def quaddobl_pade_approximants(pols, sols, idx=1, numdeg=2, dendeg=2, \
nbr=4, verbose=True):
r"""
Computes Pade approximants based on the series in quad double
precision for the polynomials in *pols*, where the leading
coefficients of the series are the solutions in *sols*.
On entry are the following seven parameters:
*pols*: a list of string representations of polynomials,
*sols*: a list of solutions of the polynomials in *pols*,
*idx*: index of the series parameter, by default equals 1,
*numdeg*: the degree of the numerator,
*dendeg*: the degree of the denominator,
*nbr*: number of steps with Newton's method,
*verbose*: whether to write intermediate output to screen or not.
On return is a list of lists of strings. Each lists of strings
represents the series solution for the variables in the list *pols*.
"""
from phcpy.solver import number_of_symbols
from phcpy.interface import store_quaddobl_solutions
from phcpy.interface import store_quaddobl_system, load_quaddobl_system
from phcpy.phcpy2c3 \
import py2c_quaddobl_Pade_approximant as Pade_approximants
from phcpy.phcpy2c3 import py2c_syspool_quaddobl_size as poolsize
from phcpy.phcpy2c3 import py2c_syspool_copy_to_quaddobl_container
from phcpy.phcpy2c3 import py2c_syspool_quaddobl_clear
nbsym = number_of_symbols(pols)
if verbose:
print("the polynomials :")
for pol in pols:
print(pol)
print("Number of variables :", nbsym)
store_quaddobl_system(pols, nbvar=nbsym)
store_quaddobl_solutions(nbsym, sols)
fail = Pade_approximants(idx, numdeg, dendeg, nbr, int(verbose))
size = (-1 if fail else poolsize())
if verbose:
if size == -1:
print("An error occurred in the Pade constructor.")
else:
print("Computed %d Pade approximants." % size)
result = []
for k in range(1, size + 1):
py2c_syspool_copy_to_quaddobl_container(k)
sersol = load_quaddobl_system()
substsersol = substitute_symbol(sersol, idx)
result.append(make_fractions(substsersol))
py2c_syspool_quaddobl_clear()
return result
def newton_step(system, solutions, precision='d', decimals=100):
"""
Applies one Newton step to the solutions of the system.
For each solution, prints its last line of diagnostics.
Four levels of precision are supported:
d : standard double precision (1.1e-15 or 2^(-53)),
dd : double double precision (4.9e-32 or 2^(-104)),
qd : quad double precision (1.2e-63 or 2^(-209)).
mp : arbitrary precision, where the number of decimal places
in the working precision is determined by decimals.
"""
dim = number_of_symbols(system)
if (precision == 'd'):
from phcpy.interface import store_standard_system
from phcpy.interface import store_standard_solutions
from phcpy.interface import load_standard_solutions
store_standard_system(system, nbvar=dim)
store_standard_solutions(dim, solutions)
from phcpy.phcpy2c3 import py2c_standard_Newton_step
py2c_standard_Newton_step()
result = load_standard_solutions()
elif (precision == 'dd'):
from phcpy.interface import store_dobldobl_system
from phcpy.interface import store_dobldobl_solutions
from phcpy.interface import load_dobldobl_solutions
store_dobldobl_system(system, nbvar=dim)
store_dobldobl_solutions(dim, solutions)
from phcpy.phcpy2c3 import py2c_dobldobl_Newton_step
py2c_dobldobl_Newton_step()
result = load_dobldobl_solutions()
elif (precision == 'qd'):
from phcpy.interface import store_quaddobl_system
from phcpy.interface import store_quaddobl_solutions
from phcpy.interface import load_quaddobl_solutions
store_quaddobl_system(system, nbvar=dim)
store_quaddobl_solutions(dim, solutions)
from phcpy.phcpy2c3 import py2c_quaddobl_Newton_step
py2c_quaddobl_Newton_step()
result = load_quaddobl_solutions()
elif (precision == 'mp'):
from phcpy.interface import store_multprec_system
from phcpy.interface import store_multprec_solutions
from phcpy.interface import load_multprec_solutions
store_multprec_system(system, decimals, nbvar=dim)
store_multprec_solutions(dim, solutions)
from phcpy.phcpy2c3 import py2c_multprec_Newton_step
py2c_multprec_Newton_step(decimals)
result = load_multprec_solutions()
else:
print('wrong argument for precision')
return None
for sol in result:
strsol = sol.split('\n')
print(strsol[-1])
return result
def newton_step(system, solutions, precision='d', decimals=100):
"""
Applies one Newton step to the solutions of the system.
For each solution, prints its last line of diagnostics.
Four levels of precision are supported:
d : standard double precision (1.1e-15 or 2^(-53)),
dd : double double precision (4.9e-32 or 2^(-104)),
qd : quad double precision (1.2e-63 or 2^(-209)).
mp : arbitrary precision, where the number of decimal places
in the working precision is determined by decimals.
"""
if(precision == 'd'):
from phcpy.interface import store_standard_system
from phcpy.interface import store_standard_solutions
from phcpy.interface import load_standard_solutions
store_standard_system(system)
store_standard_solutions(len(system), solutions)
from phcpy.phcpy2c3 import py2c_standard_Newton_step
py2c_standard_Newton_step()
result = load_standard_solutions()
elif(precision == 'dd'):
from phcpy.interface import store_dobldobl_system
from phcpy.interface import store_dobldobl_solutions
from phcpy.interface import load_dobldobl_solutions
store_dobldobl_system(system)
store_dobldobl_solutions(len(system), solutions)
from phcpy.phcpy2c3 import py2c_dobldobl_Newton_step
py2c_dobldobl_Newton_step()
result = load_dobldobl_solutions()
elif(precision == 'qd'):
from phcpy.interface import store_quaddobl_system
from phcpy.interface import store_quaddobl_solutions
from phcpy.interface import load_quaddobl_solutions
store_quaddobl_system(system)
store_quaddobl_solutions(len(system), solutions)
from phcpy.phcpy2c3 import py2c_quaddobl_Newton_step
py2c_quaddobl_Newton_step()
result = load_quaddobl_solutions()
elif(precision == 'mp'):
from phcpy.interface import store_multprec_system
from phcpy.interface import store_multprec_solutions
from phcpy.interface import load_multprec_solutions
store_multprec_system(system, decimals)
store_multprec_solutions(len(system), solutions)
from phcpy.phcpy2c3 import py2c_multprec_Newton_step
py2c_multprec_Newton_step(decimals)
result = load_multprec_solutions()
else:
print('wrong argument for precision')
return None
for sol in result:
strsol = sol.split('\n')
print(strsol[-1])
return result
def permute_quaddobl_system(pols):
"""
Permutes the equations in the list of polynomials in pols
with coefficients in quad double precision,
along the permutation used in the mixed volume computation.
