source: sasmodels/explore/precision.py @ 1ceb951

#!/usr/bin/env python
r"""
Show numerical precision of $2 J_1(x)/x$.
"""
from __future__ import division, print_function

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

import numpy as np
from numpy import pi, inf
import scipy.special
try:
    from mpmath import mp
except ImportError:
    # CRUFT: mpmath split out into its own package
    from sympy.mpmath import mp
#import matplotlib; matplotlib.use('TkAgg')
import pylab

from sasmodels import core, data, direct_model, modelinfo

class Comparator(object):
    def __init__(self, name, mp_function, np_function, ocl_function, xaxis, limits):
        self.name = name
        self.mp_function = mp_function
        self.np_function = np_function
        self.ocl_function = ocl_function
        self.xaxis = xaxis
        self.limits = limits

    def __repr__(self):
        return "Comparator(%s)"%self.name

    def call_mpmath(self, vec, bits=500):
        """
        Direct calculation using mpmath extended precision library.
        """
        with mp.workprec(bits):
            return [self.mp_function(mp.mpf(x)) for x in vec]

    def call_numpy(self, x, dtype):
        """
        Direct calculation using numpy/scipy.
        """
        x = np.asarray(x, dtype)
        return self.np_function(x)

    def call_ocl(self, x, dtype, platform='ocl'):
        """
        Calculation using sasmodels ocl libraries.
        """
        x = np.asarray(x, dtype)
        model = core.build_model(self.ocl_function, dtype=dtype)
        calculator = direct_model.DirectModel(data.empty_data1D(x), model)
        return calculator(background=0)

    def run(self, xrange="log", diff="relative"):
        r"""
        Compare accuracy of different methods for computing f.

        *xrange* is::

            log:    [10^-3,10^5]
            logq:   [10^-4, 10^1]
            linear: [1,1000]
            zoom:   [1000,1010]
            neg:    [-100,100]

        *diff* is "relative", "absolute" or "none"

        *x_bits* is the precision with which the x values are specified.  The
        default 23 should reproduce the equivalent of a single precisio
        """
        linear = not xrange.startswith("log")
        if xrange == "zoom":
            lin_min, lin_max, lin_steps = 1000, 1010, 2000
        elif xrange == "neg":
            lin_min, lin_max, lin_steps = -100.1, 100.1, 2000
        elif xrange == "linear":
            lin_min, lin_max, lin_steps = 1, 1000, 2000
            lin_min, lin_max, lin_steps = 0.001, 2, 2000
        elif xrange == "log":
            log_min, log_max, log_steps = -3, 5, 400
        elif xrange == "logq":
            log_min, log_max, log_steps = -4, 1, 400
        else:
            raise ValueError("unknown range "+xrange)
        with mp.workprec(500):
            # Note: we make sure that we are comparing apples to apples...
            # The x points are set using single precision so that we are
            # examining the accuracy of the transformation from x to f(x)
            # rather than x to f(nearest(x)) where nearest(x) is the nearest
            # value to x in the given precision.
            if linear:
                lin_min = max(lin_min, self.limits[0])
                lin_max = min(lin_max, self.limits[1])
                qrf = np.linspace(lin_min, lin_max, lin_steps, dtype='single')
                #qrf = np.linspace(lin_min, lin_max, lin_steps, dtype='double')
                qr = [mp.mpf(float(v)) for v in qrf]
                #qr = mp.linspace(lin_min, lin_max, lin_steps)
            else:
                log_min = np.log10(max(10**log_min, self.limits[0]))
                log_max = np.log10(min(10**log_max, self.limits[1]))
                qrf = np.logspace(log_min, log_max, log_steps, dtype='single')
                #qrf = np.logspace(log_min, log_max, log_steps, dtype='double')
                qr = [mp.mpf(float(v)) for v in qrf]
                #qr = [10**v for v in mp.linspace(log_min, log_max, log_steps)]

        target = self.call_mpmath(qr, bits=500)
        pylab.subplot(121)
        self.compare(qr, 'single', target, linear, diff)
        pylab.legend(loc='best')
        pylab.subplot(122)
        self.compare(qr, 'double', target, linear, diff)
        pylab.legend(loc='best')
        pylab.suptitle(self.name + " compared to 500-bit mpmath")

