This module provides a number of objects (mostly functions) useful for
dealing with polynomials, including a Polynomial class that
encapsulates the usual arithmetic operations. (General information
on how this module represents and works with polynomial objects is in
the docstring for its "parent" sub-package, numpy.polynomial).
| Class | Polynomial |
A power series class. |
| Function | polyadd |
Add one polynomial to another. |
| Function | polycompanion |
Return the companion matrix of c. |
| Function | polyder |
Differentiate a polynomial. |
| Function | polydiv |
Divide one polynomial by another. |
| Function | polyfit |
Least-squares fit of a polynomial to data. |
| Function | polyfromroots |
Generate a monic polynomial with given roots. |
| Function | polygrid2d |
Evaluate a 2-D polynomial on the Cartesian product of x and y. |
| Function | polygrid3d |
Evaluate a 3-D polynomial on the Cartesian product of x, y and z. |
| Function | polyint |
Integrate a polynomial. |
| Function | polyline |
Returns an array representing a linear polynomial. |
| Function | polymul |
Multiply one polynomial by another. |
| Function | polymulx |
Multiply a polynomial by x. |
| Function | polypow |
Raise a polynomial to a power. |
| Function | polyroots |
Compute the roots of a polynomial. |
| Function | polysub |
Subtract one polynomial from another. |
| Function | polyval |
Evaluate a polynomial at points x. |
| Function | polyval2d |
Evaluate a 2-D polynomial at points (x, y). |
| Function | polyval3d |
Evaluate a 3-D polynomial at points (x, y, z). |
| Function | polyvalfromroots |
Evaluate a polynomial specified by its roots at points x. |
| Function | polyvander |
Vandermonde matrix of given degree. |
| Function | polyvander2d |
Pseudo-Vandermonde matrix of given degrees. |
| Function | polyvander3d |
Pseudo-Vandermonde matrix of given degrees. |
| Variable | polydomain |
Undocumented |
| Variable | polyone |
Undocumented |
| Variable | polyx |
Undocumented |
| Variable | polyzero |
Undocumented |
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.
polysub, polymulx, polymul, polydiv, polypow
>>> from numpy.polynomial 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
Return the companion matrix of c.
The companion matrix for power series cannot be made symmetric by scaling the basis, so this function differs from those for the orthogonal polynomials.
Differentiate a polynomial.
Returns the polynomial coefficients c differentiated m times along
axis. At each iteration the result is multiplied by scl (the
scaling factor is for use in a linear change of variable). The
argument c is an array of coefficients from low to high degree along
each axis, e.g., [1,2,3] represents the polynomial 1 + 2*x + 3*x**2
while [[1,2],[1,2]] represents 1 + 1*x + 2*y + 2*x*y if axis=0 is
x and axis=1 is y.
scl. The end result is
multiplication by scl**m. This is for use in a linear change
of variable. (Default: 1)Axis over which the derivative is taken. (Default: 0).
polyint
>>> from numpy.polynomial import polynomial as P >>> c = (1,2,3,4) # 1 + 2x + 3x**2 + 4x**3 >>> P.polyder(c) # (d/dx)(c) = 2 + 6x + 12x**2 array([ 2., 6., 12.]) >>> P.polyder(c,3) # (d**3/dx**3)(c) = 24 array([24.]) >>> P.polyder(c,scl=-1) # (d/d(-x))(c) = -2 - 6x - 12x**2 array([ -2., -6., -12.]) >>> P.polyder(c,2,-1) # (d**2/d(-x)**2)(c) = 6 + 24x array([ 6., 24.])
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.
polyadd, polysub, polymulx, polymul, polypow
>>> from numpy.polynomial import 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])) # may vary
Least-squares fit of a polynomial to data.
Return the coefficients of a polynomial of degree deg that is the
least squares fit to the data values y given at points x. If y is
1-D the returned coefficients will also be 1-D. If y is 2-D multiple
fits are done, one for each column of y, and the resulting
coefficients are stored in the corresponding columns of a 2-D return.
The fitted polynomial(s) are in the form
where n is deg.
M,)M sample (data) points (x[i], y[i]).M,) or (M, K)polyfit by passing in for y a 2-D array that contains
one data set per column.deg is a single integer
all terms up to and including the deg'th term are included in the
fit. For NumPy versions >= 1.11.0 a list of integers specifying the
degrees of the terms to include may be used instead.rcond, relative to the largest singular value, will be
ignored. The default value is len(x)*eps, where eps is the
relative precision of the platform's float type, about 2e-16 in
most cases.M,), optionalWeights. If not None, the weight w[i] applies to the unsquared residual y[i] - y_hat[i] at x[i]. Ideally the weights are chosen so that the errors of the products w[i]*y[i] all have the same variance. When using inverse-variance weighting, use w[i] = 1/sigma(y[i]). The default value is None.
deg + 1,) or (deg + 1, K)y was 2-D,
the coefficients in column k of coef represent the polynomial
fit to the data in y's k-th column.These values are only returned if full == True
rcond.For more details, see numpy.linalg.lstsq.