"""
from phcpy.phcpy2c3 import py2c_celcon_permute_quaddobl_system
from phcpy.interface import store_quaddobl_system, load_quaddobl_system
store_quaddobl_system(pols)
py2c_celcon_permute_quaddobl_system()
return load_quaddobl_system()
def permute_quaddobl_system(pols):
"""
Permutes the equations in the list of polynomials in pols
with coefficients in quad double precision,
along the permutation used in the mixed volume computation.
"""
from phcpy.phcpy2c3 import py2c_celcon_permute_quaddobl_system
from phcpy.interface import store_quaddobl_system, load_quaddobl_system
store_quaddobl_system(pols)
py2c_celcon_permute_quaddobl_system()
return load_quaddobl_system()
def quaddobl_scale_system(pols):
"""
Applies equation and variable scaling in quad double precision
to the polynomials in the list pols.
On return is the list of scaled polynomials and the scaling coefficients.
"""
from phcpy.interface import store_quaddobl_system, load_quaddobl_system
from phcpy.phcpy2c3 import py2c_scale_quaddobl_system
store_quaddobl_system(pols)
cffs = py2c_scale_quaddobl_system(2)
spol = load_quaddobl_system()
return (spol, cffs)
def quaddobl_scale_system(pols):
"""
Applies equation and variable scaling in quad double precision
to the polynomials in the list pols.
On return is the list of scaled polynomials and the scaling coefficients.
"""
from phcpy.interface import store_quaddobl_system, load_quaddobl_system
from phcpy.phcpy2c3 import py2c_scale_quaddobl_system
store_quaddobl_system(pols)
cffs = py2c_scale_quaddobl_system(2)
spol = load_quaddobl_system()
return (spol, cffs)
def quaddobl_usolve(pol, mxi, eps):
"""
Applies the method of Durand-Kerner (aka Weierstrass)
to the polynomial in the string pol, in quad double precision
The maximum number of iterations is in mxi,
the requirement on the accuracy in eps.
"""
from phcpy.phcpy2c3 import py2c_usolve_quaddobl
from phcpy.interface import store_quaddobl_system, load_quaddobl_solutions
store_quaddobl_system([pol])
nit = py2c_usolve_quaddobl(mxi, eps)
rts = load_quaddobl_solutions()
return (nit, rts)
def quaddobl_usolve(pol, mxi, eps):
"""
Applies the method of Durand-Kerner (aka Weierstrass)
to the polynomial in the string pol, in quad double precision
The maximum number of iterations is in mxi,
the requirement on the accuracy in eps.
"""
from phcpy.phcpy2c3 import py2c_usolve_quaddobl
from phcpy.interface import store_quaddobl_system, load_quaddobl_solutions
store_quaddobl_system([pol])
nit = py2c_usolve_quaddobl(mxi, eps)
rts = load_quaddobl_solutions()
return (nit, rts)
def quaddobl_newton_power_series(pols, lser, idx=1, nbr=4, verbose=True):
r"""
Computes series in quad double precision for the polynomials
in *pols*, where the leading terms are given in the list *lser*.
On entry are the following five parameters:
*pols*: a list of string representations of polynomials,
*lser*: a list of polynomials in the series parameter (e.g.: t),
for use as start terms in Newton's method,
*idx*: index of the series parameter, by default equals 1,
*nbr*: number of steps with Newton's method,
*verbose*: whether to write intermediate output to screen or not.
On return is a list of lists of strings. Each lists of strings
represents the series solution for the variables in the list *pols*.
"""
from phcpy.solver import number_of_symbols
from phcpy.interface import store_quaddobl_system, load_quaddobl_system
from phcpy.phcpy2c2 import py2c_quaddobl_Newton_power_series as newton
from phcpy.phcpy2c2 import py2c_syspool_quaddobl_init
from phcpy.phcpy2c2 import py2c_syspool_quaddobl_create
from phcpy.phcpy2c2 import py2c_syspool_quaddobl_size as poolsize
from phcpy.phcpy2c2 import py2c_syspool_copy_to_quaddobl_container
from phcpy.phcpy2c2 import py2c_syspool_quaddobl_clear
nbsym = number_of_symbols(pols)
if verbose:
print "the polynomials :"
for pol in pols:
print pol
print "Number of variables :", nbsym
store_quaddobl_system(lser, nbvar=1)
py2c_syspool_quaddobl_init(1);
py2c_syspool_quaddobl_create(1);
store_quaddobl_system(pols, nbvar=nbsym)
fail = newton(idx, nbr, int(verbose))
size = (-1 if fail else poolsize())
if verbose:
if size == -1:
print "An error occurred in the execution of Newton's method."
else:
print "Computed one series solution."
py2c_syspool_copy_to_quaddobl_container(1)
result = load_quaddobl_system()
result = substitute_symbol(result, idx)
py2c_syspool_quaddobl_clear()
return result
def quaddobl_newton_series(pols, sols, idx=1, maxdeg=4, nbr=4, verbose=True):
r"""
Computes series in quad double precision for the polynomials
in *pols*, where the leading coefficients are the solutions in *sols*.
On entry are the following five parameters:
*pols*: a list of string representations of polynomials,
*sols*: a list of solutions of the polynomials in *pols*,
*idx*: index of the series parameter, by default equals 1,
*maxdeg*: maximal degree of the series,
*nbr*: number of steps with Newton's method,
*verbose*: whether to write intermediate output to screen or not.
On return is a list of lists of strings. Each lists of strings
represents the series solution for the variables in the list *pols*.
