>>> import numpy.polynomial as P
>>> import numpy.polynomial.chebyshev as C
>>> P.polyroots((-1,1,-1,1)) # x^3 - x^2 + x - 1 has two complex roots
array([ -4.99600361e-16-1.j, -4.99600361e-16+1.j, 1.00000e+00+0.j])
>>> C.chebroots((-1,1,-1,1)) # T3 - T2 + T1 - T0 has only real roots
array([ -5.00000000e-01, 2.60860684e-17, 1.00000000e+00])
"""
# cs is a trimmed copy
[cs] = pu.as_series([cs])
if len(cs) <= 1:
return np.array([], dtype=cs.dtype)
if len(cs) == 2:
return np.array([-cs[0] / cs[1]])
n = len(cs) - 1
cmat = np.zeros((n, n), dtype=cs.dtype)
cmat.flat[1::n + 1] = .5
cmat.flat[n::n + 1] = .5
cmat[1, 0] = 1
cmat[:, -1] -= cs[:-1] * (.5 / cs[-1])
roots = la.eigvals(cmat)
roots.sort()
return roots
#
# Chebyshev series class
#
exec polytemplate.substitute(name='Chebyshev', nick='cheb', domain='[-1,1]')
array([ -1.00000000e+00, -1.38777878e-17, 1.00000000e+00])
"""
# cs is a trimmed copy
[cs] = pu.as_series([cs])
if len(cs) <= 1 :
return np.array([], dtype=cs.dtype)
if len(cs) == 2 :
return np.array([-.5*cs[0]/cs[1]])
n = len(cs) - 1
cs /= cs[-1]
cmat = np.zeros((n,n), dtype=cs.dtype)
cmat[1, 0] = .5
for i in range(1, n):
cmat[i - 1, i] = i
if i != n - 1:
cmat[i + 1, i] = .5
else:
cmat[:, i] -= cs[:-1]*.5
roots = la.eigvals(cmat)
roots.sort()
return roots
#
# Hermite series class
#
exec polytemplate.substitute(name='Hermite', nick='herm', domain='[-1,1]')
"""
# cs is a trimmed copy
[cs] = pu.as_series([cs])
if len(cs) <= 1 :
return np.array([], dtype=cs.dtype)
if len(cs) == 2 :
return np.array([-cs[0]/cs[1]])
n = len(cs) - 1
cs /= cs[-1]
cmat = np.zeros((n,n), dtype=cs.dtype)
cmat[1, 0] = 1
for i in range(1, n):
tmp = 2*i + 1
cmat[i - 1, i] = i/tmp
if i != n - 1:
cmat[i + 1, i] = (i + 1)/tmp
else:
cmat[:, i] -= cs[:-1]*(i + 1)/tmp
roots = la.eigvals(cmat)
roots.sort()
return roots
#
# Legendre series class
#
exec polytemplate.substitute(name='Legendre', nick='leg', domain='[-1,1]')
Parameters
----------
npts : int
Number of sample points desired.
Returns
-------
pts : ndarray
The Chebyshev points of the second kind.
Notes
-----
.. versionadded:: 1.5.0
"""
_npts = int(npts)
if _npts != npts:
raise ValueError("npts must be integer")
if _npts < 2:
raise ValueError("npts must be >= 2")
x = np.linspace(-np.pi, 0, _npts)
return np.cos(x)
#
# Chebyshev series class
#
exec polytemplate.substitute(name='Chebyshev', nick='cheb', domain='[-1,1]')
"""
# cs is a trimmed copy
[cs] = pu.as_series([cs])
if len(cs) <= 1 :
return np.array([], dtype=cs.dtype)
if len(cs) == 2 :
return np.array([-cs[0]/cs[1]])
n = len(cs) - 1
cs /= cs[-1]
cmat = np.zeros((n,n), dtype=cs.dtype)
cmat[1, 0] = 1
for i in range(1, n):
tmp = 2*i + 1
cmat[i - 1, i] = i/tmp
if i != n - 1:
cmat[i + 1, i] = (i + 1)/tmp
else:
cmat[:, i] -= cs[:-1]*(i + 1)/tmp
roots = la.eigvals(cmat)
roots.sort()
return roots
#
# Legendre series class
#
exec polytemplate.substitute(name='Legendre', nick='leg', domain='[-1,1]')
"""
# cs is a trimmed copy
[cs] = pu.as_series([cs])
if len(cs) <= 1 :
return np.array([], dtype=cs.dtype)
if len(cs) == 2 :
return np.array([1 + cs[0]/cs[1]])
n = len(cs) - 1
cs /= cs[-1]
cmat = np.zeros((n,n), dtype=cs.dtype)
cmat[0, 0] = 1
cmat[1, 0] = -1
for i in range(1, n):
cmat[i - 1, i] = -i
cmat[i, i] = 2*i + 1
if i != n - 1:
cmat[i + 1, i] = -(i + 1)
else:
cmat[:, i] += cs[:-1]*(i + 1)
roots = la.eigvals(cmat)
roots.sort()
return roots
#
# Laguerre series class
#
exec polytemplate.substitute(name='Laguerre', nick='lag', domain='[-1,1]')
>>> import numpy.polynomial as P
>>> P.polyroots(P.polyfromroots((-1,0,1)))
array([-1., 0., 1.])
