def solwave_mck(c, d, M, dist = pi / 2., xtol = 1.e-12, verbose = False):
"""
Compute the collocation points and values for the d-dimensional
McKenzie solitary wave equation. n=3, m=0.
Inputs: c - soliton parameter,
d - dimension,
M - No. of points,
[dist = pi/2] - distance from the x axis to the nearest
singularity in the complex plane,
[xtol = 1.e-12] - solver tolerance,
[verbose=True/False]
Outputs: r, f - collocation points and the soliton
"""
decay = sqrt(1. - 3. / c)
h = sqrt((pi * dist) / (decay * M))
r = sinc_eo.points(M, h)
u0 = magma1d.solwave_mck(r, c) - 1.
d0 = 1.
D2 = sinc_eo.D2_e(M, h)
D1 = sinc_eo.D1_eo(M, h)
D1_r = sinc_eo.D1_x_e(M, h)
IN1 = sinc_eo.IN1_oe(M, h)
params = {'c': c, 'd': d0,
'D2': D2, 'D1':D1, 'D1_r':D1_r, 'IN': IN1}
if d < 2.:
params['d'] = d
u = fsolve(lambda v: solwave_mck_eq(v, params), u0,
fprime = lambda v:solwave_mck_eq_jac(v, params),
xtol= xtol)
else:
# Initial value for the delta dimension value.
# This can be tuned as needed
delta_dim = (d - 1.)/10.
# Loop will terminate if the delta dim value gets too
# small, this may be tunded as needed
delta_dim_min = 1.e-6
d0 = 1.
while d0 < d and delta_dim > delta_dim_min:
d1 = min([d0 + delta_dim, d])
params['d'] = d1
if verbose:
print ' computing with d = ', d1
u1, infodict, ier, mesg = \
fsolve(lambda v:solwave_mck_eq(v,params), u0, \
fprime =lambda v:solwave_mck_eq_jac(v, params), \
xtol= xtol,
full_output=1)
if ier == 1 and np.max(np.abs(u1)) > 1.e-8:
d0 = d1
u0 = u1
else:
if verbose:
if np.max(np.abs(u1)) <= 1.e-8:
print " converged to the zero solution, adjusteing delta_dim"
else:
print " solver failed to converge, adjusting delta_dim"
print " solver err = ", ier
delta_dim = delta_dim / 2.
if verbose:
jac = solwave_mck_eq_jac(u1, params)
condnum = np.linalg.cond(jac)
print " condition number = ", condnum
u = u0
if d0 < d:
print " Failed to converge to the soliton solution, last value of d =", d0
f = 0.0
else:
f = 1. + u
return r, f
def solwave_mck(c, d, M, dist = pi / 2., xtol = 1.e-12, verbose = False):
"""
Compute the collocation points and values for the d-dimensional
McKenzie solitary wave equation. n=3, m=0.
Inputs: c - soliton parameter,
d - dimension,
M - No. of points,
[dist = pi/2] - distance from the x axis to the nearest
singularity in the complex plane,
[xtol = 1.e-12] - solver tolerance,
[verbose=True/False]
Outputs: r, f - collocation points and the soliton
"""
decay = sqrt(1. - 3. / c)
h = sqrt((pi * dist) / (decay * M))
r = sinc_eo.points(M, h)
u0 = magma1d.solwave_mck(r, c) - 1.
d0 = 1.
D2 = sinc_eo.D2_e(M, h)
D1 = sinc_eo.D1_eo(M, h)
D1_r = sinc_eo.D1_x_e(M, h)
IN1 = sinc_eo.IN1_oe(M, h)
params = {'c': c, 'd': d0,
'D2': D2, 'D1':D1, 'D1_r':D1_r, 'IN': IN1}
if d < 2.:
params['d'] = d
u = fsolve(lambda v: solwave_mck_eq(v, params), u0,
fprime = lambda v:solwave_mck_eq_jac(v, params),
xtol= xtol)
else:
# Initial value for the delta dimension value.
# This can be tuned as needed
delta_dim = (d - 1.)/10.
