def chebmul(c1, c2):
"""Multiply one Chebyshev series by another.
Returns the product of two Chebyshev series `c1` * `c2`. The arguments
are sequences of coefficients ordered from low to high, i.e., [1,2,3]
is the series ``T_0 + 2*T_1 + 3*T_2.``
Parameters
----------
c1, c2 : array_like
1d arrays of chebyshev series coefficients ordered from low to
high.
Returns
-------
out : ndarray
Chebyshev series of the product.
See Also
--------
chebadd, chebsub, chebdiv, chebpow
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
z1 = _cseries_to_zseries(c1)
z2 = _cseries_to_zseries(c2)
prd = _zseries_mul(z1, z2)
ret = _zseries_to_cseries(prd)
return pu.trimseq(ret)
def chebadd(c1, c2):
"""Add one Chebyshev series to another.
Returns the sum of two Chebyshev series `c1` + `c2`. The arguments are
sequences of coefficients ordered from low to high, i.e., [1,2,3] is
the series "T_0 + 2*T_1 + 3*T_2".
Parameters
----------
c1, c2 : array_like
1d arrays of Chebyshev series coefficients ordered from low to
high.
Returns
-------
out : ndarray
Chebyshev series of the sum.
See Also
--------
chebsub, chebmul, chebdiv, chebpow
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
if len(c1) > len(c2) :
c1[:c2.size] += c2
ret = c1
else :
c2[:c1.size] += c1
ret = c2
return pu.trimseq(ret)
def chebadd(c1, c2):
"""Add one Chebyshev series to another.
Returns the sum of two Chebyshev series `c1` + `c2`. The arguments are
sequences of coefficients ordered from low to high, i.e., [1,2,3] is
the series "T_0 + 2*T_1 + 3*T_2".
Parameters
----------
c1, c2 : array_like
1d arrays of Chebyshev series coefficients ordered from low to
high.
Returns
-------
out : ndarray
Chebyshev series of the sum.
See Also
--------
chebsub, chebmul, chebdiv, chebpow
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
if len(c1) > len(c2):
c1[:c2.size] += c2
ret = c1
else:
c2[:c1.size] += c1
ret = c2
return pu.trimseq(ret)
def chebmul(c1, c2):
"""Multiply one Chebyshev series by another.
Returns the product of two Chebyshev series `c1` * `c2`. The arguments
are sequences of coefficients ordered from low to high, i.e., [1,2,3]
is the series ``T_0 + 2*T_1 + 3*T_2.``
Parameters
----------
c1, c2 : array_like
1d arrays of chebyshev series coefficients ordered from low to
high.
Returns
-------
out : ndarray
Chebyshev series of the product.
See Also
--------
chebadd, chebsub, chebdiv, chebpow
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
z1 = _cseries_to_zseries(c1)
z2 = _cseries_to_zseries(c2)
prd = _zseries_mul(z1, z2)
ret = _zseries_to_cseries(prd)
return pu.trimseq(ret)
def polymul(c1, c2):
"""Multiply one polynomial by another.
Returns the product of two polynomials `c1` * `c2`. The arguments
are sequences of coefficients ordered from low to high, i.e., [1,2,3]
is the polynomial ``1 + 2*x + 3*x**2.``
Parameters
----------
c1, c2 : array_like
1d arrays of polyyshev series coefficients ordered from low to
high.
Returns
-------
out : ndarray
polynomial of the product.
See Also
--------
polyadd, polysub, polydiv, polypow
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
ret = np.convolve(c1, c2)
return pu.trimseq(ret)
def polyadd(c1, c2):
"""Add one polynomial to another.
Returns the sum of two polynomials `c1` + `c2`. The arguments are
sequences of coefficients ordered from low to high, i.e., [1,2,3] is
the polynomial ``1 + 2*x + 3*x**2"``.
Parameters
----------
c1, c2 : array_like
1d arrays of polynomial coefficients ordered from low to
high.
Returns
-------
out : ndarray
polynomial of the sum.
See Also
--------
polysub, polymul, polydiv, polypow
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
if len(c1) > len(c2):
c1[:c2.size] += c2
ret = c1
else:
c2[:c1.size] += c1
ret = c2
return pu.trimseq(ret)
def polymul(c1, c2):
"""Multiply one polynomial by another.
Returns the product of two polynomials `c1` * `c2`. The arguments
are sequences of coefficients ordered from low to high, i.e., [1,2,3]
is the polynomial ``1 + 2*x + 3*x**2.``
Parameters
----------
c1, c2 : array_like
1d arrays of polyyshev series coefficients ordered from low to
high.
Returns
-------
out : ndarray
polynomial of the product.
See Also
--------
polyadd, polysub, polydiv, polypow
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
ret = np.convolve(c1, c2)
return pu.trimseq(ret)
def polyadd(c1, c2):
"""Add one polynomial to another.
Returns the sum of two polynomials `c1` + `c2`. The arguments are
sequences of coefficients ordered from low to high, i.e., [1,2,3] is
the polynomial ``1 + 2*x + 3*x**2"``.
