Python numpy.linalg 模块，eigvals() 实例源码
我们从Python开源项目中，提取了以下50个代码示例，用于说明如何使用numpy.linalg.eigvals()。
def do(self, a, b):
d = linalg.det(a)
(s, ld) = linalg.slogdet(a)
if asarray(a).dtype.type in (single, double):
ad = asarray(a).astype(double)
else:
ad = asarray(a).astype(cdouble)
ev = linalg.eigvals(ad)
assert_almost_equal(d, multiply.reduce(ev, axis=-1))
assert_almost_equal(s * np.exp(ld), multiply.reduce(ev, axis=-1))
s = np.atleast_1d(s)
ld = np.atleast_1d(ld)
m = (s != 0)
assert_almost_equal(np.abs(s[m]), 1)
assert_equal(ld[~m], -inf)
def do(self, a, b):
d = linalg.det(a)
(s, ld) = linalg.slogdet(a)
if asarray(a).dtype.type in (single, double):
ad = asarray(a).astype(double)
else:
ad = asarray(a).astype(cdouble)
ev = linalg.eigvals(ad)
assert_almost_equal(d, multiply.reduce(ev, axis=-1))
assert_almost_equal(s * np.exp(ld), multiply.reduce(ev, axis=-1))
s = np.atleast_1d(s)
ld = np.atleast_1d(ld)
m = (s != 0)
assert_almost_equal(np.abs(s[m]), 1)
assert_equal(ld[~m], -inf)
def do(self, a, b):
d = linalg.det(a)
(s, ld) = linalg.slogdet(a)
if asarray(a).dtype.type in (single, double):
ad = asarray(a).astype(double)
else:
ad = asarray(a).astype(cdouble)
ev = linalg.eigvals(ad)
assert_almost_equal(d, multiply.reduce(ev, axis=-1))
assert_almost_equal(s * np.exp(ld), multiply.reduce(ev, axis=-1))
s = np.atleast_1d(s)
ld = np.atleast_1d(ld)
m = (s != 0)
assert_almost_equal(np.abs(s[m]), 1)
assert_equal(ld[~m], -inf)
def do(self, a, b):
d = linalg.det(a)
(s, ld) = linalg.slogdet(a)
if asarray(a).dtype.type in (single, double):
ad = asarray(a).astype(double)
else:
ad = asarray(a).astype(cdouble)
ev = linalg.eigvals(ad)
assert_almost_equal(d, multiply.reduce(ev, axis=-1))
assert_almost_equal(s * np.exp(ld), multiply.reduce(ev, axis=-1))
s = np.atleast_1d(s)
ld = np.atleast_1d(ld)
m = (s != 0)
assert_almost_equal(np.abs(s[m]), 1)
assert_equal(ld[~m], -inf)
def do(self, a, b):
d = linalg.det(a)
(s, ld) = linalg.slogdet(a)
if asarray(a).dtype.type in (single, double):
ad = asarray(a).astype(double)
else:
ad = asarray(a).astype(cdouble)
ev = linalg.eigvals(ad)
assert_almost_equal(d, multiply.reduce(ev, axis=-1))
assert_almost_equal(s * np.exp(ld), multiply.reduce(ev, axis=-1))
s = np.atleast_1d(s)
ld = np.atleast_1d(ld)
m = (s != 0)
assert_almost_equal(np.abs(s[m]), 1)
assert_equal(ld[~m], -inf)
def do(self, a, b):
d = linalg.det(a)
(s, ld) = linalg.slogdet(a)
if asarray(a).dtype.type in (single, double):
ad = asarray(a).astype(double)
else:
ad = asarray(a).astype(cdouble)
ev = linalg.eigvals(ad)
assert_almost_equal(d, multiply.reduce(ev, axis=-1))
assert_almost_equal(s * np.exp(ld), multiply.reduce(ev, axis=-1))
s = np.atleast_1d(s)
ld = np.atleast_1d(ld)
m = (s != 0)
assert_almost_equal(np.abs(s[m]), 1)
assert_equal(ld[~m], -inf)
def do(self, a, b):
d = linalg.det(a)
(s, ld) = linalg.slogdet(a)
if asarray(a).dtype.type in (single, double):
ad = asarray(a).astype(double)
else:
ad = asarray(a).astype(cdouble)
ev = linalg.eigvals(ad)
assert_almost_equal(d, multiply.reduce(ev, axis=-1))
assert_almost_equal(s * np.exp(ld), multiply.reduce(ev, axis=-1))
s = np.atleast_1d(s)
ld = np.atleast_1d(ld)
m = (s != 0)
assert_almost_equal(np.abs(s[m]), 1)
assert_equal(ld[~m], -inf)
def calculate_beta(self):
"""
.. math::
\\beta_a = \\frac{\mathrm{Cov}(r_a,r_p)}{\mathrm{Var}(r_p)}
http://en.wikipedia.org/wiki/Beta_(finance)
"""
