def standard_littlewood_richardson_homotopies(
ndim, kdim, brackets, verbose=True, vrfcnd=False, minrep=True, tosqr=False, outputfilename=""
):
r"""
In n-dimensional space we consider k-dimensional planes,
subject to intersection conditions represented by brackets.
The parameters *ndim* and *kdim* give values for n and k respectively.
The parameter *brackets* is a list of brackets. A bracket is a list
of as many natural numbers (in the range 1..*ndim*) as *kdim*.
The Littlewood-Richardson homotopies compute k-planes that
meet the flags at spaces of dimensions prescribed by the brackets,
in standard double precision. Four options are passed as Booleans:
*verbose*: for adding extra output during computations,
*vrfcnd*: for extra diagnostic verification of Schubert conditions,
*minrep*: for a minimial representation of the problem formulation,
*tosqr*: to square the overdetermined systems.
On return is a 4-tuple. The first item of the tuple is the
formal root count, sharp for general flags, then as second
item the coordinates of the flags. The coordinates of the
flags are stored row wise in a list of real and imaginary parts.
The third and fourth item of the tuple on return are respectively
the polynomial system that has been solved and its solutions.
The length of the list of solution should match the root count.
"""
from phcpy.phcpy2c2 import py2c_solcon_clear_standard_solutions
from phcpy.phcpy2c2 import py2c_schubert_standard_littlewood_richardson_homotopies as stlrhom
from phcpy.interface import load_standard_solutions, load_standard_system
py2c_solcon_clear_standard_solutions()
nbc = len(brackets)
cds = ""
for bracket in brackets:
for num in bracket:
cds = cds + " " + str(num)
# print 'the condition string :', cds
(roco, sflags) = stlrhom(
ndim,
kdim,
nbc,
len(cds),
cds,
int(verbose),
int(vrfcnd),
int(minrep),
int(tosqr),
len(outputfilename),
outputfilename,
)
rflags = eval(sflags)
flgs = []
for k in range(len(rflags) / 2):
flgs.append(complex(rflags[2 * k], rflags[2 * k + 1]))
fsys = load_standard_system()
sols = load_standard_solutions()
return (roco, flgs, fsys, sols)
def standard_double_cascade_step(dim, embsys, esols, tasks=0):
r"""
Given in *embsys* an embedded polynomial system and
solutions with nonzero slack variables in *esols*, does one step
in the homotopy cascade, with standard double precision arithmetic.
The dimension of the solution set represented by *embsys*
and *esols* is the value of *dim*.
The number of tasks in multithreaded path tracking is given by *tasks*.
The default zero value of *tasks* indicates no multithreading.
The list on return contains witness points on
lower dimensional solution components.
"""
from phcpy.phcpy2c2 import py2c_copy_standard_container_to_start_system
from phcpy.phcpy2c2 import py2c_copy_standard_container_to_start_solutions
from phcpy.phcpy2c2 import py2c_standard_cascade_homotopy
from phcpy.phcpy2c2 import py2c_solve_by_standard_homotopy_continuation
from phcpy.phcpy2c2 import py2c_solcon_clear_standard_solutions
from phcpy.phcpy2c2 import py2c_copy_standard_target_solutions_to_container
from phcpy.interface import store_standard_witness_set
from phcpy.interface import load_standard_solutions
store_standard_witness_set(len(embsys), dim, embsys, esols)
py2c_copy_standard_container_to_start_system()
py2c_copy_standard_container_to_start_solutions()
py2c_standard_cascade_homotopy()
py2c_solve_by_standard_homotopy_continuation(tasks)
py2c_solcon_clear_standard_solutions()
py2c_copy_standard_target_solutions_to_container()
return load_standard_solutions()
def standard_solve(pols, silent=False, tasks=0):
"""
Calls the blackbox solver. On input in pols is a list of strings.
By default, the solver will print to screen the computed root counts.
To make the solver silent, set the flag silent to True.
The number of tasks for multithreading is given by tasks.
The solving happens in standard double precision arithmetic.
