def kronrod_test02 ( ):
#*****************************************************************************80
#
## KRONROD_TEST02 tests the code for the even case N = 4.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 30 April 2013
#
# Author:
#
# John Burkardt
#
import numpy
from kronrod import kronrod
n = 4
print ''
print 'KRONROD_TEST02'
print ' Request KRONROD to compute the Gauss rule'
print ' of order 4, and the Kronrod extension of'
print ' order 4+5=9.'
tol = 0.000001
[ x, w1, w2 ] = kronrod ( n, tol )
print ''
print ' KRONROD returns 3 vectors of length %d' % ( n + 1 )
print ''
print ' I X WK WG'
print ''
for i in range ( 1, n + 2 ):
print ' %4d %14f %14f %14f' % ( i, x[i-1], w1[i-1], w2[i-1] )
return
def kronrod_test02():
#*****************************************************************************80
#
## KRONROD_TEST02 tests the code for the even case N = 4.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 30 April 2013
#
# Author:
#
# John Burkardt
#
import numpy
from kronrod import kronrod
n = 4
print ''
print 'KRONROD_TEST02'
print ' Request KRONROD to compute the Gauss rule'
print ' of order 4, and the Kronrod extension of'
print ' order 4+5=9.'
tol = 0.000001
[x, w1, w2] = kronrod(n, tol)
print ''
print ' KRONROD returns 3 vectors of length %d' % (n + 1)
print ''
print ' I X WK WG'
print ''
for i in range(1, n + 2):
print ' %4d %14f %14f %14f' % (i, x[i - 1], w1[i - 1], w2[i - 1])
return
def kronrod_test01 ( ):
#*****************************************************************************80
#
## KRONROD_TEST01 tests the code for the odd case N = 3.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 30 April 2013
#
# Author:
#
# John Burkardt
#
import numpy
from kronrod import kronrod
n = 3
wg = numpy.array ( [ \
0.555555555555555555556, \
0.888888888888888888889, \
0.555555555555555555556 ] )
wk = numpy.array ( [ \
0.104656226026467265194, \
0.268488089868333440729, \
0.401397414775962222905, \
0.450916538658474142345, \
0.401397414775962222905, \
0.268488089868333440729, \
0.104656226026467265194 ] )
xg = numpy.array ( [ \
-0.77459666924148337704, \
0.0, \
0.77459666924148337704 ] )
xk = numpy.array ( [ \
-0.96049126870802028342, \
-0.77459666924148337704, \
-0.43424374934680255800, \
0.0, \
0.43424374934680255800, \
0.77459666924148337704, \
0.96049126870802028342 ] )
print ''
print 'KRONROD_TEST01'
print ' Request KRONROD to compute the Gauss rule'
print ' of order 3, and the Kronrod extension of'
print ' order 3+4=7.'
print ''
print ' Compare to exact data.'
tol = 0.000001
[ x, w1, w2 ] = kronrod ( n, tol )
print ''
print ' KRONROD returns 3 vectors of length %d' % ( n + 1 )
print ''
print ' I X WK WG'
print ''
for i in range ( 1, n + 2 ):
print ' %4d %14f %14f %14f' % ( i, x[i-1], w1[i-1], w2[i-1] )
print ''
print ' Gauss Abscissas'
print ' Exact Computed'
print ''
for i in range ( 1, n + 1 ):
if ( 2 * i <= n + 1 ):
i2 = 2 * i
s = -1.0
else:
i2 = 2 * ( n + 1 ) - 2 * i
s = +1.0
print ' %4d %14f %14f' % ( i, xg[i-1], s * x[i2-1] )
print ''
print ' Gauss Weights'
print ' Exact Computed'
print ''
for i in range ( 1, n + 1 ):
if ( 2 * i <= n + 1 ):
i2 = 2 * i
else:
i2 = 2 * ( n + 1 ) - 2 * i
print ' %4d %14f %14f' % ( i, wg[i-1], w2[i2-1] )
print ''
print ' Gauss Kronrod Abscissas'
print ' Exact Computed'
print ''
for i in range ( 1, 2 * n + 2 ):
if ( i <= n + 1 ):
i2 = i
s = -1.0
else:
i2 = 2 * ( n + 1 ) - i
s = +1.0
print ' %4d %14f %14f' % ( i, xk[i-1], s * x[i2-1] )
print ''
print ' Gauss Kronrod Weights'
print ' Exact Computed'
print ''
for i in range ( 1, 2 * n + 2 ):
if ( i <= n + 1 ):
i2 = i
else:
i2 = 2 * ( n + 1 ) - i
print ' %4d %14f %14f' % ( i, wk[i-1], w1[i2-1] )
return
def kronrod_test03 ( ):
#*****************************************************************************80
#
## KRONROD_TEST03 uses the program to estimate an integral.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 30 April 2013
#
# Author:
#
# John Burkardt
#
from f import f
from kronrod import kronrod
exact = 1.5643964440690497731
print ''
print 'KRONROD_TEST03'
print ' Call Kronrod to estimate the integral of a function.'
