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# 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

  [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.]])