package np

Get an attribute of this module as a Py.Object.t. This is useful to pass a Python function to another function.

Convert a Chebyshev series to a polynomial.

Convert an array representing the coefficients of a Chebyshev series, ordered from lowest degree to highest, to an array of the coefficients of the equivalent polynomial (relative to the 'standard' basis) ordered from lowest to highest degree.

Parameters ---------- c : array_like 1-D array containing the Chebyshev series coefficients, ordered from lowest order term to highest.

Returns ------- pol : ndarray 1-D array containing the coefficients of the equivalent polynomial (relative to the 'standard' basis) ordered from lowest order term to highest.

See Also -------- poly2cheb

Notes ----- The easy way to do conversions between polynomial basis sets is to use the convert method of a class instance.

Examples -------- >>> from numpy import polynomial as P >>> c = P.Chebyshev(range(4)) >>> c Chebyshev(0., 1., 2., 3., domain=-1, 1, window=-1, 1) >>> p = c.convert(kind=P.Polynomial) >>> p Polynomial(-2., -8., 4., 12., domain=-1., 1., window=-1., 1.) >>> P.chebyshev.cheb2poly(range(4)) array(-2., -8., 4., 12.)

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, chebmulx, 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.)

Return the scaled companion matrix of c.

The basis polynomials are scaled so that the companion matrix is symmetric when `c` is a Chebyshev basis polynomial. This provides better eigenvalue estimates than the unscaled case and for basis polynomials the eigenvalues are guaranteed to be real if `numpy.linalg.eigvalsh` is used to obtain them.

Parameters ---------- c : array_like 1-D array of Chebyshev series coefficients ordered from low to high degree.

Returns ------- mat : ndarray Scaled companion matrix of dimensions (deg, deg).

Notes -----

.. versionadded:: 1.7.0

Differentiate a Chebyshev series.

Returns the Chebyshev series 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 series ``1*T_0 + 2*T_1 + 3*T_2`` while [1,2],[1,2] represents ``1*T_0(x)*T_0(y) + 1*T_1(x)*T_0(y) + 2*T_0(x)*T_1(y) + 2*T_1(x)*T_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``.

Parameters ---------- c : array_like Array of Chebyshev series coefficients. If c is multidimensional the different axis correspond to different variables with the degree in each axis given by the corresponding index. m : int, optional Number of derivatives taken, must be non-negative. (Default: 1) scl : scalar, optional Each differentiation is multiplied by `scl`. The end result is multiplication by ``scl**m``. This is for use in a linear change of variable. (Default: 1) axis : int, optional Axis over which the derivative is taken. (Default: 0).

.. versionadded:: 1.7.0

Returns ------- der : ndarray Chebyshev series of the derivative.

See Also -------- chebint

Notes ----- In general, the result of differentiating a C-series needs to be 'reprojected' onto the C-series basis set. Thus, typically, the result of this function is 'unintuitive,' albeit correct; see Examples section below.

Examples -------- >>> from numpy.polynomial import chebyshev as C >>> c = (1,2,3,4) >>> C.chebder(c) array(14., 12., 24.) >>> C.chebder(c,3) array(96.) >>> C.chebder(c,scl=-1) array(-14., -12., -24.) >>> C.chebder(c,2,-1) array(12., 96.)

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, chemulx, 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 'reproject' the results onto said basis set, which typically produces 'unintuitive' (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.))

Least squares fit of Chebyshev series to data.

Return the coefficients of a Chebyshev series 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

.. math:: p(x) = c_0 + c_1 * T_1(x) + ... + c_n * T_n(x),

where `n` is `deg`.

Parameters ---------- x : array_like, shape (M,) x-coordinates of the M sample points ``(xi, yi)``. y : array_like, shape (M,) or (M, K) y-coordinates of the sample points. Several data sets of sample points sharing the same x-coordinates can be fitted at once by passing in a 2D-array that contains one dataset per column. deg : int or 1-D array_like Degree(s) of the fitting polynomials. If `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 : float, optional Relative condition number of the fit. Singular values smaller than this relative to the largest singular value will be ignored. The default value is len(x)*eps, where eps is the relative precision of the float type, about 2e-16 in most cases. full : bool, optional Switch determining nature of return value. When it is False (the default) just the coefficients are returned, when True diagnostic information from the singular value decomposition is also returned. w : array_like, shape (`M`,), optional Weights. If not None, the contribution of each point ``(xi,yi)`` to the fit is weighted by `wi`. Ideally the weights are chosen so that the errors of the products ``wi*yi`` all have the same variance. The default value is None.

