def Gaussian_1D_rat(t, ng):
"""
Gaussian quadrature points and weights for rational function 1/|x-t|
integration
"""
x,w = Idat.Gaussian1D(ng)
w1 = np.zeros(ng)
for i in range(ng):
for j in range(ng):
w1[i] += (2.0*j+1.0)*poly_Legendre(x[i],j)*Legendre_Qn(t,j)
w1[i] *= w[i]
return x, w1
def Gaussian_1D_log(t, ng):
"""
Gaussian quadrature points and weights for logarithmic function log|x-t|
integration
"""
x,w = Idat.Gaussian1D(ng)
w1 = np.zeros(ng)
for i in range(ng):
for j in range(1,ng-1):
w1[i] += (poly_Legendre(x[i],j-1)-poly_Legendre(x[i],j+1))*\
R_function(t,j)
w1[i] += (poly_Legendre(x[i],0)-poly_Legendre(x[i],1))*R_function(t,0)
w1[i] += poly_Legendre(x[i],ng-2)*R_function(t,ng-1)
w1[i] += poly_Legendre(x[i],ng-1)*R_function(t,ng)
w1[i] *= w[i]
return x, w1
def Gaussian_1D_Pn_Log_Rat(t, ng, m = -1):
if m <= 0:
m = math.ceil(ng/3)
x,_ = Idat.Gaussian1D(ng)
xi = np.empty((3*m,ng))
mi = np.zeros(3*m)
mi[0] = 2.0
for i in range(m):
for j in range(ng):
pol = poly_Legendre(x[j],i)
xi[i,j] = pol
xi[i+m,j] = pol*math.log(math.fabs(x[j]-t))
xi[i+2*m,j] = pol/(t-x[j])
mi[i+m] = integration_PnLog(t,i)
mi[i+2*m] = integration_PnRat(t,i)
w1,_,_,_ = np.linalg.lstsq(xi, mi)
return x, w1
def Gaussian_1D_Pn_Log(t, ng, m = -1):
"""
Gaussian quadrature points and weights for function Pn(x)*log|x-t|
"""
if m <= 0:
m = math.ceil(ng/3)
x,_ = Idat.Gaussian1D(ng)
xi = np.empty((2*m,ng))
mi = np.zeros(2*m)
mi[0] = 2.0
for i in range(m):
for j in range(ng):
pol = poly_Legendre(x[j],i)
xi[i,j] = pol
xi[i+m,j] = pol*math.log(math.fabs(x[j]-t))
mi[i+m] = integration_PnLog(t,i)
w1,_,_,_ = np.linalg.lstsq(xi, mi)
return x, w1
def Gaussian_1D_rat2(t, ng):
"""
Gaussian quadrature points and weights for rational function 1/|x-t|
integration
"""
x,w = Idat.Gaussian1D(ng)
w1 = np.zeros(ng)
for i in range(ng):
# for j in range(ng-1):
# for k in range(j,math.trunc((ng+j-3)/2)+1):
# w1[i] -= (2*j+1)*(4*k+3-2*i)*Legendre_Qn(t,j)*\
# poly_Legendre(x[i],2*k+1-(i+1))
for j in range(1,ng):
for k in range(math.trunc((j-1)/2)+1):
w1[i] -= (2*j-4*k-1)*poly_Legendre(x[i],j)*(2*j+1)*\
Legendre_Qn(t,j-2*k-1)
for j in range(ng):
w1[i] += (2.0*j+1.0)/2*poly_Legendre(x[i],j)*\
(1.0/(t-1.0) - ((-1.0)**j)/(t+1))
w1[i] *= w[i]
return x, w1