def poly(seq_of_zeros):
""" Return a sequence representing a polynomial given a sequence of roots.
If the input is a matrix, return the characteristic polynomial.
Example:
>>> b = roots([1,3,1,5,6])
>>> poly(b)
array([ 1., 3., 1., 5., 6.])
"""
seq_of_zeros = atleast_1d(seq_of_zeros)
sh = seq_of_zeros.shape
if len(sh) == 2 and sh[0] == sh[1]:
seq_of_zeros = _eigvals(seq_of_zeros)
elif len(sh) ==1:
pass
else:
raise ValueError, "input must be 1d or square 2d array."
if len(seq_of_zeros) == 0:
return 1.0
a = [1]
for k in range(len(seq_of_zeros)):
a = NX.convolve(a, [1, -seq_of_zeros[k]], mode='full')
if issubclass(a.dtype.type, NX.complexfloating):
# if complex roots are all complex conjugates, the roots are real.
roots = NX.asarray(seq_of_zeros, complex)
pos_roots = sort_complex(NX.compress(roots.imag > 0, roots))
neg_roots = NX.conjugate(sort_complex(
NX.compress(roots.imag < 0,roots)))
if (len(pos_roots) == len(neg_roots) and
NX.alltrue(neg_roots == pos_roots)):
a = a.real.copy()
return a
def poly(seq_of_zeros):
"""
Find the coefficients of a polynomial with the given sequence of roots.
Returns the coefficients of the polynomial whose leading coefficient
is one for the given sequence of zeros (multiple roots must be included
in the sequence as many times as their multiplicity; see Examples).
A square matrix (or array, which will be treated as a matrix) can also
be given, in which case the coefficients of the characteristic polynomial
of the matrix are returned.
Parameters
----------
seq_of_zeros : array_like, shape (N,) or (N, N)
A sequence of polynomial roots, or a square array or matrix object.
Returns
-------
c : ndarray
1D array of polynomial coefficients from highest to lowest degree:
``c[0] * x**(N) + c[1] * x**(N-1) + ... + c[N-1] * x + c[N]``
where c[0] always equals 1.
Raises
------
ValueError
If input is the wrong shape (the input must be a 1-D or square
2-D array).
See Also
--------
polyval : Evaluate a polynomial at a point.
roots : Return the roots of a polynomial.
polyfit : Least squares polynomial fit.
poly1d : A one-dimensional polynomial class.
Notes
-----
Specifying the roots of a polynomial still leaves one degree of
freedom, typically represented by an undetermined leading
coefficient. [1]_ In the case of this function, that coefficient -
the first one in the returned array - is always taken as one. (If
for some reason you have one other point, the only automatic way
presently to leverage that information is to use ``polyfit``.)
The characteristic polynomial, :math:`p_a(t)`, of an `n`-by-`n`
matrix **A** is given by
:math:`p_a(t) = \\mathrm{det}(t\\, \\mathbf{I} - \\mathbf{A})`,
where **I** is the `n`-by-`n` identity matrix. [2]_
References
----------
.. [1] M. Sullivan and M. Sullivan, III, "Algebra and Trignometry,
Enhanced With Graphing Utilities," Prentice-Hall, pg. 318, 1996.
.. [2] G. Strang, "Linear Algebra and Its Applications, 2nd Edition,"
Academic Press, pg. 182, 1980.
Examples
--------
Given a sequence of a polynomial's zeros:
>>> np.poly((0, 0, 0)) # Multiple root example
array([1, 0, 0, 0])
The line above represents z**3 + 0*z**2 + 0*z + 0.
>>> np.poly((-1./2, 0, 1./2))
array([ 1. , 0. , -0.25, 0. ])
The line above represents z**3 - z/4
>>> np.poly((np.random.random(1.)[0], 0, np.random.random(1.)[0]))
array([ 1. , -0.77086955, 0.08618131, 0. ]) #random
Given a square array object:
>>> P = np.array([[0, 1./3], [-1./2, 0]])
>>> np.poly(P)
array([ 1. , 0. , 0.16666667])
Or a square matrix object:
>>> np.poly(np.matrix(P))
array([ 1. , 0. , 0.16666667])
Note how in all cases the leading coefficient is always 1.