Raised if the matrix in the least-squares fit is rank deficient. The warning is only raised if full == False. The warnings can be turned off by:
>>> import warnings >>> warnings.simplefilter('ignore', np.RankWarning)
numpy.polynomial.chebyshev.chebfit numpy.polynomial.legendre.legfit numpy.polynomial.laguerre.lagfit numpy.polynomial.hermite.hermfit numpy.polynomial.hermite_e.hermefit polyval : Evaluates a polynomial. polyvander : Vandermonde matrix for powers. numpy.linalg.lstsq : Computes a least-squares fit from the matrix. scipy.interpolate.UnivariateSpline : Computes spline fits.
The solution is the coefficients of the polynomial p that minimizes
the sum of the weighted squared errors
where the wj are the weights. This problem is solved by setting up the (typically) over-determined matrix equation:
where V is the weighted pseudo Vandermonde matrix of x, c are the
coefficients to be solved for, w are the weights, and y are the
observed values. This equation is then solved using the singular value
decomposition of V.
If some of the singular values of V are so small that they are
neglected (and full == False), a RankWarning will be raised.
This means that the coefficient values may be poorly determined.
Fitting to a lower order polynomial will usually get rid of the warning
(but may not be what you want, of course; if you have independent
reason(s) for choosing the degree which isn't working, you may have to:
a) reconsider those reasons, and/or b) reconsider the quality of your
data). The rcond parameter can also be set to a value smaller than
its default, but the resulting fit may be spurious and have large
contributions from roundoff error.
Polynomial fits using double precision tend to "fail" at about (polynomial) degree 20. Fits using Chebyshev or Legendre series are generally better conditioned, but much can still depend on the distribution of the sample points and the smoothness of the data. If the quality of the fit is inadequate, splines may be a good alternative.
>>> np.random.seed(123) >>> from numpy.polynomial import polynomial as P >>> x = np.linspace(-1,1,51) # x "data": [-1, -0.96, ..., 0.96, 1] >>> y = x**3 - x + np.random.randn(len(x)) # x^3 - x + N(0,1) "noise" >>> c, stats = P.polyfit(x,y,3,full=True) >>> np.random.seed(123) >>> c # c[0], c[2] should be approx. 0, c[1] approx. -1, c[3] approx. 1 array([ 0.01909725, -1.30598256, -0.00577963, 1.02644286]) # may vary >>> stats # note the large SSR, explaining the rather poor results [array([ 38.06116253]), 4, array([ 1.38446749, 1.32119158, 0.50443316, # may vary 0.28853036]), 1.1324274851176597e-014]
Same thing without the added noise
>>> y = x**3 - x >>> c, stats = P.polyfit(x,y,3,full=True) >>> c # c[0], c[2] should be "very close to 0", c[1] ~= -1, c[3] ~= 1 array([-6.36925336e-18, -1.00000000e+00, -4.08053781e-16, 1.00000000e+00]) >>> stats # note the minuscule SSR [array([ 7.46346754e-31]), 4, array([ 1.38446749, 1.32119158, # may vary 0.50443316, 0.28853036]), 1.1324274851176597e-014]
Generate a monic polynomial with given roots.
Return the coefficients of the polynomial
where the r_n are the roots specified in roots. If a zero has
multiplicity n, then it must appear in roots n times. For instance,
if 2 is a root of multiplicity three and 3 is a root of multiplicity 2,
then roots looks something like [2, 2, 2, 3, 3]. The roots can appear
in any order.
If the returned coefficients are c, then
The coefficient of the last term is 1 for monic polynomials in this form.
out is also real, otherwise it is complex. (see
Examples below).numpy.polynomial.chebyshev.chebfromroots numpy.polynomial.legendre.legfromroots numpy.polynomial.laguerre.lagfromroots numpy.polynomial.hermite.hermfromroots numpy.polynomial.hermite_e.hermefromroots
The coefficients are determined by multiplying together linear factors of the form (x - r_i), i.e.
where n == len(roots) - 1; note that this implies that 1 is always returned for an.
