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Module « scipy.linalg »

Fonction invpascal - module scipy.linalg

Signature de la fonction invpascal

def invpascal(n, kind='symmetric', exact=True) 

Description

help(scipy.linalg.invpascal)

Returns the inverse of the n x n Pascal matrix.

The Pascal matrix is a matrix containing the binomial coefficients as
its elements.

Parameters
----------
n : int
    The size of the matrix to create; that is, the result is an n x n
    matrix.
kind : str, optional
    Must be one of 'symmetric', 'lower', or 'upper'.
    Default is 'symmetric'.
exact : bool, optional
    If `exact` is True, the result is either an array of type
    ``numpy.int64`` (if `n` <= 35) or an object array of Python integers.
    If `exact` is False, the coefficients in the matrix are computed using
    `scipy.special.comb` with `exact=False`. The result will be a floating
    point array, and for large `n`, the values in the array will not be the
    exact coefficients.

Returns
-------
invp : (n, n) ndarray
    The inverse of the Pascal matrix.

See Also
--------
pascal

Notes
-----

.. versionadded:: 0.16.0

References
----------
.. [1] "Pascal matrix", https://en.wikipedia.org/wiki/Pascal_matrix
.. [2] Cohen, A. M., "The inverse of a Pascal matrix", Mathematical
       Gazette, 59(408), pp. 111-112, 1975.

Examples
--------
>>> from scipy.linalg import invpascal, pascal
>>> invp = invpascal(5)
>>> invp
array([[  5, -10,  10,  -5,   1],
       [-10,  30, -35,  19,  -4],
       [ 10, -35,  46, -27,   6],
       [ -5,  19, -27,  17,  -4],
       [  1,  -4,   6,  -4,   1]])

>>> p = pascal(5)
>>> p.dot(invp)
array([[ 1.,  0.,  0.,  0.,  0.],
       [ 0.,  1.,  0.,  0.,  0.],
       [ 0.,  0.,  1.,  0.,  0.],
       [ 0.,  0.,  0.,  1.,  0.],
       [ 0.,  0.,  0.,  0.,  1.]])

An example of the use of `kind` and `exact`:

>>> invpascal(5, kind='lower', exact=False)
array([[ 1., -0.,  0., -0.,  0.],
       [-1.,  1., -0.,  0., -0.],
       [ 1., -2.,  1., -0.,  0.],
       [-1.,  3., -3.,  1., -0.],
       [ 1., -4.,  6., -4.,  1.]])

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