Parameters
----------
c1, c2 : array_like
1d arrays of polynomial coefficients ordered from low to
high.
Returns
-------
out : ndarray
polynomial of the sum.
See Also
--------
polysub, polymul, polydiv, polypow
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
if len(c1) > len(c2) :
c1[:c2.size] += c2
ret = c1
else :
c2[:c1.size] += c1
ret = c2
return pu.trimseq(ret)
def chebdiv(c1, c2):
"""Divide one Chebyshev series by another.
Returns the quotient of two Chebyshev series `c1` / `c2`. The arguments
are sequences of coefficients ordered from low to high, i.e., [1,2,3]
is the series ``T_0 + 2*T_1 + 3*T_2.``
Parameters
----------
c1, c2 : array_like
1d arrays of chebyshev series coefficients ordered from low to
high.
Returns
-------
[quo, rem] : ndarray
Chebyshev series of the quotient and remainder.
See Also
--------
chebadd, chebsub, chebmul, chebpow
Examples
--------
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
if c2[-1] == 0 :
raise ZeroDivisionError()
lc1 = len(c1)
lc2 = len(c2)
if lc1 < lc2 :
return c1[:1]*0, c1
elif lc2 == 1 :
return c1/c2[-1], c1[:1]*0
else :
z1 = _cseries_to_zseries(c1)
z2 = _cseries_to_zseries(c2)
quo, rem = _zseries_div(z1, z2)
quo = pu.trimseq(_zseries_to_cseries(quo))
rem = pu.trimseq(_zseries_to_cseries(rem))
return quo, rem
def chebdiv(c1, c2):
"""Divide one Chebyshev series by another.
Returns the quotient of two Chebyshev series `c1` / `c2`. The arguments
are sequences of coefficients ordered from low to high, i.e., [1,2,3]
is the series ``T_0 + 2*T_1 + 3*T_2.``
Parameters
----------
c1, c2 : array_like
1d arrays of chebyshev series coefficients ordered from low to
high.
Returns
-------
[quo, rem] : ndarray
Chebyshev series of the quotient and remainder.
See Also
--------
chebadd, chebsub, chebmul, chebpow
Examples
--------
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
if c2[-1] == 0:
raise ZeroDivisionError()
lc1 = len(c1)
lc2 = len(c2)
if lc1 < lc2:
return c1[:1] * 0, c1
elif lc2 == 1:
return c1 / c2[-1], c1[:1] * 0
else:
z1 = _cseries_to_zseries(c1)
z2 = _cseries_to_zseries(c2)
quo, rem = _zseries_div(z1, z2)
quo = pu.trimseq(_zseries_to_cseries(quo))
rem = pu.trimseq(_zseries_to_cseries(rem))
return quo, rem
def polydiv(c1, c2):
"""
Divide one polynomial by another.
Returns the quotient-with-remainder of two polynomials `c1` / `c2`.
The arguments are sequences of coefficients, from lowest order term
to highest, e.g., [1,2,3] represents ``1 + 2*x + 3*x**2``.
Parameters
----------
c1, c2 : array_like
1-d arrays of polynomial coefficients ordered from low to high.
Returns
-------
[quo, rem] : ndarrays
Of coefficient series representing the quotient and remainder.
See Also
--------
polyadd, polysub, polymul, polypow
Examples
--------
>>> import numpy.polynomial as P
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> P.polydiv(c1,c2)
(array([ 3.]), array([-8., -4.]))
>>> P.polydiv(c2,c1)
(array([ 0.33333333]), array([ 2.66666667, 1.33333333]))
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
if c2[-1] == 0 :
raise ZeroDivisionError()
len1 = len(c1)
len2 = len(c2)
if len2 == 1 :
return c1/c2[-1], c1[:1]*0
elif len1 < len2 :
return c1[:1]*0, c1
else :
dlen = len1 - len2
scl = c2[-1]
c2 = c2[:-1]/scl
i = dlen
j = len1 - 1
while i >= 0 :
c1[i:j] -= c2*c1[j]
i -= 1
j -= 1
return c1[j+1:]/scl, pu.trimseq(c1[:j+1])
def polydiv(c1, c2):
"""
Divide one polynomial by another.
Returns the quotient-with-remainder of two polynomials `c1` / `c2`.
The arguments are sequences of coefficients, from lowest order term
to highest, e.g., [1,2,3] represents ``1 + 2*x + 3*x**2``.
Parameters
----------
c1, c2 : array_like
1-d arrays of polynomial coefficients ordered from low to high.
Returns
-------
[quo, rem] : ndarrays
Of coefficient series representing the quotient and remainder.
See Also
--------
polyadd, polysub, polymul, polypow
Examples
--------
>>> import numpy.polynomial as P
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> P.polydiv(c1,c2)
(array([ 3.]), array([-8., -4.]))
