numpy.polynomial.polyutils

module documentation

(source)

Utility classes and functions for the polynomial modules.

This module provides: error and warning objects; a polynomial base class; and some routines used in both the polynomial and chebyshev modules.

Warning objects

Functions

Exception	`RankWarning`	Issued by chebfit when the design matrix is rank deficient.
Function	`as_series`	Return argument as a list of 1-d arrays.
Function	`format_float`	Undocumented
Function	`getdomain`	Return a domain suitable for given abscissae.
Function	`mapdomain`	Apply linear map to input points.
Function	`mapparms`	Linear map parameters between domains.
Function	`trimcoef`	Remove "small" "trailing" coefficients from a polynomial.
Function	`trimseq`	Remove small Poly series coefficients.
Function	`_add`	Helper function used to implement the `<type>add` functions.
Function	`_deprecate_as_int`	Like `operator.index`, but emits a deprecation warning when passed a float
Function	`_div`	Helper function used to implement the `<type>div` functions.
Function	`_fit`	Helper function used to implement the `<type>fit` functions.
Function	`_fromroots`	Helper function used to implement the `<type>fromroots` functions.
Function	`_gridnd`	Helper function used to implement the `<type>grid<n>d` functions.
Function	`_nth_slice`	Undocumented
Function	`_pow`	Helper function used to implement the `<type>pow` functions.
Function	`_sub`	Helper function used to implement the `<type>sub` functions.
Function	`_valnd`	Helper function used to implement the `<type>val<n>d` functions.
Function	`_vander_nd`	A generalization of the Vandermonde matrix for N dimensions
Function	`_vander_nd_flat`	Like `_vander_nd`, but flattens the last `len(degrees)` axes into a single axis

def as_series(alist, trim=True): (source) ¶

Return argument as a list of 1-d arrays.

The returned list contains array(s) of dtype double, complex double, or object. A 1-d argument of shape (N,) is parsed into N arrays of size one; a 2-d argument of shape (M,N) is parsed into M arrays of size N (i.e., is "parsed by row"); and a higher dimensional array raises a Value Error if it is not first reshaped into either a 1-d or 2-d array.

Examples

>>> from numpy.polynomial import polyutils as pu
>>> a = np.arange(4)
>>> pu.as_series(a)
[array([0.]), array([1.]), array([2.]), array([3.])]
>>> b = np.arange(6).reshape((2,3))
>>> pu.as_series(b)
[array([0., 1., 2.]), array([3., 4., 5.])]

>>> pu.as_series((1, np.arange(3), np.arange(2, dtype=np.float16)))
[array([1.]), array([0., 1., 2.]), array([0., 1.])]

>>> pu.as_series([2, [1.1, 0.]])
[array([2.]), array([1.1])]

>>> pu.as_series([2, [1.1, 0.]], trim=False)
[array([2.]), array([1.1, 0. ])]

Parameters
alist:`array_like`	A 1- or 2-d array_like
trim:`boolean`, optional	When True, trailing zeros are removed from the inputs. When False, the inputs are passed through intact.
Returns
`list` of 1-D arrays	[a1, a2,...] - A copy of the input data as a list of 1-d arrays.
Raises
`ValueError`	Raised when `as_series` cannot convert its input to 1-d arrays, or at least one of the resulting arrays is empty.

def format_float(x, parens=False): (source) ¶

Undocumented

def getdomain(x): (source) ¶

Return a domain suitable for given abscissae.

Find a domain suitable for a polynomial or Chebyshev series defined at the values supplied.

See Also

mapparms, mapdomain

Examples

>>> from numpy.polynomial import polyutils as pu
>>> points = np.arange(4)**2 - 5; points
array([-5, -4, -1,  4])
>>> pu.getdomain(points)
array([-5.,  4.])
>>> c = np.exp(complex(0,1)*np.pi*np.arange(12)/6) # unit circle
>>> pu.getdomain(c)
array([-1.-1.j,  1.+1.j])

Parameters
x:`array_like`	1-d array of abscissae whose domain will be determined.
Returns
`ndarray`	domain - 1-d array containing two values. If the inputs are complex, then the two returned points are the lower left and upper right corners of the smallest rectangle (aligned with the axes) in the complex plane containing the points `x`. If the inputs are real, then the two points are the ends of the smallest interval containing the points `x`.

def mapdomain(x, old, new): (source) ¶

Apply linear map to input points.

