def polynomial_print(m, o, c, e, title):
#*****************************************************************************80
#
## POLYNOMIAL_PRINT prints a polynomial.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 25 October 2014
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, integer M, the spatial dimension.
#
# Input, integer O, the "order" of the polynomial, that is,
# simply the number of terms.
#
# Input, real C(O), the coefficients.
#
# Input, integer E(O), the indices of the exponents.
#
# Input, string TITLE, a title.
#
from mono_unrank_grlex import mono_unrank_grlex
import sys
sys.stdout.write(title)
sys.stdout.write('\n')
if (o == 0):
sys.stdout.write(' 0.')
else:
for j in range(0, o):
sys.stdout.write(' ')
if (c[j] < 0):
sys.stdout.write('- ')
else:
sys.stdout.write('+ ')
sys.stdout.write(repr(abs(c[j])))
sys.stdout.write(' * x^(')
f = mono_unrank_grlex(m, e[j])
for i in range(0, m):
sys.stdout.write(repr(f[i]))
if (i < m - 1):
sys.stdout.write(',')
else:
sys.stdout.write(')')
if (j == o - 1):
sys.stdout.write('.')
sys.stdout.write('\n')
return
def polynomial_print ( m, o, c, e, title ):
#*****************************************************************************80
#
## POLYNOMIAL_PRINT prints a polynomial.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 25 October 2014
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, integer M, the spatial dimension.
#
# Input, integer O, the "order" of the polynomial, that is,
# simply the number of terms.
#
# Input, real C(O), the coefficients.
#
# Input, integer E(O), the indices of the exponents.
#
# Input, string TITLE, a title.
#
from mono_unrank_grlex import mono_unrank_grlex
import sys
sys.stdout.write ( title )
sys.stdout.write ( '\n' )
if ( o == 0 ):
sys.stdout.write ( ' 0.' )
else:
for j in range ( 0, o ):
sys.stdout.write ( ' ' )
if ( c[j] < 0 ):
sys.stdout.write ( '- ' )
else:
sys.stdout.write ( '+ ' )
sys.stdout.write ( repr ( abs ( c[j] ) ) )
sys.stdout.write ( ' * x^(' )
f = mono_unrank_grlex ( m, e[j] )
for i in range ( 0, m ):
sys.stdout.write ( repr ( f[i] ) )
if ( i < m - 1 ):
sys.stdout.write ( ',' )
else:
sys.stdout.write ( ')' )
if ( j == o - 1 ):
sys.stdout.write ( '.' )
sys.stdout.write ( '\n' )
return
def polynomial_value(m, o, c, e, nx, x):
#*****************************************************************************80
#
## POLYNOMIAL_VALUE evaluates a polynomial.
#
# Discussion:
#
# The polynomial is evaluated term by term, and no attempt is made to
# use an approach such as Horner's method to speed up the process.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 27 October 2014
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, int M, the spatial dimension.
#
# Input, integer O, the "order" of the polynomial.
#
# Input, real C[O], the coefficients of the scaled polynomial.
#
# Input, integer E[O], the indices of the exponents of
# the scaled polynomial.
#
# Input, integer NX, the number of evaluation points.
#
# Input, real X[M*NX], the coordinates of the evaluation points.
#
# Output, real V[NX], the value of the polynomial at X.
#
from mono_unrank_grlex import mono_unrank_grlex
from mono_value import mono_value
import numpy as np
p = np.zeros(nx, dtype=np.float64)
for j in range(0, o):
f = mono_unrank_grlex(m, e[j])
v = mono_value(m, nx, f, x)
for k in range(0, nx):
p[k] = p[k] + c[j] * v[k]
return p
def polynomial_value ( m, o, c, e, nx, x ):
#*****************************************************************************80
#
## POLYNOMIAL_VALUE evaluates a polynomial.
#
# Discussion:
#
# The polynomial is evaluated term by term, and no attempt is made to
# use an approach such as Horner's method to speed up the process.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 27 October 2014
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, int M, the spatial dimension.
#
# Input, integer O, the "order" of the polynomial.
#
# Input, real C[O], the coefficients of the scaled polynomial.
#
# Input, integer E[O], the indices of the exponents of
# the scaled polynomial.
#
# Input, integer NX, the number of evaluation points.
#
# Input, real X[M*NX], the coordinates of the evaluation points.
#
# Output, real V[NX], the value of the polynomial at X.
#
from mono_unrank_grlex import mono_unrank_grlex
from mono_value import mono_value
import numpy as np
p = np.zeros ( nx, dtype = np.float64 )
for j in range ( 0, o ):
f = mono_unrank_grlex ( m, e[j] )
v = mono_value ( m, nx, f, x )
for k in range ( 0, nx ):
p[k] = p[k] + c[j] * v[k]
return p
def mono_between_random(m, n1, n2, seed):
#*****************************************************************************80
#
## MONO_BETWEEN_RANDOM: random monomial with total degree between N1 and N2.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 25 October 2014
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, integer M, the spatial dimension.