>>> from numpy.polynomial import polynomial as P >>> P.polyfromroots((-1,0,1)) # x(x - 1)(x + 1) = x^3 - x array([ 0., -1., 0., 1.]) >>> j = complex(0,1) >>> P.polyfromroots((-j,j)) # complex returned, though values are real array([1.+0.j, 0.+0.j, 1.+0.j])
Evaluate a 2-D polynomial on the Cartesian product of x and y.
This function returns the values:
where the points (a, b) consist of all pairs formed by taking
a from x and b from y. The resulting points form a grid with
x in the first dimension and y in the second.
The parameters x and y are converted to arrays only if they are
tuples or a lists, otherwise they are treated as a scalars. In either
case, either x and y or their elements must support multiplication
and addition both with themselves and with the elements of c.
If c has fewer than two dimensions, ones are implicitly appended to
its shape to make it 2-D. The shape of the result will be c.shape[2:] +
x.shape + y.shape.
x and y. If x or y is a list or
tuple, it is first converted to an ndarray, otherwise it is left
unchanged and, if it isn't an ndarray, it is treated as a scalar.c has dimension
greater than two the remaining indices enumerate multiple sets of
coefficients.x and y.polyval, polyval2d, polyval3d, polygrid3d
Evaluate a 3-D polynomial on the Cartesian product of x, y and z.
This function returns the values:
where the points (a, b, c) consist of all triples formed by taking
a from x, b from y, and c from z. The resulting points form
a grid with x in the first dimension, y in the second, and z in
the third.
The parameters x, y, and z are converted to arrays only if they
are tuples or a lists, otherwise they are treated as a scalars. In
either case, either x, y, and z or their elements must support
multiplication and addition both with themselves and with the elements
of c.
If c has fewer than three dimensions, ones are implicitly appended to
its shape to make it 3-D. The shape of the result will be c.shape[3:] +
x.shape + y.shape + z.shape.
x, y, and z. If x,`y`, or z is a
list or tuple, it is first converted to an ndarray, otherwise it is
left unchanged and, if it isn't an ndarray, it is treated as a
scalar.c has dimension
greater than two the remaining indices enumerate multiple sets of
coefficients.x and y.polyval, polyval2d, polygrid2d, polyval3d
Integrate a polynomial.
Returns the polynomial coefficients c integrated m times from
lbnd along axis. At each iteration the resulting series is
multiplied by scl and an integration constant, k, is added.
The scaling factor is for use in a linear change of variable. ("Buyer
beware": note that, depending on what one is doing, one may want scl
to be the reciprocal of what one might expect; for more information,
see the Notes section below.) The argument c is an array of
coefficients, from low to high degree along each axis, e.g., [1,2,3]
represents the polynomial 1 + 2*x + 3*x**2 while [[1,2],[1,2]]
represents 1 + 1*x + 2*y + 2*x*y if axis=0 is x and axis=1 is
y.
scl
before the integration constant is added. (Default: 1)Axis over which the integral is taken. (Default: 0).
polyder
Note that the result of each integration is multiplied by scl. Why
is this important to note? Say one is making a linear change of
variable u = ax + b in an integral relative to x. Then
dx = du ⁄ a, so one will need to set scl equal to
1 ⁄ a - perhaps not what one would have first thought.
>>> from numpy.polynomial import polynomial as P >>> c = (1,2,3) >>> P.polyint(c) # should return array([0, 1, 1, 1]) array([0., 1., 1., 1.]) >>> P.polyint(c,3) # should return array([0, 0, 0, 1/6, 1/12, 1/20]) array([ 0. , 0. , 0. , 0.16666667, 0.08333333, # may vary 0.05 ]) >>> P.polyint(c,k=3) # should return array([3, 1, 1, 1]) array([3., 1., 1., 1.]) >>> P.polyint(c,lbnd=-2) # should return array([6, 1, 1, 1]) array([6., 1., 1., 1.]) >>> P.polyint(c,scl=-2) # should return array([0, -2, -2, -2]) array([ 0., -2., -2., -2.])
Returns an array representing a linear polynomial.
numpy.polynomial.chebyshev.chebline numpy.polynomial.legendre.legline numpy.polynomial.laguerre.lagline numpy.polynomial.hermite.hermline numpy.polynomial.hermite_e.hermeline
>>> from numpy.polynomial import polynomial as P >>> P.polyline(1,-1) array([ 1, -1]) >>> P.polyval(1, P.polyline(1,-1)) # should be 0 0.0
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.
polyadd, polysub, polymulx, polydiv, polypow
>>> from numpy.polynomial import polynomial as P >>> c1 = (1,2,3) >>> c2 = (3,2,1) >>> P.polymul(c1,c2) array([ 3., 8., 14., 8., 3.])