"""
from phcpy.phcpy2c2 import py2c_syscon_clear_standard_Laurent_system
from phcpy.phcpy2c2 \
import py2c_syscon_initialize_number_of_standard_Laurentials
from phcpy.phcpy2c2 import py2c_syscon_store_standard_Laurential
from phcpy.phcpy2c2 import py2c_solcon_clear_standard_solutions
from phcpy.phcpy2c2 import py2c_solve_Laurent_system
from phcpy.interface import load_standard_solutions
py2c_syscon_clear_standard_Laurent_system()
py2c_solcon_clear_standard_solutions()
dim = len(pols)
py2c_syscon_initialize_number_of_standard_Laurentials(dim)
for ind in range(0, dim):
pol = pols[ind]
nchar = len(pol)
py2c_syscon_store_standard_Laurential(nchar, dim, ind+1, pol)
py2c_solve_Laurent_system(silent, tasks)
return load_standard_solutions()
def standard_double_cascade_step(dim, embsys, esols, tasks=0):
r"""
Given in *embsys* an embedded polynomial system and
solutions with nonzero slack variables in *esols*, does one step
in the homotopy cascade, with standard double precision arithmetic.
The dimension of the solution set represented by *embsys*
and *esols* is the value of *dim*.
The number of tasks in multithreaded path tracking is given by *tasks*.
The default zero value of *tasks* indicates no multithreading.
The list on return contains witness points on
lower dimensional solution components.
"""
from phcpy.phcpy2c2 import py2c_copy_standard_container_to_start_system
from phcpy.phcpy2c2 import py2c_copy_standard_container_to_start_solutions
from phcpy.phcpy2c2 import py2c_standard_cascade_homotopy
from phcpy.phcpy2c2 import py2c_solve_by_standard_homotopy_continuation
from phcpy.phcpy2c2 import py2c_solcon_clear_standard_solutions
from phcpy.phcpy2c2 import py2c_copy_standard_target_solutions_to_container
from phcpy.interface import store_standard_witness_set
from phcpy.interface import load_standard_solutions
store_standard_witness_set(len(embsys), dim, embsys, esols)
py2c_copy_standard_container_to_start_system()
py2c_copy_standard_container_to_start_solutions()
py2c_standard_cascade_homotopy()
py2c_solve_by_standard_homotopy_continuation(tasks)
py2c_solcon_clear_standard_solutions()
py2c_copy_standard_target_solutions_to_container()
return load_standard_solutions()
def standard_start_diagonal_cascade(gamma=0, tasks=0):
r"""
Does the path tracking to start a diagonal cascade in standard double
precision. For this to work, the functions standard_diagonal_homotopy
and standard_diagonal_cascade_solutions must be executed successfully.
If *gamma* equals 0 on input, then a random gamma constant is generated,
otherwise, the given complex *gamma* will be used in the homotopy.
Multitasking is available, and activated by the *tasks* parameter.
Returns the target (system and its corresponding) solutions.
"""
from phcpy.phcpy2c2 import py2c_create_standard_homotopy
from phcpy.phcpy2c2 import py2c_create_standard_homotopy_with_gamma
from phcpy.phcpy2c2 import py2c_solve_by_standard_homotopy_continuation
from phcpy.phcpy2c2 import py2c_solcon_clear_standard_solutions
from phcpy.phcpy2c2 import py2c_syscon_clear_standard_system
from phcpy.phcpy2c2 import py2c_copy_standard_target_solutions_to_container
from phcpy.phcpy2c2 import py2c_copy_standard_target_system_to_container
from phcpy.interface import load_standard_solutions
from phcpy.interface import load_standard_system
if(gamma == 0):
py2c_create_standard_homotopy()
else:
py2c_create_standard_homotopy_with_gamma(gamma.real, gamma.imag)
py2c_solve_by_standard_homotopy_continuation(tasks)
py2c_solcon_clear_standard_solutions()
py2c_syscon_clear_standard_system()
py2c_copy_standard_target_solutions_to_container()
# from phcpy.phcpy2c2 import py2c_write_standard_target_system
# print 'the standard target system :'
# py2c_write_standard_target_system()
py2c_copy_standard_target_system_to_container()
tsys = load_standard_system()
sols = load_standard_solutions()
return (tsys, sols)
def run_pieri_homotopies(mdim, pdim, qdeg, planes, *pts, **opt):
r"""
Computes the number of *pdim*-plane producing maps of degree q*deg*
that meet *mdim*-planes at mdim*pdim + qdeq*(mdim+pdim) points.
For *qdeg* = 0, there are no interpolation points in *pts*.