# Loop will terminate if the delta dim value gets too
# small, this may be tunded as needed
delta_dim_min = 1.e-6
d0 = 1.
while d0 < d and delta_dim > delta_dim_min:
d1 = min([d0 + delta_dim, d])
params['d'] = d1
if verbose:
print ' computing with d = ', d1
u1, infodict, ier, mesg = \
fsolve(lambda v:solwave_mck_eq(v,params), u0, \
fprime =lambda v:solwave_mck_eq_jac(v, params), \
xtol= xtol,
full_output=1)
if ier == 1 and np.max(np.abs(u1)) > 1.e-8:
d0 = d1
u0 = u1
else:
if verbose:
if np.max(np.abs(u1)) <= 1.e-8:
print " converged to the zero solution, adjusteing delta_dim"
else:
print " solver failed to converge, adjusting delta_dim"
print " solver err = ", ier
delta_dim = delta_dim / 2.
if verbose:
jac = solwave_mck_eq_jac(u1, params)
condnum = np.linalg.cond(jac)
print " condition number = ", condnum
u = u0
if d0 < d:
print " Failed to converge to the soliton solution, last value of d =", d0
f = 0.0
else:
f = 1. + u
return r, f
def solwave_m1(c, n, d, M, dist = pi / 2., xtol = 1.e-12, verbose = False):
"""
Compute the collocation points and values for the general d-dimensional
solitary wave equation with n > 1, and m=1
Inputs: c - soliton parameter,
n - first nonlinearity,
d - dimension,
M - No. of points,
[dist = pi/2] - distance from the x axis to the nearest
singularity in the complex plane,
[xtol = 1.e-12] - solver tolerance,
[verbose=True/False]
Outputs: r, f - collocation points and the soliton
"""
# Iterate in soliton parameter, if neccessary, to achieve a good
# guess. This may be replaced with another algorithm if a better
# initial guess for the 1D soliton is available.
if verbose:
print " iterating in c"
# Initial value for delta c. This can be tuned as needed.
delta_c = .5 * n / c
# Loop will terminate if the delta c value gets too small, this may be
# adjusted as needed
delta_c_min = 1.e-6
c0 = n
d0 = 1.
while c0 < c and delta_c > delta_c_min:
c1 = min([c0 + delta_c, c])
decay = sqrt(1. - n / c1)
h = sqrt((pi * dist) / (decay * M))
r1 = sinc_eo.points(M, h)
D2 = sinc_eo.D2_e(M, h)
D1 = sinc_eo.D1_eo(M, h)
D1_r = sinc_eo.D1_x_e(M, h)
IN1 = sinc_eo.IN1_oe(M, h)
params = {'c': c1, 'n': n, 'd': d0,
'D2': D2, 'D1': D1, 'D1_r': D1_r, 'IN': IN1}
try:
# uguess = spline(r0, u0, r1)
uguess = sinc_eo.sincinterp_e(r0,u0,r1)
except:
uguess = 3. * (1. - n / c1) / cosh(.5 * decay * r1)**2
if verbose:
print ' computing with c = ', c1
u1, infodict, ier, mesg = \
fsolve(lambda v: solwave_m1_eq(v,params), uguess, \
# fprime = lambda v:solwave_m1_eq_jac(v, params), \
xtol= xtol,
full_output=1)
if ier == 1 and np.max(np.abs(u1)) > 1.e-10:
c0 = c1
u0 = u1
r0 = r1
else:
if verbose:
if np.max(np.abs(u1)) <= 1.e-8:
print " converged to the zero solution, adjusteing delta_c"
else:
print " solver failed to converge, adjusting delta_c"
print " solver err = ", ier
delta_c = delta_c / 2.
u = u0
r = r0
# Iterate in dimension, if neccessary
if d > 1.:
if verbose:
print " iterating in d"
delta_dim = (n - 1.) / 10.
