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Ñò
]ÐKc @ sî d Z d d k l Z d d d d d d d d
d d d
d d d d d d d d d g Z d d k Z d d k i Z d d k Z
d d k Z d d k l Z e
i
Z d „ Z d „ Z d „ Z d „ Z d „ Z d „ Z d „ Z d „ Z e i d d! g ƒ Z e i d" g ƒ Z e i d! g ƒ Z e i d" d! g ƒ Z d# „ Z d$ „ Z d% „ Z d& „ Z d' „ Z d( „ Z! d) d* „ Z" d! d! d+ „ Z# d! g d" d! d, „ Z$ d- „ Z% d. „ Z& d e( d/ „ Z) d0 „ Z* e i+ d1 d d2 d3 d4 d5 ƒ d Ud S(6 sÑ Functions for dealing with Chebyshev series.
This module provide s a number of functions that are useful in dealing with
Chebyshev series as well as a ``Chebyshev`` class that encapsuletes the usual
arithmetic operations. All the Chebyshev series are assumed to be ordered
from low to high, thus ``array([1,2,3])`` will be treated as the series
``T_0 + 2*T_1 + 3*T_2``
Constants
---------
- chebdomain -- Chebyshev series default domain
- chebzero -- Chebyshev series that evaluates to 0.
- chebone -- Chebyshev series that evaluates to 1.
- chebx -- Chebyshev series of the identity map (x).
Arithmetic
----------
- chebadd -- add a Chebyshev series to another.
- chebsub -- subtract a Chebyshev series from another.
- chebmul -- multiply a Chebyshev series by another
- chebdiv -- divide one Chebyshev series by another.
- chebval -- evaluate a Chebyshev series at given points.
Calculus
--------
- chebder -- differentiate a Chebyshev series.
- chebint -- integrate a Chebyshev series.
Misc Functions
--------------
- chebfromroots -- create a Chebyshev series with specified roots.
- chebroots -- find the roots of a Chebyshev series.
- chebvander -- Vandermode like matrix for Chebyshev polynomials.
- chebfit -- least squares fit returning a Chebyshev series.
- chebtrim -- trim leading coefficients from a Chebyshev series.
- chebline -- Chebyshev series of given straight line
- cheb2poly -- convert a Chebyshev series to a polynomial.
- poly2cheb -- convert a polynomial to a Chebyshev series.
Classes
-------
- Chebyshev -- Chebyshev series class.
Notes
-----
The implementations of multiplication, division, integration, and
differentiation use the algebraic identities:
.. math ::
T_n(x) = \frac{z^n + z^{-n}}{2} \\
z\frac{dx}{dz} = \frac{z - z^{-1}}{2}.
where
.. math :: x = \frac{z + z^{-1}}{2}.
These identities allow a Chebyshev series to be expressed as a finite,
symmetric Laurent series. These sorts of Laurent series are referred to as
z-series in this module.
iÿÿÿÿ( t divisiont chebzerot chebonet chebxt
chebdomaint cheblinet chebaddt chebsubt chebmult chebdivt chebvalt chebdert chebintt cheb2polyt poly2chebt
chebfromrootst
chebvandert chebfitt chebtrimt chebrootst ChebyshevN( t polytemplatec C sP | i } t i d | d d | i ƒ} | d | | d )| | d d d … S( sô Covert Chebyshev series to z-series.
Covert a Chebyshev series to the equivalent z-series. The result is
never an empty array. The dtype of the return is the same as that of
the input. No checks are run on the arguments as this routine is for
internal use.
Parameters
----------
cs : 1-d ndarray
Chebyshev coefficients, ordered from low to high
Returns
-------
zs : 1-d ndarray
Odd length symmetric z-series, ordered from low to high.
i i t dtypeNiÿÿÿÿ( t sizet npt zerosR ( t cst nt zs( ( s@ /usr/lib64/python2.6/site-packages/numpy/polynomial/chebyshev.pyt _cseries_to_zseriesR s c C s< | i d d } | | d i ƒ } | d | c !d 9+| S( sø Covert z-series to a Chebyshev series.
Covert a z series to the equivalent Chebyshev series. The result is
never an empty array. The dtype of the return is the same as that of
the input. No checks are run on the arguments as this routine is for
internal use.
Parameters
----------
zs : 1-d ndarray
Odd length symmetric z-series, ordered from low to high.
Returns
-------
cs : 1-d ndarray
Chebyshev coefficients, ordered from low to high.
i i ( R t copy( R R R ( ( s@ /usr/lib64/python2.6/site-packages/numpy/polynomial/chebyshev.pyt _zseries_to_cseriesj s c C s t i | | ƒ S( sÄ Multiply two z-series.
Multiply two z-series to produce a z-series.
Parameters
----------
z1, z2 : 1-d ndarray
The arrays must be 1-d but this is not checked.
Returns
-------
product : 1-d ndarray
The product z-series.