The last parameter in opt contains the value for 'verbose',
which is True by default.
"""
from phcpy.phcpy2c2 import py2c_schubert_pieri_count
from phcpy.phcpy2c2 import py2c_schubert_pieri_homotopies
from phcpy.phcpy2c2 import py2c_syscon_load_standard_polynomial
from phcpy.phcpy2c2 import py2c_solcon_number_of_standard_solutions
from phcpy.phcpy2c2 import py2c_solcon_clear_standard_solutions
from phcpy.phcpy2c2 import py2c_solcon_length_standard_solution_string
from phcpy.phcpy2c2 import py2c_solcon_write_standard_solution_string
if 'verbose' not in opt:
verbose = True # by default, the function is verbose
else:
verbose = opt['verbose']
root_count = py2c_schubert_pieri_count(mdim, pdim, qdeg)
if verbose:
print 'Pieri root count for', (mdim, pdim, qdeg), 'is', root_count
strplanes = planes_to_string(planes)
# print 'length of input data :', len(strplanes)
if (qdeg > 0):
strpts = points_to_string(pts[0])
# print 'the interpolation points :', strpts
else:
strpts = ''
# print 'calling py2c_pieri_homotopies ...'
py2c_solcon_clear_standard_solutions()
if verbose:
print 'passing %d characters for (m, p, q) = (%d, %d, %d)' \
% (len(strplanes), mdim, pdim, qdeg)
py2c_schubert_pieri_homotopies(mdim, pdim, qdeg, \
len(strplanes), strplanes, strpts)
# print 'making the system ...'
pols = []
if (qdeg == 0):
for i in range(1, mdim * pdim + 1):
pols.append(py2c_syscon_load_standard_polynomial(i))
else:
for i in range(1, mdim * pdim + qdeg * (mdim + pdim) + 1):
pols.append(py2c_syscon_load_standard_polynomial(i))
if verbose:
print 'the system :'
for poly in pols:
print poly
print 'root count :', root_count
nbsols = py2c_solcon_number_of_standard_solutions()
sols = []
for k in range(1, nbsols + 1):
lns = py2c_solcon_length_standard_solution_string(k)
sol = py2c_solcon_write_standard_solution_string(k, lns)
sols.append(sol)
if verbose:
print 'the solutions :'
for solution in sols:
print solution
return (pols, sols)
def standard_start_diagonal_cascade(gamma=0, tasks=0):
r"""
Does the path tracking to start a diagonal cascade in standard double
precision. For this to work, the functions standard_diagonal_homotopy
and standard_diagonal_cascade_solutions must be executed successfully.
If *gamma* equals 0 on input, then a random gamma constant is generated,
otherwise, the given complex *gamma* will be used in the homotopy.
Multitasking is available, and activated by the *tasks* parameter.
Returns the target (system and its corresponding) solutions.
"""
from phcpy.phcpy2c2 import py2c_create_standard_homotopy
from phcpy.phcpy2c2 import py2c_create_standard_homotopy_with_gamma
from phcpy.phcpy2c2 import py2c_solve_by_standard_homotopy_continuation
from phcpy.phcpy2c2 import py2c_solcon_clear_standard_solutions
from phcpy.phcpy2c2 import py2c_syscon_clear_standard_system
from phcpy.phcpy2c2 import py2c_copy_standard_target_solutions_to_container
from phcpy.phcpy2c2 import py2c_copy_standard_target_system_to_container
from phcpy.interface import load_standard_solutions
from phcpy.interface import load_standard_system
if (gamma == 0):
py2c_create_standard_homotopy()
else:
py2c_create_standard_homotopy_with_gamma(gamma.real, gamma.imag)
py2c_solve_by_standard_homotopy_continuation(tasks)
py2c_solcon_clear_standard_solutions()
py2c_syscon_clear_standard_system()
py2c_copy_standard_target_solutions_to_container()
# from phcpy.phcpy2c2 import py2c_write_standard_target_system
# print 'the standard target system :'
# py2c_write_standard_target_system()
py2c_copy_standard_target_system_to_container()
tsys = load_standard_system()
sols = load_standard_solutions()
return (tsys, sols)
def run_pieri_homotopies(mdim, pdim, qdeg, planes, *pts, **opt):
r"""
Computes the number of *pdim*-plane producing maps of degree q*deg*
that meet *mdim*-planes at mdim*pdim + qdeq*(mdim+pdim) points.