.. versionadded:: 1.5.0

Returns ------- coef : ndarray, shape (M,) or (M, K) Chebyshev coefficients ordered from low to high. If `y` was 2-D, the coefficients for the data in column k of `y` are in column `k`.

residuals, rank, singular_values, rcond : list These values are only returned if `full` = True

resid -- sum of squared residuals of the least squares fit rank -- the numerical rank of the scaled Vandermonde matrix sv -- singular values of the scaled Vandermonde matrix rcond -- value of `rcond`.

For more details, see `linalg.lstsq`.

Warns ----- RankWarning The rank of the coefficient matrix in the least-squares fit is deficient. The warning is only raised if `full` = False. The warnings can be turned off by

>>> import warnings >>> warnings.simplefilter('ignore', np.RankWarning)

See Also -------- polyfit, legfit, lagfit, hermfit, hermefit chebval : Evaluates a Chebyshev series. chebvander : Vandermonde matrix of Chebyshev series. chebweight : Chebyshev weight function. linalg.lstsq : Computes a least-squares fit from the matrix. scipy.interpolate.UnivariateSpline : Computes spline fits.

Notes ----- The solution is the coefficients of the Chebyshev series `p` that minimizes the sum of the weighted squared errors

.. math:: E = \sum_j w_j^2 * |y_j - p(x_j)|^2,

where :math:`w_j` are the weights. This problem is solved by setting up as the (typically) overdetermined matrix equation

.. math:: V(x) * c = w * y,

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, then a `RankWarning` will be issued. This means that the coefficient values may be poorly determined. Using a lower order fit will usually get rid of the warning. 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.

Fits using Chebyshev series are usually better conditioned than fits using power series, but much can 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.

References ---------- .. 1 Wikipedia, 'Curve fitting', https://en.wikipedia.org/wiki/Curve_fitting

Examples --------

Generate a Chebyshev series with given roots.

The function returns the coefficients of the polynomial

.. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n),

in Chebyshev form, 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

.. math:: p(x) = c_0 + c_1 * T_1(x) + ... + c_n * T_n(x)

The coefficient of the last term is not generally 1 for monic polynomials in Chebyshev form.

Parameters ---------- roots : array_like Sequence containing the roots.

Returns ------- out : ndarray 1-D array of coefficients. If all roots are real then `out` is a real array, if some of the roots are complex, then `out` is complex even if all the coefficients in the result are real (see Examples below).

See Also -------- polyfromroots, legfromroots, lagfromroots, hermfromroots, hermefromroots

Examples -------- >>> import numpy.polynomial.chebyshev as C >>> C.chebfromroots((-1,0,1)) # x^3 - x relative to the standard basis array( 0. , -0.25, 0. , 0.25) >>> j = complex(0,1) >>> C.chebfromroots((-j,j)) # x^2 + 1 relative to the standard basis array(1.5+0.j, 0. +0.j, 0.5+0.j)

Gauss-Chebyshev quadrature.

Computes the sample points and weights for Gauss-Chebyshev quadrature. These sample points and weights will correctly integrate polynomials of degree :math:`2*deg - 1` or less over the interval :math:`-1, 1` with the weight function :math:`f(x) = 1/\sqrt

- x^2

`.

Parameters ---------- deg : int Number of sample points and weights. It must be >= 1.

Returns ------- x : ndarray 1-D ndarray containing the sample points. y : ndarray 1-D ndarray containing the weights.

Notes -----

.. versionadded:: 1.7.0

The results have only been tested up to degree 100, higher degrees may be problematic. For Gauss-Chebyshev there are closed form solutions for the sample points and weights. If n = `deg`, then

.. math:: x_i = \cos(\pi (2 i - 1) / (2 n))

.. math:: w_i = \pi / n

Evaluate a 2-D Chebyshev series on the Cartesian product of x and y.

This function returns the values:

.. math:: p(a,b) = \sum_,j c_,j * T_i(a) * T_j(b),

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.shape2: + x.shape + y.shape.

Parameters ---------- x, y : array_like, compatible objects The two dimensional series is evaluated at the points in the Cartesian product of `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 : array_like Array of coefficients ordered so that the coefficient of the term of multi-degree i,j is contained in `ci,j`. If `c` has dimension greater than two the remaining indices enumerate multiple sets of coefficients.