"""
from phcpy.solver import number_of_symbols
from phcpy.interface import store_quaddobl_solutions
from phcpy.interface import store_quaddobl_system, load_quaddobl_system
from phcpy.phcpy2c2 import py2c_quaddobl_Newton_series as newton
from phcpy.phcpy2c2 import py2c_syspool_quaddobl_size as poolsize
from phcpy.phcpy2c2 import py2c_syspool_copy_to_quaddobl_container
from phcpy.phcpy2c2 import py2c_syspool_quaddobl_clear
nbsym = number_of_symbols(pols)
if verbose:
print "the polynomials :"
for pol in pols:
print pol
print "Number of variables :", nbsym
store_quaddobl_system(pols, nbvar=nbsym)
store_quaddobl_solutions(nbsym, sols)
fail = newton(idx, maxdeg, nbr, int(verbose))
size = (-1 if fail else poolsize())
if verbose:
if size == -1:
print "An error occurred in the execution of Newton's method."
else:
print "Computed %d series solutions." % size
result = []
for k in range(1, size+1):
py2c_syspool_copy_to_quaddobl_container(k)
sersol = load_quaddobl_system()
result.append(substitute_symbol(sersol, idx))
py2c_syspool_quaddobl_clear()
return result
def quaddobl_newton_power_series(pols, lser, idx=1, nbr=4, verbose=True):
r"""
Computes series in quad double precision for the polynomials
in *pols*, where the leading terms are given in the list *lser*.
On entry are the following five parameters:
*pols*: a list of string representations of polynomials,
*lser*: a list of polynomials in the series parameter (e.g.: t),
for use as start terms in Newton's method,
*idx*: index of the series parameter, by default equals 1,
*nbr*: number of steps with Newton's method,
*verbose*: whether to write intermediate output to screen or not.
On return is a list of lists of strings. Each lists of strings
represents the series solution for the variables in the list *pols*.
"""
from phcpy.solver import number_of_symbols
from phcpy.interface import store_quaddobl_system, load_quaddobl_system
from phcpy.phcpy2c3 import py2c_quaddobl_Newton_power_series as newton
from phcpy.phcpy2c3 import py2c_syspool_quaddobl_init
from phcpy.phcpy2c3 import py2c_syspool_quaddobl_create
from phcpy.phcpy2c3 import py2c_syspool_quaddobl_size as poolsize
from phcpy.phcpy2c3 import py2c_syspool_copy_to_quaddobl_container
from phcpy.phcpy2c3 import py2c_syspool_quaddobl_clear
nbsym = number_of_symbols(pols)
if verbose:
print("the polynomials :")
for pol in pols:
print(pol)
print("Number of variables :", nbsym)
store_quaddobl_system(lser, nbvar=1)
py2c_syspool_quaddobl_init(1)
py2c_syspool_quaddobl_create(1)
store_quaddobl_system(pols, nbvar=nbsym)
fail = newton(idx, nbr, int(verbose))
size = (-1 if fail else poolsize())
if verbose:
if size == -1:
print("An error occurred in the execution of Newton's method.")
else:
print("Computed one series solution.")
py2c_syspool_copy_to_quaddobl_container(1)
result = load_quaddobl_system()
result = substitute_symbol(result, idx)
py2c_syspool_quaddobl_clear()
return result
def quaddobl_newton_series(pols, sols, idx=1, nbr=4, verbose=True):
r"""
Computes series in quad double precision for the polynomials
in *pols*, where the leading coefficients are the solutions in *sols*.
On entry are the following five parameters:
*pols*: a list of string representations of polynomials,
*sols*: a list of solutions of the polynomials in *pols*,
*idx*: index of the series parameter, by default equals 1,
*nbr*: number of steps with Newton's method,
*verbose*: whether to write intermediate output to screen or not.
On return is a list of lists of strings. Each lists of strings
represents the series solution for the variables in the list *pols*.
"""
from phcpy.solver import number_of_symbols
from phcpy.interface import store_quaddobl_solutions
from phcpy.interface import store_quaddobl_system, load_quaddobl_system
from phcpy.phcpy2c2 import py2c_quaddobl_Newton_series as newton
from phcpy.phcpy2c2 import py2c_syspool_quaddobl_size as poolsize
from phcpy.phcpy2c2 import py2c_syspool_copy_to_quaddobl_container
from phcpy.phcpy2c2 import py2c_syspool_quaddobl_clear
nbsym = number_of_symbols(pols)
if verbose:
print "the polynomials :"
for pol in pols:
print pol
print "Number of variables :", nbsym
store_quaddobl_system(pols, nbvar=nbsym)
store_quaddobl_solutions(nbsym, sols)
fail = newton(idx, nbr, int(verbose))
size = (-1 if fail else poolsize())
if verbose:
if size == -1:
print "An error occurred in the execution of Newton's method."
else:
print "Computed %d series solutions." % size
result = []
for k in range(1, size+1):
py2c_syspool_copy_to_quaddobl_container(k)
sersol = load_quaddobl_system()
result.append(substitute_symbol(sersol, idx))
py2c_syspool_quaddobl_clear()
return result
def ade_tuned_quad_double_track(target, start, sols, pars, gamma=0, verbose=1):
r"""
Does path tracking with algorithm differentiation,
in quad double precision, with tuned path parameters.
On input are a target system, a start system with solutions.
The *target* is a list of strings representing the polynomials
of the target system (which has to be solved).
The *start* is a list of strings representing the polynomials
of the start system, with known solutions in *sols*.
The *sols* is a list of strings representing start solutions.
The *pars* is a tuple of tuned values for the path parameters.
On return are the string representations of the solutions
computed at the end of the paths.
If *gamma* on input equals zero, then a random complex number is generated,
otherwise the real and imaginary parts of *gamma* are used.