    def compare(self, x, precision, target, linear=False, diff="relative"):
        r"""
        Compare the different computation methods using the given precision.
        """
        if precision == 'single':
            #n=11; plotdiff(x, target, self.call_mpmath(x, n), 'mp %d bits'%n, diff=diff)
            #n=23; plotdiff(x, target, self.call_mpmath(x, n), 'mp %d bits'%n, diff=diff)
            pass
        elif precision == 'double':
            #n=53; plotdiff(x, target, self.call_mpmath(x, n), 'mp %d bits'%n, diff=diff)
            #n=83; plotdiff(x, target, self.call_mpmath(x, n), 'mp %d bits'%n, diff=diff)
            pass
        plotdiff(x, target, self.call_numpy(x, precision), 'numpy '+precision, diff=diff)
        plotdiff(x, target, self.call_ocl(x, precision, 0), 'OpenCL '+precision, diff=diff)
        pylab.xlabel(self.xaxis)
        if diff == "relative":
            pylab.ylabel("relative error")
        elif diff == "absolute":
            pylab.ylabel("absolute error")
        else:
            pylab.ylabel(self.name)
            pylab.semilogx(x, target, '-', label="true value")
        if linear:
            pylab.xscale('linear')

def plotdiff(x, target, actual, label, diff):
    """
    Plot the computed value.

    Use relative error if SHOW_DIFF, otherwise just plot the value directly.
    """
    if diff == "relative":
        err = np.array([abs((t-a)/t) for t, a in zip(target, actual)], 'd')
        #err = np.clip(err, 0, 1)
        pylab.loglog(x, err, '-', label=label)
    elif diff == "absolute":
        err = np.array([abs((t-a)) for t, a in zip(target, actual)], 'd')
        pylab.loglog(x, err, '-', label=label)
    else:
        limits = np.min(target), np.max(target)
        pylab.semilogx(x, np.clip(actual, *limits), '-', label=label)

def make_ocl(function, name, source=[]):
    class Kernel(object):
        pass
    Kernel.__file__ = name+".py"
    Kernel.name = name
    Kernel.parameters = []
    Kernel.source = source
    Kernel.Iq = function
    model_info = modelinfo.make_model_info(Kernel)
    return model_info


# =============== FUNCTION DEFINITIONS ================

FUNCTIONS = {}
def add_function(name, mp_function, np_function, ocl_function,
                 shortname=None, xaxis="x", limits=(-inf, inf)):
    if shortname is None:
        shortname = name.replace('(x)', '').replace(' ', '')
    FUNCTIONS[shortname] = Comparator(name, mp_function, np_function, ocl_function, xaxis, limits)