# Loop will terminate if the delta dim value gets too
# small, this can be adjusted as desired
delta_dim_min = 1.e-6
d0 = 1.
u0 = u
while d0 < d and delta_dim > delta_dim_min:
d1 = min([d0 + delta_dim, d])
params['d'] = d1
if verbose:
print ' computing with d = ', d1
u1, infodict, ier, mesg = \
fsolve(lambda v: solwave_m1_eq(v, params), u0, \
# fprime = lambda v:solwave_m1_eq_jac(v, params),
xtol= xtol,
full_output=1)
if ier == 1 and np.max(np.abs(u1)) > 1.e-8:
d0 = d1
u0 = u1
else:
if verbose:
if np.max(np.abs(u1)) <= 1.e-8:
print " converged to the zero solution, adjusteing delta_dim"
else:
print " solver failed to converge, adjusting delta_dim"
print " solver err = ", ier
delta_dim = delta_dim / 2.
u = u0
if d0 < d:
print " Failed to converge to the soliton solution, last value of d =", d0
f = 0.0
else:
f = 1. + u
return r, f
def solwave_m1(c, n, d, M, dist = pi / 2., xtol = 1.e-12, verbose = False):
"""
Compute the collocation points and values for the general d-dimensional
solitary wave equation with n > 1, and m=1
Inputs: c - soliton parameter,
n - first nonlinearity,
d - dimension,
M - No. of points,
[dist = pi/2] - distance from the x axis to the nearest
singularity in the complex plane,
[xtol = 1.e-12] - solver tolerance,
[verbose=True/False]
Outputs: r, f - collocation points and the soliton
"""
# Iterate in soliton parameter, if neccessary, to achieve a good
# guess. This may be replaced with another algorithm if a better
# initial guess for the 1D soliton is available.
if verbose:
print " iterating in c"
# Initial value for delta c. This can be tuned as needed.
delta_c = .5 * n / c
# Loop will terminate if the delta c value gets too small, this may be
# adjusted as needed
delta_c_min = 1.e-6
c0 = n
d0 = 1.
while c0 < c and delta_c > delta_c_min:
c1 = min([c0 + delta_c, c])
decay = sqrt(1. - n / c1)
h = sqrt((pi * dist) / (decay * M))
r1 = sinc_eo.points(M, h)
D2 = sinc_eo.D2_e(M, h)
D1 = sinc_eo.D1_eo(M, h)
D1_r = sinc_eo.D1_x_e(M, h)
IN1 = sinc_eo.IN1_oe(M, h)
params = {'c': c1, 'n': n, 'd': d0,
'D2': D2, 'D1': D1, 'D1_r': D1_r, 'IN': IN1}
try:
# uguess = spline(r0, u0, r1)
uguess = sinc_eo.sincinterp_e(r0,u0,r1)
except:
uguess = 3. * (1. - n / c1) / cosh(.5 * decay * r1)**2
if verbose:
print ' computing with c = ', c1
u1, infodict, ier, mesg = \
fsolve(lambda v: solwave_m1_eq(v,params), uguess, \
# fprime = lambda v:solwave_m1_eq_jac(v, params), \
xtol= xtol,
full_output=1)
if ier == 1 and np.max(np.abs(u1)) > 1.e-10:
c0 = c1
u0 = u1
r0 = r1
else:
if verbose:
if np.max(np.abs(u1)) <= 1.e-8:
print " converged to the zero solution, adjusteing delta_c"
else:
print " solver failed to converge, adjusting delta_c"
print " solver err = ", ier
delta_c = delta_c / 2.
u = u0
r = r0
# Iterate in dimension, if neccessary
if d > 1.:
if verbose:
print " iterating in d"
delta_dim = (n - 1.) / 10.
# Loop will terminate if the delta dim value gets too
# small, this can be adjusted as desired
delta_dim_min = 1.e-6
d0 = 1.
u0 = u
while d0 < d and delta_dim > delta_dim_min:
d1 = min([d0 + delta_dim, d])
params['d'] = d1
if verbose:
print ' computing with d = ', d1
u1, infodict, ier, mesg = \
fsolve(lambda v: solwave_m1_eq(v, params), u0, \
# fprime = lambda v:solwave_m1_eq_jac(v, params),
xtol= xtol,
full_output=1)
if ier == 1 and np.max(np.abs(u1)) > 1.e-8:
d0 = d1
u0 = u1
else:
if verbose:
if np.max(np.abs(u1)) <= 1.e-8:
print " converged to the zero solution, adjusteing delta_dim"
else:
print " solver failed to converge, adjusting delta_dim"
print " solver err = ", ier
delta_dim = delta_dim / 2.
u = u0
if d0 < d:
print " Failed to converge to the soliton solution, last value of d =", d0
f = 0.0
else:
f = 1. + u
return r, f