>>> P.polydiv(c2,c1)
(array([ 0.33333333]), array([ 2.66666667, 1.33333333]))
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
if c2[-1] == 0:
raise ZeroDivisionError()
len1 = len(c1)
len2 = len(c2)
if len2 == 1:
return c1 / c2[-1], c1[:1] * 0
elif len1 < len2:
return c1[:1] * 0, c1
else:
dlen = len1 - len2
scl = c2[-1]
c2 = c2[:-1] / scl
i = dlen
j = len1 - 1
while i >= 0:
c1[i:j] -= c2 * c1[j]
i -= 1
j -= 1
return c1[j + 1:] / scl, pu.trimseq(c1[:j + 1])
def legsub(c1, c2):
"""
Subtract one Legendre series from another.
Returns the difference of two Legendre series `c1` - `c2`. The
sequences of coefficients are from lowest order term to highest, i.e.,
[1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
Parameters
----------
c1, c2 : array_like
1-d arrays of Legendre series coefficients ordered from low to
high.
Returns
-------
out : ndarray
Of Legendre series coefficients representing their difference.
See Also
--------
legadd, legmul, legdiv, legpow
Notes
-----
Unlike multiplication, division, etc., the difference of two Legendre
series is a Legendre series (without having to "reproject" the result
onto the basis set) so subtraction, just like that of "standard"
polynomials, is simply "component-wise."
Examples
--------
>>> from numpy.polynomial import legendre as L
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> L.legsub(c1,c2)
array([-2., 0., 2.])
>>> L.legsub(c2,c1) # -C.legsub(c1,c2)
array([ 2., 0., -2.])
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
if len(c1) > len(c2) :
c1[:c2.size] -= c2
ret = c1
else :
c2 = -c2
c2[:c1.size] += c1
ret = c2
return pu.trimseq(ret)
def chebsub(c1, c2):
"""
Subtract one Chebyshev series from another.
Returns the difference of two Chebyshev series `c1` - `c2`. The
sequences of coefficients are from lowest order term to highest, i.e.,
[1,2,3] represents the series ``T_0 + 2*T_1 + 3*T_2``.
Parameters
----------
c1, c2 : array_like
1-d arrays of Chebyshev series coefficients ordered from low to
high.
Returns
-------
out : ndarray
Of Chebyshev series coefficients representing their difference.
See Also
--------
chebadd, chebmul, chebdiv, chebpow
Notes
-----
Unlike multiplication, division, etc., the difference of two Chebyshev
series is a Chebyshev series (without having to "reproject" the result
onto the basis set) so subtraction, just like that of "standard"
polynomials, is simply "component-wise."
Examples
--------
>>> from numpy.polynomial import chebyshev as C
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> C.chebsub(c1,c2)
array([-2., 0., 2.])
>>> C.chebsub(c2,c1) # -C.chebsub(c1,c2)
array([ 2., 0., -2.])
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
if len(c1) > len(c2):
c1[:c2.size] -= c2
ret = c1
else:
c2 = -c2
c2[:c1.size] += c1
ret = c2
return pu.trimseq(ret)
def polydiv(c1, c2):
"""Divide one polynomial by another.
Returns the quotient of two polynomials `c1` / `c2`. The arguments are
sequences of coefficients ordered from low to high, i.e., [1,2,3] is
the series ``1 + 2*x + 3*x**2.``
Parameters
----------
c1, c2 : array_like
1d arrays of chebyshev series coefficients ordered from low to
high.
Returns
-------
[quo, rem] : ndarray
polynomial of the quotient and remainder.
See Also
--------
polyadd, polysub, polymul, polypow
Examples
--------
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
if c2[-1] == 0 :
raise ZeroDivisionError()
len1 = len(c1)
len2 = len(c2)
if len2 == 1 :
return c1/c2[-1], c1[:1]*0
elif len1 < len2 :
return c1[:1]*0, c1
else :
dlen = len1 - len2
scl = c2[-1]
c2 = c2[:-1]/scl
i = dlen
j = len1 - 1
while i >= 0 :
c1[i:j] -= c2*c1[j]
i -= 1
j -= 1
return c1[j+1:]/scl, pu.trimseq(c1[:j+1])
def legadd(c1, c2):
"""
Add one Legendre series to another.
Returns the sum of two Legendre series `c1` + `c2`. The arguments
are sequences of coefficients ordered from lowest order term to
highest, i.e., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
Parameters
----------
c1, c2 : array_like
1-d arrays of Legendre series coefficients ordered from low to
high.
Returns
-------
out : ndarray
Array representing the Legendre series of their sum.
See Also
--------
legsub, legmul, legdiv, legpow
Notes
-----
Unlike multiplication, division, etc., the sum of two Legendre series
is a Legendre series (without having to "reproject" the result onto
the basis set) so addition, just like that of "standard" polynomials,
is simply "component-wise."
Examples
--------
>>> from numpy.polynomial import legendre as L
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> L.legadd(c1,c2)
array([ 4., 4., 4.])
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
if len(c1) > len(c2) :
c1[:c2.size] += c2
ret = c1
else :
c2[:c1.size] += c1
ret = c2
return pu.trimseq(ret)
def polydiv(c1, c2):
"""Divide one polynomial by another.