# it doesn't make much sense to calculate beta for less than two days,
# so return nan.
if len(self.algorithm_returns) < 2:
return np.nan, np.nan, np.nan, np.nan, []
returns_matrix = np.vstack([self.algorithm_returns,
self.benchmark_returns])
C = np.cov(returns_matrix, ddof=1)
# If there are missing benchmark values, then we can't calculate the
# beta.
if not np.isfinite(C).all():
return np.nan, np.nan, np.nan, np.nan, []
eigen_values = la.eigvals(C)
condition_number = max(eigen_values) / min(eigen_values)
algorithm_covariance = C[0][1]
benchmark_variance = C[1][1]
beta = algorithm_covariance / benchmark_variance
return (
beta,
algorithm_covariance,
benchmark_variance,
condition_number,
eigen_values
)
def eigenvalueconstraint(params):
sd1 = params[0]
sd2 = params[1]
cor = params[2]
bandwidth = maths.stats.choleskysqrt2d(sd1, sd2, cor)
bandwidthsq = bandwidth.dot(bandwidth.T)
return -np.min(la.eigvals(bandwidthsq))
def do(self, a, b):
ev = linalg.eigvals(a)
evalues, evectors = linalg.eig(a)
assert_almost_equal(ev, evalues)
def test_types(self):
def check(dtype):
x = np.array([[1, 0.5], [0.5, 1]], dtype=dtype)
assert_equal(linalg.eigvals(x).dtype, dtype)
x = np.array([[1, 0.5], [-1, 1]], dtype=dtype)
assert_equal(linalg.eigvals(x).dtype, get_complex_dtype(dtype))
for dtype in [single, double, csingle, cdouble]:
yield check, dtype
def do(self, a, b):
ev = linalg.eigvals(a)
evalues, evectors = linalg.eig(a)
assert_almost_equal(ev, evalues)
def test_types(self):
def check(dtype):
x = np.array([[1, 0.5], [0.5, 1]], dtype=dtype)
assert_equal(linalg.eigvals(x).dtype, dtype)
x = np.array([[1, 0.5], [-1, 1]], dtype=dtype)
assert_equal(linalg.eigvals(x).dtype, get_complex_dtype(dtype))
for dtype in [single, double, csingle, cdouble]:
yield check, dtype
def calculate_beta(self):
"""
.. math::
\\beta_a = \\frac{\mathrm{Cov}(r_a,r_p)}{\mathrm{Var}(r_p)}
http://en.wikipedia.org/wiki/Beta_(finance)
"""
# it doesn't make much sense to calculate beta for less than two days,
# so return nan.
if len(self.algorithm_returns) < 2:
return np.nan, np.nan, np.nan, np.nan, []
returns_matrix = np.vstack([self.algorithm_returns,
self.benchmark_returns])
C = np.cov(returns_matrix, ddof=1)
# If there are missing benchmark values, then we can't calculate the
# beta.
if not np.isfinite(C).all():
return np.nan, np.nan, np.nan, np.nan, []
eigen_values = la.eigvals(C)
condition_number = max(eigen_values) / min(eigen_values)
algorithm_covariance = C[0][1]
benchmark_variance = C[1][1]
beta = algorithm_covariance / benchmark_variance
return (
beta,
algorithm_covariance,
benchmark_variance,
condition_number,
eigen_values
)
def update_common_component(self, mask):
common_idx = 0
common_D = np.where(mask == common_idx)[0].shape[0]
if np.sum(common_D) == 0:
mask[-1] = 0
common_D = np.where(mask == common_idx)[0].shape[0]
common_X = self.X[:, np.where(mask == common_idx)[0]]
if common_D == 1:
covar_scale = np.var(common_X)
else:
covar_scale = np.median(LA.eigvals(np.cov(common_X.T)))
# pass
mu_scale = np.amax(common_X) - covar_scale
m_0 = common_X.mean(axis=0)
k_0 = 1.0 / self.h0
# k_0 = covar_scale**2/mu_scale**2
v_0 = common_D + 2
# S_0 = 1. / covar_scale * np.eye(common_D)
S_0 = 1. * np.eye(common_D)
common_kernel_prior = NIW(m_0, k_0, v_0, S_0)
## save for common component, unused dimensions
common_assignments = np.zeros(common_X.shape[0]) ## one component
if self.common_component_covariance_type == "full":
common_component = GaussianComponents(common_X, common_kernel_prior, common_assignments, 1)
elif self.common_component_covariance_type == "diag":
common_component = GaussianComponentsDiag(common_X, common_kernel_prior, common_assignments, 1)
elif self.common_component_covariance_type == "fixed":
common_component = GaussianComponentsFixedVar(common_X, common_kernel_prior, common_assignments, 1)
else:
assert False, "Invalid covariance type."
