quadrature.py

# coding=utf-8

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import numpy as np

def _precompute_latitudes(nlat, grid="equiangular"):
    r"""
    Convenience routine to precompute latitudes
    """

    # compute coordinates
    if grid == "legendre-gauss":
        xlg, wlg = legendre_gauss_weights(nlat)
    elif grid == "lobatto":
        xlg, wlg = lobatto_weights(nlat)
    elif grid == "equiangular":
        xlg, wlg = clenshaw_curtiss_weights(nlat)
    else:
        raise ValueError("Unknown grid")

    lats = np.flip(np.arccos(xlg)).copy()
    wlg = np.flip(wlg).copy()

    return lats, wlg

def legendre_gauss_weights(n, a=-1.0, b=1.0):
    r"""
    Helper routine which returns the Legendre-Gauss nodes and weights
    on the interval [a, b]
    """

    xlg, wlg = np.polynomial.legendre.leggauss(n)
    xlg = (b - a) * 0.5 * xlg + (b + a) * 0.5
    wlg = wlg * (b - a) * 0.5

    return xlg, wlg

def lobatto_weights(n, a=-1.0, b=1.0, tol=1e-16, maxiter=100):
    r"""
    Helper routine which returns the Legendre-Gauss-Lobatto nodes and weights
    on the interval [a, b]
    """

    wlg = np.zeros((n,))
    tlg = np.zeros((n,))
    tmp = np.zeros((n,))

    # Vandermonde Matrix
    vdm = np.zeros((n, n))
  
    # initialize Chebyshev nodes as first guess
    for i in range(n): 
        tlg[i] = -np.cos(np.pi*i / (n-1))
    
    tmp = 2.0
    
    for i in range(maxiter):
        tmp = tlg
       
        vdm[:,0] = 1.0 
        vdm[:,1] = tlg
       
        for k in range(2, n):
            vdm[:, k] = ( (2*k-1) * tlg * vdm[:, k-1] - (k-1) * vdm[:, k-2] ) / k
       
        tlg = tmp - ( tlg*vdm[:, n-1] - vdm[:, n-2] ) / ( n * vdm[:, n-1]) 
        
        if (max(abs(tlg - tmp).flatten()) < tol ):
            break 
    
    wlg = 2.0 / ( (n*(n-1))*(vdm[:, n-1]**2))

    # rescale
    tlg = (b - a) * 0.5 * tlg + (b + a) * 0.5
    wlg = wlg * (b - a) * 0.5
    
    return tlg, wlg


def clenshaw_curtiss_weights(n, a=-1.0, b=1.0):
    r"""
    Computation of the Clenshaw-Curtis quadrature nodes and weights.
    This implementation follows

    [1] Joerg Waldvogel, Fast Construction of the Fejer and Clenshaw-Curtis Quadrature Rules; BIT Numerical Mathematics, Vol. 43, No. 1, pp. 001–018.
    """

    assert(n > 1)

    tcc = np.cos(np.linspace(np.pi, 0, n))

    if n == 2:
        wcc = np.array([1., 1.])
    else:

        n1 = n - 1
        N = np.arange(1, n1, 2)
        l = len(N)
        m = n1 - l

        v = np.concatenate([2 / N / (N-2), 1 / N[-1:], np.zeros(m)])
        v = 0 - v[:-1] - v[-1:0:-1]

        g0 = -np.ones(n1)
        g0[l] = g0[l] + n1
        g0[m] = g0[m] + n1
        g = g0 / (n1**2 - 1 + (n1%2))
        wcc = np.fft.ifft(v + g).real
        wcc = np.concatenate((wcc, wcc[:1]))

    # rescale
    tcc = (b - a) * 0.5 * tcc + (b + a) * 0.5
    wcc = wcc * (b - a) * 0.5

    return tcc, wcc

def fejer2_weights(n, a=-1.0, b=1.0):
    r"""
    Computation of the Fejer quadrature nodes and weights.
    This implementation follows

    [1] Joerg Waldvogel, Fast Construction of the Fejer and Clenshaw-Curtis Quadrature Rules; BIT Numerical Mathematics, Vol. 43, No. 1, pp. 001–018.
    """

    assert(n > 2)

    tcc = np.cos(np.linspace(np.pi, 0, n))

    n1 = n - 1
    N = np.arange(1, n1, 2)
    l = len(N)
    m = n1 - l

    v = np.concatenate([2 / N / (N-2), 1 / N[-1:], np.zeros(m)])
    v = 0 - v[:-1] - v[-1:0:-1]

    wcc = np.fft.ifft(v).real
    wcc = np.concatenate((wcc, wcc[:1]))

    # rescale
    tcc = (b - a) * 0.5 * tcc + (b + a) * 0.5
    wcc = wcc * (b - a) * 0.5

    return tcc, wcc