Returns the quotient of two polynomials `c1` / `c2`. The arguments are
sequences of coefficients ordered from low to high, i.e., [1,2,3] is
the series ``1 + 2*x + 3*x**2.``
Parameters
----------
c1, c2 : array_like
1d arrays of chebyshev series coefficients ordered from low to
high.
Returns
-------
[quo, rem] : ndarray
polynomial of the quotient and remainder.
See Also
--------
polyadd, polysub, polymul, polypow
Examples
--------
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
if c2[-1] == 0:
raise ZeroDivisionError()
len1 = len(c1)
len2 = len(c2)
if len2 == 1:
return c1 / c2[-1], c1[:1] * 0
elif len1 < len2:
return c1[:1] * 0, c1
else:
dlen = len1 - len2
scl = c2[-1]
c2 = c2[:-1] / scl
i = dlen
j = len1 - 1
while i >= 0:
c1[i:j] -= c2 * c1[j]
i -= 1
j -= 1
return c1[j + 1:] / scl, pu.trimseq(c1[:j + 1])
def chebadd(c1, c2):
"""
Add one Chebyshev series to another.
Returns the sum of two Chebyshev series `c1` + `c2`. The arguments
are sequences of coefficients ordered from lowest order term to
highest, i.e., [1,2,3] represents the series ``T_0 + 2*T_1 + 3*T_2``.
Parameters
----------
c1, c2 : array_like
1-d arrays of Chebyshev series coefficients ordered from low to
high.
Returns
-------
out : ndarray
Array representing the Chebyshev series of their sum.
See Also
--------
chebsub, chebmul, chebdiv, chebpow
Notes
-----
Unlike multiplication, division, etc., the sum of two Chebyshev series
is a Chebyshev series (without having to "reproject" the result onto
the basis set) so addition, just like that of "standard" polynomials,
is simply "component-wise."
Examples
--------
>>> from numpy.polynomial import chebyshev as C
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> C.chebadd(c1,c2)
array([ 4., 4., 4.])
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
if len(c1) > len(c2) :
c1[:c2.size] += c2
ret = c1
else :
c2[:c1.size] += c1
ret = c2
return pu.trimseq(ret)
def lagadd(c1, c2):
"""
Add one Laguerre series to another.
Returns the sum of two Laguerre series `c1` + `c2`. The arguments
are sequences of coefficients ordered from lowest order term to
highest, i.e., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
Parameters
----------
c1, c2 : array_like
1-d arrays of Laguerre series coefficients ordered from low to
high.
Returns
-------
out : ndarray
Array representing the Laguerre series of their sum.
See Also
--------
lagsub, lagmul, lagdiv, lagpow
Notes
-----
Unlike multiplication, division, etc., the sum of two Laguerre series
is a Laguerre series (without having to "reproject" the result onto
the basis set) so addition, just like that of "standard" polynomials,
is simply "component-wise."
Examples
--------
>>> from numpy.polynomial.laguerre import lagadd
>>> lagadd([1, 2, 3], [1, 2, 3, 4])
array([ 2., 4., 6., 4.])
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
if len(c1) > len(c2) :
c1[:c2.size] += c2
ret = c1
else :
c2[:c1.size] += c1
ret = c2
return pu.trimseq(ret)
def hermsub(c1, c2):
"""
Subtract one Hermite series from another.
Returns the difference of two Hermite series `c1` - `c2`. The
sequences of coefficients are from lowest order term to highest, i.e.,
[1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
Parameters
----------
c1, c2 : array_like
1-d arrays of Hermite series coefficients ordered from low to
high.
Returns
-------
out : ndarray
Of Hermite series coefficients representing their difference.
See Also
--------
hermadd, hermmul, hermdiv, hermpow
Notes
-----
Unlike multiplication, division, etc., the difference of two Hermite
series is a Hermite series (without having to "reproject" the result
onto the basis set) so subtraction, just like that of "standard"
polynomials, is simply "component-wise."
Examples
--------
>>> from numpy.polynomial.hermite import hermsub
>>> hermsub([1, 2, 3, 4], [1, 2, 3])
array([ 0., 0., 0., 4.])
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
if len(c1) > len(c2) :
c1[:c2.size] -= c2
ret = c1
else :
c2 = -c2
c2[:c1.size] += c1
ret = c2
return pu.trimseq(ret)
def chebmul(c1, c2):
"""
Multiply one Chebyshev series by another.
Returns the product of two Chebyshev series `c1` * `c2`. The arguments
are sequences of coefficients, from lowest order "term" to highest,
e.g., [1,2,3] represents the series ``T_0 + 2*T_1 + 3*T_2``.
Parameters
----------
c1, c2 : array_like
1-d arrays of Chebyshev series coefficients ordered from low to
high.
Returns
-------
out : ndarray
Of Chebyshev series coefficients representing their product.
