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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! d% „ Z" e i# d& d d' d( d) d* ƒ d Ud S(+ s< Functions for dealing with polynomials.
This module provides a number of functions that are useful in dealing with
polynomials as well as a ``Polynomial`` class that encapsuletes the usual
arithmetic operations. All arrays of polynomial coefficients are assumed to
be ordered from low to high degree, thus `array([1,2,3])` will be treated
as the polynomial ``1 + 2*x + 3*x**2``
Constants
---------
- polydomain -- Polynomial default domain
- polyzero -- Polynomial that evaluates to 0.
- polyone -- Polynomial that evaluates to 1.
- polyx -- Polynomial of the identity map (x).
Arithmetic
----------
- polyadd -- add a polynomial to another.
- polysub -- subtract a polynomial from another.
- polymul -- multiply a polynomial by another
- polydiv -- divide one polynomial by another.
- polyval -- evaluate a polynomial at given points.
Calculus
--------
- polyder -- differentiate a polynomial.
- polyint -- integrate a polynomial.
Misc Functions
--------------
- polyfromroots -- create a polynomial with specified roots.
- polyroots -- find the roots of a polynomial.
- polyvander -- Vandermode like matrix for powers.
- polyfit -- least squares fit returning a polynomial.
- polytrim -- trim leading coefficients from a polynomial.
- polyline -- Polynomial of given straight line
Classes
-------
- Polynomial -- polynomial class.
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PolynomialN( t polytemplatei i c C s5 | d j o t i | | g ƒ St i | g ƒ Sd S( s2 Polynomial whose graph is a straight line.
The line has the formula ``off + scl*x``
Parameters:
-----------
off, scl : scalars
The specified line is given by ``off + scl*x``.
Returns:
--------
series : 1d ndarray
The polynomial equal to ``off + scl*x``.
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C s¶ t | ƒ d j o t i d ƒ St i | g d t ƒ\ } t i t | ƒ d d | i ƒ} d | d >> import warnings
>>> warnings.simplefilter('ignore', RankWarning)
See Also
--------
polyval : Evaluates a polynomial.
polyvander : Vandermonde matrix for powers.
chebfit : least squares fit using Chebyshev series.
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 polynomial ``P(x)``
that minimizes the squared error
``E = \sum_j |y_j - P(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 double precision and polynomials tend to fail at about
degree 20. Fits using Chebyshev 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.
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( R/ R RC R0 t ndimt TypeErrorR$ RF R1 R t finfoR t epsR t sqrtt sumt lat lstsqt Tt warningst warnR t RankWarning(
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