diff options
Diffstat (limited to 'numpy/polynomial/hermite.py')
| -rw-r--r-- | numpy/polynomial/hermite.py | 62 |
1 files changed, 52 insertions, 10 deletions
diff --git a/numpy/polynomial/hermite.py b/numpy/polynomial/hermite.py index 1fd49d774..1d3bef390 100644 --- a/numpy/polynomial/hermite.py +++ b/numpy/polynomial/hermite.py @@ -1577,13 +1577,13 @@ def hermcompanion(c): n = len(c) - 1 mat = np.zeros((n, n), dtype=c.dtype) - scl = np.hstack((1., np.sqrt(2.*np.arange(1, n)))) - scl = np.multiply.accumulate(scl) + scl = np.hstack((1., 1./np.sqrt(2.*np.arange(n - 1, 0, -1)))) + scl = np.multiply.accumulate(scl)[::-1] top = mat.reshape(-1)[1::n+1] bot = mat.reshape(-1)[n::n+1] top[...] = np.sqrt(.5*np.arange(1, n)) bot[...] = top - mat[:, -1] -= (c[:-1]/c[-1])*(scl/scl[-1])*.5 + mat[:, -1] -= scl*c[:-1]/(2.0*c[-1]) return mat @@ -1646,6 +1646,49 @@ def hermroots(c): return r +def _normed_hermite_n(x, n): + """ + Evaluate a normalized Hermite polynomial. + + Compute the value of the normalized Hermite polynomial of degree ``n`` + at the points ``x``. + + + Parameters + ---------- + x : ndarray of double. + Points at which to evaluate the function + n : int + Degree of the normalized Hermite function to be evaluated. + + Returns + ------- + values : ndarray + The shape of the return value is described above. + + Notes + ----- + .. versionadded:: 1.10.0 + + This function is needed for finding the Gauss points and integration + weights for high degrees. The values of the standard Hermite functions + overflow when n >= 207. + + """ + if n == 0: + return np.ones(x.shape)/np.sqrt(np.sqrt(np.pi)) + + c0 = 0. + c1 = 1./np.sqrt(np.sqrt(np.pi)) + nd = float(n) + for i in range(n - 1): + tmp = c0 + c0 = -c1*np.sqrt((nd - 1.)/nd) + c1 = tmp + c1*x*np.sqrt(2./nd) + nd = nd - 1.0 + return c0 + c1*x*np.sqrt(2) + + def hermgauss(deg): """ Gauss-Hermite quadrature. @@ -1688,22 +1731,21 @@ def hermgauss(deg): # first approximation of roots. We use the fact that the companion # matrix is symmetric in this case in order to obtain better zeros. - c = np.array([0]*deg + [1]) + c = np.array([0]*deg + [1], dtype=np.float64) m = hermcompanion(c) - x = la.eigvals(m) + x = la.eigvalsh(m) x.sort() # improve roots by one application of Newton - dy = hermval(x, c) - df = hermval(x, hermder(c)) + dy = _normed_hermite_n(x, ideg) + df = _normed_hermite_n(x, ideg - 1) * np.sqrt(2*ideg) x -= dy/df # compute the weights. We scale the factor to avoid possible numerical # overflow. - fm = hermval(x, c[1:]) + fm = _normed_hermite_n(x, ideg - 1) fm /= np.abs(fm).max() - df /= np.abs(df).max() - w = 1/(fm * df) + w = 1/(fm * fm) # for Hermite we can also symmetrize w = (w + w[::-1])/2 |