return common_component
def update_clustering_components(self, mask, assignments):
cluster_idx = 1
cluster_D = np.where(mask == cluster_idx)[0].shape[0]
cluster_X = self.X[:, np.where(mask == cluster_idx)[0]]
if cluster_D == 1:
covar_scale = np.var(cluster_X)
else:
covar_scale = np.median(LA.eigvals(np.cov(cluster_X.T)))
mu_scale = np.amax(cluster_X) - covar_scale
# Intialize prior
m_0 = cluster_X.mean(axis=0)
k_0 = 1.0 / self.h1
# k_0 = covar_scale ** 2 / mu_scale ** 2
v_0 = cluster_D + 2
# S_0 = 1./100 / covar_scale * np.eye(cluster_D)
S_0 = 1. * np.eye(cluster_D)
cluster_kernel_prior = NIW(m_0, k_0, v_0, S_0)
if self.covariance_type == "full":
components = GaussianComponents(cluster_X, cluster_kernel_prior, assignments, self.K_max)
elif self.covariance_type == "diag":
components = GaussianComponentsDiag(cluster_X, cluster_kernel_prior, assignments, self.K_max)
elif self.covariance_type == "fixed":
components = GaussianComponentsFixedVar(cluster_X, cluster_kernel_prior, assignments, self.K_max)
else:
assert False, "Invalid covariance type."
return components
def is_psd(m):
eigvals = linalg.eigvals(m)
return np.isreal(eigvals).all() and (eigvals >= 0).all()
def do(self, a, b):
ev = linalg.eigvals(a)
evalues, evectors = linalg.eig(a)
assert_almost_equal(ev, evalues)
def test_types(self):
def check(dtype):
x = np.array([[1, 0.5], [0.5, 1]], dtype=dtype)
assert_equal(linalg.eigvals(x).dtype, dtype)
x = np.array([[1, 0.5], [-1, 1]], dtype=dtype)
assert_equal(linalg.eigvals(x).dtype, get_complex_dtype(dtype))
for dtype in [single, double, csingle, cdouble]:
yield check, dtype
def test_stability(self, Q):
"""
Stability test for a given matrix Q.
"""
sr = np.max(np.abs(eigvals(Q)))
if not sr < 1 / self.?:
msg = "Spectral radius condition failed with radius = %f" % sr
raise ValueError(msg)
def do(self, a, b):
ev = linalg.eigvals(a)
evalues, evectors = linalg.eig(a)
assert_almost_equal(ev, evalues)
def test_types(self):
def check(dtype):
x = np.array([[1, 0.5], [0.5, 1]], dtype=dtype)
assert_equal(linalg.eigvals(x).dtype, dtype)
x = np.array([[1, 0.5], [-1, 1]], dtype=dtype)
assert_equal(linalg.eigvals(x).dtype, get_complex_dtype(dtype))
for dtype in [single, double, csingle, cdouble]:
yield check, dtype
def do(self, a, b):
ev = linalg.eigvals(a)
evalues, evectors = linalg.eig(a)
assert_almost_equal(ev, evalues)
def test_types(self):
def check(dtype):
x = np.array([[1, 0.5], [0.5, 1]], dtype=dtype)
assert_equal(linalg.eigvals(x).dtype, dtype)
x = np.array([[1, 0.5], [-1, 1]], dtype=dtype)
assert_equal(linalg.eigvals(x).dtype, get_complex_dtype(dtype))
for dtype in [single, double, csingle, cdouble]:
yield check, dtype
def do(self, a, b):
ev = linalg.eigvals(a)
evalues, evectors = linalg.eig(a)
assert_almost_equal(ev, evalues)
def test_types(self):
def check(dtype):
x = np.array([[1, 0.5], [0.5, 1]], dtype=dtype)
assert_equal(linalg.eigvals(x).dtype, dtype)
x = np.array([[1, 0.5], [-1, 1]], dtype=dtype)
assert_equal(linalg.eigvals(x).dtype, get_complex_dtype(dtype))
for dtype in [single, double, csingle, cdouble]:
yield check, dtype
def do(self, a, b):
ev = linalg.eigvals(a)
evalues, evectors = linalg.eig(a)
assert_almost_equal(ev, evalues)
def test_types(self):
def check(dtype):
x = np.array([[1, 0.5], [0.5, 1]], dtype=dtype)
assert_equal(linalg.eigvals(x).dtype, dtype)
x = np.array([[1, 0.5], [-1, 1]], dtype=dtype)
assert_equal(linalg.eigvals(x).dtype, get_complex_dtype(dtype))
for dtype in [single, double, csingle, cdouble]:
yield check, dtype
def lagroots(c):
"""
Compute the roots of a Laguerre series.
Return the roots (a.k.a. "zeros") of the polynomial
.. math:: p(x) = \\sum_i c[i] * L_i(x).
Parameters
----------
c : 1-D array_like
1-D array of coefficients.
Returns
-------
out : ndarray
Array of the roots of the series. If all the roots are real,
then `out` is also real, otherwise it is complex.
See Also
--------
polyroots, legroots, chebroots, hermroots, hermeroots
Notes
-----
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.
The Laguerre series basis polynomials aren't powers of `x` so the
results of this function may seem unintuitive.
Examples
--------
>>> from numpy.polynomial.laguerre import lagroots, lagfromroots
>>> coef = lagfromroots([0, 1, 2])
>>> coef
array([ 2., -8., 12., -6.])
>>> lagroots(coef)
array([ -4.44089210e-16, 1.00000000e+00, 2.00000000e+00])
"""
# c is a trimmed copy
[c] = pu.as_series([c])
if len(c) <= 1:
return np.array([], dtype=c.dtype)
if len(c) == 2:
return np.array([1 + c[0]/c[1]])
m = lagcompanion(c)
r = la.eigvals(m)
r.sort()
return r
def legroots(c):
"""
Compute the roots of a Legendre series.
