_e_n(x, ideg)
+ df = _normed_hermite_e_n(x, ideg - 1) * np.sqrt(ideg)
x -= dy/df
# compute the weights. We scale the factor to avoid possible numerical
# overflow.
- fm = hermeval(x, c[1:])
+ fm = _normed_hermite_e_n(x, ideg - 1)
fm /= np.abs(fm).max()
- df /= np.abs(df).max()
- w = 1/(fm * df)
+ w = 1/(fm * fm)
# for Hermite_e we can also symmetrize
w = (w + w[::-1])/2
diff --git a/numpy/polynomial/laguerre.py b/numpy/polynomial/laguerre.py
index 9d88162ce..8d2705d5d 100644
--- a/numpy/polynomial/laguerre.py
+++ b/numpy/polynomial/laguerre.py
@@ -66,17 +66,18 @@ import numpy.linalg as la
from . import polyutils as pu
from ._polybase import ABCPolyBase
-__all__ = ['lagzero', 'lagone', 'lagx', 'lagdomain', 'lagline',
- 'lagadd', 'lagsub', 'lagmulx', 'lagmul', 'lagdiv', 'lagpow',
- 'lagval', 'lagder', 'lagint', 'lag2poly', 'poly2lag', 'lagfromroots',
- 'lagvander', 'lagfit', 'lagtrim', 'lagroots', 'Laguerre', 'lagval2d',
- 'lagval3d', 'laggrid2d', 'laggrid3d', 'lagvander2d', 'lagvander3d',
- 'lagcompanion', 'laggauss', 'lagweight']
+__all__ = [
+ 'lagzero', 'lagone', 'lagx', 'lagdomain', 'lagline', 'lagadd',
+ 'lagsub', 'lagmulx', 'lagmul', 'lagdiv', 'lagpow', 'lagval', 'lagder',
+ 'lagint', 'lag2poly', 'poly2lag', 'lagfromroots', 'lagvander',
+ 'lagfit', 'lagtrim', 'lagroots', 'Laguerre', 'lagval2d', 'lagval3d',
+ 'laggrid2d', 'laggrid3d', 'lagvander2d', 'lagvander3d', 'lagcompanion',
+ 'laggauss', 'lagweight']
lagtrim = pu.trimcoef
-def poly2lag(pol) :
+def poly2lag(pol):
"""
poly2lag(pol)
@@ -117,12 +118,12 @@ def poly2lag(pol) :
[pol] = pu.as_series([pol])
deg = len(pol) - 1
res = 0
- for i in range(deg, -1, -1) :
+ for i in range(deg, -1, -1):
res = lagadd(lagmulx(res), pol[i])
return res
-def lag2poly(c) :
+def lag2poly(c):
"""
Convert a Laguerre series to a polynomial.
@@ -194,7 +195,7 @@ lagone = np.array([1])
lagx = np.array([1, -1])
-def lagline(off, scl) :
+def lagline(off, scl):
"""
Laguerre series whose graph is a straight line.
@@ -224,13 +225,13 @@ def lagline(off, scl) :
5.0
"""
- if scl != 0 :
+ if scl != 0:
return np.array([off + scl, -scl])
- else :
+ else:
return np.array([off])
-def lagfromroots(roots) :
+def lagfromroots(roots):
"""
Generate a Laguerre series with given roots.
@@ -280,9 +281,9 @@ def lagfromroots(roots) :
array([ 0.+0.j, 0.+0.j])
"""
- if len(roots) == 0 :
+ if len(roots) == 0:
return np.ones(1)
- else :
+ else:
[roots] = pu.as_series([roots], trim=False)
roots.sort()
p = [lagline(-r, 1) for r in roots]
@@ -337,10 +338,10 @@ def lagadd(c1, c2):
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
- if len(c1) > len(c2) :
+ if len(c1) > len(c2):
c1[:c2.size] += c2
ret = c1
- else :
+ else:
c2[:c1.size] += c1
ret = c2
return pu.trimseq(ret)
@@ -385,10 +386,10 @@ def lagsub(c1, c2):
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
- if len(c1) > len(c2) :
+ if len(c1) > len(c2):
c1[:c2.size] -= c2
ret = c1
- else :
+ else:
c2 = -c2
c2[:c1.size] += c1
ret = c2
@@ -499,13 +500,13 @@ def lagmul(c1, c2):
elif len(c) == 2:
c0 = c[0]*xs
c1 = c[1]*xs
- else :
+ else:
nd = len(c)
c0 = c[-2]*xs
c1 = c[-1]*xs
- for i in range(3, len(c) + 1) :
+ for i in range(3, len(c) + 1):
tmp = c0
- nd = nd - 1
+ nd = nd - 1
c0 = lagsub(c[-i]*xs, (c1*(nd - 1))/nd)
c1 = lagadd(tmp, lagsub((2*nd - 1)*c1, lagmulx(c1))/nd)
return lagadd(c0, lagsub(c1, lagmulx(c1)))
@@ -556,16 +557,16 @@ def lagdiv(c1, c2):
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
- if c2[-1] == 0 :
+ if c2[-1] == 0:
raise ZeroDivisionError()
lc1 = len(c1)
lc2 = len(c2)
- if lc1 < lc2 :
+ if lc1 < lc2:
return c1[:1]*0, c1
- elif lc2 == 1 :
+ elif lc2 == 1:
return c1/c2[-1], c1[:1]*0
- else :
+ else:
quo = np.empty(lc1 - lc2 + 1, dtype=c1.dtype)
rem = c1
for i in range(lc1 - lc2, - 1, -1):
@@ -576,7 +577,7 @@ def lagdiv(c1, c2):
return quo, pu.trimseq(rem)
-def lagpow(c, pow, maxpower=16) :
+def lagpow(c, pow, maxpower=16):
"""Raise a Laguerre series to a power.
Returns the Laguerre series `c` raised to the power `pow`. The
@@ -613,24 +614,24 @@ def lagpow(c, pow, maxpower=16) :
# c is a trimmed copy
[c] = pu.as_series([c])
power = int(pow)
- if power != pow or power < 0 :
+ if power != pow or power < 0:
raise ValueError("Power must be a non-negative integer.")
- elif maxpower is not None and power > maxpower :
+ elif maxpower is not None and power > maxpower:
raise ValueError("Power is too large")
- elif power == 0 :
+ elif power == 0:
return np.array([1], dtype=c.dtype)
- elif power == 1 :
+ elif power == 1:
return c
- else :
+ else:
# This can be made more efficient by using powers of two
# in the usual way.
prd = c
- for i in range(2, power + 1) :
+ for i in range(2, power + 1):
prd = lagmul(prd, c)
return prd
-def lagder(c, m=1, scl=1, axis=0) :
+def lagder(c, m=1, scl=1, axis=0):
"""
Differentiate a Laguerre series.
@@ -708,7 +709,7 @@ def lagder(c, m=1, scl=1, axis=0) :
n = len(c)
if cnt >= n:
c = c[:1]*0
- else :
+ else:
for i in range(cnt):
n = n - 1
c *= scl
@@ -815,9 +816,9 @@ def lagint(c, m=1, k=[], lbnd=0, scl=1, axis=0):
if cnt != m:
raise ValueError("The order of integration must be integer")
- if cnt < 0 :
+ if cnt < 0:
raise ValueError("The order of integration must be non-negative")
- if len(k) > cnt :
+ if len(k) > cnt:
raise ValueError("Too many integration constants")
if iaxis != axis:
raise ValueError("The axis must be integer")
@@ -831,7 +832,7 @@ def lagint(c, m=1, k=[], lbnd=0, scl=1, axis=0):
c = np.rollaxis(c, iaxis)
k = list(k) + [0]*(cnt - len(k))
- for i in range(cnt) :
+ for i in range(cnt):
n = len(c)
c *= scl
if n == 1 and np.all(c[0] == 0):
@@ -924,22 +925,21 @@ def lagval(x, c, tensor=True):
if isinstance(x, (tuple, list)):
x = np.asarray(x)
if isinstance(x, np.ndarray) and tensor:
- c = c.reshape(c.shape + (1,)*x.ndim)
+ c = c.reshape(c.shape + (1,)*x.ndim)
-
- if len(c) == 1 :
+ if len(c) == 1:
c0 = c[0]
c1 = 0
- elif len(c) == 2 :
+ elif len(c) == 2:
c0 = c[0]
c1 = c[1]
- else :
+ else:
nd = len(c)
c0 = c[-2]
c1 = c[-1]
- for i in range(3, len(c) + 1) :
+ for i in range(3, len(c) + 1):
tmp = c0
- nd = nd - 1
+ nd = nd - 1
c0 = c[-i] - (c1*(nd - 1))/nd
c1 = tmp + (c1*((2*nd - 1) - x))/nd
return c0 + c1*(1 - x)
@@ -1174,7 +1174,7 @@ def laggrid3d(x, y, z, c):
return c
-def lagvander(x, deg) :
+def lagvander(x, deg):
"""Pseudo-Vandermonde matrix of given degree.
Returns the pseudo-Vandermonde matrix of degree `deg` and sample points
@@ -1229,14 +1229,14 @@ def lagvander(x, deg) :
dtyp = x.dtype
v = np.empty(dims, dtype=dtyp)
v[0] = x*0 + 1
- if ideg > 0 :
+ if ideg > 0:
v[1] = 1 - x
- for i in range(2, ideg + 1) :
+ for i in range(2, ideg + 1):
v[i] = (v[i-1]*(2*i - 1 - x) - v[i-2]*(i - 1))/i
return np.rollaxis(v, 0, v.ndim)
-def lagvander2d(x, y, deg) :
+def lagvander2d(x, y, deg):
"""Pseudo-Vandermonde matrix of given degrees.
Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
@@ -1299,7 +1299,7 @@ def lagvander2d(x, y, deg) :
return v.reshape(v.shape[:-2] + (-1,))
-def lagvander3d(x, y, z, deg) :
+def lagvander3d(x, y, z, deg):
"""Pseudo-Vandermonde matrix of given degrees.
Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
@@ -1490,13 +1490,13 @@ def lagfit(x, y, deg, rcond=None, full=False, w=None):
y = np.asarray(y) + 0.0
# check arguments.
- if deg < 0 :
+ if deg < 0:
raise ValueError("expected deg >= 0")
if x.ndim != 1:
raise TypeError("expected 1D vector for x")
if x.size == 0:
raise TypeError("expected non-empty vector for x")
- if y.ndim < 1 or y.ndim > 2 :
+ if y.ndim < 1 or y.ndim > 2:
raise TypeError("expected 1D or 2D array for y")
if len(x) != len(y):
raise TypeError("expected x and y to have same length")
@@ -1516,7 +1516,7 @@ def lagfit(x, y, deg, rcond=None, full=False, w=None):
rhs = rhs * w
# set rcond
- if rcond is None :
+ if rcond is None:
rcond = len(x)*np.finfo(x.dtype).eps
# Determine the norms of the design matrix columns.
@@ -1535,9 +1535,9 @@ def lagfit(x, y, deg, rcond=None, full=False, w=None):
msg = "The fit may be poorly conditioned"
warnings.warn(msg, pu.RankWarning)
- if full :
+ if full:
return c, [resids, rank, s, rcond]
- else :
+ else:
return c
@@ -1566,7 +1566,6 @@ def lagcompanion(c):
.. versionadded::1.7.0
"""
- accprod = np.multiply.accumulate
# c is a trimmed copy
[c] = pu.as_series([c])
if len(c) < 2:
@@ -1634,9 +1633,9 @@ def lagroots(c):
"""
# c is a trimmed copy
[c] = pu.as_series([c])
- if len(c) <= 1 :
+ if len(c) <= 1:
return np.array([], dtype=c.dtype)
- if len(c) == 2 :
+ if len(c) == 2:
return np.array([1 + c[0]/c[1]])
m = lagcompanion(c)
@@ -1651,8 +1650,8 @@ def laggauss(deg):
Computes the sample points and weights for Gauss-Laguerre quadrature.
These sample points and weights will correctly integrate polynomials of
- degree :math:`2*deg - 1` or less over the interval :math:`[0, \inf]` with the
- weight function :math:`f(x) = \exp(-x)`.
+ degree :math:`2*deg - 1` or less over the interval :math:`[0, \inf]`
+ with the weight function :math:`f(x) = \exp(-x)`.
Parameters
----------
diff --git a/numpy/polynomial/legendre.py b/numpy/polynomial/legendre.py
index 58c130b7e..d2de28269 100644
--- a/numpy/polynomial/legendre.py
+++ b/numpy/polynomial/legendre.py
@@ -90,17 +90,18 @@ import numpy.linalg as la
from . import polyutils as pu
from ._polybase import ABCPolyBase
-__all__ = ['legzero', 'legone', 'legx', 'legdomain', 'legline',
- 'legadd', 'legsub', 'legmulx', 'legmul', 'legdiv', 'legpow', 'legval',
- 'legder', 'legint', 'leg2poly', 'poly2leg', 'legfromroots',
- 'legvander', 'legfit', 'legtrim', 'legroots', 'Legendre', 'legval2d',
- 'legval3d', 'leggrid2d', 'leggrid3d', 'legvander2d', 'legvander3d',
- 'legcompanion', 'leggauss', 'legweight']
+__all__ = [
+ 'legzero', 'legone', 'legx', 'legdomain', 'legline', 'legadd',
+ 'legsub', 'legmulx', 'legmul', 'legdiv', 'legpow', 'legval', 'legder',
+ 'legint', 'leg2poly', 'poly2leg', 'legfromroots', 'legvander',
+ 'legfit', 'legtrim', 'legroots', 'Legendre', 'legval2d', 'legval3d',
+ 'leggrid2d', 'leggrid3d', 'legvander2d', 'legvander3d', 'legcompanion',
+ 'leggauss', 'legweight']
legtrim = pu.trimcoef
-def poly2leg(pol) :
+def poly2leg(pol):
"""
Convert a polynomial to a Legendre series.
@@ -143,12 +144,12 @@ def poly2leg(pol) :
[pol] = pu.as_series([pol])
deg = len(pol) - 1
res = 0
- for i in range(deg, -1, -1) :
+ for i in range(deg, -1, -1):
res = legadd(legmulx(res), pol[i])
return res
-def leg2poly(c) :
+def leg2poly(c):
"""
Convert a Legendre series to a polynomial.
@@ -202,7 +203,7 @@ def leg2poly(c) :
c0 = c[-2]
c1 = c[-1]
# i is the current degree of c1
- for i in range(n - 1, 1, -1) :
+ for i in range(n - 1, 1, -1):
tmp = c0
c0 = polysub(c[i - 2], (c1*(i - 1))/i)
c1 = polyadd(tmp, (polymulx(c1)*(2*i - 1))/i)
@@ -226,7 +227,7 @@ legone = np.array([1])
legx = np.array([0, 1])
-def legline(off, scl) :
+def legline(off, scl):
"""
Legendre series whose graph is a straight line.
@@ -256,13 +257,13 @@ def legline(off, scl) :
-3.0
"""
- if scl != 0 :
+ if scl != 0:
return np.array([off, scl])
- else :
+ else:
return np.array([off])
-def legfromroots(roots) :
+def legfromroots(roots):
"""
Generate a Legendre series with given roots.
@@ -311,9 +312,9 @@ def legfromroots(roots) :
array([ 1.33333333+0.j, 0.00000000+0.j, 0.66666667+0.j])
"""
- if len(roots) == 0 :
+ if len(roots) == 0:
return np.ones(1)
- else :
+ else:
[roots] = pu.as_series([roots], trim=False)
roots.sort()
p = [legline(-r, 1) for r in roots]
@@ -369,10 +370,10 @@ def legadd(c1, c2):
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
- if len(c1) > len(c2) :
+ if len(c1) > len(c2):
c1[:c2.size] += c2
ret = c1
- else :
+ else:
c2[:c1.size] += c1
ret = c2
return pu.trimseq(ret)
@@ -421,10 +422,10 @@ def legsub(c1, c2):
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
- if len(c1) > len(c2) :
+ if len(c1) > len(c2):
c1[:c2.size] -= c2
ret = c1
- else :
+ else:
c2 = -c2
c2[:c1.size] += c1
ret = c2
@@ -533,13 +534,13 @@ def legmul(c1, c2):
elif len(c) == 2:
c0 = c[0]*xs
c1 = c[1]*xs
- else :
+ else:
nd = len(c)
c0 = c[-2]*xs
c1 = c[-1]*xs
- for i in range(3, len(c) + 1) :
+ for i in range(3, len(c) + 1):
tmp = c0
- nd = nd - 1
+ nd = nd - 1
c0 = legsub(c[-i]*xs, (c1*(nd - 1))/nd)
c1 = legadd(tmp, (legmulx(c1)*(2*nd - 1))/nd)
return legadd(c0, legmulx(c1))
@@ -593,16 +594,16 @@ def legdiv(c1, c2):
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
- if c2[-1] == 0 :
+ if c2[-1] == 0:
raise ZeroDivisionError()
lc1 = len(c1)
lc2 = len(c2)
- if lc1 < lc2 :
+ if lc1 < lc2:
return c1[:1]*0, c1
- elif lc2 == 1 :
+ elif lc2 == 1:
return c1/c2[-1], c1[:1]*0
- else :
+ else:
quo = np.empty(lc1 - lc2 + 1, dtype=c1.dtype)
rem = c1
for i in range(lc1 - lc2, - 1, -1):
@@ -613,7 +614,7 @@ def legdiv(c1, c2):
return quo, pu.trimseq(rem)
-def legpow(c, pow, maxpower=16) :
+def legpow(c, pow, maxpower=16):
"""Raise a Legendre series to a power.
Returns the Legendre series `c` raised to the power `pow`. The
@@ -647,24 +648,24 @@ def legpow(c, pow, maxpower=16) :
# c is a trimmed copy
[c] = pu.as_series([c])
power = int(pow)
- if power != pow or power < 0 :
+ if power != pow or power < 0:
raise ValueError("Power must be a non-negative integer.")
- elif maxpower is not None and power > maxpower :
+ elif maxpower is not None and power > maxpower:
raise ValueError("Power is too large")
- elif power == 0 :
+ elif power == 0:
return np.array([1], dtype=c.dtype)
- elif power == 1 :
+ elif power == 1:
return c
- else :
+ else:
# This can be made more efficient by using powers of two
# in the usual way.
prd = c
- for i in range(2, power + 1) :
+ for i in range(2, power + 1):
prd = legmul(prd, c)
return prd
-def legder(c, m=1, scl=1, axis=0) :
+def legder(c, m=1, scl=1, axis=0):
"""
Differentiate a Legendre series.
@@ -747,7 +748,7 @@ def legder(c, m=1, scl=1, axis=0) :
n = len(c)
if cnt >= n:
c = c[:1]*0
- else :
+ else:
for i in range(cnt):
n = n - 1
c *= scl
@@ -857,9 +858,9 @@ def legint(c, m=1, k=[], lbnd=0, scl=1, axis=0):
if cnt != m:
raise ValueError("The order of integration must be integer")
- if cnt < 0 :
+ if cnt < 0:
raise ValueError("The order of integration must be non-negative")
- if len(k) > cnt :
+ if len(k) > cnt:
raise ValueError("Too many integration constants")
if iaxis != axis:
raise ValueError("The axis must be integer")
@@ -873,7 +874,7 @@ def legint(c, m=1, k=[], lbnd=0, scl=1, axis=0):
c = np.rollaxis(c, iaxis)
k = list(k) + [0]*(cnt - len(k))
- for i in range(cnt) :
+ for i in range(cnt):
n = len(c)
c *= scl
if n == 1 and np.all(c[0] == 0):
@@ -964,19 +965,19 @@ def legval(x, c, tensor=True):
if isinstance(x, np.ndarray) and tensor:
c = c.reshape(c.shape + (1,)*x.ndim)
- if len(c) == 1 :
+ if len(c) == 1:
c0 = c[0]
c1 = 0
- elif len(c) == 2 :
+ elif len(c) == 2:
c0 = c[0]
c1 = c[1]
- else :
+ else:
nd = len(c)
c0 = c[-2]
c1 = c[-1]
- for i in range(3, len(c) + 1) :
+ for i in range(3, len(c) + 1):
tmp = c0
- nd = nd - 1
+ nd = nd - 1
c0 = c[-i] - (c1*(nd - 1))/nd
c1 = tmp + (c1*x*(2*nd - 1))/nd
return c0 + c1*x
@@ -1211,7 +1212,7 @@ def leggrid3d(x, y, z, c):
return c
-def legvander(x, deg) :
+def legvander(x, deg):
"""Pseudo-Vandermonde matrix of given degree.