"""
from phcpy.phcpy2c3 import py2c_copy_quaddobl_container_to_target_system
from phcpy.phcpy2c3 import py2c_copy_quaddobl_container_to_start_system
from phcpy.phcpy2c3 import py2c_ade_manypaths_qd_pars
from phcpy.interface import store_quaddobl_system
from phcpy.interface import store_quaddobl_solutions
from phcpy.interface import load_quaddobl_solutions
store_quaddobl_system(target)
py2c_copy_quaddobl_container_to_target_system()
store_quaddobl_system(start)
py2c_copy_quaddobl_container_to_start_system()
dim = len(start)
store_quaddobl_solutions(dim, sols)
if(gamma == 0):
from random import uniform
from cmath import exp, pi
angle = uniform(0, 2*pi)
gamma = exp(angle*complex(0, 1))
if(verbose > 0):
print('random gamma constant :', gamma)
if(verbose > 0):
print('the path parameters :\n', pars)
fail = py2c_ade_manypaths_qd_pars(verbose, gamma.real, gamma.imag, \
pars[0], pars[1], pars[2], pars[3], pars[4], pars[5], pars[6], \
pars[7], pars[8], pars[9], pars[10], pars[11], pars[12], pars[13])
if(fail == 0):
if(verbose > 0):
print('Path tracking with AD was a success!')
else:
print('Path tracking with AD failed!')
return load_quaddobl_solutions()
def quad_double_track(target, start, sols, gamma=0, tasks=0):
r"""
Does path tracking with quad double precision.
On input are a target system, a start system with solutions,
optionally a (random) gamma constant and the number of tasks.
The *target* is a list of strings representing the polynomials
of the target system (which has to be solved).
The *start* is a list of strings representing the polynomials
of the start system with known solutions in sols.
The *sols* is a list of strings representing start solutions.
By default, a random *gamma* constant is generated,
otherwise *gamma* must be a nonzero complex constant.
The number of tasks in the multithreading is defined by *tasks*.
The default zero value for *tasks* indicates no multithreading.
On return are the string representations of the solutions
computed at the end of the paths.
"""
from phcpy.phcpy2c3 import py2c_copy_quaddobl_container_to_target_system
from phcpy.phcpy2c3 import py2c_copy_quaddobl_container_to_start_system
from phcpy.phcpy2c3 import py2c_copy_quaddobl_container_to_start_solutions
from phcpy.phcpy2c3 import py2c_create_quaddobl_homotopy
from phcpy.phcpy2c3 import py2c_create_quaddobl_homotopy_with_gamma
from phcpy.phcpy2c3 import py2c_solve_by_quaddobl_homotopy_continuation
from phcpy.phcpy2c3 import py2c_solcon_clear_quaddobl_solutions
from phcpy.phcpy2c3 import py2c_copy_quaddobl_target_solutions_to_container
from phcpy.interface import store_quaddobl_system
from phcpy.interface import store_quaddobl_solutions
from phcpy.interface import load_quaddobl_solutions
from phcpy.solver import number_of_symbols
dim = number_of_symbols(start)
store_quaddobl_system(target, nbvar=dim)
py2c_copy_quaddobl_container_to_target_system()
store_quaddobl_system(start, nbvar=dim)
py2c_copy_quaddobl_container_to_start_system()
# py2c_clear_quaddobl_homotopy()
if(gamma == 0):
py2c_create_quaddobl_homotopy()
else:
py2c_create_quaddobl_homotopy_with_gamma(gamma.real, gamma.imag)
store_quaddobl_solutions(dim, sols)
py2c_copy_quaddobl_container_to_start_solutions()
py2c_solve_by_quaddobl_homotopy_continuation(tasks)
py2c_solcon_clear_quaddobl_solutions()
py2c_copy_quaddobl_target_solutions_to_container()
return load_quaddobl_solutions()
def quaddobl_set_homotopy(target, start, verbose=False):
"""
Initializes the homotopy with the target and start system for a
step-by-step run of the series-Pade tracker, in quad double precision.
If verbose, then extra output is written.
Returns the failure code of the homotopy initializer.
"""
from phcpy.phcpy2c3 import py2c_copy_quaddobl_container_to_target_system
from phcpy.phcpy2c3 import py2c_copy_quaddobl_container_to_start_system
from phcpy.interface import store_quaddobl_system
from phcpy.solver import number_of_symbols
dim = number_of_symbols(start)
store_quaddobl_system(target, nbvar=dim)
py2c_copy_quaddobl_container_to_target_system()
store_quaddobl_system(start, nbvar=dim)
py2c_copy_quaddobl_container_to_start_system()
from phcpy.phcpy2c3 import py2c_padcon_quaddobl_initialize_homotopy
return py2c_padcon_quaddobl_initialize_homotopy(int(verbose))
def quaddobl_deflate(system, solutions):
"""
The deflation method augments the given system with
derivatives to restore the quadratic convergence of
Newton's method at isolated singular solutions,
in quad double precision.
After application of deflation with default settings,
the new approximate solutions are returned.
"""
from phcpy.phcpy2c3 import py2c_quaddobl_deflate
from phcpy.interface import store_quaddobl_system
from phcpy.interface import store_quaddobl_solutions
from phcpy.interface import load_quaddobl_solutions
store_quaddobl_system(system)
store_quaddobl_solutions(len(system), solutions)
py2c_quaddobl_deflate()
result = load_quaddobl_solutions()
return result
def quaddobl_set_parameter_homotopy(hom, idx, verbose=False):
"""
Initializes the homotopy with the polynomials in hom for a
step-by-step run of the series-Pade tracker, in quad double precision.
The value idx gives the index of the continuation parameter
and is the index of one of the variables in the homotopy hom.
If verbose, then extra output is written.
Returns the failure code of the homotopy initializer.
"""
from phcpy.phcpy2c3 import py2c_copy_quaddobl_container_to_target_system
from phcpy.interface import store_quaddobl_system
from phcpy.solver import number_of_symbols
dim = number_of_symbols(hom)
store_quaddobl_system(hom, nbvar=dim)
py2c_copy_quaddobl_container_to_target_system()
from phcpy.phcpy2c3 \
import py2c_padcon_quaddobl_initialize_parameter_homotopy
vrb = int(verbose)
return py2c_padcon_quaddobl_initialize_parameter_homotopy(idx, vrb)
def initialize_quaddobl_tracker(target, start, fixedgamma=True):
r"""
Initializes a path tracker with a generator for a *target*
and *start* system given in quad double precision.
If *fixedgamma*, then gamma will be a fixed default value,
otherwise, a random complex constant for gamma is generated.