Return the roots (a.k.a. "zeros") of the polynomial
.. math:: p(x) = \\sum_i c[i] * L_i(x).
Parameters
----------
c : 1-D array_like
1-D array of coefficients.
Returns
-------
out : ndarray
Array of the roots of the series. If all the roots are real,
then `out` is also real, otherwise it is complex.
See Also
--------
polyroots, chebroots, lagroots, hermroots, hermeroots
Notes
-----
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.
The Legendre series basis polynomials aren't powers of ``x`` so the
results of this function may seem unintuitive.
Examples
--------
>>> import numpy.polynomial.legendre as leg
>>> leg.legroots((1, 2, 3, 4)) # 4L_3 + 3L_2 + 2L_1 + 1L_0, all real roots
array([-0.85099543, -0.11407192, 0.51506735])
"""
# c is a trimmed copy
[c] = pu.as_series([c])
if len(c) < 2:
return np.array([], dtype=c.dtype)
if len(c) == 2:
return np.array([-c[0]/c[1]])
m = legcompanion(c)
r = la.eigvals(m)
r.sort()
return r
def polyroots(c):
"""
Compute the roots of a polynomial.
Return the roots (a.k.a. "zeros") of the polynomial
.. math:: p(x) = \\sum_i c[i] * x^i.
Parameters
----------
c : 1-D array_like
1-D array of polynomial coefficients.
Returns
-------
out : ndarray
Array of the roots of the polynomial. If all the roots are real,
then `out` is also real, otherwise it is complex.
See Also
--------
chebroots
Notes
-----
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 power 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.
Examples
--------
>>> import numpy.polynomial.polynomial as poly
>>> poly.polyroots(poly.polyfromroots((-1,0,1)))
array([-1., 0., 1.])
>>> poly.polyroots(poly.polyfromroots((-1,0,1))).dtype
dtype('float64')
>>> j = complex(0,1)
>>> poly.polyroots(poly.polyfromroots((-j,0,j)))
array([ 0.00000000e+00+0.j, 0.00000000e+00+1.j, 2.77555756e-17-1.j])
"""
# c is a trimmed copy
[c] = pu.as_series([c])
if len(c) < 2:
return np.array([], dtype=c.dtype)
if len(c) == 2:
return np.array([-c[0]/c[1]])
m = polycompanion(c)
r = la.eigvals(m)
r.sort()
return r
#
# polynomial class
#
def chebroots(c):
"""
Compute the roots of a Chebyshev series.
Return the roots (a.k.a. "zeros") of the polynomial
.. math:: p(x) = \\sum_i c[i] * T_i(x).
Parameters
----------
c : 1-D array_like
1-D array of coefficients.
Returns
-------
out : ndarray
Array of the roots of the series. If all the roots are real,
then `out` is also real, otherwise it is complex.
See Also
--------
polyroots, legroots, lagroots, hermroots, hermeroots
Notes
-----
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.
The Chebyshev series basis polynomials aren't powers of `x` so the
results of this function may seem unintuitive.
Examples
--------
>>> import numpy.polynomial.chebyshev as cheb
>>> cheb.chebroots((-1, 1,-1, 1)) # T3 - T2 + T1 - T0 has real roots
array([ -5.00000000e-01, 2.60860684e-17, 1.00000000e+00])
"""
# c is a trimmed copy
[c] = pu.as_series([c])
if len(c) < 2:
return np.array([], dtype=c.dtype)
if len(c) == 2:
return np.array([-c[0]/c[1]])
m = chebcompanion(c)
r = la.eigvals(m)
r.sort()
return r
def hermroots(c):
"""
Compute the roots of a Hermite series.
Return the roots (a.k.a. "zeros") of the polynomial
.. math:: p(x) = \\sum_i c[i] * H_i(x).
Parameters
----------
c : 1-D array_like
1-D array of coefficients.
Returns
-------
out : ndarray
Array of the roots of the series. If all the roots are real,
then `out` is also real, otherwise it is complex.
See Also
--------
polyroots, legroots, lagroots, chebroots, hermeroots
Notes
-----
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.
The Hermite series basis polynomials aren't powers of `x` so the
results of this function may seem unintuitive.
Examples
--------
>>> from numpy.polynomial.hermite import hermroots, hermfromroots
>>> coef = hermfromroots([-1, 0, 1])
>>> coef
array([ 0. , 0.25 , 0. , 0.125])
>>> hermroots(coef)
array([ -1.00000000e+00, -1.38777878e-17, 1.00000000e+00])
"""
# c is a trimmed copy
[c] = pu.as_series([c])
if len(c) <= 1:
return np.array([], dtype=c.dtype)
if len(c) == 2:
return np.array([-.5*c[0]/c[1]])
m = hermcompanion(c)
r = la.eigvals(m)
r.sort()
return r
def lagroots(c):
"""
Compute the roots of a Laguerre series.