Returns the pseudo-Vandermonde matrix of degree `deg` and sample points
@@ -1259,14 +1260,14 @@ def legvander(x, deg) :
# Use forward recursion to generate the entries. This is not as accurate
# as reverse recursion in this application but it is more efficient.
v[0] = x*0 + 1
- if ideg > 0 :
+ if ideg > 0:
v[1] = x
- for i in range(2, ideg + 1) :
+ for i in range(2, ideg + 1):
v[i] = (v[i-1]*x*(2*i - 1) - v[i-2]*(i - 1))/i
return np.rollaxis(v, 0, v.ndim)
-def legvander2d(x, y, deg) :
+def legvander2d(x, y, deg):
"""Pseudo-Vandermonde matrix of given degrees.
Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
@@ -1329,7 +1330,7 @@ def legvander2d(x, y, deg) :
return v.reshape(v.shape[:-2] + (-1,))
-def legvander3d(x, y, z, deg) :
+def legvander3d(x, y, z, deg):
"""Pseudo-Vandermonde matrix of given degrees.
Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
@@ -1515,13 +1516,13 @@ def legfit(x, y, deg, rcond=None, full=False, w=None):
y = np.asarray(y) + 0.0
# check arguments.
- if deg < 0 :
+ if deg < 0:
raise ValueError("expected deg >= 0")
if x.ndim != 1:
raise TypeError("expected 1D vector for x")
if x.size == 0:
raise TypeError("expected non-empty vector for x")
- if y.ndim < 1 or y.ndim > 2 :
+ if y.ndim < 1 or y.ndim > 2:
raise TypeError("expected 1D or 2D array for y")
if len(x) != len(y):
raise TypeError("expected x and y to have same length")
@@ -1541,7 +1542,7 @@ def legfit(x, y, deg, rcond=None, full=False, w=None):
rhs = rhs * w
# set rcond
- if rcond is None :
+ if rcond is None:
rcond = len(x)*np.finfo(x.dtype).eps
# Determine the norms of the design matrix columns.
@@ -1560,9 +1561,9 @@ def legfit(x, y, deg, rcond=None, full=False, w=None):
msg = "The fit may be poorly conditioned"
warnings.warn(msg, pu.RankWarning)
- if full :
+ if full:
return c, [resids, rank, s, rcond]
- else :
+ else:
return c
@@ -1637,11 +1638,11 @@ def legroots(c):
-----
The root estimates are obtained as the eigenvalues of the companion
matrix, Roots far from the origin of the complex plane may have large
- errors due to the numerical instability of the series for such
- values. Roots with multiplicity greater than 1 will also show larger
- errors as the value of the series near such points is relatively
- insensitive to errors in the roots. Isolated roots near the origin can
- be improved by a few iterations of Newton's method.
+ errors due to the numerical instability of the series for such values.
+ Roots with multiplicity greater than 1 will also show larger errors as
+ the value of the series near such points is relatively insensitive to
+ errors in the roots. Isolated roots near the origin can be improved by
+ a few iterations of Newton's method.
The Legendre series basis polynomials aren't powers of ``x`` so the
results of this function may seem unintuitive.
@@ -1649,7 +1650,7 @@ def legroots(c):
Examples
--------
>>> import numpy.polynomial.legendre as leg
- >>> leg.legroots((1, 2, 3, 4)) # 4L_3 + 3L_2 + 2L_1 + 1L_0 has only real roots
+ >>> leg.legroots((1, 2, 3, 4)) # 4L_3 + 3L_2 + 2L_1 + 1L_0, all real roots
array([-0.85099543, -0.11407192, 0.51506735])
"""
diff --git a/numpy/polynomial/polynomial.py b/numpy/polynomial/polynomial.py
index 60aaff83f..60e339a1d 100644
--- a/numpy/polynomial/polynomial.py
+++ b/numpy/polynomial/polynomial.py
@@ -55,11 +55,12 @@ See Also
"""
from __future__ import division, absolute_import, print_function
-__all__ = ['polyzero', 'polyone', 'polyx', 'polydomain', 'polyline',
- 'polyadd', 'polysub', 'polymulx', 'polymul', 'polydiv', 'polypow',
- 'polyval', 'polyder', 'polyint', 'polyfromroots', 'polyvander',
- 'polyfit', 'polytrim', 'polyroots', 'Polynomial', 'polyval2d',
- 'polyval3d', 'polygrid2d', 'polygrid3d', 'polyvander2d', 'polyvander3d']
+__all__ = [
+ 'polyzero', 'polyone', 'polyx', 'polydomain', 'polyline', 'polyadd',
+ 'polysub', 'polymulx', 'polymul', 'polydiv', 'polypow', 'polyval',
+ 'polyder', 'polyint', 'polyfromroots', 'polyvander', 'polyfit',
+ 'polytrim', 'polyroots', 'Polynomial', 'polyval2d', 'polyval3d',
+ 'polygrid2d', 'polygrid3d', 'polyvander2d', 'polyvander3d']
import warnings
import numpy as np
@@ -92,7 +93,7 @@ polyx = np.array([0, 1])
#
-def polyline(off, scl) :
+def polyline(off, scl):
"""
Returns an array representing a linear polynomial.
@@ -113,20 +114,20 @@ def polyline(off, scl) :
Examples
--------
- >>> from numpy import polynomial as P
+ >>> from numpy.polynomial import polynomial as P
>>> P.polyline(1,-1)
array([ 1, -1])
>>> P.polyval(1, P.polyline(1,-1)) # should be 0
0.0
"""
- if scl != 0 :
+ if scl != 0:
return np.array([off, scl])
- else :
+ else:
return np.array([off])
-def polyfromroots(roots) :
+def polyfromroots(roots):
"""
Generate a monic polynomial with given roots.
@@ -176,7 +177,7 @@ def polyfromroots(roots) :
Examples
--------
- >>> import numpy.polynomial as P
+ >>> from numpy.polynomial import polynomial as P
>>> P.polyfromroots((-1,0,1)) # x(x - 1)(x + 1) = x^3 - x
array([ 0., -1., 0., 1.])
>>> j = complex(0,1)
@@ -184,9 +185,9 @@ def polyfromroots(roots) :
array([ 1.+0.j, 0.+0.j, 1.+0.j])
"""
- if len(roots) == 0 :
+ if len(roots) == 0:
return np.ones(1)
- else :
+ else:
[roots] = pu.as_series([roots], trim=False)
roots.sort()
p = [polyline(-r, 1) for r in roots]
@@ -225,7 +226,7 @@ def polyadd(c1, c2):
Examples
--------
- >>> from numpy import polynomial as P
+ >>> from numpy.polynomial import polynomial as P
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> sum = P.polyadd(c1,c2); sum
@@ -236,10 +237,10 @@ def polyadd(c1, c2):
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
- if len(c1) > len(c2) :
+ if len(c1) > len(c2):
c1[:c2.size] += c2
ret = c1
- else :
+ else:
c2[:c1.size] += c1
ret = c2
return pu.trimseq(ret)
@@ -270,7 +271,7 @@ def polysub(c1, c2):
Examples
--------
- >>> from numpy import polynomial as P
+ >>> from numpy.polynomial import polynomial as P
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> P.polysub(c1,c2)
@@ -281,10 +282,10 @@ def polysub(c1, c2):
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
- if len(c1) > len(c2) :
+ if len(c1) > len(c2):
c1[:c2.size] -= c2
ret = c1
- else :
+ else:
c2 = -c2
c2[:c1.size] += c1
ret = c2
@@ -352,7 +353,7 @@ def polymul(c1, c2):
Examples
--------
- >>> import numpy.polynomial as P
+ >>> from numpy.polynomial import polynomial as P
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> P.polymul(c1,c2)
@@ -389,7 +390,7 @@ def polydiv(c1, c2):
Examples
--------
- >>> import numpy.polynomial as P
+ >>> from numpy.polynomial import polynomial as P
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> P.polydiv(c1,c2)
@@ -400,29 +401,29 @@ def polydiv(c1, c2):
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
- if c2[-1] == 0 :
+ if c2[-1] == 0:
raise ZeroDivisionError()
len1 = len(c1)
len2 = len(c2)
- if len2 == 1 :
+ if len2 == 1:
return c1/c2[-1], c1[:1]*0
- elif len1 < len2 :
+ elif len1 < len2:
return c1[:1]*0, c1
- else :
+ else:
dlen = len1 - len2
scl = c2[-1]
- c2 = c2[:-1]/scl
+ c2 = c2[:-1]/scl
i = dlen
j = len1 - 1
- while i >= 0 :
+ while i >= 0:
c1[i:j] -= c2*c1[j]
i -= 1
j -= 1
return c1[j+1:]/scl, pu.trimseq(c1[:j+1])
-def polypow(c, pow, maxpower=None) :
+def polypow(c, pow, maxpower=None):
"""Raise a polynomial to a power.
Returns the polynomial `c` raised to the power `pow`. The argument
@@ -456,19 +457,19 @@ def polypow(c, pow, maxpower=None) :
# c is a trimmed copy
[c] = pu.as_series([c])
power = int(pow)
- if power != pow or power < 0 :
+ if power != pow or power < 0:
raise ValueError("Power must be a non-negative integer.")
- elif maxpower is not None and power > maxpower :
+ elif maxpower is not None and power > maxpower:
raise ValueError("Power is too large")
- elif power == 0 :
+ elif power == 0:
return np.array([1], dtype=c.dtype)
- elif power == 1 :
+ elif power == 1:
return c
- else :
+ else:
# This can be made more efficient by using powers of two
# in the usual way.
prd = c
- for i in range(2, power + 1) :
+ for i in range(2, power + 1):
prd = np.convolve(prd, c)
return prd
@@ -513,7 +514,7 @@ def polyder(c, m=1, scl=1, axis=0):
Examples
--------
- >>> from numpy import polynomial as P
+ >>> from numpy.polynomial import polynomial as P
>>> c = (1,2,3,4) # 1 + 2x + 3x**2 + 4x**3
>>> P.polyder(c) # (d/dx)(c) = 2 + 6x + 12x**2
array([ 2., 6., 12.])
@@ -550,7 +551,7 @@ def polyder(c, m=1, scl=1, axis=0):
n = len(c)
if cnt >= n:
c = c[:1]*0
- else :
+ else:
for i in range(cnt):
n = n - 1
c *= scl
@@ -624,7 +625,7 @@ def polyint(c, m=1, k=[], lbnd=0, scl=1, axis=0):
Examples
--------
- >>> from numpy import polynomial as P
+ >>> from numpy.polynomial import polynomial as P
>>> c = (1,2,3)
>>> P.polyint(c) # should return array([0, 1, 1, 1])
array([ 0., 1., 1., 1.])