"""
from phcpy.phcpy2c3 import py2c_copy_quaddobl_container_to_target_system
from phcpy.phcpy2c3 import py2c_copy_quaddobl_container_to_start_system
from phcpy.phcpy2c3 import py2c_initialize_quaddobl_homotopy
from phcpy.interface import store_quaddobl_system
store_quaddobl_system(target)
py2c_copy_quaddobl_container_to_target_system()
store_quaddobl_system(start)
py2c_copy_quaddobl_container_to_start_system()
if fixedgamma:
return py2c_initialize_quaddobl_homotopy(1)
else:
return py2c_initialize_quaddobl_homotopy(0)
def initialize_quaddobl_tracker(target, start, fixedgamma=True):
"""
Initializes a path tracker with a generator for a target
and start system given in quad double precision.
If fixedgamma, then gamma will be a fixed default value,
otherwise, a random complex constant for gamma is generated.
"""
from phcpy.phcpy2c2 import py2c_copy_quaddobl_container_to_target_system
from phcpy.phcpy2c2 import py2c_copy_quaddobl_container_to_start_system
from phcpy.phcpy2c2 import py2c_initialize_quaddobl_homotopy
from phcpy.interface import store_quaddobl_system
store_quaddobl_system(target)
py2c_copy_quaddobl_container_to_target_system()
store_quaddobl_system(start)
py2c_copy_quaddobl_container_to_start_system()
if fixedgamma:
return py2c_initialize_quaddobl_homotopy(1)
else:
return py2c_initialize_quaddobl_homotopy(0)
def gpu_quad_double_track(target, start, sols, gamma=0, verbose=1):
"""
GPU accelerated path tracking with algorithm differentiation,
in quad double precision.
On input are a target system, a start system with solutions.
The target is a list of strings representing the polynomials
of the target system (which has to be solved).
The start is a list of strings representing the polynomials
of the start system, with known solutions in sols.
The sols is a list of strings representing start solutions.
On return are the string representations of the solutions
computed at the end of the paths.
If gamma on input equals zero, then a random complex number is generated,
otherwise the real and imaginary parts of gamma are used.
"""
from phcpy.phcpy2c2 import py2c_copy_quaddobl_container_to_target_system
from phcpy.phcpy2c2 import py2c_copy_quaddobl_container_to_start_system
from phcpy.phcpy2c2 import py2c_gpu_manypaths_qd
from phcpy.interface import store_quaddobl_system
from phcpy.interface import store_quaddobl_solutions
from phcpy.interface import load_quaddobl_solutions
store_quaddobl_system(target)
py2c_copy_quaddobl_container_to_target_system()
store_quaddobl_system(start)
py2c_copy_quaddobl_container_to_start_system()
dim = len(start)
store_quaddobl_solutions(dim, sols)
if gamma == 0:
from random import uniform
from cmath import exp, pi
angle = uniform(0, 2 * pi)
gamma = exp(angle * complex(0, 1))
if verbose > 0:
print "random gamma constant :", gamma
fail = py2c_gpu_manypaths_qd(2, verbose, gamma.real, gamma.imag)
if fail == 0:
if verbose > 0:
print "Path tracking on the GPU was a success!"
else:
print "Path tracking on the GPU failed!"
return load_quaddobl_solutions()
def quad_double_track(target, start, sols, gamma=0, tasks=0):
"""
Does path tracking with quad double precision.
On input are a target system, a start system with solutions,
optionally a (random) gamma constant and the number of tasks.
The target is a list of strings representing the polynomials
of the target system (which has to be solved).
The start is a list of strings representing the polynomials
of the start system with known solutions in sols.
The sols is a list of strings representing start solutions.
By default, a random gamma constant is generated,
otherwise gamma must be a nonzero complex constant.
On return are the string representations of the solutions
computed at the end of the paths.
"""
from phcpy.phcpy2c2 import py2c_copy_quaddobl_container_to_target_system
from phcpy.phcpy2c2 import py2c_copy_quaddobl_container_to_start_system
from phcpy.phcpy2c2 import py2c_copy_quaddobl_container_to_start_solutions
from phcpy.phcpy2c2 import py2c_create_quaddobl_homotopy
from phcpy.phcpy2c2 import py2c_create_quaddobl_homotopy_with_gamma
from phcpy.phcpy2c2 import py2c_solve_by_quaddobl_homotopy_continuation
from phcpy.phcpy2c2 import py2c_solcon_clear_quaddobl_solutions
from phcpy.phcpy2c2 import py2c_copy_quaddobl_target_solutions_to_container
from phcpy.interface import store_quaddobl_system
from phcpy.interface import store_quaddobl_solutions
from phcpy.interface import load_quaddobl_solutions
from phcpy.solver import number_of_symbols
dim = number_of_symbols(start)
store_quaddobl_system(target, nbvar=dim)
py2c_copy_quaddobl_container_to_target_system()
store_quaddobl_system(start, nbvar=dim)
py2c_copy_quaddobl_container_to_start_system()
# py2c_clear_quaddobl_homotopy()
if(gamma == 0):
py2c_create_quaddobl_homotopy()
else:
py2c_create_quaddobl_homotopy_with_gamma(gamma.real, gamma.imag)
store_quaddobl_solutions(dim, sols)
py2c_copy_quaddobl_container_to_start_solutions()
py2c_solve_by_quaddobl_homotopy_continuation(tasks)
py2c_solcon_clear_quaddobl_solutions()
py2c_copy_quaddobl_target_solutions_to_container()
return load_quaddobl_solutions()
def quaddobl_deflate(system, solutions):
"""
The deflation method augments the given system with
derivatives to restore the quadratic convergence of
Newton's method at isolated singular solutions,
in quad double precision.
After application of deflation with default settings,
the new approximate solutions are returned.
"""
from phcpy.phcpy2c3 import py2c_quaddobl_deflate
from phcpy.interface import store_quaddobl_system
from phcpy.interface import store_quaddobl_solutions
from phcpy.interface import load_quaddobl_solutions
dim = number_of_symbols(system)
store_quaddobl_system(system, nbvar=dim)
store_quaddobl_solutions(dim, solutions)
py2c_quaddobl_deflate()
result = load_quaddobl_solutions()
return result
def gpu_quad_double_track(target, start, sols, gamma=0, verbose=1):
"""
GPU accelerated path tracking with algorithm differentiation,
in quad double precision.