Return the roots (a.k.a. "zeros") of the polynomial
.. math:: p(x) = \\sum_i c[i] * L_i(x).
Parameters
----------
c : 1-D array_like
1-D array of coefficients.
Returns
-------
out : ndarray
Array of the roots of the series. If all the roots are real,
then `out` is also real, otherwise it is complex.
See Also
--------
polyroots, legroots, chebroots, hermroots, hermeroots
Notes
-----
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.
The Laguerre series basis polynomials aren't powers of `x` so the
results of this function may seem unintuitive.
Examples
--------
>>> from numpy.polynomial.laguerre import lagroots, lagfromroots
>>> coef = lagfromroots([0, 1, 2])
>>> coef
array([ 2., -8., 12., -6.])
>>> lagroots(coef)
array([ -4.44089210e-16, 1.00000000e+00, 2.00000000e+00])
"""
# c is a trimmed copy
[c] = pu.as_series([c])
if len(c) <= 1:
return np.array([], dtype=c.dtype)
if len(c) == 2:
return np.array([1 + c[0]/c[1]])
m = lagcompanion(c)
r = la.eigvals(m)
r.sort()
return r
def legroots(c):
"""
Compute the roots of a Legendre series.
Return the roots (a.k.a. "zeros") of the polynomial
.. math:: p(x) = \\sum_i c[i] * L_i(x).
Parameters
----------
c : 1-D array_like
1-D array of coefficients.
Returns
-------
out : ndarray
Array of the roots of the series. If all the roots are real,
then `out` is also real, otherwise it is complex.
See Also
--------
polyroots, chebroots, lagroots, hermroots, hermeroots
Notes
-----
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.
The Legendre series basis polynomials aren't powers of ``x`` so the
results of this function may seem unintuitive.
Examples
--------
>>> import numpy.polynomial.legendre as leg
>>> leg.legroots((1, 2, 3, 4)) # 4L_3 + 3L_2 + 2L_1 + 1L_0, all real roots
array([-0.85099543, -0.11407192, 0.51506735])
"""
# c is a trimmed copy
[c] = pu.as_series([c])
if len(c) < 2:
return np.array([], dtype=c.dtype)
if len(c) == 2:
return np.array([-c[0]/c[1]])
m = legcompanion(c)
r = la.eigvals(m)
r.sort()
return r
def polyroots(c):
"""
Compute the roots of a polynomial.
Return the roots (a.k.a. "zeros") of the polynomial
.. math:: p(x) = \\sum_i c[i] * x^i.
Parameters
----------
c : 1-D array_like
1-D array of polynomial coefficients.
Returns
-------
out : ndarray
Array of the roots of the polynomial. If all the roots are real,
then `out` is also real, otherwise it is complex.
See Also
--------
chebroots
Notes
-----
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 power 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.
Examples
--------
>>> import numpy.polynomial.polynomial as poly
>>> poly.polyroots(poly.polyfromroots((-1,0,1)))
array([-1., 0., 1.])
>>> poly.polyroots(poly.polyfromroots((-1,0,1))).dtype
dtype('float64')
>>> j = complex(0,1)
>>> poly.polyroots(poly.polyfromroots((-j,0,j)))
array([ 0.00000000e+00+0.j, 0.00000000e+00+1.j, 2.77555756e-17-1.j])
"""
# c is a trimmed copy
[c] = pu.as_series([c])
if len(c) < 2:
return np.array([], dtype=c.dtype)
if len(c) == 2:
return np.array([-c[0]/c[1]])
m = polycompanion(c)
r = la.eigvals(m)
r.sort()
return r
#
# polynomial class
#
def chebroots(c):
"""
Compute the roots of a Chebyshev series.
Return the roots (a.k.a. "zeros") of the polynomial
.. math:: p(x) = \\sum_i c[i] * T_i(x).
Parameters
----------
c : 1-D array_like
1-D array of coefficients.
Returns
-------
out : ndarray
Array of the roots of the series. If all the roots are real,
then `out` is also real, otherwise it is complex.
See Also
--------
polyroots, legroots, lagroots, hermroots, hermeroots
Notes
-----
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.
The Chebyshev series basis polynomials aren't powers of `x` so the
results of this function may seem unintuitive.
Examples
--------
>>> import numpy.polynomial.chebyshev as cheb
>>> cheb.chebroots((-1, 1,-1, 1)) # T3 - T2 + T1 - T0 has real roots
array([ -5.00000000e-01, 2.60860684e-17, 1.00000000e+00])
"""
# c is a trimmed copy
[c] = pu.as_series([c])
if len(c) < 2:
return np.array([], dtype=c.dtype)
if len(c) == 2:
return np.array([-c[0]/c[1]])
m = chebcompanion(c)
r = la.eigvals(m)
r.sort()
return r
def hermroots(c):
"""
Compute the roots of a Hermite series.
Return the roots (a.k.a. "zeros") of the polynomial
.. math:: p(x) = \\sum_i c[i] * H_i(x).
Parameters
----------
c : 1-D array_like
1-D array of coefficients.
Returns
-------
out : ndarray
Array of the roots of the series. If all the roots are real,
then `out` is also real, otherwise it is complex.