See Also
--------
chebadd, chebsub, chebdiv, chebpow
Notes
-----
In general, the (polynomial) product of two C-series results in terms
that are not in the Chebyshev polynomial basis set. Thus, to express
the product as a C-series, it is typically necessary to "re-project"
the product onto said basis set, which typically produces
"un-intuitive" (but correct) results; see Examples section below.
Examples
--------
>>> from numpy.polynomial import chebyshev as C
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> C.chebmul(c1,c2) # multiplication requires "reprojection"
array([ 6.5, 12. , 12. , 4. , 1.5])
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
z1 = _cseries_to_zseries(c1)
z2 = _cseries_to_zseries(c2)
prd = _zseries_mul(z1, z2)
ret = _zseries_to_cseries(prd)
return pu.trimseq(ret)
def chebmul(c1, c2):
"""
Multiply one Chebyshev series by another.
Returns the product of two Chebyshev series `c1` * `c2`. The arguments
are sequences of coefficients, from lowest order "term" to highest,
e.g., [1,2,3] represents the series ``T_0 + 2*T_1 + 3*T_2``.
Parameters
----------
c1, c2 : array_like
1-d arrays of Chebyshev series coefficients ordered from low to
high.
Returns
-------
out : ndarray
Of Chebyshev series coefficients representing their product.
See Also
--------
chebadd, chebsub, chebdiv, chebpow
Notes
-----
In general, the (polynomial) product of two C-series results in terms
that are not in the Chebyshev polynomial basis set. Thus, to express
the product as a C-series, it is typically necessary to "re-project"
the product onto said basis set, which typically produces
"un-intuitive" (but correct) results; see Examples section below.
Examples
--------
>>> from numpy.polynomial import chebyshev as C
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> C.chebmul(c1,c2) # multiplication requires "reprojection"
array([ 6.5, 12. , 12. , 4. , 1.5])
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
z1 = _cseries_to_zseries(c1)
z2 = _cseries_to_zseries(c2)
prd = _zseries_mul(z1, z2)
ret = _zseries_to_cseries(prd)
return pu.trimseq(ret)
def polysub(c1, c2):
"""
Subtract one polynomial from another.
Returns the difference of two polynomials `c1` - `c2`. The arguments
are sequences of coefficients from lowest order term to highest, i.e.,
[1,2,3] represents the polynomial ``1 + 2*x + 3*x**2``.
Parameters
----------
c1, c2 : array_like
1-d arrays of polynomial coefficients ordered from low to
high.
Returns
-------
out : ndarray
Of coefficients representing their difference.
See Also
--------
polyadd, polymul, polydiv, polypow
Examples
--------
>>> from numpy import polynomial as P
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> P.polysub(c1,c2)
array([-2., 0., 2.])
>>> P.polysub(c2,c1) # -P.polysub(c1,c2)
array([ 2., 0., -2.])
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
if len(c1) > len(c2) :
c1[:c2.size] -= c2
ret = c1
else :
c2 = -c2
c2[:c1.size] += c1
ret = c2
return pu.trimseq(ret)
def polysub(c1, c2):
"""
Subtract one polynomial from another.
Returns the difference of two polynomials `c1` - `c2`. The arguments
are sequences of coefficients from lowest order term to highest, i.e.,
[1,2,3] represents the polynomial ``1 + 2*x + 3*x**2``.
Parameters
----------
c1, c2 : array_like
1-d arrays of polynomial coefficients ordered from low to
high.
Returns
-------
out : ndarray
Of coefficients representing their difference.
See Also
--------
polyadd, polymul, polydiv, polypow
Examples
--------
>>> from numpy import polynomial as P
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> P.polysub(c1,c2)
array([-2., 0., 2.])
>>> P.polysub(c2,c1) # -P.polysub(c1,c2)
array([ 2., 0., -2.])
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
if len(c1) > len(c2):
c1[:c2.size] -= c2
ret = c1
else:
c2 = -c2
c2[:c1.size] += c1
ret = c2
return pu.trimseq(ret)
def polyadd(c1, c2):
"""
Add one polynomial to another.
Returns the sum of two polynomials `c1` + `c2`. The arguments are
sequences of coefficients from lowest order term to highest, i.e.,
[1,2,3] represents the polynomial ``1 + 2*x + 3*x**2"``.
Parameters
----------
c1, c2 : array_like
1-d arrays of polynomial coefficients ordered from low to high.
Returns
-------
out : ndarray
The coefficient array representing their sum.
See Also
--------
polysub, polymul, polydiv, polypow
Examples
--------
>>> from numpy import polynomial as P
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> sum = P.polyadd(c1,c2); sum
array([ 4., 4., 4.])
>>> P.polyval(2, sum) # 4 + 4(2) + 4(2**2)
28.0
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
if len(c1) > len(c2):
c1[:c2.size] += c2
ret = c1
else:
c2[:c1.size] += c1
ret = c2
return pu.trimseq(ret)
def polyadd(c1, c2):
"""
Add one polynomial to another.