Returns ------- values : ndarray, compatible object The values of the two dimensional Chebyshev series at points in the Cartesian product of `x` and `y`.

See Also -------- chebval, chebval2d, chebval3d, chebgrid3d

Notes -----

.. versionadded:: 1.7.0

Evaluate a 3-D Chebyshev series on the Cartesian product of x, y, and z.

This function returns the values:

.. math:: p(a,b,c) = \sum_,j,k c_,j,k * T_i(a) * T_j(b) * T_k(c)

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.shape3: + x.shape + y.shape + z.shape.

Parameters ---------- x, y, z : array_like, compatible objects The three dimensional series is evaluated at the points in the Cartesian product of `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 : array_like Array of coefficients ordered so that the coefficients for terms of degree i,j are contained in ``ci,j``. If `c` has dimension greater than two the remaining indices enumerate multiple sets of coefficients.

Returns ------- values : ndarray, compatible object The values of the two dimensional polynomial at points in the Cartesian product of `x` and `y`.

See Also -------- chebval, chebval2d, chebgrid2d, chebval3d

Notes -----

.. versionadded:: 1.7.0

Integrate a Chebyshev series.

Returns the Chebyshev series 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 series ``T_0 + 2*T_1 + 3*T_2`` while [1,2],[1,2] represents ``1*T_0(x)*T_0(y) + 1*T_1(x)*T_0(y) + 2*T_0(x)*T_1(y) + 2*T_1(x)*T_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``.

Parameters ---------- c : array_like Array of Chebyshev series coefficients. If c is multidimensional the different axis correspond to different variables with the degree in each axis given by the corresponding index. m : int, optional Order of integration, must be positive. (Default: 1) k :

], list, scalar}, optional
    Integration constant(s).  The value of the first integral at zero
    is the first value in the list, the value of the second integral
    at zero is the second value, etc.  If ``k == []`` (the default),
    all constants are set to zero.  If ``m == 1``, a single scalar can
    be given instead of a list.
lbnd : scalar, optional
    The lower bound of the integral. (Default: 0)
scl : scalar, optional
    Following each integration the result is *multiplied* by `scl`
    before the integration constant is added. (Default: 1)
axis : int, optional
    Axis over which the integral is taken. (Default: 0).

    .. versionadded:: 1.7.0

Returns
-------
S : ndarray
    C-series coefficients of the integral.

Raises
------
ValueError
    If ``m < 1``, ``len(k) > m``, ``np.ndim(lbnd) != 0``, or
    ``np.ndim(scl) != 0``.

See Also
--------
chebder

Notes
-----
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 :math:`u = ax + b` in an integral relative to `x`.  Then
:math:`dx = du/a`, so one will need to set `scl` equal to
:math:`1/a`- perhaps not what one would have first thought.

Also note that, in general, the result of integrating a C-series needs
to be 'reprojected' onto the C-series basis set.  Thus, typically,
the result of this function is 'unintuitive,' albeit correct; see
Examples section below.

Examples
--------
>>> from numpy.polynomial import chebyshev as C
>>> c = (1,2,3)
>>> C.chebint(c)
array([ 0.5, -0.5,  0.5,  0.5])
>>> C.chebint(c,3)
array([ 0.03125   , -0.1875    ,  0.04166667, -0.05208333,  0.01041667, # may vary
    0.00625   ])
>>> C.chebint(c, k=3)
array([ 3.5, -0.5,  0.5,  0.5])
>>> C.chebint(c,lbnd=-2)
array([ 8.5, -0.5,  0.5,  0.5])
>>> C.chebint(c,scl=-2)
array([-1.,  1., -1., -1.])

Interpolate a function at the Chebyshev points of the first kind.

Returns the Chebyshev series that interpolates `func` at the Chebyshev points of the first kind in the interval -1, 1. The interpolating series tends to a minmax approximation to `func` with increasing `deg` if the function is continuous in the interval.

.. versionadded:: 1.14.0

Parameters ---------- func : function The function to be approximated. It must be a function of a single variable of the form ``f(x, a, b, c...)``, where ``a, b, c...`` are extra arguments passed in the `args` parameter. deg : int Degree of the interpolating polynomial args : tuple, optional Extra arguments to be used in the function call. Default is no extra arguments.

Returns ------- coef : ndarray, shape (deg + 1,) Chebyshev coefficients of the interpolating series ordered from low to high.

