scipy.linalg.invpascal

scipy.linalg.invpascal(n, kind='symmetric', exact=True)

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

New in version 0.16.0.

References

[R126]

“Pascal matrix”, http://en.wikipedia.org/wiki/Pascal_matrix

[R127]

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.]])