Returns the sum of two polynomials `c1` + `c2`. The arguments are
sequences of coefficients from lowest order term to highest, i.e.,
[1,2,3] represents the polynomial ``1 + 2*x + 3*x**2"``.
Parameters
----------
c1, c2 : array_like
1-d arrays of polynomial coefficients ordered from low to high.
Returns
-------
out : ndarray
The coefficient array representing their sum.
See Also
--------
polysub, polymul, polydiv, polypow
Examples
--------
>>> from numpy import polynomial as P
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> sum = P.polyadd(c1,c2); sum
array([ 4., 4., 4.])
>>> P.polyval(2, sum) # 4 + 4(2) + 4(2**2)
28.0
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
if len(c1) > len(c2) :
c1[:c2.size] += c2
ret = c1
else :
c2[:c1.size] += c1
ret = c2
return pu.trimseq(ret)
def polymul(c1, c2):
"""
Multiply one polynomial by another.
Returns the product of two polynomials `c1` * `c2`. The arguments are
sequences of coefficients, from lowest order term to highest, e.g.,
[1,2,3] represents the polynomial ``1 + 2*x + 3*x**2.``
Parameters
----------
c1, c2 : array_like
1-d arrays of coefficients representing a polynomial, relative to the
"standard" basis, and ordered from lowest order term to highest.
Returns
-------
out : ndarray
Of the coefficients of their product.
See Also
--------
polyadd, polysub, polydiv, polypow
Examples
--------
>>> import numpy.polynomial as P
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> P.polymul(c1,c2)
array([ 3., 8., 14., 8., 3.])
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
ret = np.convolve(c1, c2)
return pu.trimseq(ret)
def polymul(c1, c2):
"""
Multiply one polynomial by another.
Returns the product of two polynomials `c1` * `c2`. The arguments are
sequences of coefficients, from lowest order term to highest, e.g.,
[1,2,3] represents the polynomial ``1 + 2*x + 3*x**2.``
Parameters
----------
c1, c2 : array_like
1-d arrays of coefficients representing a polynomial, relative to the
"standard" basis, and ordered from lowest order term to highest.
Returns
-------
out : ndarray
Of the coefficients of their product.
See Also
--------
polyadd, polysub, polydiv, polypow
Examples
--------
>>> import numpy.polynomial as P
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> P.polymul(c1,c2)
array([ 3., 8., 14., 8., 3.])
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
ret = np.convolve(c1, c2)
return pu.trimseq(ret)
def polysub(c1, c2):
"""Subtract one polynomial from another.
Returns the difference of two polynomials `c1` - `c2`. The arguments
are sequences of coefficients ordered from low to high, i.e., [1,2,3]
is the polynomial ``1 + 2*x + 3*x**2``.
Parameters
----------
c1, c2 : array_like
1d arrays of polynomial coefficients ordered from low to
high.
Returns
-------
out : ndarray
polynomial of the difference.
See Also
--------
polyadd, polymul, polydiv, polypow
Examples
--------
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
if len(c1) > len(c2) :
c1[:c2.size] -= c2
ret = c1
else :
c2 = -c2
c2[:c1.size] += c1
ret = c2
return pu.trimseq(ret)
def polysub(c1, c2):
"""Subtract one polynomial from another.
Returns the difference of two polynomials `c1` - `c2`. The arguments
are sequences of coefficients ordered from low to high, i.e., [1,2,3]
is the polynomial ``1 + 2*x + 3*x**2``.
Parameters
----------
c1, c2 : array_like
1d arrays of polynomial coefficients ordered from low to
high.
Returns
-------
out : ndarray
polynomial of the difference.
See Also
--------
polyadd, polymul, polydiv, polypow
Examples
--------
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
if len(c1) > len(c2):
c1[:c2.size] -= c2
ret = c1
else:
c2 = -c2
c2[:c1.size] += c1
ret = c2
return pu.trimseq(ret)
def chebsub(c1, c2):
"""Subtract one Chebyshev series from another.
Returns the difference of two Chebyshev series `c1` - `c2`. The
sequences of coefficients are ordered from low to high, i.e., [1,2,3]
is the series ``T_0 + 2*T_1 + 3*T_2.``
Parameters
----------
c1, c2 : array_like
1d arrays of Chebyshev series coefficients ordered from low to
high.
Returns
-------
out : ndarray
Chebyshev series of the difference.
See Also
--------
chebadd, chebmul, chebdiv, chebpow
Examples
--------
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
if len(c1) > len(c2):
c1[:c2.size] -= c2
ret = c1
else:
c2 = -c2
c2[:c1.size] += c1
ret = c2
return pu.trimseq(ret)
def chebsub(c1, c2):
"""Subtract one Chebyshev series from another.
Returns the difference of two Chebyshev series `c1` - `c2`. The
sequences of coefficients are ordered from low to high, i.e., [1,2,3]
is the series ``T_0 + 2*T_1 + 3*T_2.``
Parameters
----------
c1, c2 : array_like
1d arrays of Chebyshev series coefficients ordered from low to
high.