For *qdeg* = 0, there are no interpolation points in *pts*.
The last parameter in opt contains the value for 'verbose',
which is True by default.
"""
from phcpy.phcpy2c2 import py2c_schubert_pieri_count
from phcpy.phcpy2c2 import py2c_schubert_pieri_homotopies
from phcpy.phcpy2c2 import py2c_syscon_load_standard_polynomial
from phcpy.phcpy2c2 import py2c_solcon_number_of_standard_solutions
from phcpy.phcpy2c2 import py2c_solcon_clear_standard_solutions
from phcpy.phcpy2c2 import py2c_solcon_length_standard_solution_string
from phcpy.phcpy2c2 import py2c_solcon_write_standard_solution_string
if 'verbose' not in opt:
verbose = True # by default, the function is verbose
else:
verbose = opt['verbose']
root_count = py2c_schubert_pieri_count(mdim, pdim, qdeg)
if verbose:
print 'Pieri root count for', (mdim, pdim, qdeg), 'is', root_count
strplanes = planes_to_string(planes)
# print 'length of input data :', len(strplanes)
if(qdeg > 0):
strpts = points_to_string(pts[0])
# print 'the interpolation points :', strpts
else:
strpts = ''
# print 'calling py2c_pieri_homotopies ...'
py2c_solcon_clear_standard_solutions()
if verbose:
print 'passing %d characters for (m, p, q) = (%d, %d, %d)' \
% (len(strplanes), mdim, pdim, qdeg)
py2c_schubert_pieri_homotopies(mdim, pdim, qdeg, \
len(strplanes), strplanes, strpts)
# print 'making the system ...'
pols = []
if(qdeg == 0):
for i in range(1, mdim*pdim+1):
pols.append(py2c_syscon_load_standard_polynomial(i))
else:
for i in range(1, mdim*pdim+qdeg*(mdim+pdim)+1):
pols.append(py2c_syscon_load_standard_polynomial(i))
if verbose:
print 'the system :'
for poly in pols:
print poly
print 'root count :', root_count
nbsols = py2c_solcon_number_of_standard_solutions()
sols = []
for k in range(1, nbsols+1):
lns = py2c_solcon_length_standard_solution_string(k)
sol = py2c_solcon_write_standard_solution_string(k, lns)
sols.append(sol)
if verbose:
print 'the solutions :'
for solution in sols:
print solution
return (pols, sols)
def standard_littlewood_richardson_homotopies(ndim, kdim, brackets, \
verbose=True, vrfcnd=False, minrep=True, tosqr=False, outputfilename=''):
r"""
In n-dimensional space we consider k-dimensional planes,
subject to intersection conditions represented by brackets.
The parameters *ndim* and *kdim* give values for n and k respectively.
The parameter *brackets* is a list of brackets. A bracket is a list
of as many natural numbers (in the range 1..*ndim*) as *kdim*.
The Littlewood-Richardson homotopies compute k-planes that
meet the flags at spaces of dimensions prescribed by the brackets,
in standard double precision. Four options are passed as Booleans:
*verbose*: for adding extra output during computations,
*vrfcnd*: for extra diagnostic verification of Schubert conditions,
*minrep*: for a minimial representation of the problem formulation,
*tosqr*: to square the overdetermined systems.
On return is a 4-tuple. The first item of the tuple is the
formal root count, sharp for general flags, then as second
item the coordinates of the flags. The coordinates of the
flags are stored row wise in a list of real and imaginary parts.
The third and fourth item of the tuple on return are respectively
the polynomial system that has been solved and its solutions.
The length of the list of solution should match the root count.
"""
from phcpy.phcpy2c2 import py2c_solcon_clear_standard_solutions
from phcpy.phcpy2c2 \
import py2c_schubert_standard_littlewood_richardson_homotopies \
as stlrhom
from phcpy.interface import load_standard_solutions, load_standard_system
py2c_solcon_clear_standard_solutions()
nbc = len(brackets)
cds = ''
for bracket in brackets:
for num in bracket:
cds = cds + ' ' + str(num)
# print 'the condition string :', cds
(roco, sflags) = stlrhom(ndim, kdim, nbc, len(cds), cds, \
int(verbose), int(vrfcnd), int(minrep), int(tosqr), \
len(outputfilename), outputfilename)
rflags = eval(sflags)
flgs = []
for k in range(len(rflags) / 2):
flgs.append(complex(rflags[2 * k], rflags[2 * k + 1]))
fsys = load_standard_system()
sols = load_standard_solutions()
return (roco, flgs, fsys, sols)
def store_standard_solutions(nvar, sols):
"""
Stores the solutions in the list sols, represented as strings
in PHCpack format into the container for solutions
with standard double precision.