The linear map offset + scale*x that maps the domain old to the domain new is applied to the points x.

See Also

getdomain, mapparms

Notes

Effectively, this implements:

x_out = new[0] + m(x − old[0])

where

m = (new[1] − new[0])/(old[1] − old[0])

Examples

>>> from numpy.polynomial import polyutils as pu
>>> old_domain = (-1,1)
>>> new_domain = (0,2*np.pi)
>>> x = np.linspace(-1,1,6); x
array([-1. , -0.6, -0.2,  0.2,  0.6,  1. ])
>>> x_out = pu.mapdomain(x, old_domain, new_domain); x_out
array([ 0.        ,  1.25663706,  2.51327412,  3.76991118,  5.02654825, # may vary
        6.28318531])
>>> x - pu.mapdomain(x_out, new_domain, old_domain)
array([0., 0., 0., 0., 0., 0.])

Also works for complex numbers (and thus can be used to map any line in the complex plane to any other line therein).

>>> i = complex(0,1)
>>> old = (-1 - i, 1 + i)
>>> new = (-1 + i, 1 - i)
>>> z = np.linspace(old[0], old[1], 6); z
array([-1. -1.j , -0.6-0.6j, -0.2-0.2j,  0.2+0.2j,  0.6+0.6j,  1. +1.j ])
>>> new_z = pu.mapdomain(z, old, new); new_z
array([-1.0+1.j , -0.6+0.6j, -0.2+0.2j,  0.2-0.2j,  0.6-0.6j,  1.0-1.j ]) # may vary

Parameters
x:`array_like`	Points to be mapped. If `x` is a subtype of ndarray the subtype will be preserved.
old:`array_like`	The two domains that determine the map. Each must (successfully) convert to 1-d arrays containing precisely two values.
new:`array_like`	The two domains that determine the map. Each must (successfully) convert to 1-d arrays containing precisely two values.
Returns
`ndarray`	x_out - Array of points of the same shape as `x`, after application of the linear map between the two domains.

def mapparms(old, new): (source) ¶

Linear map parameters between domains.

Return the parameters of the linear map offset + scale*x that maps old to new such that old[i] -> new[i], i = 0, 1.

See Also

getdomain, mapdomain

Notes

Also works for complex numbers, and thus can be used to calculate the parameters required to map any line in the complex plane to any other line therein.

Examples

>>> from numpy.polynomial import polyutils as pu
>>> pu.mapparms((-1,1),(-1,1))
(0.0, 1.0)
>>> pu.mapparms((1,-1),(-1,1))
(-0.0, -1.0)
>>> i = complex(0,1)
>>> pu.mapparms((-i,-1),(1,i))
((1+1j), (1-0j))

Parameters
old:`array_like`	Domains. Each domain must (successfully) convert to a 1-d array containing precisely two values.
new:`array_like`	Domains. Each domain must (successfully) convert to a 1-d array containing precisely two values.
Returns
`scalars`	offset, scale - The map `L(x) = offset + scale*x` maps the first domain to the second.

def trimcoef(c, tol=0): (source) ¶

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."

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])

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
`ndarray`	trimmed - 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

def trimseq(seq): (source) ¶

Remove small Poly series coefficients.

Notes

Do not lose the type info if the sequence contains unknown objects.

Parameters
seq:`sequence`	Sequence of Poly series coefficients. This routine fails for empty sequences.
Returns
`sequence`	series - Subsequence with trailing zeros removed. If the resulting sequence would be empty, return the first element. The returned sequence may or may not be a view.

def _add(c1, c2): (source) ¶

Helper function used to implement the <type>add functions.

def _deprecate_as_int(x, desc): (source) ¶

Like operator.index, but emits a deprecation warning when passed a float

:raises TypeError : if x is a non-integral float or non-numeric: :raises DeprecationWarning : if x is an integral float:

Parameters
x:`int-like`, or float with integral value	Value to interpret as an integer
desc:`str`	description to include in any error message

def _div(mul_f, c1, c2): (source) ¶

Helper function used to implement the <type>div functions.