Multiply a polynomial by x.
Multiply the polynomial c by x, where x is the independent
variable.
polyadd, polysub, polymul, polydiv, polypow
Raise a polynomial to a power.
Returns the polynomial c raised to the power pow. The argument
c is a sequence of coefficients ordered from low to high. i.e.,
[1,2,3] is the series 1 + 2*x + 3*x**2.
polyadd, polysub, polymulx, polymul, polydiv
>>> from numpy.polynomial import polynomial as P >>> P.polypow([1,2,3], 2) array([ 1., 4., 10., 12., 9.])
Compute the roots of a polynomial.
Return the roots (a.k.a. "zeros") of the polynomial
out is also real, otherwise it is complex.numpy.polynomial.chebyshev.chebroots numpy.polynomial.legendre.legroots numpy.polynomial.laguerre.lagroots numpy.polynomial.hermite.hermroots numpy.polynomial.hermite_e.hermeroots
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.
>>> 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]) # may vary
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.
polyadd, polymulx, polymul, polydiv, polypow
>>> from numpy.polynomial 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.])
Evaluate a polynomial at points x.
If c is of length n + 1, this function returns the value
The parameter x is converted to an array only if it is a tuple or a
list, otherwise it is treated as a scalar. In either case, either x
or its elements must support multiplication and addition both with
themselves and with the elements of c.
If c is a 1-D array, then p(x) will have the same shape as x. If
c is multidimensional, then the shape of the result depends on the
value of tensor. If tensor is true the shape will be c.shape[1:] +
x.shape. If tensor is false the shape will be c.shape[1:]. Note that
scalars have shape (,).
Trailing zeros in the coefficients will be used in the evaluation, so they should be avoided if efficiency is a concern.
x is a list or tuple, it is converted to an ndarray, otherwise
it is left unchanged and treated as a scalar. In either case, x
or its elements must support addition and multiplication with
with themselves and with the elements of c.c is multidimensional the
remaining indices enumerate multiple polynomials. In the two
dimensional case the coefficients may be thought of as stored in
the columns of c.If True, the shape of the coefficient array is extended with ones
on the right, one for each dimension of x. Scalars have dimension 0
for this action. The result is that every column of coefficients in
c is evaluated for every element of x. If False, x is broadcast
over the columns of c for the evaluation. This keyword is useful
when c is multidimensional. The default value is True.
polyval2d, polygrid2d, polyval3d, polygrid3d
The evaluation uses Horner's method.
>>> from numpy.polynomial.polynomial import polyval >>> polyval(1, [1,2,3]) 6.0 >>> a = np.arange(4).reshape(2,2) >>> a array([[0, 1], [2, 3]]) >>> polyval(a, [1,2,3]) array([[ 1., 6.], [17., 34.]]) >>> coef = np.arange(4).reshape(2,2) # multidimensional coefficients >>> coef array([[0, 1], [2, 3]]) >>> polyval([1,2], coef, tensor=True) array([[2., 4.], [4., 7.]]) >>> polyval([1,2], coef, tensor=False) array([2., 7.])
Evaluate a 2-D polynomial at points (x, y).
This function returns the value
The parameters x and y are converted to arrays only if they are
tuples or a lists, otherwise they are treated as a scalars and they
must have the same shape after conversion. In either case, either x
and y or their elements must support multiplication and addition both
with themselves and with the elements of c.
If c has fewer than two dimensions, ones are implicitly appended to
its shape to make it 2-D. The shape of the result will be c.shape[2:] +
x.shape.
(x, y),
where x and y must have the same shape. If x or y is a list
or tuple, it is first converted to an ndarray, otherwise it is left
unchanged and, if it isn't an ndarray, it is treated as a scalar.c[i,j]. If c has
dimension greater than two the remaining indices enumerate multiple
sets of coefficients.x and y.polyval, polygrid2d, polyval3d, polygrid3d
Evaluate a 3-D polynomial at points (x, y, z).
This function returns the values:
The parameters x, y, and z are converted to arrays only if
they are tuples or a lists, otherwise they are treated as a scalars and
they must have the same shape after conversion. In either case, either
x, y, and z or their elements must support multiplication and
addition both with themselves and with the elements of c.
If c has fewer than 3 dimensions, ones are implicitly appended to its
shape to make it 3-D. The shape of the result will be c.shape[3:] +
x.shape.
(x, y, z), where x, y, and z must have the same shape. If
any of x, y, or z is a list or tuple, it is first converted
to an ndarray, otherwise it is left unchanged and if it isn't an
ndarray it is treated as a scalar.c has dimension
greater than 3 the remaining indices enumerate multiple sets of
coefficients.x, y, and z.polyval, polyval2d, polygrid2d, polygrid3d
Evaluate a polynomial specified by its roots at points x.
