def gramsmidtcheby(n):
P=ansatz(n)
for m in range(len(P)):
for j in range(len(P[m])-1):
for i in range(len(P[j])):
tmp = fivepoint(lambda x:evalp(P[m],x)*evalp(P[j],x)/((1-x*x)**0.5),-1,1,10)
P[m][i]-=tmp*P[j][i]
P[m]=normalisecheby(P[m])
return P
def gramschmidt(n):
#generate n polynumials using the gram-schmidt orthonormal method
#P array containig all the polinomial coefficients
P=ansatz(n)
for m in range(len(P)):
for j in range(len(P[m])):
for i in range(len(P[j])):
P[m][i]-=fivepoint(lambda x:evalp(P[m],x)*evalp(P[j],x),-1,1,10)*P[j][i]
P[m]=normalise(P[m])
return P
def fivepoint(f, a, b, B = 1):
''' 5-point open-type Newton-Cotes integration with non-equidistant
points of support at [-5, -4, -2, 0, 2, 4, 5] where the boundary
points -5, 5 are ignored because of the open type '''
from math import fsum
def phi(t):
''' the interval transformation -5..5 -> a..b '''
return a + 0.1*(float(t)+5.)*(b-a)
if B > 1:
return fsum(map(lambda i: fivepoint(f, a+i*(b-a)/float(B), a+(i+1)*(b-a)/float(B)), xrange(B)))
return fsum(map(lambda x: coefficients[abs(x)//2]*f(phi(float(x))), [-4, -2, 0, 2, 4]))*(b-a)*0.1
def normalise(P): #normalise a polinomial, P contains the coefficients of the polynomial
for j in range(len(P)):
P/=fivepoint(lambda x:evalp(P,x)**2,-1,1,10)**0.5
return P
def func_prod(lhs, rhs, weightf=lambda x: 1., B=100, i=(-1.,1.)):
from fivepoint import fivepoint
return fivepoint(lambda x: lhs(x)*rhs(x)*weightf(x), i[0], i[1], B)
''' 5-point open-type Newton-Cotes integration with non-equidistant
points of support at [-5, -4, -2, 0, 2, 4, 5] where the boundary
points -5, 5 are ignored because of the open type '''
from math import fsum
def phi(t):
''' the interval transformation -5..5 -> a..b '''
return a + 0.1*(float(t)+5.)*(b-a)
if B > 1:
return fsum(map(lambda i: fivepoint(f, a+i*(b-a)/float(B), a+(i+1)*(b-a)/float(B)), xrange(B)))
return fsum(map(lambda x: coefficients[abs(x)//2]*f(phi(float(x))), [-4, -2, 0, 2, 4]))*(b-a)*0.1
def func_prod(lhs, rhs, weightf=lambda x: 1., B=100, i=(-1.,1.)):
from fivepoint import fivepoint
return fivepoint(lambda x: lhs(x)*rhs(x)*weightf(x), i[0], i[1], B)
def gaussian_error_function_part_normalized(t):
''' this is the gaussian error function function part
normalized to the range 0..1 (instead of -inf..inf)
using the chainrule for the transformation t -> 1/(1-t) - 1 '''
from math import exp
return exp(-(1./(1.-t) -1.)*(1./(1.-t) -1.))/((1-t)*(1-t))
if __name__ == '__main__':
from numpy import pi, sqrt
print "int(exp(-x^2)) from -inf..inf =", 2.*fivepoint(gaussian_error_function_part_normalized, 0., 1., 100)
print "sqrt(pi) =", sqrt(pi)
def normalisecheby(P):
for j in range(len(P)):
P/=fivepoint(lambda x:evalp(P,x)**2/((1-x*x)**0.5),-1,1,10)**0.5
return P