add_function(
    name="J0(x)",
    mp_function=mp.j0,
    np_function=scipy.special.j0,
    ocl_function=make_ocl("return sas_J0(q);", "sas_J0", ["lib/polevl.c", "lib/sas_J0.c"]),
)
add_function(
    name="J1(x)",
    mp_function=mp.j1,
    np_function=scipy.special.j1,
    ocl_function=make_ocl("return sas_J1(q);", "sas_J1", ["lib/polevl.c", "lib/sas_J1.c"]),
)
add_function(
    name="JN(-3, x)",
    mp_function=lambda x: mp.besselj(-3, x),
    np_function=lambda x: scipy.special.jn(-3, x),
    ocl_function=make_ocl("return sas_JN(-3, q);", "sas_JN",
                          ["lib/polevl.c", "lib/sas_J0.c", "lib/sas_J1.c", "lib/sas_JN.c"]),
    shortname="J-3",
)
add_function(
    name="JN(3, x)",
    mp_function=lambda x: mp.besselj(3, x),
    np_function=lambda x: scipy.special.jn(3, x),
    ocl_function=make_ocl("return sas_JN(3, q);", "sas_JN",
                          ["lib/polevl.c", "lib/sas_J0.c", "lib/sas_J1.c", "lib/sas_JN.c"]),
    shortname="J3",
)
add_function(
    name="JN(2, x)",
    mp_function=lambda x: mp.besselj(2, x),
    np_function=lambda x: scipy.special.jn(2, x),
    ocl_function=make_ocl("return sas_JN(2, q);", "sas_JN",
                          ["lib/polevl.c", "lib/sas_J0.c", "lib/sas_J1.c", "lib/sas_JN.c"]),
    shortname="J2",
)
add_function(
    name="2 J1(x)/x",
    mp_function=lambda x: 2*mp.j1(x)/x,
    np_function=lambda x: 2*scipy.special.j1(x)/x,
    ocl_function=make_ocl("return sas_2J1x_x(q);", "sas_2J1x_x", ["lib/polevl.c", "lib/sas_J1.c"]),
)
add_function(
    name="J1(x)",
    mp_function=mp.j1,
    np_function=scipy.special.j1,
    ocl_function=make_ocl("return sas_J1(q);", "sas_J1", ["lib/polevl.c", "lib/sas_J1.c"]),
)
add_function(
    name="Si(x)",
    mp_function=mp.si,
    np_function=lambda x: scipy.special.sici(x)[0],
    ocl_function=make_ocl("return sas_Si(q);", "sas_Si", ["lib/sas_Si.c"]),
)
#import fnlib
#add_function(
#    name="fnlibJ1",
#    mp_function=mp.j1,
#    np_function=fnlib.J1,
#    ocl_function=make_ocl("return sas_J1(q);", "sas_J1", ["lib/polevl.c", "lib/sas_J1.c"]),
#)
add_function(
    name="sin(x)",
    mp_function=mp.sin,
    np_function=np.sin,
    #ocl_function=make_ocl("double sn, cn; SINCOS(q,sn,cn); return sn;", "sas_sin"),
    ocl_function=make_ocl("return sin(q);", "sas_sin"),
)
add_function(
    name="sin(x)/x",
    mp_function=lambda x: mp.sin(x)/x if x != 0 else 1,
    ## scipy sinc function is inaccurate and has an implied pi*x term
    #np_function=lambda x: scipy.special.sinc(x/pi),
    ## numpy sin(x)/x needs to check for x=0
    np_function=lambda x: np.sin(x)/x,
    ocl_function=make_ocl("return sas_sinx_x(q);", "sas_sinc"),
)
add_function(
    name="cos(x)",
    mp_function=mp.cos,
    np_function=np.cos,
    #ocl_function=make_ocl("double sn, cn; SINCOS(q,sn,cn); return cn;", "sas_cos"),
    ocl_function=make_ocl("return cos(q);", "sas_cos"),
)
add_function(
    name="gamma(x)",
    mp_function=mp.gamma,
    np_function=scipy.special.gamma,
    ocl_function=make_ocl("return sas_gamma(q);", "sas_gamma", ["lib/sas_gamma.c"]),
    limits=(-3.1, 10),
)
add_function(
    name="erf(x)",
    mp_function=mp.erf,
    np_function=scipy.special.erf,
    ocl_function=make_ocl("return sas_erf(q);", "sas_erf", ["lib/polevl.c", "lib/sas_erf.c"]),
    limits=(-5., 5.),
)
add_function(
    name="erfc(x)",
    mp_function=mp.erfc,
    np_function=scipy.special.erfc,
    ocl_function=make_ocl("return sas_erfc(q);", "sas_erfc", ["lib/polevl.c", "lib/sas_erf.c"]),
    limits=(-5., 5.),
)
add_function(
    name="arctan(x)",
    mp_function=mp.atan,
    np_function=np.arctan,
    ocl_function=make_ocl("return atan(q);", "sas_arctan"),
)
add_function(
    name="3 j1(x)/x",
    mp_function=lambda x: 3*(mp.sin(x)/x - mp.cos(x))/(x*x),
    # Note: no taylor expansion near 0
    np_function=lambda x: 3*(np.sin(x)/x - np.cos(x))/(x*x),
    ocl_function=make_ocl("return sas_3j1x_x(q);", "sas_j1c", ["lib/sas_3j1x_x.c"]),
)
add_function(
    name="(1-cos(x))/x^2",
    mp_function=lambda x: (1 - mp.cos(x))/(x*x),
    np_function=lambda x: (1 - np.cos(x))/(x*x),
    ocl_function=make_ocl("return (1-cos(q))/q/q;", "sas_1mcosx_x2"),
)
add_function(
    name="(1-sin(x)/x)/x",
    mp_function=lambda x: 1/x - mp.sin(x)/(x*x),
    np_function=lambda x: 1/x - np.sin(x)/(x*x),
    ocl_function=make_ocl("return (1-sas_sinx_x(q))/q;", "sas_1msinx_x_x"),
)
add_function(
    name="(1/2+(1-cos(x))/x^2-sin(x)/x)/x",
    mp_function=lambda x: (0.5 - mp.sin(x)/x + (1-mp.cos(x))/(x*x))/x,
    np_function=lambda x: (0.5 - np.sin(x)/x + (1-np.cos(x))/(x*x))/x,
    ocl_function=make_ocl("return (0.5-sin(q)/q + (1-cos(q))/q/q)/q;", "sas_T2"),
)
add_function(
    name="fmod_2pi",
    mp_function=lambda x: mp.fmod(x, 2*mp.pi),
    np_function=lambda x: np.fmod(x, 2*np.pi),
    ocl_function=make_ocl("return fmod(q, 2*M_PI);", "sas_fmod"),
)
add_function(
    name="debye",
    mp_function=lambda x: 2*(mp.exp(-x**2) + x**2 - 1)/x**4,
    np_function=lambda x: 2*(np.expm1(-x**2) + x**2)/x**4,
    ocl_function=make_ocl("""
    const double qsq = q*q;
    if (qsq < 1.0) { // Pade approximation
        const double x = qsq;
        if (0) { // 0.36 single
            // PadeApproximant[2*Exp[-x^2] + x^2-1)/x^4, {x, 0, 4}]
            return (x*x/180. + 1.)/((1./30.*x + 1./3.)*x + 1);
        } else if (0) { // 1.0 for single
            // padeapproximant[2*exp[-x^2] + x^2-1)/x^4, {x, 0, 6}]
            const double A1=1./24., A2=1./84, A3=-1./3360;
            const double B1=3./8., B2=3./56., B3=1./336.;
            return (((A3*x + A2)*x + A1)*x + 1.)/(((B3*x + B2)*x + B1)*x + 1.);
        } else if (1) { // 1.0 for single, 0.25 for double
            // PadeApproximant[2*Exp[-x^2] + x^2-1)/x^4, {x, 0, 8}]
            const double A1=1./15., A2=1./60, A3=0., A4=1./75600.;
            const double B1=2./5., B2=1./15., B3=1./180., B4=1./5040.;
            return ((((A4*x + A3)*x + A2)*x + A1)*x + 1.)
                  /((((B4*x + B3)*x + B2)*x + B1)*x + 1.);
        } else { // 1.0 for single, 0.5 for double
            // PadeApproximant[2*Exp[-x^2] + x^2-1)/x^4, {x, 0, 8}]
            const double A1=1./12., A2=2./99., A3=1./2640., A4=1./23760., A5=-1./1995840.;
            const double B1=5./12., B2=5./66., B3=1./132., B4=1./2376., B5=1./95040.;
            return (((((A5*x + A4)*x + A3)*x + A2)*x + A1)*x + 1.)
                  /(((((B5*x + B4)*x + B3)*x + B2)*x + B1)*x + 1.);
        }
    } else if (qsq < 1.) { // Taylor series; 0.9 for single, 0.25 for double
        const double x = qsq;
        const double C0 = +1.;
        const double C1 = -1./3.;
        const double C2 = +1./12.;
        const double C3 = -1./60.;
        const double C4 = +1./360.;
        const double C5 = -1./2520.;
        const double C6 = +1./20160.;
        const double C7 = -1./181440.;
        //return ((((C5*x + C4)*x + C3)*x + C2)*x + C1)*x + C0;
        //return (((((C6*x + C5)*x + C4)*x + C3)*x + C2)*x + C1)*x + C0;
        return ((((((C7*x + C6)*x + C5)*x + C4)*x + C3)*x + C2)*x + C1)*x + C0;
    } else {
        return 2.*(expm1(-qsq) + qsq)/(qsq*qsq);
    }
    """, "sas_debye"),
)