Returns
-------
out : ndarray
Chebyshev series of the difference.
See Also
--------
chebadd, chebmul, chebdiv, chebpow
Examples
--------
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
if len(c1) > len(c2) :
c1[:c2.size] -= c2
ret = c1
else :
c2 = -c2
c2[:c1.size] += c1
ret = c2
return pu.trimseq(ret)
def chebdiv(c1, c2):
"""
Divide one Chebyshev series by another.
Returns the quotient-with-remainder of two Chebyshev series
`c1` / `c2`. The arguments are sequences of coefficients from lowest
order "term" to highest, e.g., [1,2,3] represents the series
``T_0 + 2*T_1 + 3*T_2``.
Parameters
----------
c1, c2 : array_like
1-d arrays of Chebyshev series coefficients ordered from low to
high.
Returns
-------
[quo, rem] : ndarrays
Of Chebyshev series coefficients representing the quotient and
remainder.
See Also
--------
chebadd, chebsub, chebmul, chebpow
Notes
-----
In general, the (polynomial) division of one C-series by another
results in quotient and remainder terms that are not in the Chebyshev
polynomial basis set. Thus, to express these results as C-series, it
is typically necessary to "re-project" the results onto said basis
set, which typically produces "un-intuitive" (but correct) results;
see Examples section below.
Examples
--------
>>> from numpy.polynomial import chebyshev as C
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> C.chebdiv(c1,c2) # quotient "intuitive," remainder not
(array([ 3.]), array([-8., -4.]))
>>> c2 = (0,1,2,3)
>>> C.chebdiv(c2,c1) # neither "intuitive"
(array([ 0., 2.]), array([-2., -4.]))
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
if c2[-1] == 0:
raise ZeroDivisionError()
lc1 = len(c1)
lc2 = len(c2)
if lc1 < lc2:
return c1[:1] * 0, c1
elif lc2 == 1:
return c1 / c2[-1], c1[:1] * 0
else:
z1 = _cseries_to_zseries(c1)
z2 = _cseries_to_zseries(c2)
quo, rem = _zseries_div(z1, z2)
quo = pu.trimseq(_zseries_to_cseries(quo))
rem = pu.trimseq(_zseries_to_cseries(rem))
return quo, rem
def chebdiv(c1, c2):
"""
Divide one Chebyshev series by another.
Returns the quotient-with-remainder of two Chebyshev series
`c1` / `c2`. The arguments are sequences of coefficients from lowest
order "term" to highest, e.g., [1,2,3] represents the series
``T_0 + 2*T_1 + 3*T_2``.
Parameters
----------
c1, c2 : array_like
1-d arrays of Chebyshev series coefficients ordered from low to
high.
Returns
-------
[quo, rem] : ndarrays
Of Chebyshev series coefficients representing the quotient and
remainder.
See Also
--------
chebadd, chebsub, chebmul, chebpow
Notes
-----
In general, the (polynomial) division of one C-series by another
results in quotient and remainder terms that are not in the Chebyshev
polynomial basis set. Thus, to express these results as C-series, it
is typically necessary to "re-project" the results onto said basis
set, which typically produces "un-intuitive" (but correct) results;
see Examples section below.
Examples
--------
>>> from numpy.polynomial import chebyshev as C
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> C.chebdiv(c1,c2) # quotient "intuitive," remainder not
(array([ 3.]), array([-8., -4.]))
>>> c2 = (0,1,2,3)
>>> C.chebdiv(c2,c1) # neither "intuitive"
(array([ 0., 2.]), array([-2., -4.]))
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
if c2[-1] == 0 :
raise ZeroDivisionError()
lc1 = len(c1)
lc2 = len(c2)
if lc1 < lc2 :
return c1[:1]*0, c1
elif lc2 == 1 :
return c1/c2[-1], c1[:1]*0
else :
z1 = _cseries_to_zseries(c1)
z2 = _cseries_to_zseries(c2)
quo, rem = _zseries_div(z1, z2)
quo = pu.trimseq(_zseries_to_cseries(quo))
rem = pu.trimseq(_zseries_to_cseries(rem))
return quo, rem
def legdiv(c1, c2):
"""
Divide one Legendre series by another.
Returns the quotient-with-remainder of two Legendre series
`c1` / `c2`. The arguments are sequences of coefficients from lowest
order "term" to highest, e.g., [1,2,3] represents the series
``P_0 + 2*P_1 + 3*P_2``.
Parameters
----------
c1, c2 : array_like
1-D arrays of Legendre series coefficients ordered from low to
high.
Returns
-------
quo, rem : ndarrays
Of Legendre series coefficients representing the quotient and
remainder.
See Also
--------
legadd, legsub, legmul, legpow
Notes
-----
In general, the (polynomial) division of one Legendre series by another
results in quotient and remainder terms that are not in the Legendre
polynomial basis set. Thus, to express these results as a Legendre
series, it is necessary to "re-project" the results onto the Legendre
basis set, which may produce "un-intuitive" (but correct) results; see
Examples section below.
