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

nint

The size of the matrix to create; that is, the result is an n x n matrix.

kindstr, optional

Must be one of ‘symmetric’, ‘lower’, or ‘upper’. Default is ‘symmetric’.

exactbool, 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

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.]])