Examples -------- >>> import numpy.polynomial.chebyshev as C >>> C.chebfromfunction(lambda x: np.tanh(x) + 0.5, 8) array( 5.00000000e-01, 8.11675684e-01, -9.86864911e-17, -5.42457905e-02, -2.71387850e-16, 4.51658839e-03, 2.46716228e-17, -3.79694221e-04, -3.26899002e-16)

Notes -----

The Chebyshev polynomials used in the interpolation are orthogonal when sampled at the Chebyshev points of the first kind. If it is desired to constrain some of the coefficients they can simply be set to the desired value after the interpolation, no new interpolation or fit is needed. This is especially useful if it is known apriori that some of coefficients are zero. For instance, if the function is even then the coefficients of the terms of odd degree in the result can be set to zero.

Chebyshev series whose graph is a straight line.

Parameters ---------- off, scl : scalars The specified line is given by ``off + scl*x``.

Returns ------- y : ndarray This module's representation of the Chebyshev series for ``off + scl*x``.

See Also -------- polyline

Examples -------- >>> import numpy.polynomial.chebyshev as C >>> C.chebline(3,2) array(3, 2) >>> C.chebval(-3, C.chebline(3,2)) # should be -3 -3.0

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, chebmulx, 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 'reproject' the product onto said basis set, which typically produces 'unintuitive live' (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)

Multiply a Chebyshev series by x.

Multiply the polynomial `c` by x, where x is the independent variable.

Parameters ---------- c : array_like 1-D array of Chebyshev series coefficients ordered from low to high.

Returns ------- out : ndarray Array representing the result of the multiplication.

Notes -----

.. versionadded:: 1.5.0

Examples -------- >>> from numpy.polynomial import chebyshev as C >>> C.chebmulx(1,2,3) array(1. , 2.5, 1. , 1.5)

Raise a Chebyshev series to a power.

Returns the Chebyshev series `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 ``T_0 + 2*T_1 + 3*T_2.``

Parameters ---------- c : array_like 1-D array of Chebyshev series coefficients ordered from low to high. pow : integer Power to which the series will be raised maxpower : integer, optional Maximum power allowed. This is mainly to limit growth of the series to unmanageable size. Default is 16

Returns ------- coef : ndarray Chebyshev series of power.

See Also -------- chebadd, chebsub, chebmulx, chebmul, chebdiv

Examples -------- >>> from numpy.polynomial import chebyshev as C >>> C.chebpow(1, 2, 3, 4, 2) array(15.5, 22. , 16. , ..., 12.5, 12. , 8. )

Chebyshev points of the first kind.

The Chebyshev points of the first kind are the points ``cos(x)``, where ``x = pi*(k + .5)/npts for k in range(npts)``.

Parameters ---------- npts : int Number of sample points desired.

Returns ------- pts : ndarray The Chebyshev points of the first kind.

See Also -------- chebpts2

Notes -----

.. versionadded:: 1.5.0

Chebyshev points of the second kind.

The Chebyshev points of the second kind are the points ``cos(x)``, where ``x = pi*k/(npts - 1) for k in range(npts)``.

Parameters ---------- npts : int Number of sample points desired.

Returns ------- pts : ndarray The Chebyshev points of the second kind.

Notes -----

.. versionadded:: 1.5.0

Compute the roots of a Chebyshev series.

Return the roots (a.k.a. 'zeros') of the polynomial

.. math:: p(x) = \sum_i ci * 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) # may vary

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, chebmulx, 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.)

Remove 'small' 'trailing' coefficients from a polynomial.

'Small' means 'small in absolute value' and is controlled by the parameter `tol`; 'trailing' means highest order coefficient(s), e.g., in ``0, 1, 1, 0, 0`` (which represents ``0 + x + x**2 + 0*x**3 + 0*x**4``) both the 3-rd and 4-th order coefficients would be 'trimmed.'

Parameters ---------- c : array_like 1-d array of coefficients, ordered from lowest order to highest. tol : number, optional Trailing (i.e., highest order) elements with absolute value less than or equal to `tol` (default value is zero) are removed.

Returns ------- trimmed : ndarray 1-d array with trailing zeros removed. If the resulting series would be empty, a series containing a single zero is returned.