@@ -650,9 +651,9 @@ def polyint(c, m=1, k=[], lbnd=0, scl=1, axis=0):
if cnt != m:
raise ValueError("The order of integration must be integer")
- if cnt < 0 :
+ if cnt < 0:
raise ValueError("The order of integration must be non-negative")
- if len(k) > cnt :
+ if len(k) > cnt:
raise ValueError("Too many integration constants")
if iaxis != axis:
raise ValueError("The axis must be integer")
@@ -661,7 +662,6 @@ def polyint(c, m=1, k=[], lbnd=0, scl=1, axis=0):
if iaxis < 0:
iaxis += c.ndim
-
if cnt == 0:
return c
@@ -775,7 +775,7 @@ def polyval(x, c, tensor=True):
c = c.reshape(c.shape + (1,)*x.ndim)
c0 = c[-1] + x*0
- for i in range(2, len(c) + 1) :
+ for i in range(2, len(c) + 1):
c0 = c[-i] + c0*x
return c0
@@ -1010,7 +1010,7 @@ def polygrid3d(x, y, z, c):
return c
-def polyvander(x, deg) :
+def polyvander(x, deg):
"""Vandermonde matrix of given degree.
Returns the Vandermonde matrix of degree `deg` and sample points
@@ -1059,14 +1059,14 @@ def polyvander(x, deg) :
dtyp = x.dtype
v = np.empty(dims, dtype=dtyp)
v[0] = x*0 + 1
- if ideg > 0 :
+ if ideg > 0:
v[1] = x
- for i in range(2, ideg + 1) :
+ for i in range(2, ideg + 1):
v[i] = v[i-1]*x
return np.rollaxis(v, 0, v.ndim)
-def polyvander2d(x, y, deg) :
+def polyvander2d(x, y, deg):
"""Pseudo-Vandermonde matrix of given degrees.
Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
@@ -1126,7 +1126,7 @@ def polyvander2d(x, y, deg) :
return v.reshape(v.shape[:-2] + (-1,))
-def polyvander3d(x, y, z, deg) :
+def polyvander3d(x, y, z, deg):
"""Pseudo-Vandermonde matrix of given degrees.
Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
@@ -1254,7 +1254,7 @@ def polyfit(x, y, deg, rcond=None, full=False, w=None):
rcond -- value of `rcond`.
For more details, see `linalg.lstsq`.
-
+
Raises
------
RankWarning
@@ -1310,7 +1310,7 @@ def polyfit(x, y, deg, rcond=None, full=False, w=None):
Examples
--------
- >>> from numpy import polynomial as P
+ >>> from numpy.polynomial import polynomial as P
>>> x = np.linspace(-1,1,51) # x "data": [-1, -0.96, ..., 0.96, 1]
>>> y = x**3 - x + np.random.randn(len(x)) # x^3 - x + N(0,1) "noise"
>>> c, stats = P.polyfit(x,y,3,full=True)
@@ -1337,13 +1337,13 @@ def polyfit(x, y, deg, rcond=None, full=False, w=None):
y = np.asarray(y) + 0.0
# check arguments.
- if deg < 0 :
+ if deg < 0:
raise ValueError("expected deg >= 0")
if x.ndim != 1:
raise TypeError("expected 1D vector for x")
if x.size == 0:
raise TypeError("expected non-empty vector for x")
- if y.ndim < 1 or y.ndim > 2 :
+ if y.ndim < 1 or y.ndim > 2:
raise TypeError("expected 1D or 2D array for y")
if len(x) != len(y):
raise TypeError("expected x and y to have same length")
@@ -1363,7 +1363,7 @@ def polyfit(x, y, deg, rcond=None, full=False, w=None):
rhs = rhs * w
# set rcond
- if rcond is None :
+ if rcond is None:
rcond = len(x)*np.finfo(x.dtype).eps
# Determine the norms of the design matrix columns.
@@ -1382,9 +1382,9 @@ def polyfit(x, y, deg, rcond=None, full=False, w=None):
msg = "The fit may be poorly conditioned"
warnings.warn(msg, pu.RankWarning)
- if full :
+ if full:
return c, [resids, rank, s, rcond]
- else :
+ else:
return c
@@ -1415,7 +1415,7 @@ def polycompanion(c):
"""
# c is a trimmed copy
[c] = pu.as_series([c])
- if len(c) < 2 :
+ if len(c) < 2:
raise ValueError('Series must have maximum degree of at least 1.')
if len(c) == 2:
return np.array([[-c[0]/c[1]]])
diff --git a/numpy/polynomial/polytemplate.py b/numpy/polynomial/polytemplate.py
deleted file mode 100644
index e68dd18ef..000000000
--- a/numpy/polynomial/polytemplate.py
+++ /dev/null
@@ -1,927 +0,0 @@
-"""
-Template for the Chebyshev and Polynomial classes.
-
-This module houses a Python string module Template object (see, e.g.,
-http://docs.python.org/library/string.html#template-strings) used by
-the `polynomial` and `chebyshev` modules to implement their respective
-`Polynomial` and `Chebyshev` classes. It provides a mechanism for easily
-creating additional specific polynomial classes (e.g., Legendre, Jacobi,
-etc.) in the future, such that all these classes will have a common API.
-
-"""
-from __future__ import division, absolute_import, print_function
-
-import string
-import sys
-import warnings
-from number import Number
-
-from numpy import ModuleDeprecationWarning
-
-warnings.warn("The polytemplate module will be removed in Numpy 1.10.0.",
- ModuleDeprecationWarning)
-
-polytemplate = string.Template('''
-from __future__ import division, absolute_import, print_function
-import numpy as np
-import warnings
-from . import polyutils as pu
-
-class $name(pu.PolyBase) :
- """A $name series class.
-
- $name instances provide the standard Python numerical methods '+',
- '-', '*', '//', '%', 'divmod', '**', and '()' as well as the listed
- methods.
-
- Parameters
- ----------
- coef : array_like
- $name coefficients, in increasing order. For example,
- ``(1, 2, 3)`` implies ``P_0 + 2P_1 + 3P_2`` where the
- ``P_i`` are a graded polynomial basis.
- domain : (2,) array_like, optional
- Domain to use. The interval ``[domain[0], domain[1]]`` is mapped to
- the interval ``[window[0], window[1]]`` by shifting and scaling.
- The default value is $domain.
- window : (2,) array_like, optional
- Window, see ``domain`` for its use. The default value is $domain.
- .. versionadded:: 1.6.0
-
- Attributes
- ----------
- coef : (N,) ndarray
- $name coefficients, from low to high.
- domain : (2,) ndarray
- Domain that is mapped to ``window``.
- window : (2,) ndarray
- Window that ``domain`` is mapped to.
-
- Class Attributes
- ----------------
- maxpower : int
- Maximum power allowed, i.e., the largest number ``n`` such that
- ``p(x)**n`` is allowed. This is to limit runaway polynomial size.
- domain : (2,) ndarray
- Default domain of the class.
- window : (2,) ndarray
- Default window of the class.
-
- Notes
- -----
- It is important to specify the domain in many cases, for instance in
- fitting data, because many of the important properties of the
- polynomial basis only hold in a specified interval and consequently
- the data must be mapped into that interval in order to benefit.
-
- Examples
- --------
-
- """
- # Limit runaway size. T_n^m has degree n*2^m
- maxpower = 16
- # Default domain
- domain = np.array($domain)
- # Default window
- window = np.array($domain)
- # Don't let participate in array operations. Value doesn't matter.
- __array_priority__ = 1000
- # Not hashable
- __hash__ = None
-
- def has_samecoef(self, other):
- """Check if coefficients match.
-
- Parameters
- ----------
- other : class instance
- The other class must have the ``coef`` attribute.
-
- Returns
- -------
- bool : boolean
- True if the coefficients are the same, False otherwise.
-
- Notes
- -----
- .. versionadded:: 1.6.0
-
- """
- if len(self.coef) != len(other.coef):
- return False
- elif not np.all(self.coef == other.coef):
- return False
- else:
- return True
-
- def has_samedomain(self, other):
- """Check if domains match.
-
- Parameters
- ----------
- other : class instance
- The other class must have the ``domain`` attribute.
-
- Returns
- -------
- bool : boolean
- True if the domains are the same, False otherwise.
-
- Notes
- -----
- .. versionadded:: 1.6.0
-
- """
- return np.all(self.domain == other.domain)
-
- def has_samewindow(self, other):
- """Check if windows match.
-
- Parameters
- ----------
- other : class instance
- The other class must have the ``window`` attribute.
-
- Returns
- -------
- bool : boolean
- True if the windows are the same, False otherwise.
-
- Notes
- -----
- .. versionadded:: 1.6.0
-
- """
- return np.all(self.window == other.window)
-
- def has_sametype(self, other):
- """Check if types match.
-
- Parameters
- ----------
- other : object
- Class instance.
-
- Returns
- -------
- bool : boolean
- True if other is same class as self
-
- Notes
- -----
- .. versionadded:: 1.7.0
-
- """
- return isinstance(other, self.__class__)
-
- def __init__(self, coef, domain=$domain, window=$domain) :
- [coef, dom, win] = pu.as_series([coef, domain, window], trim=False)
- if len(dom) != 2 :
- raise ValueError("Domain has wrong number of elements.")
- if len(win) != 2 :
- raise ValueError("Window has wrong number of elements.")
- self.coef = coef
- self.domain = dom
- self.window = win
-
- def __repr__(self):
- format = "%s(%s, %s, %s)"
- coef = repr(self.coef)[6:-1]
- domain = repr(self.domain)[6:-1]
- window = repr(self.window)[6:-1]
- return format % ('$name', coef, domain, window)
-
- def __str__(self) :
- format = "%s(%s)"
- coef = str(self.coef)
- return format % ('$nick', coef)
-
- # Pickle and copy
-
- def __getstate__(self) :
- ret = self.__dict__.copy()
- ret['coef'] = self.coef.copy()
- ret['domain'] = self.domain.copy()
- ret['window'] = self.window.copy()
- return ret
-
- def __setstate__(self, dict) :
- self.__dict__ = dict
-
- # Call
-
- def __call__(self, arg) :
- off, scl = pu.mapparms(self.domain, self.window)
- arg = off + scl*arg
- return ${nick}val(arg, self.coef)
-
- def __iter__(self) :
- return iter(self.coef)
-
- def __len__(self) :
- return len(self.coef)
-
- # Numeric properties.