>>> P.polyroots(P.polyfromroots((-1,0,1))).dtype
dtype('float64')
>>> j = complex(0,1)
>>> P.polyroots(P.polyfromroots((-j,0,j)))
array([ 0.00000000e+00+0.j, 0.00000000e+00+1.j, 2.77555756e-17-1.j])
"""
# cs is a trimmed copy
[cs] = pu.as_series([cs])
if len(cs) <= 1:
return np.array([], dtype=cs.dtype)
if len(cs) == 2:
return np.array([-cs[0] / cs[1]])
n = len(cs) - 1
cmat = np.zeros((n, n), dtype=cs.dtype)
cmat.flat[n::n + 1] = 1
cmat[:, -1] -= cs[:-1] / cs[-1]
roots = la.eigvals(cmat)
roots.sort()
return roots
#
# polynomial class
#
exec polytemplate.substitute(name='Polynomial', nick='poly', domain='[-1,1]')
array([-1., 0., 1.])
>>> P.polyroots(P.polyfromroots((-1,0,1))).dtype
dtype('float64')
>>> j = complex(0,1)
>>> P.polyroots(P.polyfromroots((-j,0,j)))
array([ 0.00000000e+00+0.j, 0.00000000e+00+1.j, 2.77555756e-17-1.j])
"""
# cs is a trimmed copy
[cs] = pu.as_series([cs])
if len(cs) <= 1 :
return np.array([], dtype=cs.dtype)
if len(cs) == 2 :
return np.array([-cs[0]/cs[1]])
n = len(cs) - 1
cmat = np.zeros((n,n), dtype=cs.dtype)
cmat.flat[n::n+1] = 1
cmat[:,-1] -= cs[:-1]/cs[-1]
roots = la.eigvals(cmat)
roots.sort()
return roots
#
# polynomial class
#
exec polytemplate.substitute(name='Polynomial', nick='poly', domain='[-1,1]')
"""
# cs is a trimmed copy
[cs] = pu.as_series([cs])
if len(cs) <= 1:
return np.array([], dtype=cs.dtype)
if len(cs) == 2:
return np.array([1 + cs[0] / cs[1]])
n = len(cs) - 1
cs /= cs[-1]
cmat = np.zeros((n, n), dtype=cs.dtype)
cmat[0, 0] = 1
cmat[1, 0] = -1
for i in range(1, n):
cmat[i - 1, i] = -i
cmat[i, i] = 2 * i + 1
if i != n - 1:
cmat[i + 1, i] = -(i + 1)
else:
cmat[:, i] += cs[:-1] * (i + 1)
roots = la.eigvals(cmat)
roots.sort()
return roots
#
# Laguerre series class
#
exec polytemplate.substitute(name='Laguerre', nick='lag', domain='[-1,1]')
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])
"""
# cs is a trimmed copy
[cs] = pu.as_series([cs])
if len(cs) < 2:
return np.array([], dtype=cs.dtype)
if len(cs) == 2:
return np.array([-cs[0] / cs[1]])
m = polycompanion(cs)
r = la.eigvals(m)
r.sort()
return r
#
# polynomial class
#
exec polytemplate.substitute(name="Polynomial", nick="poly", domain="[-1,1]")