On input are a target system, a start system with solutions.
The target is a list of strings representing the polynomials
of the target system (which has to be solved).
The start is a list of strings representing the polynomials
of the start system, with known solutions in sols.
The sols is a list of strings representing start solutions.
On return are the string representations of the solutions
computed at the end of the paths.
If gamma on input equals zero, then a random complex number is generated,
otherwise the real and imaginary parts of gamma are used.
"""
from phcpy.phcpy2c2 import py2c_copy_quaddobl_container_to_target_system
from phcpy.phcpy2c2 import py2c_copy_quaddobl_container_to_start_system
from phcpy.phcpy2c2 import py2c_gpu_manypaths_qd
from phcpy.interface import store_quaddobl_system
from phcpy.interface import store_quaddobl_solutions
from phcpy.interface import load_quaddobl_solutions
store_quaddobl_system(target)
py2c_copy_quaddobl_container_to_target_system()
store_quaddobl_system(start)
py2c_copy_quaddobl_container_to_start_system()
dim = len(start)
store_quaddobl_solutions(dim, sols)
if(gamma == 0):
from random import uniform
from cmath import exp, pi
angle = uniform(0, 2*pi)
gamma = exp(angle*complex(0, 1))
if(verbose > 0):
print 'random gamma constant :', gamma
fail = py2c_gpu_manypaths_qd(2, verbose, gamma.real, gamma.imag)
if(fail == 0):
if(verbose > 0):
print 'Path tracking on the GPU was a success!'
else:
print 'Path tracking on the GPU failed!'
return load_quaddobl_solutions()
def quaddobl_track(target, start, sols, filename="", verbose=False):
"""
Wraps the tracker for Pade continuation in quad double precision.
On input are a target system, a start system with solutions,
optionally: a string *filename* and the *verbose* flag.
The *target* is a list of strings representing the polynomials
of the target system (which has to be solved).
The *start* is a list of strings representing the polynomials
of the start system, with known solutions in *sols*.
The *sols* is a list of strings representing start solutions.
On return are the string representations of the solutions
computed at the end of the paths.
"""
from phcpy.phcpy2c2 import py2c_copy_quaddobl_container_to_target_system
from phcpy.phcpy2c2 import py2c_copy_quaddobl_container_to_start_system
from phcpy.phcpy2c2 import py2c_copy_quaddobl_container_to_start_solutions
from phcpy.phcpy2c2 import py2c_create_quaddobl_homotopy_with_gamma
from phcpy.phcpy2c2 import py2c_clear_quaddobl_homotopy
from phcpy.phcpy2c2 import py2c_clear_quaddobl_operations_data
from phcpy.interface import store_quaddobl_system
from phcpy.interface import store_quaddobl_solutions
from phcpy.interface import load_quaddobl_solutions
from phcpy.solver import number_of_symbols
from phcpy.phcpy2c2 import py2c_padcon_quaddobl_track
dim = number_of_symbols(start)
store_quaddobl_system(target, nbvar=dim)
py2c_copy_quaddobl_container_to_target_system()
store_quaddobl_system(start, nbvar=dim)
py2c_copy_quaddobl_container_to_start_system()
(regamma, imgamma) = get_homotopy_continuation_parameter(1)
store_quaddobl_solutions(dim, sols)
py2c_copy_quaddobl_container_to_start_solutions()
(regamma, imgamma) = get_homotopy_continuation_parameter(1)
py2c_create_quaddobl_homotopy_with_gamma(regamma, imgamma)
nbc = len(filename)
fail = py2c_padcon_quaddobl_track(nbc, filename, int(verbose))
# py2c_clear_quaddobl_homotopy()
# py2c_clear_quaddobl_operations_data()
return load_quaddobl_solutions()
def quaddobl_polysys_solve(pols, topdim=-1, \
filter=True, factor=True, tasks=0, verbose=True):
"""
Runs the cascades of homotopies on the polynomial system in pols
in quad double precision. The default top dimension topdim
is the number of variables in pols minus one.
"""
from phcpy.phcpy2c3 import py2c_quaddobl_polysys_solve
from phcpy.phcpy2c3 import py2c_copy_quaddobl_polysys_witset
from phcpy.solver import number_of_symbols
from phcpy.interface import store_quaddobl_system
from phcpy.interface import load_quaddobl_system, load_quaddobl_solutions
dim = number_of_symbols(pols)
if(topdim == -1):
topdim = dim - 1
fail = store_quaddobl_system(pols, nbvar=dim)
fail = py2c_quaddobl_polysys_solve(tasks,topdim, \
int(filter),int(factor),int(verbose))
witsols = []
for soldim in range(0, topdim+1):
fail = py2c_copy_quaddobl_polysys_witset(soldim)
witset = (load_quaddobl_system(), load_quaddobl_solutions())
witsols.append(witset)
return witsols
def quaddobl_polysys_solve(pols, topdim=-1, \
filter=True, factor=True, tasks=0, verbose=True):
"""
Runs the cascades of homotopies on the polynomial system in pols
in quad double precision. The default top dimension topdim
is the number of variables in pols minus one.
"""
from phcpy.phcpy2c3 import py2c_quaddobl_polysys_solve
from phcpy.phcpy2c3 import py2c_copy_quaddobl_polysys_witset
from phcpy.solver import number_of_symbols
from phcpy.interface import store_quaddobl_system
from phcpy.interface import load_quaddobl_system, load_quaddobl_solutions
dim = number_of_symbols(pols)
if (topdim == -1):
topdim = dim - 1
fail = store_quaddobl_system(pols, nbvar=dim)
fail = py2c_quaddobl_polysys_solve(tasks,topdim, \
int(filter),int(factor),int(verbose))
witsols = []
for soldim in range(0, topdim + 1):
fail = py2c_copy_quaddobl_polysys_witset(soldim)
witset = (load_quaddobl_system(), load_quaddobl_solutions())
witsols.append(witset)
return witsols
def quaddobl_newton_power_series(pols, lser, idx=1, maxdeg=4, nbr=4, \
checkin=True, verbose=True):
r"""
Computes series in quad double precision for the polynomials
in *pols*, where the leading terms are given in the list *lser*.