-
- def __neg__(self) :
- return self.__class__(-self.coef, self.domain, self.window)
-
- def __pos__(self) :
- return self
-
- def __add__(self, other) :
- """Returns sum"""
- if isinstance(other, pu.PolyBase):
- if not self.has_sametype(other):
- raise TypeError("Polynomial types differ")
- elif not self.has_samedomain(other):
- raise TypeError("Domains differ")
- elif not self.has_samewindow(other):
- raise TypeError("Windows differ")
- else:
- coef = ${nick}add(self.coef, other.coef)
- else :
- try :
- coef = ${nick}add(self.coef, other)
- except :
- return NotImplemented
- return self.__class__(coef, self.domain, self.window)
-
- def __sub__(self, other) :
- """Returns difference"""
- if isinstance(other, pu.PolyBase):
- if not self.has_sametype(other):
- raise TypeError("Polynomial types differ")
- elif not self.has_samedomain(other):
- raise TypeError("Domains differ")
- elif not self.has_samewindow(other):
- raise TypeError("Windows differ")
- else:
- coef = ${nick}sub(self.coef, other.coef)
- else :
- try :
- coef = ${nick}sub(self.coef, other)
- except :
- return NotImplemented
- return self.__class__(coef, self.domain, self.window)
-
- def __mul__(self, other) :
- """Returns product"""
- if isinstance(other, pu.PolyBase):
- if not self.has_sametype(other):
- raise TypeError("Polynomial types differ")
- elif not self.has_samedomain(other):
- raise TypeError("Domains differ")
- elif not self.has_samewindow(other):
- raise TypeError("Windows differ")
- else:
- coef = ${nick}mul(self.coef, other.coef)
- else :
- try :
- coef = ${nick}mul(self.coef, other)
- except :
- return NotImplemented
- return self.__class__(coef, self.domain, self.window)
-
- def __div__(self, other):
- # set to __floordiv__, /, for now.
- return self.__floordiv__(other)
-
- def __truediv__(self, other) :
- # there is no true divide if the rhs is not a Number, although it
- # could return the first n elements of an infinite series.
- # It is hard to see where n would come from, though.
- if not isinstance(other, Number) or isinstance(other, bool):
- form = "unsupported types for true division: '%s', '%s'"
- raise TypeError(form % (type(self), type(other)))
- return self.__floordiv__(other)
-
- def __floordiv__(self, other) :
- """Returns the quotient."""
- if isinstance(other, pu.PolyBase):
- if not self.has_sametype(other):
- raise TypeError("Polynomial types differ")
- elif not self.has_samedomain(other):
- raise TypeError("Domains differ")
- elif not self.has_samewindow(other):
- raise TypeError("Windows differ")
- else:
- quo, rem = ${nick}div(self.coef, other.coef)
- else :
- try :
- quo, rem = ${nick}div(self.coef, other)
- except :
- return NotImplemented
- return self.__class__(quo, self.domain, self.window)
-
- def __mod__(self, other) :
- """Returns the remainder."""
- if isinstance(other, pu.PolyBase):
- if not self.has_sametype(other):
- raise TypeError("Polynomial types differ")
- elif not self.has_samedomain(other):
- raise TypeError("Domains differ")
- elif not self.has_samewindow(other):
- raise TypeError("Windows differ")
- else:
- quo, rem = ${nick}div(self.coef, other.coef)
- else :
- try :
- quo, rem = ${nick}div(self.coef, other)
- except :
- return NotImplemented
- return self.__class__(rem, self.domain, self.window)
-
- def __divmod__(self, other) :
- """Returns quo, remainder"""
- if isinstance(other, self.__class__) :
- if not self.has_samedomain(other):
- raise TypeError("Domains are not equal")
- elif not self.has_samewindow(other):
- raise TypeError("Windows are not equal")
- else:
- quo, rem = ${nick}div(self.coef, other.coef)
- else :
- try :
- quo, rem = ${nick}div(self.coef, other)
- except :
- return NotImplemented
- quo = self.__class__(quo, self.domain, self.window)
- rem = self.__class__(rem, self.domain, self.window)
- return quo, rem
-
- def __pow__(self, other) :
- try :
- coef = ${nick}pow(self.coef, other, maxpower = self.maxpower)
- except :
- raise
- return self.__class__(coef, self.domain, self.window)
-
- def __radd__(self, other) :
- try :
- coef = ${nick}add(other, self.coef)
- except :
- return NotImplemented
- return self.__class__(coef, self.domain, self.window)
-
- def __rsub__(self, other):
- try :
- coef = ${nick}sub(other, self.coef)
- except :
- return NotImplemented
- return self.__class__(coef, self.domain, self.window)
-
- def __rmul__(self, other) :
- try :
- coef = ${nick}mul(other, self.coef)
- except :
- return NotImplemented
- return self.__class__(coef, self.domain, self.window)
-
- def __rdiv__(self, other):
- # set to __floordiv__ /.
- return self.__rfloordiv__(other)
-
- def __rtruediv__(self, other) :
- # An instance of PolyBase is not considered a
- # Number.
- return NotImplemented
-
- def __rfloordiv__(self, other) :
- try :
- quo, rem = ${nick}div(other, self.coef)
- except:
- return NotImplemented
- return self.__class__(quo, self.domain, self.window)
-
- def __rmod__(self, other) :
- try :
- quo, rem = ${nick}div(other, self.coef)
- except :
- return NotImplemented
- return self.__class__(rem, self.domain, self.window)
-
- def __rdivmod__(self, other) :
- try :
- quo, rem = ${nick}div(other, self.coef)
- except :
- return NotImplemented
- quo = self.__class__(quo, self.domain, self.window)
- rem = self.__class__(rem, self.domain, self.window)
- return quo, rem
-
- # Enhance me
- # some augmented arithmetic operations could be added here
-
- def __eq__(self, other) :
- res = isinstance(other, self.__class__) \
- and self.has_samecoef(other) \
- and self.has_samedomain(other) \
- and self.has_samewindow(other)
- return res
-
- def __ne__(self, other) :
- return not self.__eq__(other)
-
- #
- # Extra methods.
- #
-
- def copy(self) :
- """Return a copy.
-
- Return a copy of the current $name instance.
-
- Returns
- -------
- new_instance : $name
- Copy of current instance.
-
- """
- return self.__class__(self.coef, self.domain, self.window)
-
- def degree(self) :
- """The degree of the series.
-
- Notes
- -----
- .. versionadded:: 1.5.0
-
- """
- return len(self) - 1
-
- def cutdeg(self, deg) :
- """Truncate series to the given degree.
-
- Reduce the degree of the $name series to `deg` by discarding the
- high order terms. If `deg` is greater than the current degree a
- copy of the current series is returned. This can be useful in least
- squares where the coefficients of the high degree terms may be very
- small.
-
- Parameters
- ----------
- deg : non-negative int
- The series is reduced to degree `deg` by discarding the high
- order terms. The value of `deg` must be a non-negative integer.
-
- Returns
- -------
- new_instance : $name
- New instance of $name with reduced degree.
-
- Notes
- -----
- .. versionadded:: 1.5.0
-
- """
- return self.truncate(deg + 1)
-
- def trim(self, tol=0) :
- """Remove small leading coefficients
-
- Remove leading coefficients until a coefficient is reached whose
- absolute value greater than `tol` or the beginning of the series is
- reached. If all the coefficients would be removed the series is set to
- ``[0]``. A new $name instance is returned with the new coefficients.
- The current instance remains unchanged.
-
- Parameters
- ----------
- tol : non-negative number.
- All trailing coefficients less than `tol` will be removed.
-
- Returns
- -------
- new_instance : $name
- Contains the new set of coefficients.
-
- """
- coef = pu.trimcoef(self.coef, tol)
- return self.__class__(coef, self.domain, self.window)
-
- def truncate(self, size) :
- """Truncate series to length `size`.
-
- Reduce the $name series to length `size` by discarding the high
- degree terms. The value of `size` must be a positive integer. This
- can be useful in least squares where the coefficients of the
- high degree terms may be very small.
-
- Parameters
- ----------
- size : positive int
- The series is reduced to length `size` by discarding the high
- degree terms. The value of `size` must be a positive integer.
-
- Returns
- -------
- new_instance : $name
- New instance of $name with truncated coefficients.
-
- """
- isize = int(size)
- if isize != size or isize < 1 :
- raise ValueError("size must be a positive integer")
- if isize >= len(self.coef) :
- coef = self.coef
- else :
- coef = self.coef[:isize]
- return self.__class__(coef, self.domain, self.window)
-
- def convert(self, domain=None, kind=None, window=None) :
- """Convert to different class and/or domain.
-
- Parameters
- ----------
- domain : array_like, optional
- The domain of the converted series. If the value is None,
- the default domain of `kind` is used.
- kind : class, optional
- The polynomial series type class to which the current instance
- should be converted. If kind is None, then the class of the
- current instance is used.
- window : array_like, optional
- The window of the converted series. If the value is None,
- the default window of `kind` is used.
-
- Returns
- -------
- new_series_instance : `kind`
- The returned class can be of different type than the current
- instance and/or have a different domain.
-
- Notes
- -----
- Conversion between domains and class types can result in
- numerically ill defined series.
-
- Examples
- --------
-
- """
- if kind is None:
- kind = $name
- if domain is None:
- domain = kind.domain
- if window is None:
- window = kind.window
- return self(kind.identity(domain, window=window))
-
- def mapparms(self) :
- """Return the mapping parameters.
-
- The returned values define a linear map ``off + scl*x`` that is
- applied to the input arguments before the series is evaluated. The
- map depends on the ``domain`` and ``window``; if the current
- ``domain`` is equal to the ``window`` the resulting map is the
- identity. If the coefficients of the ``$name`` instance are to be
- used by themselves outside this class, then the linear function
- must be substituted for the ``x`` in the standard representation of
- the base polynomials.
-
- Returns
- -------
- off, scl : floats or complex
- The mapping function is defined by ``off + scl*x``.
-
- Notes
- -----
- If the current domain is the interval ``[l_1, r_1]`` and the window
- is ``[l_2, r_2]``, then the linear mapping function ``L`` is
- defined by the equations::
-
- L(l_1) = l_2
- L(r_1) = r_2
-
- """
- return pu.mapparms(self.domain, self.window)
-
- def integ(self, m=1, k=[], lbnd=None) :
- """Integrate.
