def newton_laurent_step(system, solutions, precision='d', decimals=100):
"""
Applies one Newton step to the solutions of the Laurent 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_laurent_system
from phcpy.interface import store_standard_solutions
from phcpy.interface import load_standard_solutions
store_standard_laurent_system(system, nbvar=dim)
store_standard_solutions(dim, solutions)
from phcpy.phcpy2c3 import py2c_standard_Newton_Laurent_step
py2c_standard_Newton_Laurent_step()
result = load_standard_solutions()
elif (precision == 'dd'):
from phcpy.interface import store_dobldobl_laurent_system
from phcpy.interface import store_dobldobl_solutions
from phcpy.interface import load_dobldobl_solutions
store_dobldobl_laurent_system(system, nbvar=dim)
store_dobldobl_solutions(dim, solutions)
from phcpy.phcpy2c3 import py2c_dobldobl_Newton_Laurent_step
py2c_dobldobl_Newton_Laurent_step()
result = load_dobldobl_solutions()
elif (precision == 'qd'):
from phcpy.interface import store_quaddobl_laurent_system
from phcpy.interface import store_quaddobl_solutions
from phcpy.interface import load_quaddobl_solutions
store_quaddobl_laurent_system(system, nbvar=dim)
store_quaddobl_solutions(dim, solutions)
from phcpy.phcpy2c3 import py2c_quaddobl_Newton_Laurent_step
py2c_quaddobl_Newton_Laurent_step()
result = load_quaddobl_solutions()
elif (precision == 'mp'):
from phcpy.interface import store_multprec_laurent_system
from phcpy.interface import store_multprec_solutions
from phcpy.interface import load_multprec_solutions
store_multprec_laurent_system(system, decimals, nbvar=dim)
store_multprec_solutions(dim, solutions)
from phcpy.phcpy2c3 import py2c_multprec_Newton_Laurent_step
py2c_multprec_Newton_Laurent_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_laurent_step(system, solutions, precision='d', decimals=100):
"""
Applies one Newton step to the solutions of the Laurent 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_laurent_system
from phcpy.interface import store_standard_solutions
from phcpy.interface import load_standard_solutions
store_standard_laurent_system(system)
store_standard_solutions(len(system), solutions)
from phcpy.phcpy2c3 import py2c_standard_Newton_Laurent_step
py2c_standard_Newton_Laurent_step()
result = load_standard_solutions()
elif(precision == 'dd'):
from phcpy.interface import store_dobldobl_laurent_system
from phcpy.interface import store_dobldobl_solutions
from phcpy.interface import load_dobldobl_solutions
store_dobldobl_laurent_system(system)
store_dobldobl_solutions(len(system), solutions)
from phcpy.phcpy2c3 import py2c_dobldobl_Newton_Laurent_step
py2c_dobldobl_Newton_Laurent_step()
result = load_dobldobl_solutions()
elif(precision == 'qd'):
from phcpy.interface import store_quaddobl_laurent_system
from phcpy.interface import store_quaddobl_solutions
from phcpy.interface import load_quaddobl_solutions
store_quaddobl_laurent_system(system)
store_quaddobl_solutions(len(system), solutions)
from phcpy.phcpy2c3 import py2c_quaddobl_Newton_Laurent_step
py2c_quaddobl_Newton_Laurent_step()
result = load_quaddobl_solutions()
elif(precision == 'mp'):
from phcpy.interface import store_multprec_laurent_system
from phcpy.interface import store_multprec_solutions
from phcpy.interface import load_multprec_solutions
store_multprec_laurent_system(system, decimals)
store_multprec_solutions(len(system), solutions)
from phcpy.phcpy2c3 import py2c_multprec_Newton_Laurent_step
py2c_multprec_Newton_Laurent_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 multprec_usolve(pol, mxi, eps, decimals):
"""
Applies the method of Durand-Kerner (aka Weierstrass)
to the polynomial in the string pol, in arbitrary multiprecision,
the number of decimal places in the precision is in decimals.
The maximum number of iterations is in mxi,
the requirement on the accuracy in eps.
"""
from phcpy.phcpy2c3 import py2c_usolve_multprec
from phcpy.interface import store_multprec_system, load_multprec_solutions
store_multprec_system([pol], decimals)
nit = py2c_usolve_multprec(decimals, mxi, eps)
rts = load_multprec_solutions()
return (nit, rts)
def multprec_usolve(pol, mxi, eps, decimals):
"""
Applies the method of Durand-Kerner (aka Weierstrass)
to the polynomial in the string pol, in arbitrary multiprecision,
the number of decimal places in the precision is in decimals.
The maximum number of iterations is in mxi,
the requirement on the accuracy in eps.
"""
from phcpy.phcpy2c3 import py2c_usolve_multprec
from phcpy.interface import store_multprec_system, load_multprec_solutions
store_multprec_system([pol], decimals)
nit = py2c_usolve_multprec(decimals, mxi, eps)
rts = load_multprec_solutions()
return (nit, rts)
def multiprecision_track(target, start, sols, gamma=0, decimals=80):
"""
Does path tracking with multiprecision.
On input are a target system, a start system with solutions,
and optionally a (random) gamma constant.
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 decimal places in the working precision is
given by the value of decimals.