"""
seq_of_zeros = atleast_1d(seq_of_zeros)
sh = seq_of_zeros.shape
if len(sh) == 2 and sh[0] == sh[1] and sh[0] != 0:
seq_of_zeros = eigvals(seq_of_zeros)
elif len(sh) == 1:
dt = seq_of_zeros.dtype
# Let object arrays slip through, e.g. for arbitrary precision
if dt != object:
seq_of_zeros = seq_of_zeros.astype(mintypecode(dt.char))
else:
raise ValueError("input must be 1d or non-empty square 2d array.")
if len(seq_of_zeros) == 0:
return 1.0
dt = seq_of_zeros.dtype
a = ones((1, ), dtype=dt)
for k in range(len(seq_of_zeros)):
a = NX.convolve(a, array([1, -seq_of_zeros[k]], dtype=dt), mode='full')
if issubclass(a.dtype.type, NX.complexfloating):
# if complex roots are all complex conjugates, the roots are real.
roots = NX.asarray(seq_of_zeros, complex)
pos_roots = sort_complex(NX.compress(roots.imag > 0, roots))
neg_roots = NX.conjugate(
sort_complex(NX.compress(roots.imag < 0, roots)))
if (len(pos_roots) == len(neg_roots)
and NX.alltrue(neg_roots == pos_roots)):
a = a.real.copy()
return a
def poly(seq_of_zeros):
"""
Find the coefficients of a polynomial with the given sequence of roots.
Returns the coefficients of the polynomial whose leading coefficient
is one for the given sequence of zeros (multiple roots must be included
in the sequence as many times as their multiplicity; see Examples).
A square matrix (or array, which will be treated as a matrix) can also
be given, in which case the coefficients of the characteristic polynomial
of the matrix are returned.
Parameters
----------
seq_of_zeros : array_like, shape (N,) or (N, N)
A sequence of polynomial roots, or a square array or matrix object.
Returns
-------
c : ndarray
1D array of polynomial coefficients from highest to lowest degree:
``c[0] * x**(N) + c[1] * x**(N-1) + ... + c[N-1] * x + c[N]``
where c[0] always equals 1.
Raises
------
ValueError
If input is the wrong shape (the input must be a 1-D or square
2-D array).
See Also
--------
polyval : Evaluate a polynomial at a point.
roots : Return the roots of a polynomial.
polyfit : Least squares polynomial fit.
poly1d : A one-dimensional polynomial class.
Notes
-----
Specifying the roots of a polynomial still leaves one degree of
freedom, typically represented by an undetermined leading
coefficient. [1]_ In the case of this function, that coefficient -
the first one in the returned array - is always taken as one. (If
for some reason you have one other point, the only automatic way
presently to leverage that information is to use ``polyfit``.)
The characteristic polynomial, :math:`p_a(t)`, of an `n`-by-`n`
matrix **A** is given by
:math:`p_a(t) = \\mathrm{det}(t\\, \\mathbf{I} - \\mathbf{A})`,
where **I** is the `n`-by-`n` identity matrix. [2]_
References
----------
.. [1] M. Sullivan and M. Sullivan, III, "Algebra and Trignometry,
Enhanced With Graphing Utilities," Prentice-Hall, pg. 318, 1996.
.. [2] G. Strang, "Linear Algebra and Its Applications, 2nd Edition,"
Academic Press, pg. 182, 1980.
Examples
--------
Given a sequence of a polynomial's zeros:
>>> np.poly((0, 0, 0)) # Multiple root example
array([1, 0, 0, 0])
The line above represents z**3 + 0*z**2 + 0*z + 0.
>>> np.poly((-1./2, 0, 1./2))
array([ 1. , 0. , -0.25, 0. ])
The line above represents z**3 - z/4
>>> np.poly((np.random.random(1.)[0], 0, np.random.random(1.)[0]))
array([ 1. , -0.77086955, 0.08618131, 0. ]) #random
Given a square array object:
>>> P = np.array([[0, 1./3], [-1./2, 0]])
>>> np.poly(P)
array([ 1. , 0. , 0.16666667])
Or a square matrix object:
>>> np.poly(np.matrix(P))
array([ 1. , 0. , 0.16666667])
Note how in all cases the leading coefficient is always 1.
"""
seq_of_zeros = atleast_1d(seq_of_zeros)
sh = seq_of_zeros.shape
if len(sh) == 2 and sh[0] == sh[1] and sh[0] != 0:
seq_of_zeros = eigvals(seq_of_zeros)
elif len(sh) == 1:
pass
else:
raise ValueError("input must be 1d or square 2d array.")
if len(seq_of_zeros) == 0:
return 1.0
a = [1]
for k in range(len(seq_of_zeros)):
a = NX.convolve(a, [1, -seq_of_zeros[k]], mode='full')
if issubclass(a.dtype.type, NX.complexfloating):
# if complex roots are all complex conjugates, the roots are real.
roots = NX.asarray(seq_of_zeros, complex)
pos_roots = sort_complex(NX.compress(roots.imag > 0, roots))
neg_roots = NX.conjugate(sort_complex(
NX.compress(roots.imag < 0,roots)))
if (len(pos_roots) == len(neg_roots) and
NX.alltrue(neg_roots == pos_roots)):
a = a.real.copy()
return a
def poly(seq_of_zeros):
"""
Return polynomial coefficients given a sequence of roots.