-
- Return an instance of $name that is the definite integral of the
- current series. Refer to `${nick}int` for full documentation.
-
- Parameters
- ----------
- m : non-negative int
- The number of integrations to perform.
- k : array_like
- Integration constants. The first constant is applied to the
- first integration, the second to the second, and so on. The
- list of values must less than or equal to `m` in length and any
- missing values are set to zero.
- lbnd : Scalar
- The lower bound of the definite integral.
-
- Returns
- -------
- integral : $name
- The integral of the series using the same domain.
-
- See Also
- --------
- ${nick}int : similar function.
- ${nick}der : similar function for derivative.
-
- """
- off, scl = self.mapparms()
- if lbnd is None :
- lbnd = 0
- else :
- lbnd = off + scl*lbnd
- coef = ${nick}int(self.coef, m, k, lbnd, 1./scl)
- return self.__class__(coef, self.domain, self.window)
-
- def deriv(self, m=1):
- """Differentiate.
-
- Return an instance of $name that is the derivative of the current
- series. Refer to `${nick}der` for full documentation.
-
- Parameters
- ----------
- m : non-negative int
- The number of integrations to perform.
-
- Returns
- -------
- derivative : $name
- The derivative of the series using the same domain.
-
- See Also
- --------
- ${nick}der : similar function.
- ${nick}int : similar function for integration.
-
- """
- off, scl = self.mapparms()
- coef = ${nick}der(self.coef, m, scl)
- return self.__class__(coef, self.domain, self.window)
-
- def roots(self) :
- """Return list of roots.
-
- Return ndarray of roots for this series. See `${nick}roots` for
- full documentation. Note that the accuracy of the roots is likely to
- decrease the further outside the domain they lie.
-
- See Also
- --------
- ${nick}roots : similar function
- ${nick}fromroots : function to go generate series from roots.
-
- """
- roots = ${nick}roots(self.coef)
- return pu.mapdomain(roots, self.window, self.domain)
-
- def linspace(self, n=100, domain=None):
- """Return x,y values at equally spaced points in domain.
-
- Returns x, y values at `n` linearly spaced points across domain.
- Here y is the value of the polynomial at the points x. By default
- the domain is the same as that of the $name instance. This method
- is intended mostly as a plotting aid.
-
- Parameters
- ----------
- n : int, optional
- Number of point pairs to return. The default value is 100.
- domain : {None, array_like}
- If not None, the specified domain is used instead of that of
- the calling instance. It should be of the form ``[beg,end]``.
- The default is None.
-
- Returns
- -------
- x, y : ndarrays
- ``x`` is equal to linspace(self.domain[0], self.domain[1], n)
- ``y`` is the polynomial evaluated at ``x``.
-
- .. versionadded:: 1.5.0
-
- """
- if domain is None:
- domain = self.domain
- x = np.linspace(domain[0], domain[1], n)
- y = self(x)
- return x, y
-
-
-
- @staticmethod
- def fit(x, y, deg, domain=None, rcond=None, full=False, w=None,
- window=$domain):
- """Least squares fit to data.
-
- Return a `$name` instance that is the least squares fit to the data
- `y` sampled at `x`. Unlike `${nick}fit`, the domain of the returned
- instance can be specified and this will often result in a superior
- fit with less chance of ill conditioning. Support for NA was added
- in version 1.7.0. See `${nick}fit` for full documentation of the
- implementation.
-
- Parameters
- ----------
- x : array_like, shape (M,)
- x-coordinates of the M sample points ``(x[i], y[i])``.
- y : array_like, shape (M,) or (M, K)
- y-coordinates of the sample points. Several data sets of sample
- points sharing the same x-coordinates can be fitted at once by
- passing in a 2D-array that contains one dataset per column.
- deg : int
- Degree of the fitting polynomial.
- domain : {None, [beg, end], []}, optional
- Domain to use for the returned $name instance. If ``None``,
- then a minimal domain that covers the points `x` is chosen. If
- ``[]`` the default domain ``$domain`` is used. The default
- value is $domain in numpy 1.4.x and ``None`` in later versions.
- The ``[]`` value was added in numpy 1.5.0.
- rcond : float, optional
- Relative condition number of the fit. Singular values smaller
- than this relative to the largest singular value will be
- ignored. The default value is len(x)*eps, where eps is the
- relative precision of the float type, about 2e-16 in most
- cases.
- full : bool, optional
- Switch determining nature of return value. When it is False
- (the default) just the coefficients are returned, when True
- diagnostic information from the singular value decomposition is
- also returned.
- w : array_like, shape (M,), optional
- Weights. If not None the contribution of each point
- ``(x[i],y[i])`` to the fit is weighted by `w[i]`. Ideally the
- weights are chosen so that the errors of the products
- ``w[i]*y[i]`` all have the same variance. The default value is
- None.
- .. versionadded:: 1.5.0
- window : {[beg, end]}, optional
- Window to use for the returned $name instance. The default
- value is ``$domain``
- .. versionadded:: 1.6.0
-
- Returns
- -------
- least_squares_fit : instance of $name
- The $name instance is the least squares fit to the data and
- has the domain specified in the call.
-
- [residuals, rank, singular_values, rcond] : only if `full` = True
- Residuals of the least squares fit, the effective rank of the
- scaled Vandermonde matrix and its singular values, and the
- specified value of `rcond`. For more details, see
- `linalg.lstsq`.
-
- See Also
- --------
- ${nick}fit : similar function
-
- """
- if domain is None:
- domain = pu.getdomain(x)
- elif type(domain) is list and len(domain) == 0:
- domain = $domain
-
- if type(window) is list and len(window) == 0:
- window = $domain
-
- xnew = pu.mapdomain(x, domain, window)
- res = ${nick}fit(xnew, y, deg, w=w, rcond=rcond, full=full)
- if full :
- [coef, status] = res
- return $name(coef, domain=domain, window=window), status
- else :
- coef = res
- return $name(coef, domain=domain, window=window)
-
- @staticmethod
- def fromroots(roots, domain=$domain, window=$domain) :
- """Return $name instance with specified roots.
-
- Returns an instance of $name representing the product
- ``(x - r[0])*(x - r[1])*...*(x - r[n-1])``, where ``r`` is the
- list of roots.
-
- Parameters
- ----------
- roots : array_like
- List of roots.
- domain : {array_like, None}, optional
- Domain for the resulting instance of $name. If none the domain
- is the interval from the smallest root to the largest. The
- default is $domain.
- window : array_like, optional
- Window for the resulting instance of $name. The default value
- is $domain.
-
- Returns
- -------
- object : $name instance
- Series with the specified roots.
-
- See Also
- --------
- ${nick}fromroots : equivalent function
-
- """
- [roots] = pu.as_series([roots], trim=False)
- if domain is None :
- domain = pu.getdomain(roots)
- deg = len(roots)
- off, scl = pu.mapparms(domain, window)
- rnew = off + scl*roots
- coef = ${nick}fromroots(rnew) / scl**deg
- return $name(coef, domain=domain, window=window)
-
- @staticmethod
- def identity(domain=$domain, window=$domain) :
- """Identity function.
-
- If ``p`` is the returned $name object, then ``p(x) == x`` for all
- values of x.
-
- Parameters
- ----------
- domain : array_like
- The resulting array must be of the form ``[beg, end]``, where
- ``beg`` and ``end`` are the endpoints of the domain.
- window : array_like
- The resulting array must be if the form ``[beg, end]``, where
- ``beg`` and ``end`` are the endpoints of the window.
-
- Returns
- -------
- identity : $name instance
-
- """
- off, scl = pu.mapparms(window, domain)
- coef = ${nick}line(off, scl)
- return $name(coef, domain, window)
-
- @staticmethod
- def basis(deg, domain=$domain, window=$domain):
- """$name polynomial of degree `deg`.
-
- Returns an instance of the $name polynomial of degree `d`.
-
- Parameters
- ----------
- deg : int
- Degree of the $name polynomial. Must be >= 0.
- domain : array_like
- The resulting array must be of the form ``[beg, end]``, where
- ``beg`` and ``end`` are the endpoints of the domain.
- window : array_like
- The resulting array must be if the form ``[beg, end]``, where
- ``beg`` and ``end`` are the endpoints of the window.
-
- Returns
- p : $name instance
-
- Notes
- -----
- .. versionadded:: 1.7.0
-
- """
- ideg = int(deg)
- if ideg != deg or ideg < 0:
- raise ValueError("deg must be non-negative integer")
- return $name([0]*ideg + [1], domain, window)
-
- @staticmethod
- def cast(series, domain=$domain, window=$domain):
- """Convert instance to equivalent $name series.
-
- The `series` is expected to be an instance of some polynomial
- series of one of the types supported by by the numpy.polynomial
- module, but could be some other class that supports the convert
- method.
-
- Parameters
- ----------
- series : series
- The instance series to be converted.
- domain : array_like
- The resulting array must be of the form ``[beg, end]``, where
- ``beg`` and ``end`` are the endpoints of the domain.
- window : array_like
- The resulting array must be if the form ``[beg, end]``, where
- ``beg`` and ``end`` are the endpoints of the window.
-
- Returns
- p : $name instance
- A $name series equal to the `poly` series.
-
- See Also
- --------
- convert -- similar instance method
-
- Notes
- -----
- .. versionadded:: 1.7.0
-
- """
- return series.convert(domain, $name, window)
-
-''')
diff --git a/numpy/polynomial/polyutils.py b/numpy/polynomial/polyutils.py
index 99f508521..9348559ed 100644
--- a/numpy/polynomial/polyutils.py
+++ b/numpy/polynomial/polyutils.py
@@ -45,27 +45,25 @@ Functions
"""
from __future__ import division, absolute_import, print_function
-__all__ = ['RankWarning', 'PolyError', 'PolyDomainError', 'as_series',
- 'trimseq', 'trimcoef', 'getdomain', 'mapdomain', 'mapparms',
- 'PolyBase']
-
-import warnings
import numpy as np
-import sys
+
+__all__ = [
+ 'RankWarning', 'PolyError', 'PolyDomainError', 'as_series', 'trimseq',
+ 'trimcoef', 'getdomain', 'mapdomain', 'mapparms', 'PolyBase']
#
# Warnings and Exceptions
#
-class RankWarning(UserWarning) :
+class RankWarning(UserWarning):
"""Issued by chebfit when the design matrix is rank deficient."""
pass
-class PolyError(Exception) :
+class PolyError(Exception):
"""Base class for errors in this module."""
pass
-class PolyDomainError(PolyError) :
+class PolyDomainError(PolyError):
"""Issued by the generic Poly class when two domains don't match.