See Also
--------
polyroots, legroots, lagroots, chebroots, hermeroots
Notes
-----
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.
The Hermite series basis polynomials aren't powers of `x` so the
results of this function may seem unintuitive.
Examples
--------
>>> from numpy.polynomial.hermite import hermroots, hermfromroots
>>> coef = hermfromroots([-1, 0, 1])
>>> coef
array([ 0. , 0.25 , 0. , 0.125])
>>> hermroots(coef)
array([ -1.00000000e+00, -1.38777878e-17, 1.00000000e+00])
"""
# c is a trimmed copy
[c] = pu.as_series([c])
if len(c) <= 1:
return np.array([], dtype=c.dtype)
if len(c) == 2:
return np.array([-.5*c[0]/c[1]])
m = hermcompanion(c)
r = la.eigvals(m)
r.sort()
return r
def lagroots(c):
"""
Compute the roots of a Laguerre series.
Return the roots (a.k.a. "zeros") of the polynomial
.. math:: p(x) = \\sum_i c[i] * L_i(x).
Parameters
----------
c : 1-D array_like
1-D array of coefficients.
Returns
-------
out : ndarray
Array of the roots of the series. If all the roots are real,
then `out` is also real, otherwise it is complex.
See Also
--------
polyroots, legroots, chebroots, hermroots, hermeroots
Notes
-----
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.
The Laguerre series basis polynomials aren't powers of `x` so the
results of this function may seem unintuitive.
Examples
--------
>>> from numpy.polynomial.laguerre import lagroots, lagfromroots
>>> coef = lagfromroots([0, 1, 2])
>>> coef
array([ 2., -8., 12., -6.])
>>> lagroots(coef)
array([ -4.44089210e-16, 1.00000000e+00, 2.00000000e+00])
"""
# c is a trimmed copy
[c] = pu.as_series([c])
if len(c) <= 1:
return np.array([], dtype=c.dtype)
if len(c) == 2:
return np.array([1 + c[0]/c[1]])
m = lagcompanion(c)
r = la.eigvals(m)
r.sort()
return r
def hermeroots(c):
"""
Compute the roots of a HermiteE series.
Return the roots (a.k.a. "zeros") of the polynomial
.. math:: p(x) = \\sum_i c[i] * He_i(x).
Parameters
----------
c : 1-D array_like
1-D array of coefficients.
Returns
-------
out : ndarray
Array of the roots of the series. If all the roots are real,
then `out` is also real, otherwise it is complex.
See Also
--------
polyroots, legroots, lagroots, hermroots, chebroots
Notes
-----
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.
The HermiteE series basis polynomials aren't powers of `x` so the
results of this function may seem unintuitive.
Examples
--------
>>> from numpy.polynomial.hermite_e import hermeroots, hermefromroots
>>> coef = hermefromroots([-1, 0, 1])
>>> coef
array([ 0., 2., 0., 1.])
>>> hermeroots(coef)
array([-1., 0., 1.])
"""
# c is a trimmed copy
[c] = pu.as_series([c])
if len(c) <= 1:
return np.array([], dtype=c.dtype)
if len(c) == 2:
return np.array([-c[0]/c[1]])
m = hermecompanion(c)
r = la.eigvals(m)
r.sort()
return r
def polyroots(c):
"""
Compute the roots of a polynomial.
Return the roots (a.k.a. "zeros") of the polynomial
.. math:: p(x) = \\sum_i c[i] * x^i.
Parameters
----------
c : 1-D array_like
1-D array of polynomial coefficients.
Returns
-------
out : ndarray
Array of the roots of the polynomial. If all the roots are real,
then `out` is also real, otherwise it is complex.
See Also
--------
chebroots
Notes
-----
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 power 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.
Examples
--------
>>> import numpy.polynomial.polynomial as poly
>>> poly.polyroots(poly.polyfromroots((-1,0,1)))
array([-1., 0., 1.])
>>> poly.polyroots(poly.polyfromroots((-1,0,1))).dtype
dtype('float64')
>>> j = complex(0,1)
>>> poly.polyroots(poly.polyfromroots((-j,0,j)))
array([ 0.00000000e+00+0.j, 0.00000000e+00+1.j, 2.77555756e-17-1.j])
"""
# c is a trimmed copy
[c] = pu.as_series([c])
if len(c) < 2:
return np.array([], dtype=c.dtype)
if len(c) == 2:
return np.array([-c[0]/c[1]])
m = polycompanion(c)
r = la.eigvals(m)
r.sort()
return r
#
# polynomial class
#
def chebroots(c):
"""
Compute the roots of a Chebyshev series.
Return the roots (a.k.a. "zeros") of the polynomial
.. math:: p(x) = \\sum_i c[i] * T_i(x).
Parameters
----------
c : 1-D array_like
1-D array of coefficients.
Returns
-------
out : ndarray
Array of the roots of the series. If all the roots are real,
then `out` is also real, otherwise it is complex.
See Also
--------
polyroots, legroots, lagroots, hermroots, hermeroots
Notes
-----
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.
The Chebyshev series basis polynomials aren't powers of `x` so the
results of this function may seem unintuitive.