On return are the string representations of the solutions
computed at the end of the paths.
"""
from phcpy.phcpy2c2 import py2c_copy_multprec_container_to_target_system
from phcpy.phcpy2c2 import py2c_copy_multprec_container_to_start_system
from phcpy.phcpy2c2 import py2c_copy_multprec_container_to_start_solutions
from phcpy.phcpy2c2 import py2c_create_multprec_homotopy
from phcpy.phcpy2c2 import py2c_create_multprec_homotopy_with_gamma
from phcpy.phcpy2c2 import py2c_solve_by_multprec_homotopy_continuation
from phcpy.phcpy2c2 import py2c_solcon_clear_multprec_solutions
from phcpy.phcpy2c2 import py2c_copy_multprec_target_solutions_to_container
from phcpy.interface import store_multprec_system
from phcpy.interface import store_multprec_solutions
from phcpy.interface import load_multprec_solutions
store_multprec_system(target, decimals)
py2c_copy_multprec_container_to_target_system()
store_multprec_system(start, decimals)
py2c_copy_multprec_container_to_start_system()
# py2c_clear_multprec_homotopy()
if gamma == 0:
py2c_create_multprec_homotopy()
else:
py2c_create_multprec_homotopy_with_gamma(gamma.real, gamma.imag)
dim = len(start)
store_multprec_solutions(dim, sols)
py2c_copy_multprec_container_to_start_solutions()
py2c_solve_by_multprec_homotopy_continuation(decimals)
py2c_solcon_clear_multprec_solutions()
py2c_copy_multprec_target_solutions_to_container()
return load_multprec_solutions()
def multiprecision_track(target, start, sols, gamma=0, decimals=80):
"""
Does path tracking with multiprecision.
On input are a target system, a start system with solutions,
and optionally a (random) gamma constant.
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 decimal places in the working precision is
given by the value of decimals.
On return are the string representations of the solutions
computed at the end of the paths.
"""
from phcpy.phcpy2c2 import py2c_copy_multprec_container_to_target_system
from phcpy.phcpy2c2 import py2c_copy_multprec_container_to_start_system
from phcpy.phcpy2c2 import py2c_copy_multprec_container_to_start_solutions
from phcpy.phcpy2c2 import py2c_create_multprec_homotopy
from phcpy.phcpy2c2 import py2c_create_multprec_homotopy_with_gamma
from phcpy.phcpy2c2 import py2c_solve_by_multprec_homotopy_continuation
from phcpy.phcpy2c2 import py2c_solcon_clear_multprec_solutions
from phcpy.phcpy2c2 import py2c_copy_multprec_target_solutions_to_container
from phcpy.interface import store_multprec_system
from phcpy.interface import store_multprec_solutions
from phcpy.interface import load_multprec_solutions
store_multprec_system(target, decimals)
py2c_copy_multprec_container_to_target_system()
store_multprec_system(start, decimals)
py2c_copy_multprec_container_to_start_system()
# py2c_clear_multprec_homotopy()
if(gamma == 0):
py2c_create_multprec_homotopy()
else:
py2c_create_multprec_homotopy_with_gamma(gamma.real, gamma.imag)
dim = len(start)
store_multprec_solutions(dim, sols)
py2c_copy_multprec_container_to_start_solutions()
py2c_solve_by_multprec_homotopy_continuation(decimals)
py2c_solcon_clear_multprec_solutions()
py2c_copy_multprec_target_solutions_to_container()
return load_multprec_solutions()
def newton_steps(system, solutions, accuracy=8, maxsteps=4, maxprec=256):
"""
Runs a sequence of variable precision Newton steps to approximate
solutions accurate up to a specified number of decimal places.
In addition to the system and solutions, there are three parameters:
accuracy : number of decimal places wanted to be accurate,
maxsteps : maximum number of Newton steps,
maxprec : maximum number of decimal places in the precision used
to estimate the condition numbers.
"""
from phcpy.phcpy2c3 import py2c_varbprec_Newton_Laurent_steps as vmpnewt
from phcpy.interface import store_multprec_solutions
from phcpy.interface import load_multprec_solutions
dim = len(system)
store_multprec_solutions(dim, solutions)
pols = ""
for polynomial in system:
pols = pols + polynomial
vmpnewt(dim, accuracy, maxsteps, maxprec, len(pols), pols)
result = load_multprec_solutions()
return result
def newton_steps(system, solutions, accuracy=8, maxsteps=4, maxprec=256):
"""
Runs a sequence of variable precision Newton steps to approximate
solutions accurate up to a specified number of decimal places.
In addition to the system and solutions, there are three parameters:
accuracy : number of decimal places wanted to be accurate,
maxsteps : maximum number of Newton steps,
maxprec : maximum number of decimal places in the precision used
to estimate the condition numbers.
"""
from phcpy.phcpy2c3 import py2c_varbprec_Newton_Laurent_steps as vmpnewt
from phcpy.interface import store_multprec_solutions
from phcpy.interface import load_multprec_solutions
dim = len(system)
store_multprec_solutions(dim, solutions)
pols = ""
for polynomial in system:
pols = pols + polynomial
vmpnewt(dim, accuracy, maxsteps, maxprec, len(pols), pols)
result = load_multprec_solutions()
return result