This is raised when an binary operation is passed Poly objects with
@@ -78,7 +76,7 @@ class PolyDomainError(PolyError) :
# Base class for all polynomial types
#
-class PolyBase(object) :
+class PolyBase(object):
"""
Base class for all polynomial types.
@@ -93,7 +91,7 @@ class PolyBase(object) :
#
# Helper functions to convert inputs to 1-D arrays
#
-def trimseq(seq) :
+def trimseq(seq):
"""Remove small Poly series coefficients.
Parameters
@@ -114,16 +112,16 @@ def trimseq(seq) :
Do not lose the type info if the sequence contains unknown objects.
"""
- if len(seq) == 0 :
+ if len(seq) == 0:
return seq
- else :
- for i in range(len(seq) - 1, -1, -1) :
- if seq[i] != 0 :
+ else:
+ for i in range(len(seq) - 1, -1, -1):
+ if seq[i] != 0:
break
return seq[:i+1]
-def as_series(alist, trim=True) :
+def as_series(alist, trim=True):
"""
Return argument as a list of 1-d arrays.
@@ -165,32 +163,32 @@ def as_series(alist, trim=True) :
"""
arrays = [np.array(a, ndmin=1, copy=0) for a in alist]
- if min([a.size for a in arrays]) == 0 :
+ if min([a.size for a in arrays]) == 0:
raise ValueError("Coefficient array is empty")
- if any([a.ndim != 1 for a in arrays]) :
+ if any([a.ndim != 1 for a in arrays]):
raise ValueError("Coefficient array is not 1-d")
- if trim :
+ if trim:
arrays = [trimseq(a) for a in arrays]
- if any([a.dtype == np.dtype(object) for a in arrays]) :
+ if any([a.dtype == np.dtype(object) for a in arrays]):
ret = []
- for a in arrays :
- if a.dtype != np.dtype(object) :
+ for a in arrays:
+ if a.dtype != np.dtype(object):
tmp = np.empty(len(a), dtype=np.dtype(object))
tmp[:] = a[:]
ret.append(tmp)
- else :
+ else:
ret.append(a.copy())
- else :
- try :
+ else:
+ try:
dtype = np.common_type(*arrays)
- except :
+ except:
raise ValueError("Coefficient arrays have no common type")
ret = [np.array(a, copy=1, dtype=dtype) for a in arrays]
return ret
-def trimcoef(c, tol=0) :
+def trimcoef(c, tol=0):
"""
Remove "small" "trailing" coefficients from a polynomial.
@@ -234,17 +232,17 @@ def trimcoef(c, tol=0) :
array([ 0.0003+0.j , 0.0010-0.001j])
"""
- if tol < 0 :
+ if tol < 0:
raise ValueError("tol must be non-negative")
[c] = as_series([c])
[ind] = np.where(np.abs(c) > tol)
- if len(ind) == 0 :
+ if len(ind) == 0:
return c[:1]*0
- else :
+ else:
return c[:ind[-1] + 1].copy()
-def getdomain(x) :
+def getdomain(x):
"""
Return a domain suitable for given abscissae.
@@ -283,14 +281,14 @@ def getdomain(x) :
"""
[x] = as_series([x], trim=False)
- if x.dtype.char in np.typecodes['Complex'] :
+ if x.dtype.char in np.typecodes['Complex']:
rmin, rmax = x.real.min(), x.real.max()
imin, imax = x.imag.min(), x.imag.max()
return np.array((complex(rmin, imin), complex(rmax, imax)))
- else :
+ else:
return np.array((x.min(), x.max()))
-def mapparms(old, new) :
+def mapparms(old, new):
"""
Linear map parameters between domains.
@@ -337,7 +335,7 @@ def mapparms(old, new) :
scl = newlen/oldlen
return off, scl
-def mapdomain(x, old, new) :
+def mapdomain(x, old, new):
"""
Apply linear map to input points.
diff --git a/numpy/polynomial/tests/test_chebyshev.py b/numpy/polynomial/tests/test_chebyshev.py
index 82c3ba9ea..a596905f6 100644
--- a/numpy/polynomial/tests/test_chebyshev.py
+++ b/numpy/polynomial/tests/test_chebyshev.py
@@ -400,14 +400,14 @@ class TestFitting(TestCase):
return x*(x - 1)*(x - 2)
# Test exceptions
- assert_raises(ValueError, cheb.chebfit, [1], [1], -1)
- assert_raises(TypeError, cheb.chebfit, [[1]], [1], 0)
- assert_raises(TypeError, cheb.chebfit, [], [1], 0)
- assert_raises(TypeError, cheb.chebfit, [1], [[[1]]], 0)
- assert_raises(TypeError, cheb.chebfit, [1, 2], [1], 0)
- assert_raises(TypeError, cheb.chebfit, [1], [1, 2], 0)
- assert_raises(TypeError, cheb.chebfit, [1], [1], 0, w=[[1]])
- assert_raises(TypeError, cheb.chebfit, [1], [1], 0, w=[1, 1])
+ assert_raises(ValueError, cheb.chebfit, [1], [1], -1)
+ assert_raises(TypeError, cheb.chebfit, [[1]], [1], 0)
+ assert_raises(TypeError, cheb.chebfit, [], [1], 0)
+ assert_raises(TypeError, cheb.chebfit, [1], [[[1]]], 0)
+ assert_raises(TypeError, cheb.chebfit, [1, 2], [1], 0)
+ assert_raises(TypeError, cheb.chebfit, [1], [1, 2], 0)
+ assert_raises(TypeError, cheb.chebfit, [1], [1], 0, w=[[1]])
+ assert_raises(TypeError, cheb.chebfit, [1], [1], 0, w=[1, 1])
# Test fit
x = np.linspace(0, 2)
@@ -532,7 +532,7 @@ class TestMisc(TestCase):
assert_almost_equal(cheb.chebpts1(2), tgt)
tgt = [-0.86602540378443871, 0, 0.86602540378443871]
assert_almost_equal(cheb.chebpts1(3), tgt)
- tgt = [-0.9238795325, -0.3826834323, 0.3826834323, 0.9238795325]
+ tgt = [-0.9238795325, -0.3826834323, 0.3826834323, 0.9238795325]
assert_almost_equal(cheb.chebpts1(4), tgt)
def test_chebpts2(self):
diff --git a/numpy/polynomial/tests/test_classes.py b/numpy/polynomial/tests/test_classes.py
index f9134b8c1..cd5a54687 100644
--- a/numpy/polynomial/tests/test_classes.py
+++ b/numpy/polynomial/tests/test_classes.py
@@ -10,12 +10,10 @@ from numbers import Number
import numpy as np
from numpy.polynomial import (
- Polynomial, Legendre, Chebyshev, Laguerre,
- Hermite, HermiteE)
+ Polynomial, Legendre, Chebyshev, Laguerre, Hermite, HermiteE)
from numpy.testing import (
- TestCase, assert_almost_equal, assert_raises,
- assert_equal, assert_, run_module_suite, dec)
-from numpy.testing.noseclasses import KnownFailure
+ assert_almost_equal, assert_raises, assert_equal, assert_,
+ run_module_suite)
from numpy.compat import long
@@ -410,6 +408,9 @@ def check_roots(Poly):
d = Poly.domain + random((2,))*.25
w = Poly.window + random((2,))*.25
tgt = np.sort(random((5,)))
+ res = np.sort(Poly.fromroots(tgt, domain=d, window=w).roots())
+ assert_almost_equal(res, tgt)
+ # default domain and window
res = np.sort(Poly.fromroots(tgt).roots())
assert_almost_equal(res, tgt)
@@ -468,6 +469,12 @@ def check_deriv(Poly):
p3 = p1.integ(1, k=[1])
assert_almost_equal(p2.deriv(1).coef, p3.coef)
assert_almost_equal(p2.deriv(2).coef, p1.coef)
+ # default domain and window
+ p1 = Poly([1, 2, 3])
+ p2 = p1.integ(2, k=[1, 2])
+ p3 = p1.integ(1, k=[1])
+ assert_almost_equal(p2.deriv(1).coef, p3.coef)
+ assert_almost_equal(p2.deriv(2).coef, p1.coef)
def check_linspace(Poly):
@@ -491,11 +498,18 @@ def check_linspace(Poly):
def check_pow(Poly):
d = Poly.domain + random((2,))*.25
w = Poly.window + random((2,))*.25
- tgt = Poly([1], domain=d, window=d)
- tst = Poly([1, 2, 3], domain=d, window=d)
+ tgt = Poly([1], domain=d, window=w)
+ tst = Poly([1, 2, 3], domain=d, window=w)
+ for i in range(5):
+ assert_poly_almost_equal(tst**i, tgt)
+ tgt = tgt * tst
+ # default domain and window
+ tgt = Poly([1])
+ tst = Poly([1, 2, 3])
for i in range(5):
assert_poly_almost_equal(tst**i, tgt)
tgt = tgt * tst
+ # check error for invalid powers
assert_raises(ValueError, op.pow, tgt, 1.5)
assert_raises(ValueError, op.pow, tgt, -1)
diff --git a/numpy/polynomial/tests/test_hermite.py b/numpy/polynomial/tests/test_hermite.py
index ac60007d1..e67625a88 100644
--- a/numpy/polynomial/tests/test_hermite.py
+++ b/numpy/polynomial/tests/test_hermite.py
@@ -119,7 +119,6 @@ class TestEvaluation(TestCase):
y = [polyval(x, c) for c in Hlist]
for i in range(10):
msg = "At i=%d" % i
- ser = np.zeros
tgt = y[i]
res = herm.hermval(x, [0]*i + [1])
assert_almost_equal(res, tgt, err_msg=msg)
@@ -389,14 +388,14 @@ class TestFitting(TestCase):
return x*(x - 1)*(x - 2)
# Test exceptions
- assert_raises(ValueError, herm.hermfit, [1], [1], -1)
- assert_raises(TypeError, herm.hermfit, [[1]], [1], 0)
- assert_raises(TypeError, herm.hermfit, [], [1], 0)
- assert_raises(TypeError, herm.hermfit, [1], [[[1]]], 0)
- assert_raises(TypeError, herm.hermfit, [1, 2], [1], 0)