The number nvar equals the number of variables.
"""
from phcpy.phcpy2c2 import py2c_solcon_clear_standard_solutions
from phcpy.phcpy2c2 import py2c_solcon_append_standard_solution_string
py2c_solcon_clear_standard_solutions()
for ind in range(0, len(sols)):
fail = py2c_solcon_append_standard_solution_string\
(nvar, len(sols[ind]), sols[ind])
if(fail != 0):
break
return fail
def store_standard_solutions(nvar, sols):
"""
Stores the solutions in the list sols, represented as strings
in PHCpack format into the container for solutions
with standard double precision.
The number nvar equals the number of variables.
"""
from phcpy.phcpy2c2 import py2c_solcon_clear_standard_solutions
from phcpy.phcpy2c2 import py2c_solcon_append_standard_solution_string
py2c_solcon_clear_standard_solutions()
for ind in range(0, len(sols)):
fail = py2c_solcon_append_standard_solution_string\
(nvar, len(sols[ind]), sols[ind])
if(fail != 0):
break
return fail
def standard_double_track(target, start, sols, gamma=0, tasks=0):
r"""
Does path tracking with standard double precision.
On input are a target system, a start system with solutions,
optionally: a (random) *gamma* constant and the number of *tasks*.
The default value zero for *tasks* indicates no multithreading.
The number of tasks in the multithreading is given by *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_standard_container_to_target_system
from phcpy.phcpy2c2 import py2c_copy_standard_container_to_start_system
from phcpy.phcpy2c2 import py2c_copy_standard_container_to_start_solutions
from phcpy.phcpy2c2 import py2c_create_standard_homotopy
from phcpy.phcpy2c2 import py2c_create_standard_homotopy_with_gamma
from phcpy.phcpy2c2 import py2c_solve_by_standard_homotopy_continuation
from phcpy.phcpy2c2 import py2c_solcon_clear_standard_solutions
from phcpy.phcpy2c2 import py2c_copy_standard_target_solutions_to_container
from phcpy.interface import store_standard_system
from phcpy.interface import store_standard_solutions
from phcpy.interface import load_standard_solutions
from phcpy.solver import number_of_symbols
dim = number_of_symbols(start)
store_standard_system(target, nbvar=dim)
py2c_copy_standard_container_to_target_system()
store_standard_system(start, nbvar=dim)
py2c_copy_standard_container_to_start_system()
# py2c_clear_standard_homotopy()
if(gamma == 0):
py2c_create_standard_homotopy()
else:
py2c_create_standard_homotopy_with_gamma(gamma.real, gamma.imag)
store_standard_solutions(dim, sols)
py2c_copy_standard_container_to_start_solutions()
py2c_solve_by_standard_homotopy_continuation(tasks)
py2c_solcon_clear_standard_solutions()
py2c_copy_standard_target_solutions_to_container()
return load_standard_solutions()
def store_standard_solutions(nvar, sols):
r"""
Stores the solutions in the list *sols*, represented as strings
in PHCpack format into the container for solutions
with standard double precision.
The number *nvar* equals the number of variables.
"""
from phcpy.phcpy2c2 import py2c_solcon_clear_standard_solutions
from phcpy.phcpy2c2 import py2c_solcon_append_standard_solution_string
py2c_solcon_clear_standard_solutions()
fail = 0
for ind in range(0, len(sols)):
fail = py2c_solcon_append_standard_solution_string\
(nvar, len(sols[ind]), sols[ind])
if(fail != 0):
# break
print 'Solution at position', ind, 'is not appended.'
return fail
def store_standard_solutions(nvar, sols):
r"""
Stores the solutions in the list *sols*, represented as strings
in PHCpack format into the container for solutions
with standard double precision.
The number *nvar* equals the number of variables.