If r is of length N, this function returns the value
The parameter x is converted to an array only if it is a tuple or a
list, otherwise it is treated as a scalar. In either case, either x
or its elements must support multiplication and addition both with
themselves and with the elements of r.
If r is a 1-D array, then p(x) will have the same shape as x. If r
is multidimensional, then the shape of the result depends on the value of
tensor. If tensor is ``True` the shape will be r.shape[1:] + x.shape;
that is, each polynomial is evaluated at every value of x. If tensor is
False, the shape will be r.shape[1:]; that is, each polynomial is
evaluated only for the corresponding broadcast value of x. Note that
scalars have shape (,).
x is a list or tuple, it is converted to an ndarray, otherwise
it is left unchanged and treated as a scalar. In either case, x
or its elements must support addition and multiplication with
with themselves and with the elements of r.r is multidimensional the first index is the
root index, while the remaining indices enumerate multiple
polynomials. For instance, in the two dimensional case the roots
of each polynomial may be thought of as stored in the columns of r.x. Scalars have dimension 0 for this
action. The result is that every column of coefficients in r is
evaluated for every element of x. If False, x is broadcast over the
columns of r for the evaluation. This keyword is useful when r is
multidimensional. The default value is True.polyroots, polyfromroots, polyval
>>> from numpy.polynomial.polynomial import polyvalfromroots >>> polyvalfromroots(1, [1,2,3]) 0.0 >>> a = np.arange(4).reshape(2,2) >>> a array([[0, 1], [2, 3]]) >>> polyvalfromroots(a, [-1, 0, 1]) array([[-0., 0.], [ 6., 24.]]) >>> r = np.arange(-2, 2).reshape(2,2) # multidimensional coefficients >>> r # each column of r defines one polynomial array([[-2, -1], [ 0, 1]]) >>> b = [-2, 1] >>> polyvalfromroots(b, r, tensor=True) array([[-0., 3.], [ 3., 0.]]) >>> polyvalfromroots(b, r, tensor=False) array([-0., 0.])
Vandermonde matrix of given degree.
Returns the Vandermonde matrix of degree deg and sample points
x. The Vandermonde matrix is defined by
where 0 <= i <= deg. The leading indices of V index the elements of
x and the last index is the power of x.
If c is a 1-D array of coefficients of length n + 1 and V is the
matrix V = polyvander(x, n), then np.dot(V, c) and
polyval(x, c) are the same up to roundoff. This equivalence is
useful both for least squares fitting and for the evaluation of a large
number of polynomials of the same degree and sample points.
x is
scalar it is converted to a 1-D array.x.
The dtype will be the same as the converted x.polyvander2d, polyvander3d
Pseudo-Vandermonde matrix of given degrees.
Returns the pseudo-Vandermonde matrix of degrees deg and sample
points (x, y). The pseudo-Vandermonde matrix is defined by
where 0 <= i <= deg[0] and 0 <= j <= deg[1]. The leading indices of
V index the points (x, y) and the last index encodes the powers of
x and y.
If V = polyvander2d(x, y, [xdeg, ydeg]), then the columns of V
correspond to the elements of a 2-D coefficient array c of shape
(xdeg + 1, ydeg + 1) in the order
and np.dot(V, c.flat) and polyval2d(x, y, c) will be the same up to roundoff. This equivalence is useful both for least squares fitting and for the evaluation of a large number of 2-D polynomials of the same degrees and sample points.
x and y.polyvander, polyvander3d, polyval2d, polyval3d
Pseudo-Vandermonde matrix of given degrees.
Returns the pseudo-Vandermonde matrix of degrees deg and sample
points (x, y, z). If l, m, n are the given degrees in x, y, z,
then The pseudo-Vandermonde matrix is defined by
where 0 <= i <= l, 0 <= j <= m, and 0 <= j <= n. The leading
indices of V index the points (x, y, z) and the last index encodes
the powers of x, y, and z.
If V = polyvander3d(x, y, z, [xdeg, ydeg, zdeg]), then the columns
of V correspond to the elements of a 3-D coefficient array c of
shape (xdeg + 1, ydeg + 1, zdeg + 1) in the order
and np.dot(V, c.flat) and polyval3d(x, y, z, c) will be the same up to roundoff. This equivalence is useful both for least squares fitting and for the evaluation of a large number of 3-D polynomials of the same degrees and sample points.
x, y, and z.polyvander, polyvander3d, polyval2d, polyval3d