source: sasmodels/explore/symint.py @ 57eb6a4

"""
Explore integration of rotationally symmetric shapes
"""

from __future__ import print_function, division

import os, sys
sys.path.insert(0, os.path.dirname(os.path.dirname(__file__)))

import numpy as np
from numpy import pi, sin, cos, sqrt, exp, expm1, degrees, log10
from scipy.integrate import dblquad, simps, romb, romberg
from scipy.special.orthogonal import p_roots
import pylab

from sasmodels.special import square
from sasmodels.special import Gauss20Wt, Gauss20Z
from sasmodels.special import Gauss76Wt, Gauss76Z
from sasmodels.special import Gauss150Wt, Gauss150Z
from sasmodels.special import sas_2J1x_x, sas_sinx_x, sas_3j1x_x

SLD = 3.0
SLD_SOLVENT = 6.3
CONTRAST = SLD - SLD_SOLVENT

def make_cylinder(radius, length):
    def cylinder(qab, qc):
        return sas_2J1x_x(qab*radius) * sas_sinx_x(qc*0.5*length)
    cylinder.__doc__ = "cylinder radius=%g, length=%g"%(radius, length)
    volume = pi*radius**2*length
    norm = 1e-4*volume*CONTRAST**2
    return norm, cylinder

def make_long_cylinder(radius, length):
    def long_cylinder(q):
        return norm/q * sas_2J1x_x(q*radius)**2
    long_cylinder.__doc__ = "long cylinder radius=%g, length=%g"%(radius, length)
    volume = pi*radius**2*length
    norm = 1e-4*volume*CONTRAST**2*pi/length
    return long_cylinder

def make_sphere(radius):
    def sphere(qab, qc):
        q = sqrt(qab**2 + qc**2)
        return sas_3j1x_x(q*radius)
    sphere.__doc__ = "sphere radius=%g"%(radius,)
    volume = 4*pi*radius**3/3
    norm = 1e-4*volume*CONTRAST**2
    return norm, sphere

THETA_LOW, THETA_HIGH = 0, pi/2
SCALE = 1


def kernel(q, theta):
    """
    S(q) kernel for paracrystal forms.
    """
    qab = q*sin(theta)
    qc = q*cos(theta)
    return NORM*KERNEL(qab, qc)**2


def gauss_quad(q, n=150):
    """
    Compute the integral using gaussian quadrature for n = 20, 76 or 150.
    """
    z, w = p_roots(n)
    theta = (THETA_HIGH-THETA_LOW)*(z + 1)/2 + THETA_LOW
    sin_theta = abs(sin(theta))
    Zq = kernel(q=q, theta=theta)
    return np.sum(Zq*w*sin_theta)*(THETA_HIGH-THETA_LOW)/2


def gridded_integrals(q, n=300):
    """
    Compute the integral on a regular grid using rectangular, trapezoidal,
    simpsons, and romberg integration.  Romberg integration requires that
    the grid be of size n = 2**k + 1.
    """
    theta = np.linspace(THETA_LOW, THETA_HIGH, n)
    Zq = kernel(q=q, theta=theta)
    Zq *= abs(sin(theta))
    dx = theta[1]-theta[0]
    print("rect", n, np.sum(Zq)*dx*SCALE)
    print("trapz", n, np.trapz(Zq, dx=dx)*SCALE)
    print("simpson", n, simps(Zq, dx=dx)*SCALE)
    print("romb", n, romb(Zq, dx=dx)*SCALE)

def scipy_romberg(q):
    """
    Compute the integral using romberg integration.  This function does not
    complete in a reasonable time.  No idea if it is accurate.
    """
    evals = [0]
    def outer(theta):
        evals[0] += 1
        return kernel(q, theta=theta)*abs(sin(theta))
    result = romberg(outer, THETA_LOW, THETA_HIGH, divmax=100)*SCALE
    print("scipy romberg", evals[0], result)

def plot(q, n=300):
    """
    Plot the 2D surface that needs to be integrated in order to compute
    the BCC S(q) at a particular q, dnn and d_factor.  *n* is the number
    of points in the grid.
    """
    theta = np.linspace(THETA_LOW, THETA_HIGH, n)
    Zq = kernel(q=q, theta=theta)
    Zq *= abs(sin(theta))
    pylab.semilogy(degrees(theta), np.fmax(Zq, 1.e-6), label="Q=%g"%q)
    pylab.title("%s I(q, theta) sin(theta)" % (KERNEL.__doc__,))
    pylab.xlabel("theta (degrees)")
    pylab.ylabel("Iq 1/cm")

def Iq_trapz(q, n):
    theta = np.linspace(THETA_LOW, THETA_HIGH, n)
    Zq = kernel(q=q, theta=theta)
    Zq *= abs(sin(theta))
    dx = theta[1]-theta[0]
    return np.trapz(Zq, dx=dx)*SCALE

def plot_Iq(q, n, form="trapz"):
    if form == "trapz":
        Iq = np.array([Iq_trapz(qk, n) for qk in q])
    elif form == "gauss":
        Iq = np.array([gauss_quad(qk, n) for qk in q])
    pylab.loglog(q, Iq, label="%s, n=%d"%(form, n))
    pylab.xlabel("q (1/A)")
    pylab.ylabel("Iq (1/cm)")
    pylab.title(KERNEL.__doc__ + " I(q) circular average")
    return Iq

radius = 10.
length = 1e5
NORM, KERNEL = make_cylinder(radius=radius, length=length)
long_cyl = make_long_cylinder(radius=radius, length=length)
#NORM, KERNEL = make_sphere(radius=50.)


if __name__ == "__main__":
    Q = 0.386
    for n in (20, 76, 150, 300, 1000): #, 10000, 30000):
        print("gauss", n, gauss_quad(Q, n=n))
    for k in (8, 10, 13, 16, 19):
        gridded_integrals(Q, n=2**k+1)
    #print("inf cyl", 0, long_cyl(Q))
    #scipy_romberg(Q)

    plot(0.386, n=2000)
    plot(0.5, n=2000)
    plot(0.8, n=2000)
    pylab.legend()
    pylab.figure()

    q = np.logspace(-3, 0, 400)
    I1 = long_cyl(q)
    I2 = plot_Iq(q, n=2**19+1, form="trapz")
    #plot_Iq(q, n=2**16+1, form="trapz")
    #plot_Iq(q, n=2**10+1, form="trapz")
    plot_Iq(q, n=1024, form="gauss")
    #plot_Iq(q, n=300, form="gauss")
    #plot_Iq(q, n=150, form="gauss")
    #plot_Iq(q, n=76, form="gauss")
    pylab.loglog(q, long_cyl(q), label="limit")
    pylab.legend()

    pylab.figure()
    pylab.semilogx(q, (I2 - I1)/I1)

    pylab.show()