Raises ------ ValueError If `tol` < 0

See Also -------- trimseq

Examples -------- >>> from numpy.polynomial import polyutils as pu >>> pu.trimcoef((0,0,3,0,5,0,0)) array(0., 0., 3., 0., 5.) >>> pu.trimcoef((0,0,1e-3,0,1e-5,0,0),1e-3) # item == tol is trimmed array(0.) >>> i = complex(0,1) # works for complex >>> pu.trimcoef((3e-4,1e-3*(1-i),5e-4,2e-5*(1+i)), 1e-3) array(0.0003+0.j , 0.001 -0.001j)

Evaluate a Chebyshev series at points x.

If `c` is of length `n + 1`, this function returns the value:

.. math:: p(x) = c_0 * T_0(x) + c_1 * T_1(x) + ... + c_n * T_n(x)

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.shape1: + x.shape. If `tensor` is false the shape will be c.shape1:. 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.

Parameters ---------- x : array_like, compatible object If `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 : array_like Array of coefficients ordered so that the coefficients for terms of degree n are contained in cn. If `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`. tensor : boolean, optional 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.

.. versionadded:: 1.7.0

Returns ------- values : ndarray, algebra_like The shape of the return value is described above.

See Also -------- chebval2d, chebgrid2d, chebval3d, chebgrid3d

Notes ----- The evaluation uses Clenshaw recursion, aka synthetic division.

Examples --------

Evaluate a 2-D Chebyshev series at points (x, y).

This function returns the values:

.. math:: p(x,y) = \sum_,j c_,j * T_i(x) * T_j(y)

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` is a 1-D array a one is implicitly appended to its shape to make it 2-D. The shape of the result will be c.shape2: + x.shape.

Parameters ---------- x, y : array_like, compatible objects The two dimensional series is evaluated at the points `(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 : array_like Array of coefficients ordered so that the coefficient of the term of multi-degree i,j is contained in ``ci,j``. If `c` has dimension greater than 2 the remaining indices enumerate multiple sets of coefficients.

Returns ------- values : ndarray, compatible object The values of the two dimensional Chebyshev series at points formed from pairs of corresponding values from `x` and `y`.

See Also -------- chebval, chebgrid2d, chebval3d, chebgrid3d

Notes -----

.. versionadded:: 1.7.0

Evaluate a 3-D Chebyshev series at points (x, y, z).

This function returns the values:

.. math:: p(x,y,z) = \sum_,j,k c_,j,k * T_i(x) * T_j(y) * T_k(z)

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.shape3: + x.shape.

Parameters ---------- x, y, z : array_like, compatible object The three dimensional series is evaluated at the points `(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 : array_like Array of coefficients ordered so that the coefficient of the term of multi-degree i,j,k is contained in ``ci,j,k``. If `c` has dimension greater than 3 the remaining indices enumerate multiple sets of coefficients.

Returns ------- values : ndarray, compatible object The values of the multidimensional polynomial on points formed with triples of corresponding values from `x`, `y`, and `z`.

See Also -------- chebval, chebval2d, chebgrid2d, chebgrid3d

Notes -----

.. versionadded:: 1.7.0

Pseudo-Vandermonde matrix of given degree.

Returns the pseudo-Vandermonde matrix of degree `deg` and sample points `x`. The pseudo-Vandermonde matrix is defined by

.. math:: V..., i = T_i(x),

where `0 <= i <= deg`. The leading indices of `V` index the elements of `x` and the last index is the degree of the Chebyshev polynomial.

If `c` is a 1-D array of coefficients of length `n + 1` and `V` is the matrix ``V = chebvander(x, n)``, then ``np.dot(V, c)`` and ``chebval(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 Chebyshev series of the same degree and sample points.

Parameters ---------- x : array_like Array of points. The dtype is converted to float64 or complex128 depending on whether any of the elements are complex. If `x` is scalar it is converted to a 1-D array. deg : int Degree of the resulting matrix.

Returns ------- vander : ndarray The pseudo Vandermonde matrix. The shape of the returned matrix is ``x.shape + (deg + 1,)``, where The last index is the degree of the corresponding Chebyshev polynomial. The dtype will be the same as the converted `x`.

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

.. math:: V..., (deg[1] + 1)*i + j = T_i(x) * T_j(y),

where `0 <= i <= deg0` and `0 <= j <= deg1`. The leading indices of `V` index the points `(x, y)` and the last index encodes the degrees of the Chebyshev polynomials.

If ``V = chebvander2d(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

.. math:: c_

, c_

, c_

... , c_

, c_

, c_

...

and ``np.dot(V, c.flat)`` and ``chebval2d(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 Chebyshev series of the same degrees and sample points.