RADIUS=3000
LENGTH=30
THETA=45
def mp_cyl(x):
    f = mp.mpf
    theta = f(THETA)*mp.pi/f(180)
    qr = x * f(RADIUS)*mp.sin(theta)
    qh = x * f(LENGTH)/f(2)*mp.cos(theta)
    be = f(2)*mp.j1(qr)/qr
    si = mp.sin(qh)/qh
    background = f(0)
    #background = f(1)/f(1000)
    volume = mp.pi*f(RADIUS)**f(2)*f(LENGTH)
    contrast = f(5)
    units = f(1)/f(10000)
    #return be
    #return si
    return units*(volume*contrast*be*si)**f(2)/volume + background
def np_cyl(x):
    f = np.float64 if x.dtype == np.float64 else np.float32
    theta = f(THETA)*f(np.pi)/f(180)
    qr = x * f(RADIUS)*np.sin(theta)
    qh = x * f(LENGTH)/f(2)*np.cos(theta)
    be = f(2)*scipy.special.j1(qr)/qr
    si = np.sin(qh)/qh
    background = f(0)
    #background = f(1)/f(1000)
    volume = f(np.pi)*f(RADIUS)**2*f(LENGTH)
    contrast = f(5)
    units = f(1)/f(10000)
    #return be
    #return si
    return units*(volume*contrast*be*si)**f(2)/volume + background
ocl_cyl = """\
    double THETA = %(THETA).15e*M_PI_180;
    double qr = q*%(RADIUS).15e*sin(THETA);
    double qh = q*0.5*%(LENGTH).15e*cos(THETA);
    double be = sas_2J1x_x(qr);
    double si = sas_sinx_x(qh);
    double background = 0;
    //double background = 0.001;
    double volume = M_PI*square(%(RADIUS).15e)*%(LENGTH).15e;
    double contrast = 5.0;
    double units = 1e-4;
    //return be;
    //return si;
    return units*square(volume*contrast*be*si)/volume + background;
"""%{"LENGTH":LENGTH, "RADIUS": RADIUS, "THETA": THETA}
add_function(
    name="cylinder(r=%g, l=%g, theta=%g)"%(RADIUS, LENGTH, THETA),
    mp_function=mp_cyl,
    np_function=np_cyl,
    ocl_function=make_ocl(ocl_cyl, "ocl_cyl", ["lib/polevl.c", "lib/sas_J1.c"]),
    shortname="cylinder",
    xaxis="$q/A^{-1}$",
)