On entry are the following five parameters:
*pols*: a list of string representations of polynomials,
*lser*: a list of polynomials in the series parameter (e.g.: t),
for use as start terms in Newton's method,
*idx*: index of the series parameter, by default equals 1,
*maxdeg*: maximal degree of the series,
*nbr*: number of steps with Newton's method,
*checkin*: checks whether the number of symbols in pols matches
the length of the list lser if idx == 0, or is one less than
the length of the list lser if idx != 0. If the conditions are
not satisfied, then an error message is printed and lser is returned.
*verbose*: whether to write intermediate output to screen or not.
On return is a list of lists of strings. Each lists of strings
represents the series solution for the variables in the list *pols*.
"""
from phcpy.solver import number_of_symbols
from phcpy.interface import store_quaddobl_system, load_quaddobl_system
from phcpy.phcpy2c2 import py2c_quaddobl_Newton_power_series as newton
from phcpy.phcpy2c2 import py2c_syspool_quaddobl_init
from phcpy.phcpy2c2 import py2c_syspool_quaddobl_create
from phcpy.phcpy2c2 import py2c_syspool_quaddobl_size as poolsize
from phcpy.phcpy2c2 import py2c_syspool_copy_to_quaddobl_container
from phcpy.phcpy2c2 import py2c_syspool_quaddobl_clear
nbsym = number_of_symbols(pols)
if verbose:
print "the polynomials :"
for pol in pols:
print pol
print "Number of variables :", nbsym
if checkin:
if not checkin_newton_power_series(nbsym, lser, idx):
return lser
store_quaddobl_system(lser, nbvar=1)
py2c_syspool_quaddobl_init(1);
py2c_syspool_quaddobl_create(1);
store_quaddobl_system(pols, nbvar=nbsym)
fail = newton(idx, maxdeg, nbr, int(verbose))
size = (-1 if fail else poolsize())
if verbose:
if size == -1:
print "An error occurred in the execution of Newton's method."
else:
print "Computed one series solution."
py2c_syspool_copy_to_quaddobl_container(1)
result = load_quaddobl_system()
result = substitute_symbol(result, idx)
py2c_syspool_quaddobl_clear()
return result
def quaddobl_monodromy_breakup(embsys, esols, dim, \
islaurent=0, verbose=True, nbloops=0):
r"""
Applies the monodromy breakup algorithm in quad double precision
to factor the *dim*-dimensional algebraic set represented by the embedded
system *embsys* and its solutions *esols*.
If the embedded polynomial system is a Laurent system,
then islaurent must equal one, the default is zero.
If *verbose* is False, then no output is written.
If *nbloops* equals zero, then the user is prompted to give
the maximum number of loops.
"""
from phcpy.phcpy2c3 import py2c_factor_set_quaddobl_to_mute
from phcpy.phcpy2c3 import py2c_factor_set_quaddobl_to_verbose
from phcpy.phcpy2c3 import py2c_factor_quaddobl_assign_labels
from phcpy.phcpy2c3 import py2c_factor_initialize_quaddobl_monodromy
from phcpy.phcpy2c3 import py2c_factor_initialize_quaddobl_sampler
from phcpy.phcpy2c3 import py2c_factor_initialize_quaddobl_Laurent_sampler
from phcpy.phcpy2c3 import py2c_factor_quaddobl_trace_grid_diagnostics
from phcpy.phcpy2c3 import py2c_factor_set_quaddobl_trace_slice
from phcpy.phcpy2c3 import py2c_factor_store_quaddobl_gammas
from phcpy.phcpy2c3 import py2c_factor_quaddobl_track_paths
from phcpy.phcpy2c3 import py2c_factor_store_quaddobl_solutions
from phcpy.phcpy2c3 import py2c_factor_restore_quaddobl_solutions
from phcpy.phcpy2c3 import py2c_factor_new_quaddobl_slices
from phcpy.phcpy2c3 import py2c_factor_swap_quaddobl_slices
from phcpy.phcpy2c3 import py2c_factor_permutation_after_quaddobl_loop
from phcpy.phcpy2c3 import py2c_factor_number_of_quaddobl_components
from phcpy.phcpy2c3 import py2c_factor_update_quaddobl_decomposition
from phcpy.phcpy2c3 import py2c_solcon_write_quaddobl_solutions
from phcpy.phcpy2c3 import py2c_solcon_clear_quaddobl_solutions
from phcpy.interface import store_quaddobl_solutions
from phcpy.interface import store_quaddobl_system
from phcpy.interface import store_quaddobl_laurent_system
if(verbose):
print('... applying monodromy factorization with quad doubles ...')
py2c_factor_set_quaddobl_to_verbose()
else:
py2c_factor_set_quaddobl_to_mute()
deg = len(esols)
nvar = len(embsys)
if(verbose):
print('dim =', dim)
store_quaddobl_solutions(nvar, esols)
if(islaurent == 1):
store_quaddobl_laurent_system(embsys)
py2c_factor_quaddobl_assign_labels(nvar, deg)
py2c_factor_initialize_quaddobl_Laurent_sampler(dim)
else:
store_quaddobl_system(embsys)
py2c_factor_quaddobl_assign_labels(nvar, deg)
py2c_factor_initialize_quaddobl_sampler(dim)
if(verbose):
py2c_solcon_write_quaddobl_solutions()
if(nbloops == 0):
strnbloops = input('give the maximum number of loops : ')
nbloops = int(strnbloops)
py2c_factor_initialize_quaddobl_monodromy(nbloops, deg, dim)
py2c_factor_store_quaddobl_solutions()
if(verbose):
print('... initializing the grid ...')
