def triangle01_poly_integral(d, p):
#*****************************************************************************80
#
## TRIANGLE01_POLY_INTEGRAL: polynomial integral over the unit triangle.
#
# Location:
#
# http://people.sc.fsu.edu/~jburkardt/py_src/triangle_integrals/triangle01_poly_integral.py
#
# Discussion:
#
# The unit triangle is T = ( (0,0), (1,0), (0,1) ).
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 23 April 2015
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, integer D, the degree of the polynomial.
#
# Input, real P(M), the polynomial coefficients.
# M = ((D+1)*(D+2))/2.
#
# Output, real Q, the integral.
#
from i4_to_pascal import i4_to_pascal
from triangle01_monomial_integral import triangle01_monomial_integral
m = ((d + 1) * (d + 2)) / 2
q = 0.0
for k in range(1, m + 1):
km1 = k - 1
i, j = i4_to_pascal(k)
qk = triangle01_monomial_integral(i, j)
q = q + p[km1] * qk
return q
def triangle01_poly_integral ( d, p ):
#*****************************************************************************80
#
## TRIANGLE01_POLY_INTEGRAL: polynomial integral over the unit triangle.
#
# Location:
#
# http://people.sc.fsu.edu/~jburkardt/py_src/triangle_integrals/triangle01_poly_integral.py
#
# Discussion:
#
# The unit triangle is T = ( (0,0), (1,0), (0,1) ).
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 23 April 2015
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, integer D, the degree of the polynomial.
#
# Input, real P(M), the polynomial coefficients.
# M = ((D+1)*(D+2))/2.
#
# Output, real Q, the integral.
#
from i4_to_pascal import i4_to_pascal
from triangle01_monomial_integral import triangle01_monomial_integral
m = ( ( d + 1 ) * ( d + 2 ) ) / 2
q = 0.0
for k in range ( 1, m + 1 ):
km1 = k - 1
i, j = i4_to_pascal ( k )
qk = triangle01_monomial_integral ( i, j )
q = q + p[km1] * qk
return q
def triangle01_poly_integral_test():
#*****************************************************************************80
#
## TRIANGLE01_POLY_INTEGRAL_TEST: polynomial integrals over the unit triangle.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 23 April 2015
#
# Author:
#
# John Burkardt
#
import numpy as np
from i4_to_pascal import i4_to_pascal
from poly_print import poly_print
from triangle01_monomial_integral import triangle01_monomial_integral
d_max = 6
k_max = ((d_max + 1) * (d_max + 2)) / 2
qm = np.zeros(k_max)
for k in range(1, k_max + 1):
km1 = k - 1
i, j = i4_to_pascal(k)
qm[km1] = triangle01_monomial_integral(i, j)
print ''
print 'TRIANGLE01_POLY_INTEGRAL_TEST'
print ' TRIANGLE01_POLY_INTEGRAL returns the integral Q of'
print ' a polynomial P(X,Y) over the interior of the unit triangle.'
d = 1
m = ((d + 1) * (d + 2)) / 2
p = np.array([1.0, 2.0, 3.0])
print ''
poly_print(d, p, ' p(x,y)')
q = triangle01_poly_integral(d, p)
print ''
print ' Q = %g' % (q)
q2 = np.dot(p, qm[0:m])
print ' Q (exact) = %g' % (q2)
d = 2
m = ((d + 1) * (d + 2)) / 2
p = np.array([0.0, 0.0, 0.0, 0.0, 1.0, 0.0])
print ''
poly_print(d, p, ' p(x,y)')
q = triangle01_poly_integral(d, p)
print ''
print ' Q = %g' % (q)
q2 = np.dot(p, qm[0:m])
print ' Q (exact) = %g' % (q2)
d = 2
m = ((d + 1) * (d + 2)) / 2
p = np.array([1.0, -2.0, 3.0, -4.0, 5.0, -6.0])
print ''
poly_print(d, p, ' p(x,y)')
q = triangle01_poly_integral(d, p)
print ''
print ' Q = %g' % (q)
q2 = np.dot(p, qm[0:m])
print ' Q (exact) = %g' % (q2)
#
# Terminate.
#
print ''
print 'TRIANGLE01_POLY_INTEGRAL_TEST'
print ' Normal end of execution.'
return
def poly_print ( d, p, title ):
#*****************************************************************************80
#
## POLY_PRINT prints an XY polynomial.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 17 April 2015
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, integer D, the degree of the polynomial.