#
# Input, integer N1, N2, the minimum and maximum degrees.
# 0 <= N1 <= N2.
#
# Input, integer SEED, the random number seed.
#
# Output integer X[M], the random monomial.
#
# Output integer RANK, the rank of the monomial.
#
# Output, integer SEED, the updated random number seed.
#
from i4_uniform_ab import i4_uniform_ab
from mono_unrank_grlex import mono_unrank_grlex
from mono_upto_enum import mono_upto_enum
n1_copy = max(n1, 0)
rank_min = mono_upto_enum(m, n1_copy - 1) + 1
rank_max = mono_upto_enum(m, n2)
rank, seed = i4_uniform_ab(rank_min, rank_max, seed)
x = mono_unrank_grlex(m, rank)
return x, rank, seed
def mono_total_random(m, n, seed):
#*****************************************************************************80
#
## MONO_TOTAL_RANDOM: random monomial with total degree equal to N.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 21 November 2013
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, integer M, the spatial dimension.
#
# Input, integer N, the degree.
# 0 <= N.
#
# Input, integer SEED, the random number seed.
#
# Output, integer X[M], the random monomial.
#
# Output, integer RANK, the rank of the monomial.
#
# Output, integer SEED, the random number seed.
#
from i4_uniform_ab import i4_uniform_ab
from mono_unrank_grlex import mono_unrank_grlex
from mono_upto_enum import mono_upto_enum
rank_min = mono_upto_enum(m, n - 1) + 1
rank_max = mono_upto_enum(m, n)
rank, seed = i4_uniform_ab(rank_min, rank_max, seed)
x = mono_unrank_grlex(m, rank)
return x, rank, seed
def mono_upto_random ( m, n, seed ): #*****************************************************************************80 # ## MONO_UPTO_RANDOM: random monomial with total degree less than or equal to N. # # Licensing: # # This code is distributed under the GNU LGPL license. # # Modified: # # 25 October 2014 # # Author: # # John Burkardt # # Parameters: # # Input, integer M, the spatial dimension. # # Input, integer N, the degree. # 0 <= N. # # Input, integer SEED, the random number seed. # # Output, integer X[M], the random monomial. # # Output, integer RANK, the rank of the monomial. # # Output, integer SEED, the random number seed. # from i4_uniform_ab import i4_uniform_ab from mono_unrank_grlex import mono_unrank_grlex from mono_upto_enum import mono_upto_enum rank_min = 1 rank_max = mono_upto_enum ( m, n ) rank, seed = i4_uniform_ab ( rank_min, rank_max, seed ) x = mono_unrank_grlex ( m, rank ) return x, rank, seed
def lpp_to_polynomial(m, l, o_max):
#*****************************************************************************80
#
## LPP_TO_POLYNOMIAL writes a Legendre Product Polynomial as a polynomial.
#
# Discussion:
#
# For example, if
# M = 3,
# L = ( 1, 0, 2 ),
# then
# L(1,0,2)(X,Y,Z)
# = L(1)(X) * L(0)(Y) * L(2)(Z)
# = X * 1 * ( 3Z^2-1)/2
# = - 1/2 X + (3/2) X Z^2
# so
# O = 2 (2 nonzero terms)
# C = -0.5
# 1.5
# E = 4 <-- index in 3-space of exponent (1,0,0)
# 15 <-- index in 3-space of exponent (1,0,2)
#
# The output value of O is no greater than
# O_MAX = product ( 1 <= I <= M ) (L(I)+2)/2
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 31 October 2014
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, int M, the spatial dimension.