Examples
--------
>>> import numpy.polynomial.chebyshev as cheb
>>> cheb.chebroots((-1, 1,-1, 1)) # T3 - T2 + T1 - T0 has real roots
array([ -5.00000000e-01, 2.60860684e-17, 1.00000000e+00])
"""
# c is a trimmed copy
[c] = pu.as_series([c])
if len(c) < 2:
return np.array([], dtype=c.dtype)
if len(c) == 2:
return np.array([-c[0]/c[1]])
m = chebcompanion(c)
r = la.eigvals(m)
r.sort()
return r
def hermroots(c):
"""
Compute the roots of a Hermite series.
Return the roots (a.k.a. "zeros") of the polynomial
.. math:: p(x) = \\sum_i c[i] * H_i(x).
Parameters
----------
c : 1-D array_like
1-D array of coefficients.
Returns
-------
out : ndarray
Array of the roots of the series. If all the roots are real,
then `out` is also real, otherwise it is complex.
See Also
--------
polyroots, legroots, lagroots, chebroots, hermeroots
Notes
-----
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.
The Hermite series basis polynomials aren't powers of `x` so the
results of this function may seem unintuitive.
Examples
--------
>>> from numpy.polynomial.hermite import hermroots, hermfromroots
>>> coef = hermfromroots([-1, 0, 1])
>>> coef
array([ 0. , 0.25 , 0. , 0.125])
>>> hermroots(coef)
array([ -1.00000000e+00, -1.38777878e-17, 1.00000000e+00])
"""
# c is a trimmed copy
[c] = pu.as_series([c])
if len(c) <= 1:
return np.array([], dtype=c.dtype)
if len(c) == 2:
return np.array([-.5*c[0]/c[1]])
m = hermcompanion(c)
r = la.eigvals(m)
r.sort()
return r
def lagroots(c):
"""
Compute the roots of a Laguerre series.
Return the roots (a.k.a. "zeros") of the polynomial
.. math:: p(x) = \\sum_i c[i] * L_i(x).
Parameters
----------
c : 1-D array_like
1-D array of coefficients.
Returns
-------
out : ndarray
Array of the roots of the series. If all the roots are real,
then `out` is also real, otherwise it is complex.
See Also
--------
polyroots, legroots, chebroots, hermroots, hermeroots
Notes
-----
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.
The Laguerre series basis polynomials aren't powers of `x` so the
results of this function may seem unintuitive.
Examples
--------
>>> from numpy.polynomial.laguerre import lagroots, lagfromroots
>>> coef = lagfromroots([0, 1, 2])
>>> coef
array([ 2., -8., 12., -6.])
>>> lagroots(coef)
array([ -4.44089210e-16, 1.00000000e+00, 2.00000000e+00])
"""
# c is a trimmed copy
[c] = pu.as_series([c])
if len(c) <= 1:
return np.array([], dtype=c.dtype)
if len(c) == 2:
return np.array([1 + c[0]/c[1]])
m = lagcompanion(c)
r = la.eigvals(m)
r.sort()
return r
def legroots(c):
"""
Compute the roots of a Legendre series.
Return the roots (a.k.a. "zeros") of the polynomial
.. math:: p(x) = \\sum_i c[i] * L_i(x).
Parameters
----------
c : 1-D array_like
1-D array of coefficients.
Returns
-------
out : ndarray
Array of the roots of the series. If all the roots are real,
then `out` is also real, otherwise it is complex.
See Also
--------
polyroots, chebroots, lagroots, hermroots, hermeroots
Notes
-----
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.
The Legendre series basis polynomials aren't powers of ``x`` so the
results of this function may seem unintuitive.
Examples
--------
>>> import numpy.polynomial.legendre as leg
>>> leg.legroots((1, 2, 3, 4)) # 4L_3 + 3L_2 + 2L_1 + 1L_0, all real roots
array([-0.85099543, -0.11407192, 0.51506735])
"""
# c is a trimmed copy
[c] = pu.as_series([c])
if len(c) < 2:
return np.array([], dtype=c.dtype)
if len(c) == 2:
return np.array([-c[0]/c[1]])
m = legcompanion(c)
r = la.eigvals(m)
r.sort()
return r
def polyroots(c):
"""
Compute the roots of a polynomial.
Return the roots (a.k.a. "zeros") of the polynomial
.. math:: p(x) = \\sum_i c[i] * x^i.
Parameters
----------
c : 1-D array_like
1-D array of polynomial coefficients.
Returns
-------
out : ndarray
Array of the roots of the polynomial. If all the roots are real,
then `out` is also real, otherwise it is complex.
See Also
--------
chebroots
Notes
-----
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 power 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.
Examples
--------
>>> import numpy.polynomial.polynomial as poly
>>> poly.polyroots(poly.polyfromroots((-1,0,1)))
array([-1., 0., 1.])