#
# Output, real P(M), the coefficients of all monomials of degree 0 through D.
# P must contain ((D+1)*(D+2))/2 entries.
#
# Input, string TITLE, a title string.
#
from i4_to_pascal import i4_to_pascal
m = ( ( d + 1 ) * ( d + 2 ) ) / 2
allzero = True
for i in range ( 0, m ):
if ( p[i] != 0.0 ):
allzero = False
if ( allzero ):
print '%s = 0.0' % ( title )
else:
print '%s = ' % ( title )
for k in range ( 1, m + 1 ):
i, j = i4_to_pascal ( k )
di = i + j
km1 = k - 1
if ( p[km1] != 0.0 ):
if ( p[km1] < 0.0 ):
print ' -%g' % ( abs ( p[km1] ) ),
else:
print ' +%g' % ( p[km1] ),
if ( di != 0 ):
print '',
#
# "PASS" does nothing, but Python requires a statement here.
#
# Python idiotically insists on putting a space between successive prints,
# and I don't feel like calling sys.stdout.write ( string ) in order to
# suppress something I shouldn't have to deal with in the first place,
# so pretend you like it.
#
if ( i == 0 ):
pass
elif ( i == 1 ):
print 'x',
else:
print 'x^%d' % ( i ),
if ( j == 0 ):
pass
elif ( j == 1 ):
print 'y',
else:
print 'y^%d' % ( j ),
print ''
return
def triangle01_poly_integral_test ( ):
#*****************************************************************************80
#
## TRIANGLE01_POLY_INTEGRAL_TEST: polynomial integrals over the unit triangle.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 23 April 2015
#
# Author:
#
# John Burkardt
#
import numpy as np
from i4_to_pascal import i4_to_pascal
from poly_print import poly_print
from triangle01_monomial_integral import triangle01_monomial_integral
d_max = 6
k_max = ( ( d_max + 1 ) * ( d_max + 2 ) ) / 2
qm = np.zeros ( k_max )
for k in range ( 1, k_max + 1 ):
km1 = k - 1
i, j = i4_to_pascal ( k )
qm[km1] = triangle01_monomial_integral ( i, j )
print ''
print 'TRIANGLE01_POLY_INTEGRAL_TEST'
print ' TRIANGLE01_POLY_INTEGRAL returns the integral Q of'
print ' a polynomial P(X,Y) over the interior of the unit triangle.'
d = 1
m = ( ( d + 1 ) * ( d + 2 ) ) / 2
p = np.array ( [ 1.0, 2.0, 3.0 ] )
print ''
poly_print ( d, p, ' p(x,y)' )
q = triangle01_poly_integral ( d, p )
print ''
print ' Q = %g' % ( q )
q2 = np.dot ( p, qm[0:m] )
print ' Q (exact) = %g' % ( q2 )
d = 2
m = ( ( d + 1 ) * ( d + 2 ) ) / 2
p = np.array ( [ 0.0, 0.0, 0.0, 0.0, 1.0, 0.0 ] )
print ''
poly_print ( d, p, ' p(x,y)' )
q = triangle01_poly_integral ( d, p )
print ''
print ' Q = %g' % ( q )
q2 = np.dot ( p, qm[0:m] )
print ' Q (exact) = %g' % ( q2 )
d = 2
m = ( ( d + 1 ) * ( d + 2 ) ) / 2
p = np.array ( [ 1.0, -2.0, 3.0, -4.0, 5.0, -6.0 ] )
print ''
poly_print ( d, p, ' p(x,y)' )
q = triangle01_poly_integral ( d, p )
print ''
print ' Q = %g' % ( q )
q2 = np.dot ( p, qm[0:m] )
print ' Q (exact) = %g' % ( q2 )
#
# Terminate.
#
print ''
print 'TRIANGLE01_POLY_INTEGRAL_TEST'
print ' Normal end of execution.'
return
def poly_product ( d1, p1, d2, p2 ):
#*****************************************************************************80
#
#% POLY_PRODUCT computes P3(x,y) = P1(x,y) * P2(x,y) for polynomials.