#
# Input, int L[M], the index of each Legendre product
# polynomial factor. 0 <= L(*).
#
# Input, int O_MAX, an upper limit on the size of the
# output arrays.
# O_MAX = product ( 1 <= I <= M ) (L(I)+2)/2.
#
# Output, int O, the "order" of the polynomial product.
#
# Output, double C[O], the coefficients of the polynomial product.
#
# Output, int E[O], the indices of the exponents of the
# polynomial product.
#
from lp_coefficients import lp_coefficients
from mono_rank_grlex import mono_rank_grlex
from mono_unrank_grlex import mono_unrank_grlex
from polynomial_compress import polynomial_compress
from polynomial_sort import polynomial_sort
import numpy as np
c1 = np.zeros(o_max, dtype=np.float64)
c2 = np.zeros(o_max, dtype=np.float64)
e1 = np.zeros(o_max, dtype=np.int32)
e2 = np.zeros(o_max, dtype=np.int32)
f2 = np.zeros(o_max, dtype=np.int32)
pp = np.zeros(m, dtype=np.int32)
o1 = 1
c1[0] = 1.0
e1[0] = 1
#
# Implicate one factor at a time.
#
for i in range(0, m):
o2, c2, f2 = lp_coefficients(l[i])
o = 0
c = np.zeros(o_max, dtype=np.float64)
e = np.zeros(o_max, dtype=np.int32)
for j2 in range(0, o2):
for j1 in range(0, o1):
c[o] = c1[j1] * c2[j2]
if (0 < i):
p = mono_unrank_grlex(i, e1[j1])
for i2 in range(0, i):
pp[i2] = p[i2]
pp[i] = f2[j2]
e[o] = mono_rank_grlex(i + 1, pp)
o = o + 1
c, e = polynomial_sort(o, c, e)
o, c, e = polynomial_compress(o, c, e)
o1 = o
for i1 in range(0, o):
c1[i1] = c[i1]
e1[i1] = e[i1]
return o1, c1, e1
def lpp_to_polynomial ( m, l, o_max ):
#*****************************************************************************80
#
## LPP_TO_POLYNOMIAL writes a Legendre Product Polynomial as a polynomial.
#
# Discussion:
#
# For example, if
# M = 3,
# L = ( 1, 0, 2 ),
# then
# L(1,0,2)(X,Y,Z)
# = L(1)(X) * L(0)(Y) * L(2)(Z)
# = X * 1 * ( 3Z^2-1)/2
# = - 1/2 X + (3/2) X Z^2
# so
# O = 2 (2 nonzero terms)
# C = -0.5
# 1.5
# E = 4 <-- index in 3-space of exponent (1,0,0)
# 15 <-- index in 3-space of exponent (1,0,2)
#
# The output value of O is no greater than
# O_MAX = product ( 1 <= I <= M ) (L(I)+2)/2
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 31 October 2014
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, int M, the spatial dimension.
#
# Input, int L[M], the index of each Legendre product
# polynomial factor. 0 <= L(*).
#
# Input, int O_MAX, an upper limit on the size of the
# output arrays.
# O_MAX = product ( 1 <= I <= M ) (L(I)+2)/2.
#
# Output, int O, the "order" of the polynomial product.
#
# Output, double C[O], the coefficients of the polynomial product.
#
# Output, int E[O], the indices of the exponents of the
# polynomial product.
#
from lp_coefficients import lp_coefficients
from mono_rank_grlex import mono_rank_grlex
from mono_unrank_grlex import mono_unrank_grlex
from polynomial_compress import polynomial_compress
from polynomial_sort import polynomial_sort
import numpy as np
c1 = np.zeros ( o_max, dtype = np.float64 )
c2 = np.zeros ( o_max, dtype = np.float64 )
e1 = np.zeros ( o_max, dtype = np.int32 )
e2 = np.zeros ( o_max, dtype = np.int32 )
f2 = np.zeros ( o_max, dtype = np.int32 )
pp = np.zeros ( m, dtype = np.int32 )
o1 = 1
c1[0] = 1.0
e1[0] = 1
#
# Implicate one factor at a time.
#
for i in range ( 0, m ):
o2, c2, f2 = lp_coefficients ( l[i] );
o = 0;
c = np.zeros ( o_max, dtype = np.float64 )
e = np.zeros ( o_max, dtype = np.int32 )
for j2 in range ( 0, o2 ):
for j1 in range ( 0, o1 ):
c[o] = c1[j1] * c2[j2]
if ( 0 < i ):
p = mono_unrank_grlex ( i, e1[j1] )
for i2 in range ( 0, i ):
pp[i2] = p[i2]
pp[i] = f2[j2]
e[o] = mono_rank_grlex ( i + 1, pp )
o = o + 1
c, e = polynomial_sort ( o, c, e )
o, c, e = polynomial_compress ( o, c, e )
o1 = o;
for i1 in range ( 0, o ):
c1[i1] = c[i1]
e1[i1] = e[i1]
return o1, c1, e1