>>> poly.polyroots(poly.polyfromroots((-1,0,1))).dtype
dtype('float64')
>>> j = complex(0,1)
>>> poly.polyroots(poly.polyfromroots((-j,0,j)))
array([ 0.00000000e+00+0.j, 0.00000000e+00+1.j, 2.77555756e-17-1.j])
"""
# c is a trimmed copy
[c] = pu.as_series([c])
if len(c) < 2:
return np.array([], dtype=c.dtype)
if len(c) == 2:
return np.array([-c[0]/c[1]])
m = polycompanion(c)
r = la.eigvals(m)
r.sort()
return r
#
# polynomial class
#
def chebroots(c):
"""
Compute the roots of a Chebyshev series.
Return the roots (a.k.a. "zeros") of the polynomial
.. math:: p(x) = \\sum_i c[i] * T_i(x).
Parameters
----------
c : 1-D array_like
1-D array of coefficients.
Returns
-------
out : ndarray
Array of the roots of the series. If all the roots are real,
then `out` is also real, otherwise it is complex.
See Also
--------
polyroots, legroots, lagroots, hermroots, hermeroots
Notes
-----
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.
The Chebyshev series basis polynomials aren't powers of `x` so the
results of this function may seem unintuitive.
Examples
--------
>>> import numpy.polynomial.chebyshev as cheb
>>> cheb.chebroots((-1, 1,-1, 1)) # T3 - T2 + T1 - T0 has real roots
array([ -5.00000000e-01, 2.60860684e-17, 1.00000000e+00])
"""
# c is a trimmed copy
[c] = pu.as_series([c])
if len(c) < 2:
return np.array([], dtype=c.dtype)
if len(c) == 2:
return np.array([-c[0]/c[1]])
m = chebcompanion(c)
r = la.eigvals(m)
r.sort()
return r
def hermroots(c):
"""
Compute the roots of a Hermite series.
Return the roots (a.k.a. "zeros") of the polynomial
.. math:: p(x) = \\sum_i c[i] * H_i(x).
Parameters
----------
c : 1-D array_like
1-D array of coefficients.
Returns
-------
out : ndarray
Array of the roots of the series. If all the roots are real,
then `out` is also real, otherwise it is complex.
See Also
--------
polyroots, legroots, lagroots, chebroots, hermeroots
Notes
-----
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.
The Hermite series basis polynomials aren't powers of `x` so the
results of this function may seem unintuitive.
Examples
--------
>>> from numpy.polynomial.hermite import hermroots, hermfromroots
>>> coef = hermfromroots([-1, 0, 1])
>>> coef
array([ 0. , 0.25 , 0. , 0.125])
>>> hermroots(coef)
array([ -1.00000000e+00, -1.38777878e-17, 1.00000000e+00])
"""
# c is a trimmed copy
[c] = pu.as_series([c])
if len(c) <= 1:
return np.array([], dtype=c.dtype)
if len(c) == 2:
return np.array([-.5*c[0]/c[1]])
m = hermcompanion(c)
r = la.eigvals(m)
r.sort()
return r
def lagroots(c):
"""
Compute the roots of a Laguerre series.
Return the roots (a.k.a. "zeros") of the polynomial
.. math:: p(x) = \\sum_i c[i] * L_i(x).
Parameters
----------
c : 1-D array_like
1-D array of coefficients.
Returns
-------
out : ndarray
Array of the roots of the series. If all the roots are real,
then `out` is also real, otherwise it is complex.
See Also
--------
polyroots, legroots, chebroots, hermroots, hermeroots
Notes
-----
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.
The Laguerre series basis polynomials aren't powers of `x` so the
results of this function may seem unintuitive.
Examples
--------
>>> from numpy.polynomial.laguerre import lagroots, lagfromroots
>>> coef = lagfromroots([0, 1, 2])
>>> coef
array([ 2., -8., 12., -6.])
>>> lagroots(coef)
array([ -4.44089210e-16, 1.00000000e+00, 2.00000000e+00])
"""
# c is a trimmed copy
[c] = pu.as_series([c])
if len(c) <= 1:
return np.array([], dtype=c.dtype)
if len(c) == 2:
return np.array([1 + c[0]/c[1]])
m = lagcompanion(c)
r = la.eigvals(m)
r.sort()
return r
def legroots(c):
"""
Compute the roots of a Legendre series.
Return the roots (a.k.a. "zeros") of the polynomial
.. math:: p(x) = \\sum_i c[i] * L_i(x).
Parameters
----------
c : 1-D array_like
1-D array of coefficients.
Returns
-------
out : ndarray
Array of the roots of the series. If all the roots are real,
then `out` is also real, otherwise it is complex.
See Also
--------
polyroots, chebroots, lagroots, hermroots, hermeroots
Notes
-----
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.
The Legendre series basis polynomials aren't powers of ``x`` so the
results of this function may seem unintuitive.
Examples
--------
>>> import numpy.polynomial.legendre as leg
>>> leg.legroots((1, 2, 3, 4)) # 4L_3 + 3L_2 + 2L_1 + 1L_0, all real roots
array([-0.85099543, -0.11407192, 0.51506735])
"""
# c is a trimmed copy
[c] = pu.as_series([c])
if len(c) < 2:
return np.array([], dtype=c.dtype)
if len(c) == 2:
return np.array([-c[0]/c[1]])
m = legcompanion(c)
r = la.eigvals(m)
r.sort()
return r