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Generate a Hermite series with given roots.

The function returns the coefficients of the polynomial

.. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n),

in Hermite form, where the `r_n` are the roots specified in `roots`.
If a zero has multiplicity n, then it must appear in `roots` n times.
For instance, if 2 is a root of multiplicity three and 3 is a root of
multiplicity 2, then `roots` looks something like [2, 2, 2, 3, 3]. The(more...)

        def hermfromroots(roots) :
    """
    Generate a Hermite series with given roots.

    The function returns the coefficients of the polynomial

    .. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n),

    in Hermite form, where the `r_n` are the roots specified in `roots`.
    If a zero has multiplicity n, then it must appear in `roots` n times.
    For instance, if 2 is a root of multiplicity three and 3 is a root of
    multiplicity 2, then `roots` looks something like [2, 2, 2, 3, 3]. The
    roots can appear in any order.

    If the returned coefficients are `c`, then

    .. math:: p(x) = c_0 + c_1 * H_1(x) + ... +  c_n * H_n(x)

    The coefficient of the last term is not generally 1 for monic
    polynomials in Hermite form.

    Parameters
    ----------
    roots : array_like
        Sequence containing the roots.

    Returns
    -------
    out : ndarray
        1-D array of coefficients.  If all roots are real then `out` is a
        real array, if some of the roots are complex, then `out` is complex
        even if all the coefficients in the result are real (see Examples
        below).

    See Also
    --------
    polyfromroots, legfromroots, lagfromroots, chebfromroots,
    hermefromroots.

    Examples
    --------
    >>> from numpy.polynomial.hermite import hermfromroots, hermval
    >>> coef = hermfromroots((-1, 0, 1))
    >>> hermval((-1, 0, 1), coef)
    array([ 0.,  0.,  0.])
    >>> coef = hermfromroots((-1j, 1j))
    >>> hermval((-1j, 1j), coef)
    array([ 0.+0.j,  0.+0.j])

    """
    if len(roots) == 0 :
        return np.ones(1)
    else :
        [roots] = pu.as_series([roots], trim=False)
        roots.sort()
        p = [hermline(-r, 1) for r in roots]
        n = len(p)
        while n > 1:
            m, r = divmod(n, 2)
            tmp = [hermmul(p[i], p[i+m]) for i in range(m)]
            if r:
                tmp[0] = hermmul(tmp[0], p[-1])
            p = tmp
            n = m
        return p[0]
        


src/n/u/nupic-linux64-HEAD/lib64/python2.6/site-packages/numpy/polynomial/tests/test_hermite.py   nupic-linux64(Download)
    def test_hermfromroots(self) :
        res = herm.hermfromroots([])
        assert_almost_equal(trim(res), [1])
        for i in range(1,5) :
            roots = np.cos(np.linspace(-np.pi, 0, 2*i + 1)[1::2])
            pol = herm.hermfromroots(roots)
        for i in range(2,5) :
            tgt = np.linspace(-1, 1, i)
            res = herm.hermroots(herm.hermfromroots(tgt))
            assert_almost_equal(trim(res), trim(tgt))
 

src/n/u/numpy-1.8.1/numpy/polynomial/tests/test_hermite.py   numpy(Download)
    def test_hermfromroots(self) :
        res = herm.hermfromroots([])
        assert_almost_equal(trim(res), [1])
        for i in range(1, 5) :
            roots = np.cos(np.linspace(-np.pi, 0, 2*i + 1)[1::2])
            pol = herm.hermfromroots(roots)
        for i in range(2, 5) :
            tgt = np.linspace(-1, 1, i)
            res = herm.hermroots(herm.hermfromroots(tgt))
            assert_almost_equal(trim(res), trim(tgt))