#
# Location:
#
# http://people.sc.fsu.edu/~jburkardt/py_src/triangle_integrals/poly_product.py
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 22 April 2015
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, integer D1, the degree of factor 1.
#
# Input, real P1(M1), the factor 1 coefficients.
# M1 = ((D1+1)*(D1+2))/2.
#
# Input, integer D2, the degree of factor 2.
#
# Input, real P2(M2), the factor2 coefficients.
# M2 = ((D2+1)*(D2+2))/2.
#
# Output, integer D3, the degree of the result.
#
# Output, real P3(M3), the result coefficients.
# M3 = ((D3+1)*(D3+2))/2.
#
import numpy as np
from i4_to_pascal import i4_to_pascal
from pascal_to_i4 import pascal_to_i4
d3 = d1 + d2
m3 = ( ( d3 + 1 ) * ( d3 + 2 ) ) / 2
p3 = np.zeros ( m3 )
#
# Consider each entry in P1:
# P1(K1) * X^I1 * Y^J1
# and multiply it by each entry in P2:
# P2(K2) * X^I2 * Y^J2
# getting
# P3(K3) = P3(K3) + P1(K1) * P2(X2) * X^(I1+I2) * Y(J1+J2)
#
m1 = ( ( d1 + 1 ) * ( d1 + 2 ) ) / 2
m2 = ( ( d2 + 1 ) * ( d2 + 2 ) ) / 2
for k1m1 in range ( 0, m1 ):
k1 = k1m1 + 1
i1, j1 = i4_to_pascal ( k1 )
for k2m1 in range ( 0, m2 ):
k2 = k2m1 + 1
i2, j2 = i4_to_pascal ( k2 )
i3 = i1 + i2
j3 = j1 + j2
k3 = pascal_to_i4 ( i3, j3 )
k3m1 = k3 - 1
p3[k3m1] = p3[k3m1] + p1[k1m1] * p2[k2m1]
return d3, p3
def poly_product(d1, p1, d2, p2):
#*****************************************************************************80
#
#% POLY_PRODUCT computes P3(x,y) = P1(x,y) * P2(x,y) for polynomials.
#
# Location:
#
# http://people.sc.fsu.edu/~jburkardt/py_src/triangle_integrals/poly_product.py
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 22 April 2015
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, integer D1, the degree of factor 1.
#
# Input, real P1(M1), the factor 1 coefficients.
# M1 = ((D1+1)*(D1+2))/2.
#
# Input, integer D2, the degree of factor 2.
#
# Input, real P2(M2), the factor2 coefficients.
# M2 = ((D2+1)*(D2+2))/2.
#
# Output, integer D3, the degree of the result.
#
# Output, real P3(M3), the result coefficients.
# M3 = ((D3+1)*(D3+2))/2.
#
import numpy as np
from i4_to_pascal import i4_to_pascal
from pascal_to_i4 import pascal_to_i4
d3 = d1 + d2
m3 = ((d3 + 1) * (d3 + 2)) / 2
p3 = np.zeros(m3)
#
# Consider each entry in P1:
# P1(K1) * X^I1 * Y^J1
# and multiply it by each entry in P2:
# P2(K2) * X^I2 * Y^J2
# getting
# P3(K3) = P3(K3) + P1(K1) * P2(X2) * X^(I1+I2) * Y(J1+J2)
#
m1 = ((d1 + 1) * (d1 + 2)) / 2
m2 = ((d2 + 1) * (d2 + 2)) / 2
for k1m1 in range(0, m1):
k1 = k1m1 + 1
i1, j1 = i4_to_pascal(k1)
for k2m1 in range(0, m2):
k2 = k2m1 + 1
i2, j2 = i4_to_pascal(k2)
i3 = i1 + i2
j3 = j1 + j2
k3 = pascal_to_i4(i3, j3)
k3m1 = k3 - 1
p3[k3m1] = p3[k3m1] + p1[k1m1] * p2[k2m1]
return d3, p3