Implementation uses repeated subtraction of c2 multiplied by the nth basis. For some polynomial types, a more efficient approach may be possible.

Parameters
mul_f:`function(array_like`, `array_like)` -> array_like	The `<type>mul` function, such as `polymul`
c1	See the `<type>div` functions for more detail
c2	See the `<type>div` functions for more detail

def _fit(vander_f, x, y, deg, rcond=None, full=False, w=None): (source) ¶

Helper function used to implement the <type>fit functions.

Parameters
vander_f:`function(array_like`, `int)` -> ndarray	The 1d vander function, such as `polyvander`
x	Undocumented
y	Undocumented
deg	Undocumented
rcond	Undocumented
full	Undocumented
w	Undocumented
c1	See the `<type>fit` functions for more detail
c2	See the `<type>fit` functions for more detail

def _fromroots(line_f, mul_f, roots): (source) ¶

Helper function used to implement the <type>fromroots functions.

Parameters
line_f:`function(float`, `float)` -> ndarray	The `<type>line` function, such as `polyline`
mul_f:`function(array_like`, `array_like)` -> ndarray	The `<type>mul` function, such as `polymul`
roots	See the `<type>fromroots` functions for more detail

def _gridnd(val_f, c, *args): (source) ¶

Helper function used to implement the <type>grid<n>d functions.

Parameters
val_f:`function(array_like`, `array_like`, `tensor`: `bool)` -> array_like	The `<type>val` function, such as `polyval`
c	See the `<type>grid<n>d` functions for more detail
*args	See the `<type>grid<n>d` functions for more detail

def _nth_slice(i, ndim): (source) ¶

Undocumented

def _pow(mul_f, c, pow, maxpower): (source) ¶

Helper function used to implement the <type>pow functions.

Parameters
mul_f:`function(array_like`, `array_like)` -> ndarray	The `<type>mul` function, such as `polymul`
c:`array_like`	1-D array of array of series coefficients
pow	See the `<type>pow` functions for more detail
maxpower	See the `<type>pow` functions for more detail

def _sub(c1, c2): (source) ¶

Helper function used to implement the <type>sub functions.

def _valnd(val_f, c, *args): (source) ¶

Helper function used to implement the <type>val<n>d functions.

Parameters
val_f:`function(array_like`, `array_like`, `tensor`: `bool)` -> array_like	The `<type>val` function, such as `polyval`
c	See the `<type>val<n>d` functions for more detail
*args	See the `<type>val<n>d` functions for more detail

def _vander_nd(vander_fs, points, degrees): (source) ¶

A generalization of the Vandermonde matrix for N dimensions

The result is built by combining the results of 1d Vandermonde matrices,

W[i₀, …, i_M, j₀, …, j_N] = ^N∏_k = 0V_k(x_k)[i₀, …, i_M, j_k]

where

N = len(points) = len(degrees) = len(vander_fs) M = points[k].ndim V_k = vander_fs[k] x_k = points[k] 0 ≤ j_k ≤ degrees[k]

Expanding the one-dimensional V_k functions gives:

W[i₀, …, i_M, j₀, …, j_N] = ^N∏_k = 0B_{k, j_k}(x_k[i₀, …, i_M])

where B_k, m is the m'th basis of the polynomial construction used along dimension k. For a regular polynomial, B_k, m(x) = P_m(x) = x^m.

Parameters
vander_fs:`Sequence[function(array_like`, `int)` -> ndarray ]	The 1d vander function to use for each axis, such as `polyvander`
points:`Sequence[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. This must be the same length as `vander_fs`.
degrees:`Sequence[int]`	The maximum degree (inclusive) to use for each axis. This must be the same length as `vander_fs`.
Returns
`ndarray`	vander_nd - An array of shape `points[0].shape + tuple(d + 1 for d in degrees)`.

def _vander_nd_flat(vander_fs, points, degrees): (source) ¶

Like _vander_nd, but flattens the last len(degrees) axes into a single axis

Used to implement the public <type>vander<n>d functions.