lanczos_gamma = """\
    const double coeff[] = {
            76.18009172947146,     -86.50532032941677,
            24.01409824083091,     -1.231739572450155,
            0.1208650973866179e-2,-0.5395239384953e-5
            };
    const double x = q;
    double tmp  = x + 5.5;
    tmp -= (x + 0.5)*log(tmp);
    double ser = 1.000000000190015;
    for (int k=0; k < 6; k++) ser += coeff[k]/(x + k+1);
    return -tmp + log(2.5066282746310005*ser/x);
"""
add_function(
    name="log gamma(x)",
    mp_function=mp.loggamma,
    np_function=scipy.special.gammaln,
    ocl_function=make_ocl(lanczos_gamma, "lgamma"),
)

# Alternate versions of 3 j1(x)/x, for posterity
def taylor_3j1x_x(x):
    """
    Calculation using taylor series.
    """
    # Generate coefficients using the precision of the target value.
    n = 5
    cinv = [3991680, -45360, 840, -30, 3]
    three = x.dtype.type(3)
    p = three/np.array(cinv, x.dtype)
    return np.polyval(p[-n:], x*x)
add_function(
    name="3 j1(x)/x: taylor",
    mp_function=lambda x: 3*(mp.sin(x)/x - mp.cos(x))/(x*x),
    np_function=taylor_3j1x_x,
    ocl_function=make_ocl("return sas_3j1x_x(q);", "sas_j1c", ["lib/sas_3j1x_x.c"]),
)
def trig_3j1x_x(x):
    r"""
    Direct calculation using linear combination of sin/cos.

    Use the following trig identity:

    .. math::

        a \sin(x) + b \cos(x) = c \sin(x + \phi)