Calculate the coefficients of a polynomial given the zeros
of the polynomial.
If a square matrix is given, then the coefficients for
characteristic equation of the matrix, defined by
:math:`\\mathrm{det}(\\mathbf{A} - \\lambda \\mathbf{I})`,
are returned.
Parameters
----------
seq_of_zeros : ndarray
A sequence of polynomial roots or a square matrix.
Returns
-------
coefs : ndarray
A sequence of polynomial coefficients representing the polynomial
:math:`\\mathrm{coefs}[0] x^{n-1} + \\mathrm{coefs}[1] x^{n-2} +
... + \\mathrm{coefs}[2] x + \\mathrm{coefs}[n]`
See Also
--------
numpy.poly1d : A one-dimensional polynomial class.
numpy.roots : Return the roots of the polynomial coefficients in p
numpy.polyfit : Least squares polynomial fit
Examples
--------
Given a sequence of polynomial zeros,
>>> b = np.roots([1, 3, 1, 5, 6])
>>> np.poly(b)
array([ 1., 3., 1., 5., 6.])
Given a square matrix,
>>> P = np.array([[19, 3], [-2, 26]])
>>> np.poly(P)
array([ 1., -45., 500.])
"""
seq_of_zeros = atleast_1d(seq_of_zeros)
sh = seq_of_zeros.shape
if len(sh) == 2 and sh[0] == sh[1]:
seq_of_zeros = eigvals(seq_of_zeros)
elif len(sh) ==1:
pass
else:
raise ValueError, "input must be 1d or square 2d array."
if len(seq_of_zeros) == 0:
return 1.0
a = [1]
for k in range(len(seq_of_zeros)):
a = NX.convolve(a, [1, -seq_of_zeros[k]], mode='full')
if issubclass(a.dtype.type, NX.complexfloating):
# if complex roots are all complex conjugates, the roots are real.
roots = NX.asarray(seq_of_zeros, complex)
pos_roots = sort_complex(NX.compress(roots.imag > 0, roots))
neg_roots = NX.conjugate(sort_complex(
NX.compress(roots.imag < 0,roots)))
if (len(pos_roots) == len(neg_roots) and
NX.alltrue(neg_roots == pos_roots)):
a = a.real.copy()
return a
def poly(seq_of_zeros):
"""
Return polynomial coefficients given a sequence of roots.
Calculate the coefficients of a polynomial given the zeros
of the polynomial.
If a square matrix is given, then the coefficients for
characteristic equation of the matrix, defined by
:math:`\\mathrm{det}(\\mathbf{A} - \\lambda \\mathbf{I})`,
are returned.
Parameters
----------
seq_of_zeros : ndarray
A sequence of polynomial roots or a square matrix.
Returns
-------
coefs : ndarray
A sequence of polynomial coefficients representing the polynomial
:math:`\\mathrm{coefs}[0] x^{n-1} + \\mathrm{coefs}[1] x^{n-2} +
... + \\mathrm{coefs}[2] x + \\mathrm{coefs}[n]`
See Also
--------
numpy.poly1d : A one-dimensional polynomial class.
numpy.roots : Return the roots of the polynomial coefficients in p
numpy.polyfit : Least squares polynomial fit
Examples
--------
Given a sequence of polynomial zeros,
>>> b = np.roots([1, 3, 1, 5, 6])
>>> np.poly(b)
array([ 1., 3., 1., 5., 6.])
Given a square matrix,
>>> P = np.array([[19, 3], [-2, 26]])
>>> np.poly(P)
array([ 1., -45., 500.])
"""
seq_of_zeros = atleast_1d(seq_of_zeros)
sh = seq_of_zeros.shape
if len(sh) == 2 and sh[0] == sh[1]:
seq_of_zeros = eigvals(seq_of_zeros)
elif len(sh) == 1:
pass
else:
raise ValueError, "input must be 1d or square 2d array."
if len(seq_of_zeros) == 0:
return 1.0
a = [1]
for k in range(len(seq_of_zeros)):
a = NX.convolve(a, [1, -seq_of_zeros[k]], mode='full')
if issubclass(a.dtype.type, NX.complexfloating):
# if complex roots are all complex conjugates, the roots are real.
roots = NX.asarray(seq_of_zeros, complex)
pos_roots = sort_complex(NX.compress(roots.imag > 0, roots))
neg_roots = NX.conjugate(
sort_complex(NX.compress(roots.imag < 0, roots)))
if (len(pos_roots) == len(neg_roots)
and NX.alltrue(neg_roots == pos_roots)):
a = a.real.copy()
return a
