def check1():
I1, n1 = romberg(f1, 0, 1)
I2, n2 = romberg(f2, 0, 4)
print("Comparing Solutions\n")
print("The integral for the 1st equation using romberg's is:", I1, "and the number of evaluations are:", n1 )
print("The integral for the 2nd equation using romberg's is:", I2, "and the number of evaluations are:", n2, "\n" )
print("The percentage error in the 1st solution for 10^2 is", math.fabs((((I1 - result11) / I1 )*100)))
print("The percentage error in the 1st solution for 10^4 is", math.fabs((((I1 - result12) / I1 )*100)))
print("The percentage error in the 1st solution for 10^6 is", math.fabs((((I1 - result13) / I1 )*100)))
print("The percentage error in the 2nd solution for 10^2 is", math.fabs((((I2 - result21) / I2 )*100)))
print("The percentage error in the 2nd solution for 10^4 is", math.fabs((((I2 - result22) / I2 )*100)))
print("The percentage error in the 2nd solution for 10^6 is", math.fabs((((I2 - result23) / I2 )*100)))
print("\n")
def problem22():
Iold = 0.0
a = 0.0
b = 1.0
integral, npan = romberg(f2, 0, 4)
print("n", "\t", "h", "\t", "T_h(f)", "\t", "e_h", "\t", "e_2h/e_h")
for k in range(1,9):
Inew = trapezoid(f1, a, b, Iold, k)
if (k > 1) and (abs(Inew - Iold)) < 1.0e-10: break
eh2 = ((integral - Iold) / (integral - Inew))
Iold = Inew
n1 = (2**(k-1))
print(2**(k-1), "\t", '{0:.4f}'.format(((b-a)/n1)), "\t", '{0:.4f}'.format(Iold), "\t",'{0:.4f}'.format(math.fabs(integral - Iold)), "\t", '{0:.4f}'.format((eh2)), "\n")
# The u values that the function will be evaluated with
u = arange(0, 1.01, 0.05)
print(" u\t g(u)")
gu = [] #list that will contain all of g(u)s (y-coordinates for your plot)
# Iterating through the u list
for i in u:
# if the current i value is 0 then the integral will evaluate to 0
if i == 0:
g = 0.0
# Otherwise perform romberg integration
else:
I, nPanels = romberg(
f, 0,
1 / i) # perform romberg integration on f here (get result in
# I, nPanels is the number of panels used but is not
# used for output).
# g(i) is i^3 multiplied by the romberg integration of f(x)
g = I * i**3 # evaluate g(i) here
print('{:6.2f}{:13.6f}'.format(i, g))
gu.append(g)
#
# Place the code that creates the required plot using pylab here.
# Be sure to label axes and provide the same title as shown
# in the "prob6_1-14.png" image file on BB
# Setting the y axis label
plt.ylabel('g(u)')
# Setting the x axis label
from romberg import *
import math
def f(x):
return math.sin(x)**-.5
def F(t):
return 2 * (1 - t**4)**-.5
I, n = romberg(F, math.sin(0)**.5, math.sin(math.pi / 4)**.5)
print("Integral =", I)
print("nPanels =", n)
'''
::: CODE RUN :::
cheng@Cheng:~/code/na/romberg$ python3 main.py
Integral = 1.7911613389539645
nPanels = 64
'''
return 2 * sp.jv(3, 2.7 * x) * sp.jv(3, 2.7 * x) * x
def integ2(x):
return 2 * pow(abs(sp.jv(3, 2.7) / sp.kv(3, 1.2)), 2) * sp.kv(3, 1.2 * x) * sp.kv(3, 1.2 * x) * x
def total_integ(x):
if x <= 1:
return integ1(x)
else:
return integ2(x)
ss, j, y, dy, index = romberg(total_integ, 0, 15, full=True)
ANS = 0.046038860370705266
y = array(y)
temp = [ANS] * len(index)
temp = array(temp)
dy = array(dy)
index = array(index)
loglog(pow(2, index), abs(dy), "k", color="r")
ss, j, y, dy, index = qromb3(total_integ, 0, 15, full=True)
y = array(y)
temp = [ANS] * len(index)
temp = array(temp)
dy = array(dy)
index = array(index)
## example6_7
from math import cos, sqrt, pi
from romberg import *
def f(x):
return 2.0 * (x**2) * cos(x**2)
I, n = romberg(f, 0, sqrt(pi))
print "Integral =", I
print "numEvals =", n
raw_input("\nPress return to exit")
#Imports
from romberg import * # Have the romberg file and the trapezoid integration file provided on study direct in the same folder as your code.
#Constants
LowerLimit = 0
UpperLimit = 10
#Defining Function
def function(x):
return (x**2)
#Returns the integral and the number of panels used. Unless stated otherwise, leave the last term as "tol=1.0e-6". tol stands for tolerance, and controls the size of the error.
#Answer[0] = Numerical value of definite integration.
#Answer[1] = Number of panels (Number of trapeziums used).