    where $c = \surd(a^2+b^2)$ and $\phi = \tan^{-1}(b/a) to calculate the
    numerator $\sin(x) - x\cos(x)$.
    """
    one = x.dtype.type(1)
    three = x.dtype.type(3)
    c = np.sqrt(one + x*x)
    phi = np.arctan2(-x, one)
    return three*(c*np.sin(x+phi))/(x*x*x)
add_function(
    name="3 j1(x)/x: trig",
    mp_function=lambda x: 3*(mp.sin(x)/x - mp.cos(x))/(x*x),
    np_function=trig_3j1x_x,
    ocl_function=make_ocl("return sas_3j1x_x(q);", "sas_j1c", ["lib/sas_3j1x_x.c"]),
)
def np_2J1x_x(x):
    """
    numpy implementation of 2J1(x)/x using single precision algorithm
    """
    # pylint: disable=bad-continuation
    f = x.dtype.type
    ax = abs(x)
    if ax < f(8.0):
        y = x*x
        ans1 = f(2)*(f(72362614232.0)
                  + y*(f(-7895059235.0)
                  + y*(f(242396853.1)
                  + y*(f(-2972611.439)
                  + y*(f(15704.48260)
                  + y*(f(-30.16036606)))))))
        ans2 = (f(144725228442.0)
                  + y*(f(2300535178.0)
                  + y*(f(18583304.74)
                  + y*(f(99447.43394)
                  + y*(f(376.9991397)
                  + y)))))
        return ans1/ans2
    else:
        y = f(64.0)/(ax*ax)
        xx = ax - f(2.356194491)
        ans1 = (f(1.0)
                  + y*(f(0.183105e-2)
                  + y*(f(-0.3516396496e-4)
                  + y*(f(0.2457520174e-5)
                  + y*f(-0.240337019e-6)))))
        ans2 = (f(0.04687499995)
                  + y*(f(-0.2002690873e-3)
                  + y*(f(0.8449199096e-5)
                  + y*(f(-0.88228987e-6)
                  + y*f(0.105787412e-6)))))
        sn, cn = np.sin(xx), np.cos(xx)
        ans = np.sqrt(f(0.636619772)/ax) * (cn*ans1 - (f(8.0)/ax)*sn*ans2) * f(2)/x
        return -ans if (x < f(0.0)) else ans
add_function(
    name="2 J1(x)/x:alt",
    mp_function=lambda x: 2*mp.j1(x)/x,
    np_function=lambda x: np.asarray([np_2J1x_x(v) for v in x], x.dtype),
    ocl_function=make_ocl("return sas_2J1x_x(q);", "sas_2J1x_x", ["lib/polevl.c", "lib/sas_J1.c"]),
)

ALL_FUNCTIONS = set(FUNCTIONS.keys())
ALL_FUNCTIONS.discard("loggamma")  # OCL version not ready yet
ALL_FUNCTIONS.discard("3j1/x:taylor")
ALL_FUNCTIONS.discard("3j1/x:trig")
ALL_FUNCTIONS.discard("2J1/x:alt")

# =============== MAIN PROGRAM ================

def usage():
    names = ", ".join(sorted(ALL_FUNCTIONS))
    print("""\
usage: precision.py [-f/a/r] [-x<range>] name...
where
    -f indicates that the function value should be plotted,
    -a indicates that the absolute error should be plotted,
    -r indicates that the relative error should be plotted (default),
    -x<range> indicates the steps in x, where <range> is one of the following
      log indicates log stepping in [10^-3, 10^5] (default)
      logq indicates log stepping in [10^-4, 10^1]
      linear indicates linear stepping in [1, 1000]
      zoom indicates linear stepping in [1000, 1010]
      neg indicates linear stepping in [-100.1, 100.1]
and name is "all [first]" or one of:
    """+names)
    sys.exit(1)

def main():
    import sys
    diff = "relative"
    xrange = "log"
    options = [v for v in sys.argv[1:] if v.startswith('-')]
    for opt in options:
        if opt == '-f':
            diff = "none"
        elif opt == '-r':
            diff = "relative"
        elif opt == '-a':
            diff = "absolute"
        elif opt.startswith('-x'):
            xrange = opt[2:]
        else:
            usage()

    names = [v for v in sys.argv[1:] if not v.startswith('-')]
    if not names:
        usage()

    if names[0] == "all":
        cutoff = names[1] if len(names) > 1 else ""
        names = list(sorted(ALL_FUNCTIONS))
        names = [k for k in names if k >= cutoff]
    if any(k not in FUNCTIONS for k in names):
        usage()
    multiple = len(names) > 1
    pylab.interactive(multiple)
    for k in names:
        pylab.clf()
        comparator = FUNCTIONS[k]
        comparator.run(xrange=xrange, diff=diff)
        if multiple:
            raw_input()
    if not multiple:
        pylab.show()

if __name__ == "__main__":
    main()