for i in range(1, 3):
py2c_factor_set_quaddobl_trace_slice(i)
py2c_factor_store_quaddobl_gammas(nvar)
py2c_factor_quaddobl_track_paths(islaurent)
py2c_factor_store_quaddobl_solutions()
py2c_factor_restore_quaddobl_solutions()
py2c_factor_swap_quaddobl_slices()
(err, dis) = py2c_factor_quaddobl_trace_grid_diagnostics()
if(verbose):
print('The diagnostics of the trace grid :')
print(' largest error on the samples :', err)
print(' smallest distance between the samples :', dis)
for i in range(1, nbloops+1):
if(verbose):
print('... starting loop %d ...' % i)
py2c_factor_new_quaddobl_slices(dim, nvar)
py2c_factor_store_quaddobl_gammas(nvar)
py2c_factor_quaddobl_track_paths(islaurent)
py2c_solcon_clear_quaddobl_solutions()
py2c_factor_store_quaddobl_gammas(nvar)
py2c_factor_quaddobl_track_paths(islaurent)
py2c_factor_store_quaddobl_solutions()
sprm = py2c_factor_permutation_after_quaddobl_loop(deg)
if(verbose):
perm = eval(sprm)
print('the permutation :', perm)
nb0 = py2c_factor_number_of_quaddobl_components()
done = py2c_factor_update_quaddobl_decomposition(deg, len(sprm), sprm)
nb1 = py2c_factor_number_of_quaddobl_components()
if(verbose):
print('number of factors : %d -> %d' % (nb0, nb1))
deco = decomposition(deg, 'qd')
print('decomposition :', deco)
if(done == 1):
break
py2c_factor_restore_quaddobl_solutions()
def quaddobl_monodromy_breakup(embsys, esols, dim, \
islaurent=0, verbose=True, nbloops=0):
r"""
Applies the monodromy breakup algorithm in quad double precision
to factor the *dim*-dimensional algebraic set represented by the embedded
system *embsys* and its solutions *esols*.
If the embedded polynomial system is a Laurent system,
then islaurent must equal one, the default is zero.
If *verbose* is False, then no output is written.
If *nbloops* equals zero, then the user is prompted to give
the maximum number of loops.
"""
from phcpy.phcpy2c3 import py2c_factor_set_quaddobl_to_mute
from phcpy.phcpy2c3 import py2c_factor_set_quaddobl_to_verbose
from phcpy.phcpy2c3 import py2c_factor_quaddobl_assign_labels
from phcpy.phcpy2c3 import py2c_factor_initialize_quaddobl_monodromy
from phcpy.phcpy2c3 import py2c_factor_initialize_quaddobl_sampler
from phcpy.phcpy2c3 import py2c_factor_initialize_quaddobl_Laurent_sampler
from phcpy.phcpy2c3 import py2c_factor_quaddobl_trace_grid_diagnostics
from phcpy.phcpy2c3 import py2c_factor_set_quaddobl_trace_slice
from phcpy.phcpy2c3 import py2c_factor_store_quaddobl_gammas
from phcpy.phcpy2c3 import py2c_factor_quaddobl_track_paths
from phcpy.phcpy2c3 import py2c_factor_store_quaddobl_solutions
from phcpy.phcpy2c3 import py2c_factor_restore_quaddobl_solutions
from phcpy.phcpy2c3 import py2c_factor_new_quaddobl_slices
from phcpy.phcpy2c3 import py2c_factor_swap_quaddobl_slices
from phcpy.phcpy2c3 import py2c_factor_permutation_after_quaddobl_loop
from phcpy.phcpy2c3 import py2c_factor_number_of_quaddobl_components
from phcpy.phcpy2c3 import py2c_factor_update_quaddobl_decomposition
from phcpy.phcpy2c3 import py2c_solcon_write_quaddobl_solutions
from phcpy.phcpy2c3 import py2c_solcon_clear_quaddobl_solutions
from phcpy.interface import store_quaddobl_solutions
from phcpy.interface import store_quaddobl_system
from phcpy.interface import store_quaddobl_laurent_system
if (verbose):
print('... applying monodromy factorization with quad doubles ...')
py2c_factor_set_quaddobl_to_verbose()
else:
py2c_factor_set_quaddobl_to_mute()
deg = len(esols)
nvar = len(embsys)
if (verbose):
print('dim =', dim)
store_quaddobl_solutions(nvar, esols)
if (islaurent == 1):
store_quaddobl_laurent_system(embsys)
py2c_factor_quaddobl_assign_labels(nvar, deg)
py2c_factor_initialize_quaddobl_Laurent_sampler(dim)
else:
store_quaddobl_system(embsys)
py2c_factor_quaddobl_assign_labels(nvar, deg)
py2c_factor_initialize_quaddobl_sampler(dim)
if (verbose):
py2c_solcon_write_quaddobl_solutions()
if (nbloops == 0):
strnbloops = input('give the maximum number of loops : ')
nbloops = int(strnbloops)
py2c_factor_initialize_quaddobl_monodromy(nbloops, deg, dim)
py2c_factor_store_quaddobl_solutions()
if (verbose):
print('... initializing the grid ...')
for i in range(1, 3):
py2c_factor_set_quaddobl_trace_slice(i)
py2c_factor_store_quaddobl_gammas(nvar)
py2c_factor_quaddobl_track_paths(islaurent)
py2c_factor_store_quaddobl_solutions()
py2c_factor_restore_quaddobl_solutions()
py2c_factor_swap_quaddobl_slices()
(err, dis) = py2c_factor_quaddobl_trace_grid_diagnostics()
if (verbose):
print('The diagnostics of the trace grid :')
print(' largest error on the samples :', err)
print(' smallest distance between the samples :', dis)
for i in range(1, nbloops + 1):
if (verbose):
print('... starting loop %d ...' % i)
py2c_factor_new_quaddobl_slices(dim, nvar)
py2c_factor_store_quaddobl_gammas(nvar)
py2c_factor_quaddobl_track_paths(islaurent)
py2c_solcon_clear_quaddobl_solutions()
py2c_factor_store_quaddobl_gammas(nvar)
py2c_factor_quaddobl_track_paths(islaurent)
py2c_factor_store_quaddobl_solutions()
sprm = py2c_factor_permutation_after_quaddobl_loop(deg)
if (verbose):
perm = eval(sprm)
print('the permutation :', perm)
nb0 = py2c_factor_number_of_quaddobl_components()
done = py2c_factor_update_quaddobl_decomposition(deg, len(sprm), sprm)
nb1 = py2c_factor_number_of_quaddobl_components()
if (verbose):
print('number of factors : %d -> %d' % (nb0, nb1))
deco = decomposition(deg, 'qd')
print('decomposition :', deco)
if (done == 1):
break
py2c_factor_restore_quaddobl_solutions()