Notes
-----
This is simply convolution. If symmetic/anti-symmetric z-series are
denoted by S/A then the following rules apply:
S*S, A*A -> S
S*A, A*S -> A
( R t convolve( t z1t z2( ( s@ /usr/lib64/python2.6/site-packages/numpy/polynomial/chebyshev.pyt _zseries_mul‚ s c C s³ | i ƒ } | i ƒ } t | ƒ } t | ƒ } | d j o | | } | | d d f S| | j o | d d | f S| | } | d } | | } t i | d d | i ƒ} d } | } x„ | | j ov | | } | | | | <| | | | <| | }
| | | | c !|
8+| | | | c !|
8+| d 7} | d 8} qà W| | } | | | <| | }
| | | | c !|
8+| | } | | d | d | !i ƒ } | | f Sd S( s¹ Divide the first z-series by the second.
Divide `z1` by `z2` and return the quotient and remainder as z-series.
Warning: this implementation only applies when both z1 and z2 have the
same symmetry, which is sufficient for present purposes.
Parameters
----------
z1, z2 : 1-d ndarray
The arrays must be 1-d and have the same symmetry, but this is not
checked.
Returns
-------
(quotient, remainder) : 1-d ndarrays
Quotient and remainder as z-series.
Notes
-----
This is not the same as polynomial division on account of the desired form
of the remainder. If symmetic/anti-symmetric z-series are denoted by S/A
then the following rules apply:
S/S -> S,S
A/A -> S,A
The restriction to types of the same symmetry could be fixed but seems like
uneeded generality. There is no natural form for the remainder in the case
where there is no symmetry.
i i R N( R t lenR t emptyR ( R! R" t len1t len2t dlent sclt quot it jt rt tmpt rem( ( s@ /usr/lib64/python2.6/site-packages/numpy/polynomial/chebyshev.pyt _zseries_divœ s@ !
c C si t | ƒ d } t i d d d g d | i ƒ} | t i | | d ƒ d 9} t | | ƒ \ } } | S( sŒ Differentiate a z-series.
The derivative is with respect to x, not z. This is achieved using the
chain rule and the value of dx/dz given in the module notes.
Parameters
----------
zs : z-series
The z-series to differentiate.
Returns
-------
derivative : z-series
The derivative
Notes
-----
The zseries for x (ns) has been multiplied by two in order to avoid
using floats that are incompatible with Decimal and likely other
specialized scalar types. This scaling has been compensated by
multiplying the value of zs by two also so that the two cancels in the
division.
i iÿÿÿÿi i R ( R$ R t arrayR t arangeR0 ( R R t nst dR- ( ( s@ /usr/lib64/python2.6/site-packages/numpy/polynomial/chebyshev.pyt _zseries_derÞ s
! c C s d t | ƒ d } t i d d d g d | i ƒ} t | | ƒ } t i | | d ƒ d } | | c | | *| | d c | | d )d | | <| S( sM Integrate a z-series.
The integral is with respect to x, not z. This is achieved by a change
of variable using dx/dz given in the module notes.
Parameters
----------
zs : z-series
The z-series to integrate
Returns
-------
integral : z-series
The indefinite integral
Notes
-----
The zseries for x (ns) has been multiplied by two in order to avoid
using floats that are incompatible with Decimal and likely other
specialized scalar types. This scaling has been compensated by
dividing the resulting zs by two.
i i iÿÿÿÿi R ( R$ R R1 R R# R2 ( R R R3 t div( ( s@ /usr/lib64/python2.6/site-packages/numpy/polynomial/chebyshev.pyt _zseries_intý s !
c C s¦ t i | g ƒ \ } | d d d … } | d i ƒ } t i d d d g d | i ƒ} x@ t d t | ƒ ƒ D]) } t | | ƒ } | | c | | 7>> import warnings
>>> warnings.simplefilter('ignore', RankWarning)
See Also
--------
chebval : Evaluates a Chebyshev series.
chebvander : Vandermonde matrix of Chebyshev series.
polyfit : least squares fit using polynomials.
linalg.lstsq : Computes a least-squares fit from the matrix.
scipy.interpolate.UnivariateSpline : Computes spline fits.
Notes
-----
The solution are the coefficients ``c[i]`` of the Chebyshev series
``T(x)`` that minimizes the squared error
``E = \sum_j |y_j - T(x_j)|^2``.
This problem is solved by setting up as the overdetermined matrix
equation
``V(x)*c = y``,
where ``V`` is the Vandermonde matrix of `x`, the elements of ``c`` are
the coefficients to be solved for, and the elements of `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
coeficient 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",
http://en.wikipedia.org/wiki/Curve_fitting
Examples
--------
i g i s expected deg >= 0s expected 1D vector for xs expected non-empty vector for xi s expected 1D or 2D array for ys$ expected x and y to have same lengths! The fit may be poorly conditionedN( RK R RY RL t ndimt TypeErrorR R\ RM R$ t finfoR t epsR t sqrtt sumt lat lstsqt Tt warningst warnR8 t RankWarning(
R< t yR] t rcondt fullR^ t AR) t ct residst rankt st msg( ( s@ /usr/lib64/python2.6/site-packages/numpy/polynomial/chebyshev.pyR I s2 _
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