Answer = romberg(function, LowerLimit, UpperLimit, tol=1.0e-6)
print Answer
def f(x): return eval(fninput.getText())
print(type(f))
Iold=0.0
a=eval(lower.getText())
print(type(a))
b=eval(upper.getText())
print(type(b))
for k in range(1,21):
Inew=trapezoid(f,a,b,Iold,k)
if (k>1) and (abs(Inew-Iold))<1.0e-6: break
Iold=Inew
print("Trapezoidal Integral =",Inew)
answer=Inew
showAnswer()
#Romberg
if go.y>310 and go.y<385:
from romberg import *
def f(x): return eval(fninput.getText())
print(type(f))
I,n=romberg(f,eval(lower.getText()),eval(upper.getText()))
print("Romberg Integral =",I)
answer=I
showAnswer()
#Gaussian
if go.y>420 and go.y<495:
pass
## else:
from numpy import *
import scipy.weave as weave
from trapz import *
from week0 import *
from romberg import *
from trapz import *
import scipy.special as sp
from matplotlib.pyplot import *
def integ1(x) :
return 2*sp.jv(3,2.7*x)*sp.jv(3,2.7*x)*x
def integ2(x):
return 2*pow(abs(sp.jv(3,2.7)/sp.kv(3,1.2)),2)*sp.kv(3,1.2*x)*sp.kv(3,1.2*x)*x
def total_integ(x):
if x<=1 :
return integ1(x)
else:
return integ2(x)
def integ1_mod(x):
return sp.jv(3,2.7*x**0.5)*sp.jv(3,2.7*x**0.5)
print romberg(integ1,0,1,EPS =1e-15)
print romberg(integ1_mod,0,1,EPS = 1e-15)
import numpy as np
func = lambda x: np.sqrt(x) * np.cos(x)
## Recursive Trapezoidal Rule
from trapezoid import *
Iold = 0.0
a = 0.0
b = np.pi
for k in range(1, 10):
Inew = trapezoid(func, a, b, Iold, k)
Iold = Inew
print('Inew : ', Inew, ' Panel : ', 2**(k - 1))
## Romberg Integration
from romberg import *
print(romberg(func, a, b, tol=1.0e-06))
import scipy.special as sp
from matplotlib.pyplot import *
def integ1(x) :
return 2*sp.jv(3,2.7*x)*sp.jv(3,2.7*x)*x
def integ2(x):
return 2*pow(abs(sp.jv(3,2.7)/sp.kv(3,1.2)),2)*sp.kv(3,1.2*x)*sp.kv(3,1.2*x)*x
def total_integ(x):
if x<=1 :
return integ1(x)
else:
return integ2(x)
romb = romberg(total_integ,0,15)
print "integration value = %f , no. of function calls = %d" % (romb[0],pow(2,romb[1]))
a = romberg(integ1,0,1)
b = romberg(integ2,1,15)
ans = a[0]+b[0]
calls = pow(2,a[1]-1)+pow(2,b[1]-1)
print "integration value with splitting = %f no . of function calls = %d "%(ans,calls)
ss,j,y,dy,index = romberg(total_integ,0,15,full = True)
ANS = 0.046038860370705266
print index, dy
y = array(y)
temp = [ANS]*len(index)
temp = array(temp)
dy = array(dy)
index = array(index)
# Evaluates the integrand at the provided value of x
def f(x):
if x == 0: return 0
else: return (x**4 * exp(x)) / ((exp(x) - 1)**2)
# Sets initial data and creates a list for the values of g(u)
u = arange(0, 1.01, 0.05)
print(" u\t g(u)")
gu = []
for i in u:
if i == 0: g = 0.0
else:
# Performs Romberg integration on f
I, nPanels = romberg(f, 0, 1 / i)
# Evaluates g(i)
g = i**3 * I
# Prints the values of u and g(u)
print('{:6.2f}{:13.6f}'.format(i, g))
gu.append(g)
# Plots g(u) vs u and saves the plot to prob6_1_14.png
plt.plot(u, gu, "b-")
plt.xlabel("u")
plt.ylabel("g(u)")
plt.xlim(xmin=0.0, xmax=1.0)
plt.ylim(ymin=0.0)
plt.title("Problem 6.1.14")
plt.savefig("prob6_1_14.png")
plt.show()
## example6_7
from math import cos,sqrt,pi
from romberg import *
def f(x): return 2.0*(x**2)*cos(x**2)
I,n = romberg(f,0,sqrt(pi))
print "Integral =",I
print "numEvals =",n
raw_input("\nPress return to exit")
#Imports from romberg import * # Have the romberg file and the trapezoid integration file provided on study direct in the same folder as your code. #Constants LowerLimit = 0 UpperLimit = 10 #Defining Function def function(x): return (x**2) #Returns the integral and the number of panels used. Unless stated otherwise, leave the last term as "tol=1.0e-6". tol stands for tolerance, and controls the size of the error. #Answer[0] = Numerical value of definite integration. #Answer[1] = Number of panels (Number of trapeziums used). Answer = romberg(function,LowerLimit,UpperLimit,tol=1.0e-6) print Answer