- assert_raises(TypeError, herm.hermfit, [1], [1, 2], 0)
- assert_raises(TypeError, herm.hermfit, [1], [1], 0, w=[[1]])
- assert_raises(TypeError, herm.hermfit, [1], [1], 0, w=[1, 1])
+ assert_raises(ValueError, herm.hermfit, [1], [1], -1)
+ assert_raises(TypeError, herm.hermfit, [[1]], [1], 0)
+ assert_raises(TypeError, herm.hermfit, [], [1], 0)
+ assert_raises(TypeError, herm.hermfit, [1], [[[1]]], 0)
+ assert_raises(TypeError, herm.hermfit, [1, 2], [1], 0)
+ assert_raises(TypeError, herm.hermfit, [1], [1, 2], 0)
+ assert_raises(TypeError, herm.hermfit, [1], [1], 0, w=[[1]])
+ assert_raises(TypeError, herm.hermfit, [1], [1], 0, w=[1, 1])
# Test fit
x = np.linspace(0, 2)
diff --git a/numpy/polynomial/tests/test_hermite_e.py b/numpy/polynomial/tests/test_hermite_e.py
index 5341dc7ff..f8601a828 100644
--- a/numpy/polynomial/tests/test_hermite_e.py
+++ b/numpy/polynomial/tests/test_hermite_e.py
@@ -6,7 +6,9 @@ from __future__ import division, absolute_import, print_function
import numpy as np
import numpy.polynomial.hermite_e as herme
from numpy.polynomial.polynomial import polyval
-from numpy.testing import *
+from numpy.testing import (
+ TestCase, assert_almost_equal, assert_raises,
+ assert_equal, assert_, run_module_suite)
He0 = np.array([1])
He1 = np.array([0, 1])
@@ -117,7 +119,6 @@ class TestEvaluation(TestCase):
y = [polyval(x, c) for c in Helist]
for i in range(10):
msg = "At i=%d" % i
- ser = np.zeros
tgt = y[i]
res = herme.hermeval(x, [0]*i + [1])
assert_almost_equal(res, tgt, err_msg=msg)
@@ -388,14 +389,14 @@ class TestFitting(TestCase):
return x*(x - 1)*(x - 2)
# Test exceptions
- assert_raises(ValueError, herme.hermefit, [1], [1], -1)
- assert_raises(TypeError, herme.hermefit, [[1]], [1], 0)
- assert_raises(TypeError, herme.hermefit, [], [1], 0)
- assert_raises(TypeError, herme.hermefit, [1], [[[1]]], 0)
- assert_raises(TypeError, herme.hermefit, [1, 2], [1], 0)
- assert_raises(TypeError, herme.hermefit, [1], [1, 2], 0)
- assert_raises(TypeError, herme.hermefit, [1], [1], 0, w=[[1]])
- assert_raises(TypeError, herme.hermefit, [1], [1], 0, w=[1, 1])
+ assert_raises(ValueError, herme.hermefit, [1], [1], -1)
+ assert_raises(TypeError, herme.hermefit, [[1]], [1], 0)
+ assert_raises(TypeError, herme.hermefit, [], [1], 0)
+ assert_raises(TypeError, herme.hermefit, [1], [[[1]]], 0)
+ assert_raises(TypeError, herme.hermefit, [1, 2], [1], 0)
+ assert_raises(TypeError, herme.hermefit, [1], [1, 2], 0)
+ assert_raises(TypeError, herme.hermefit, [1], [1], 0, w=[[1]])
+ assert_raises(TypeError, herme.hermefit, [1], [1], 0, w=[1, 1])
# Test fit
x = np.linspace(0, 2)
diff --git a/numpy/polynomial/tests/test_laguerre.py b/numpy/polynomial/tests/test_laguerre.py
index b3d8fe5ee..1dc57a960 100644
--- a/numpy/polynomial/tests/test_laguerre.py
+++ b/numpy/polynomial/tests/test_laguerre.py
@@ -116,7 +116,6 @@ class TestEvaluation(TestCase):
y = [polyval(x, c) for c in Llist]
for i in range(7):
msg = "At i=%d" % i
- ser = np.zeros
tgt = y[i]
res = lag.lagval(x, [0]*i + [1])
assert_almost_equal(res, tgt, err_msg=msg)
@@ -386,14 +385,14 @@ class TestFitting(TestCase):
return x*(x - 1)*(x - 2)
# Test exceptions
- assert_raises(ValueError, lag.lagfit, [1], [1], -1)
- assert_raises(TypeError, lag.lagfit, [[1]], [1], 0)
- assert_raises(TypeError, lag.lagfit, [], [1], 0)
- assert_raises(TypeError, lag.lagfit, [1], [[[1]]], 0)
- assert_raises(TypeError, lag.lagfit, [1, 2], [1], 0)
- assert_raises(TypeError, lag.lagfit, [1], [1, 2], 0)
- assert_raises(TypeError, lag.lagfit, [1], [1], 0, w=[[1]])
- assert_raises(TypeError, lag.lagfit, [1], [1], 0, w=[1, 1])
+ assert_raises(ValueError, lag.lagfit, [1], [1], -1)
+ assert_raises(TypeError, lag.lagfit, [[1]], [1], 0)
+ assert_raises(TypeError, lag.lagfit, [], [1], 0)
+ assert_raises(TypeError, lag.lagfit, [1], [[[1]]], 0)
+ assert_raises(TypeError, lag.lagfit, [1, 2], [1], 0)
+ assert_raises(TypeError, lag.lagfit, [1], [1, 2], 0)
+ assert_raises(TypeError, lag.lagfit, [1], [1], 0, w=[[1]])
+ assert_raises(TypeError, lag.lagfit, [1], [1], 0, w=[1, 1])
# Test fit
x = np.linspace(0, 2)
diff --git a/numpy/polynomial/tests/test_legendre.py b/numpy/polynomial/tests/test_legendre.py
index e248f005d..8ac1feb58 100644
--- a/numpy/polynomial/tests/test_legendre.py
+++ b/numpy/polynomial/tests/test_legendre.py
@@ -120,7 +120,6 @@ class TestEvaluation(TestCase):
y = [polyval(x, c) for c in Llist]
for i in range(10):
msg = "At i=%d" % i
- ser = np.zeros
tgt = y[i]
res = leg.legval(x, [0]*i + [1])
assert_almost_equal(res, tgt, err_msg=msg)
@@ -390,14 +389,14 @@ class TestFitting(TestCase):
return x*(x - 1)*(x - 2)
# Test exceptions
- assert_raises(ValueError, leg.legfit, [1], [1], -1)
- assert_raises(TypeError, leg.legfit, [[1]], [1], 0)
- assert_raises(TypeError, leg.legfit, [], [1], 0)
- assert_raises(TypeError, leg.legfit, [1], [[[1]]], 0)
- assert_raises(TypeError, leg.legfit, [1, 2], [1], 0)
- assert_raises(TypeError, leg.legfit, [1], [1, 2], 0)
- assert_raises(TypeError, leg.legfit, [1], [1], 0, w=[[1]])
- assert_raises(TypeError, leg.legfit, [1], [1], 0, w=[1, 1])
+ assert_raises(ValueError, leg.legfit, [1], [1], -1)
+ assert_raises(TypeError, leg.legfit, [[1]], [1], 0)
+ assert_raises(TypeError, leg.legfit, [], [1], 0)
+ assert_raises(TypeError, leg.legfit, [1], [[[1]]], 0)
+ assert_raises(TypeError, leg.legfit, [1, 2], [1], 0)
+ assert_raises(TypeError, leg.legfit, [1], [1, 2], 0)
+ assert_raises(TypeError, leg.legfit, [1], [1], 0, w=[[1]])
+ assert_raises(TypeError, leg.legfit, [1], [1], 0, w=[1, 1])
# Test fit
x = np.linspace(0, 2)
diff --git a/numpy/polynomial/tests/test_polynomial.py b/numpy/polynomial/tests/test_polynomial.py
index 77092cd2f..c806a8497 100644
--- a/numpy/polynomial/tests/test_polynomial.py
+++ b/numpy/polynomial/tests/test_polynomial.py
@@ -420,14 +420,14 @@ class TestMisc(TestCase):
return x*(x - 1)*(x - 2)
# Test exceptions
- assert_raises(ValueError, poly.polyfit, [1], [1], -1)
- assert_raises(TypeError, poly.polyfit, [[1]], [1], 0)
- assert_raises(TypeError, poly.polyfit, [], [1], 0)
- assert_raises(TypeError, poly.polyfit, [1], [[[1]]], 0)
- assert_raises(TypeError, poly.polyfit, [1, 2], [1], 0)
- assert_raises(TypeError, poly.polyfit, [1], [1, 2], 0)
- assert_raises(TypeError, poly.polyfit, [1], [1], 0, w=[[1]])
- assert_raises(TypeError, poly.polyfit, [1], [1], 0, w=[1, 1])
+ assert_raises(ValueError, poly.polyfit, [1], [1], -1)
+ assert_raises(TypeError, poly.polyfit, [[1]], [1], 0)
+ assert_raises(TypeError, poly.polyfit, [], [1], 0)
+ assert_raises(TypeError, poly.polyfit, [1], [[[1]]], 0)
+ assert_raises(TypeError, poly.polyfit, [1, 2], [1], 0)
+ assert_raises(TypeError, poly.polyfit, [1], [1, 2], 0)
+ assert_raises(TypeError, poly.polyfit, [1], [1], 0, w=[[1]])
+ assert_raises(TypeError, poly.polyfit, [1], [1], 0, w=[1, 1])
# Test fit
x = np.linspace(0, 2)
diff --git a/numpy/polynomial/tests/test_polyutils.py b/numpy/polynomial/tests/test_polyutils.py
index c77ee2435..974e2e09a 100644
--- a/numpy/polynomial/tests/test_polyutils.py
+++ b/numpy/polynomial/tests/test_polyutils.py
@@ -5,7 +5,9 @@ from __future__ import division, absolute_import, print_function
import numpy as np
import numpy.polynomial.polyutils as pu
-from numpy.testing import *
+from numpy.testing import (
+ TestCase, assert_almost_equal, assert_raises,
+ assert_equal, assert_, run_module_suite)
class TestMisc(TestCase):
@@ -101,3 +103,7 @@ class TestDomain(TestCase):
tgt = [-1 + 1j, 1 - 1j]
res = pu.mapparms(dom1, dom2)
assert_almost_equal(res, tgt)
+
+
+if __name__ == "__main__":
+ run_module_suite()
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