## Description

This interact illustrates numerical integration using Gaussian quadrature. The user may select how many points will be used to estimate the definite integral and a function to be integrated. The vertical bars are shaded to reflect the relative weights of the points, where a darker bar means a greater weight. The error is printed out and compared on a bar graph.

## Sage Cell

#### Code

import scipy
import numpy
from scipy.special.orthogonal import p_roots, t_roots, u_roots
from scipy.integrate import quad, trapz, simps
from sage.ext.fast_eval import fast_float
from numpy import linspace, asanyarray, diff
show_weight_graph=False
#  'Hermite': {'w': e**(-x**2), 'xmin': -numpy.inf, 'xmax': numpy.inf, 'func': h_roots},
#  'Laguerre': {'w': e**(-x), 'xmin': 0, 'xmax': numpy.inf, 'func': l_roots},

methods = {'Legendre': {'w': 1, 'xmin': -1, 'xmax': 1, 'func': p_roots},
'Chebyshev': {'w': 1/sqrt(1-x**2), 'xmin': -1, 'xmax': 1, 'func': t_roots},
'Chebyshev2': {'w': sqrt(1-x**2), 'xmin': -1, 'xmax': 1, 'func': u_roots},
'Trapezoid': {'w': 1, 'xmin': -1, 'xmax': 1,
'func': lambda n: (linspace(-1r,1,n), numpy.array([1.0r]+[2.0r]*(n-2)+[1.0r])*1.0r/n)},
'Simpson': {'w': 1, 'xmin': -1, 'xmax': 1,
'func': lambda n: (linspace(-1r,1,n),
numpy.array([1.0r]+[4.0r,2.0r]*int((n-3.0r)/2.0r)+[4.0r,1.0r])*2.0r/(3.0r*n))}}
var("x")
def box(center, height, area,**kwds):
width2 = 1.0*area/height/2.0
return polygon([(center-width2,0),
(center+width2,0),(center+width2,height),(center-width2,height)],**kwds)

@interact
def weights(n=slider(1,30,1,default=10),f=input_box(default=3*x+cos(10*x),type=SR),
show_method=["Legendre", "Chebyshev", "Chebyshev2", "Trapezoid","Simpson"]):
ff = fast_float(f,'x')
method = methods[show_method]
xcoords,w = (method['func'])(int(n))
xmin = method['xmin']
xmax = method['xmax']
plot_min = max(xmin, -10)
plot_max = min(xmax, 10)
scaled_func = f*method['w']
scaled_ff = fast_float(scaled_func, 'x')

coords = zip(xcoords,w)
max_weight = max(w)
coords_scaled = zip(xcoords,w/max_weight)

f_graph = plot(scaled_func,plot_min,plot_max)
boxes = sum(box(x,ff(x),w*ff(x),rgbcolor=(0.5,0.5,0.5),alpha=0.3) for x,w in coords)
stems = sum(line([(x,0),(x,scaled_ff(x))],rgbcolor=(1-y,1-y,1-y),
thickness=2,markersize=6,alpha=y) for x,y in coords_scaled)
points = sum([point([(x,0),
(x,scaled_ff(x))],rgbcolor='black',pointsize=30) for x,_ in coords])
graph = stems+points+f_graph+boxes
if show_weight_graph:
graph += line([(x,y) for x,y in coords_scaled], rgbcolor='green',alpha=0.4)

show(graph,xmin=plot_min,xmax=plot_max,aspect_ratio="auto")

approximation = sum([w*ff(x) for x,w in coords])
x_val = linspace(min(xcoords), max(xcoords),n)
y_val = [*map(scaled_ff,x_val)]
trapezoid = integral-trapz(y_val, x_val)
simpson = integral-simps(y_val, x_val)
pretty_print(html(r"$$\sum_{i=1}^{i=%s}w_i\left(%s\right)= %s\approx %s =\int_{-1}^{1}%s \,dx$$"%(n,
latex(f), approximation, integral, latex(scaled_func))))
error_data = [trapezoid, simpson, integral-approximation,integral_error]
print("Trapezoid: %s, Simpson: %s, \nMethod: %s, Real: %s" % tuple(error_data))
show(bar_chart(error_data,width=1),ymin=min(error_data), ymax=max(error_data))


none

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