print ' Keep trying until the error is small.'
#
# TOL just tells KRONROD how carefully it must compute X, W1 and W2.
# It is NOT a statement about the accuracy of your integral estimate!
#
tol = 0.000001
#
# Start the process with a 1 point rule.
#
n = 1
while ( 1 ):
[ x, w1, w2 ] = kronrod ( n, tol )
#
# Compute the estimates.
# There are two complications here:
#
# 1) Both rules use all the points. However, the lower order rule uses
# a zero weight for the points it doesn't need.
#
# 2) The points X are all positive, and are listed in descending order.
# this means that 0 is always in the list, and always occurs as the
# last member. Therefore, the integral estimates should use the
# function value at 0 once, and the function values at the other
# X values "twice", that is, once at X and once at -X.
#
i1 = w1[n+1-1] * f ( x[n+1-1] )
i2 = w2[n+1-1] * f ( x[n+1-1] )
for i in range ( 1, n + 1 ):
i1 = i1 + w1[i-1] * ( f ( - x[i-1] ) + f ( x[i-1] ) )
i2 = i2 + w2[i-1] * ( f ( - x[i-1] ) + f ( x[i-1] ) )
if ( abs ( i1 - i2 ) < 0.0001 ):
print ''
print ' Error tolerance satisfied with N = %d' % ( n )
print ' Coarse integral estimate = %14.6g' % ( i1 )
print ' Fine integral estimate = %14.6g' % ( i2 )
print ' Error estimate = %g' % ( abs ( i2 - i1 ) )
print ' Actual error = %g' % ( abs ( exact - i2 ) )
break
if ( 25 < n ):
print ''
print ' Error tolerance failed even for n = %d' % ( n )
print ' Canceling iteration, and accepting bad estimates!'
print ' Coarse integral estimate = %14.6g' % ( i1 )
print ' Fine integral estimate = %14.6g' % ( i2 )
print ' Error estimate = %g' % ( abs ( i2 - i1 ) )
print ' Actual error = %g' % ( abs ( exact - i2 ) )
break
n = 2 * n + 1
return
def F(x):
return np.exp(-x**2)
n = 1 # Setting the Order
tol = 1e-6 # Tolerance
Err = 1 # Error of the difference of Kronrod and Gauss
while Err > tol:
abscissa = [] # an array for abscissa
WK = [] # an array for Kronrod Weights
WG = [] # an array for Gauss Weights
# Calling Kronrod to get abscissa [0,1] and weights
abscissa, WK, WG = kronrod(n, tol)
abscissa_rev = abscissa[::-1] # make a reverse of abscissa
# making the abscissa to [-1,1]
for i in range(n):
abscissa = np.append(abscissa,
2 * abscissa_rev[0] - abscissa_rev[i + 1])
WG_ = []
WK_ = []
for i in range(n):
WG_ = np.append(WG_, WG[i])
WK_ = np.append(WK_, WK[i])
WG_rev = WG_[::-1]
WK_rev = WK_[::-1]
for i in range(n):
WK = np.append(WK, WK_rev[i])
WG = np.append(WG, WG_rev[i])