Examples
--------
>>> from numpy.polynomial import legendre as L
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> L.legdiv(c1,c2) # quotient "intuitive," remainder not
(array([ 3.]), array([-8., -4.]))
>>> c2 = (0,1,2,3)
>>> L.legdiv(c2,c1) # neither "intuitive"
(array([-0.07407407, 1.66666667]), array([-1.03703704, -2.51851852]))
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
if c2[-1] == 0 :
raise ZeroDivisionError()
lc1 = len(c1)
lc2 = len(c2)
if lc1 < lc2 :
return c1[:1]*0, c1
elif lc2 == 1 :
return c1/c2[-1], c1[:1]*0
else :
quo = np.empty(lc1 - lc2 + 1, dtype=c1.dtype)
rem = c1
for i in range(lc1 - lc2, - 1, -1):
p = legmul([0]*i + [1], c2)
q = rem[-1]/p[-1]
rem = rem[:-1] - q*p[:-1]
quo[i] = q
return quo, pu.trimseq(rem)
def legdiv(c1, c2):
"""
Divide one Legendre series by another.
Returns the quotient-with-remainder of two Legendre series
`c1` / `c2`. The arguments are sequences of coefficients from lowest
order "term" to highest, e.g., [1,2,3] represents the series
``P_0 + 2*P_1 + 3*P_2``.
Parameters
----------
c1, c2 : array_like
1-D arrays of Legendre series coefficients ordered from low to
high.
Returns
-------
quo, rem : ndarrays
Of Legendre series coefficients representing the quotient and
remainder.
See Also
--------
legadd, legsub, legmul, legpow
Notes
-----
In general, the (polynomial) division of one Legendre series by another
results in quotient and remainder terms that are not in the Legendre
polynomial basis set. Thus, to express these results as a Legendre
series, it is necessary to "re-project" the results onto the Legendre
basis set, which may produce "un-intuitive" (but correct) results; see
Examples section below.
Examples
--------
>>> from numpy.polynomial import legendre as L
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> L.legdiv(c1,c2) # quotient "intuitive," remainder not
(array([ 3.]), array([-8., -4.]))
>>> c2 = (0,1,2,3)
>>> L.legdiv(c2,c1) # neither "intuitive"
(array([-0.07407407, 1.66666667]), array([-1.03703704, -2.51851852]))
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
if c2[-1] == 0 :
raise ZeroDivisionError()
lc1 = len(c1)
lc2 = len(c2)
if lc1 < lc2 :
return c1[:1]*0, c1
elif lc2 == 1 :
return c1/c2[-1], c1[:1]*0
else :
quo = np.empty(lc1 - lc2 + 1, dtype=c1.dtype)
rem = c1
for i in range(lc1 - lc2, - 1, -1):
p = legmul([0]*i + [1], c2)
q = rem[-1]/p[-1]
rem = rem[:-1] - q*p[:-1]
quo[i] = q
return quo, pu.trimseq(rem)
def hermdiv(c1, c2):
"""
Divide one Hermite series by another.
Returns the quotient-with-remainder of two Hermite series
`c1` / `c2`. The arguments are sequences of coefficients from lowest
order "term" to highest, e.g., [1,2,3] represents the series
``P_0 + 2*P_1 + 3*P_2``.
Parameters
----------
c1, c2 : array_like
1-d arrays of Hermite series coefficients ordered from low to
high.
Returns
-------
[quo, rem] : ndarrays
Of Hermite series coefficients representing the quotient and
remainder.
See Also
--------
hermadd, hermsub, hermmul, hermpow
Notes
-----
In general, the (polynomial) division of one Hermite series by another
results in quotient and remainder terms that are not in the Hermite
polynomial basis set. Thus, to express these results as a Hermite
series, it is necessary to "re-project" the results onto the Hermite
basis set, which may produce "un-intuitive" (but correct) results; see
Examples section below.
Examples
--------
>>> from numpy.polynomial.hermite import hermdiv
>>> hermdiv([ 52., 29., 52., 7., 6.], [0, 1, 2])
(array([ 1., 2., 3.]), array([ 0.]))
>>> hermdiv([ 54., 31., 52., 7., 6.], [0, 1, 2])
(array([ 1., 2., 3.]), array([ 2., 2.]))
>>> hermdiv([ 53., 30., 52., 7., 6.], [0, 1, 2])
(array([ 1., 2., 3.]), array([ 1., 1.]))
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
if c2[-1] == 0 :
raise ZeroDivisionError()
lc1 = len(c1)
lc2 = len(c2)
if lc1 < lc2 :
return c1[:1]*0, c1
elif lc2 == 1 :
return c1/c2[-1], c1[:1]*0
else :
quo = np.empty(lc1 - lc2 + 1, dtype=c1.dtype)
rem = c1
for i in range(lc1 - lc2, - 1, -1):
p = hermmul([0]*i + [1], c2)
q = rem[-1]/p[-1]
rem = rem[:-1] - q*p[:-1]
quo[i] = q
return quo, pu.trimseq(rem)