Parameters ---------- x, y : array_like Arrays of point coordinates, all of the same shape. The dtypes will be converted to either float64 or complex128 depending on whether any of the elements are complex. Scalars are converted to 1-D arrays. deg : list of ints List of maximum degrees of the form x_deg, y_deg.

Returns ------- vander2d : ndarray The shape of the returned matrix is ``x.shape + (order,)``, where :math:`order = (deg0+1)*(deg(1+1)`. The dtype will be the same as the converted `x` and `y`.

See Also -------- chebvander, chebvander3d, chebval2d, chebval3d

Notes -----

.. versionadded:: 1.7.0

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

.. math:: V..., (m+1)(n+1)i + (n+1)j + k = T_i(x)*T_j(y)*T_k(z),

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 degrees of the Chebyshev polynomials.

If ``V = chebvander3d(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

.. math:: c_

, c_

, c_

,... , c_

, c_

, c_

,...

and ``np.dot(V, c.flat)`` and ``chebval3d(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 Chebyshev series of the same degrees and sample points.

Parameters ---------- x, y, z : array_like Arrays of point coordinates, all of the same shape. The dtypes will be converted to either float64 or complex128 depending on whether any of the elements are complex. Scalars are converted to 1-D arrays. deg : list of ints List of maximum degrees of the form x_deg, y_deg, z_deg.

Returns ------- vander3d : ndarray The shape of the returned matrix is ``x.shape + (order,)``, where :math:`order = (deg0+1)*(deg(1+1)*(deg2+1)`. The dtype will be the same as the converted `x`, `y`, and `z`.

See Also -------- chebvander, chebvander3d, chebval2d, chebval3d

Notes -----

.. versionadded:: 1.7.0

The weight function of the Chebyshev polynomials.

The weight function is :math:`1/\sqrt

- x^2

` and the interval of integration is :math:`-1, 1`. The Chebyshev polynomials are orthogonal, but not normalized, with respect to this weight function.

Parameters ---------- x : array_like Values at which the weight function will be computed.

Returns ------- w : ndarray The weight function at `x`.

Notes -----

.. versionadded:: 1.7.0

normalize_axis_index(axis, ndim, msg_prefix=None)

Normalizes an axis index, `axis`, such that is a valid positive index into the shape of array with `ndim` dimensions. Raises an AxisError with an appropriate message if this is not possible.

Used internally by all axis-checking logic.

.. versionadded:: 1.13.0

Parameters ---------- axis : int The un-normalized index of the axis. Can be negative ndim : int The number of dimensions of the array that `axis` should be normalized against msg_prefix : str A prefix to put before the message, typically the name of the argument

Returns ------- normalized_axis : int The normalized axis index, such that `0 <= normalized_axis < ndim`

Raises ------ AxisError If the axis index is invalid, when `-ndim <= axis < ndim` is false.

Examples -------- >>> normalize_axis_index(0, ndim=3) 0 >>> normalize_axis_index(1, ndim=3) 1 >>> normalize_axis_index(-1, ndim=3) 2

>>> normalize_axis_index(3, ndim=3) Traceback (most recent call last): ... AxisError: axis 3 is out of bounds for array of dimension 3 >>> normalize_axis_index(-4, ndim=3, msg_prefix='axes_arg') Traceback (most recent call last): ... AxisError: axes_arg: axis -4 is out of bounds for array of dimension 3

Convert a polynomial to a Chebyshev series.

Convert an array representing the coefficients of a polynomial (relative to the 'standard' basis) ordered from lowest degree to highest, to an array of the coefficients of the equivalent Chebyshev series, ordered from lowest to highest degree.

Parameters ---------- pol : array_like 1-D array containing the polynomial coefficients

Returns ------- c : ndarray 1-D array containing the coefficients of the equivalent Chebyshev series.

See Also -------- cheb2poly

Notes ----- The easy way to do conversions between polynomial basis sets is to use the convert method of a class instance.

Examples -------- >>> from numpy import polynomial as P >>> p = P.Polynomial(range(4)) >>> p Polynomial(0., 1., 2., 3., domain=-1, 1, window=-1, 1) >>> c = p.convert(kind=P.Chebyshev) >>> c Chebyshev(1. , 3.25, 1. , 0.75, domain=-1., 1., window=-1., 1.) >>> P.chebyshev.poly2cheb(range(4)) array(1. , 3.25, 1. , 0.75)