"""
from phcpy.phcpy2c2 import py2c_solcon_clear_standard_solutions
from phcpy.phcpy2c2 import py2c_solcon_append_standard_solution_string
py2c_solcon_clear_standard_solutions()
fail = 0
for ind in range(0, len(sols)):
fail = py2c_solcon_append_standard_solution_string\
(nvar, len(sols[ind]), sols[ind])
if (fail != 0):
# break
print 'Solution at position', ind, 'is not appended.'
return fail
def standard_double_track(target, start, sols, gamma=0, tasks=0):
"""
Does path tracking with standard 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_standard_container_to_target_system
from phcpy.phcpy2c2 import py2c_copy_standard_container_to_start_system
from phcpy.phcpy2c2 import py2c_copy_standard_container_to_start_solutions
from phcpy.phcpy2c2 import py2c_create_standard_homotopy
from phcpy.phcpy2c2 import py2c_create_standard_homotopy_with_gamma
from phcpy.phcpy2c2 import py2c_solve_by_standard_homotopy_continuation
from phcpy.phcpy2c2 import py2c_solcon_clear_standard_solutions
from phcpy.phcpy2c2 import py2c_copy_standard_target_solutions_to_container
from phcpy.interface import store_standard_system
from phcpy.interface import store_standard_solutions
from phcpy.interface import load_standard_solutions
from phcpy.solver import number_of_symbols
dim = number_of_symbols(start)
store_standard_system(target, nbvar=dim)
py2c_copy_standard_container_to_target_system()
store_standard_system(start, nbvar=dim)
py2c_copy_standard_container_to_start_system()
# py2c_clear_standard_homotopy()
if(gamma == 0):
py2c_create_standard_homotopy()
else:
py2c_create_standard_homotopy_with_gamma(gamma.real, gamma.imag)
store_standard_solutions(dim, sols)
py2c_copy_standard_container_to_start_solutions()
py2c_solve_by_standard_homotopy_continuation(tasks)
py2c_solcon_clear_standard_solutions()
py2c_copy_standard_target_solutions_to_container()
return load_standard_solutions()
def standard_littlewood_richardson_homotopies(ndim, kdim, brackets, \
verbose=True, vrfcnd=False, outputfilename='/tmp/output'):
"""
In n-dimensional space we consider k-dimensional planes,
subject to intersection conditions represented by brackets.
The parameters ndim and kdim give values for n and k respectively.
The parameter brackets is a list of brackets. A bracket is a list
of as many natural numbers (in the range 1..ndim) as kdim.
The Littlewood-Richardson homotopies compute k-planes that
meet the flags at spaces of dimensions prescribed by the brackets,
in standard double precision.
On return is a 4-tuple. The first item of the tuple is the
formal root count, sharp for general flags, then as second
item the coordinates of the flags. The coordinates of the
flags are stored row wise in a list of real and imaginary parts.
The third and fourth item of the tuple on return are respectively
the polynomial system that has been solved and its solutions.
The length of the list of solution should match the root count.
"""
from phcpy.phcpy2c2 import py2c_solcon_clear_standard_solutions
from phcpy.phcpy2c2 \
import py2c_schubert_standard_littlewood_richardson_homotopies \
as stlrhom
from phcpy.interface import load_standard_solutions, load_standard_system
py2c_solcon_clear_standard_solutions()
nbc = len(brackets)
cds = ''
for bracket in brackets:
for num in bracket:
cds = cds + ' ' + str(num)
# print 'the condition string :', cds
(roco, sflags) = stlrhom(ndim, kdim, nbc, len(cds), cds, int(verbose), \
int(vrfcnd), len(outputfilename), outputfilename)
rflags = eval(sflags)
flgs = []
for k in range(len(rflags) / 2):
flgs.append(complex(rflags[2 * k], rflags[2 * k + 1]))
fsys = load_standard_system()
sols = load_standard_solutions()
return (roco, flgs, fsys, sols)
def standard_littlewood_richardson_homotopies(ndim, kdim, brackets, \
verbose=True, vrfcnd=False, outputfilename='/tmp/output'):
"""
In n-dimensional space we consider k-dimensional planes,
subject to intersection conditions represented by brackets.
The parameters ndim and kdim give values for n and k respectively.
The parameter brackets is a list of brackets. A bracket is a list
of as many natural numbers (in the range 1..ndim) as kdim.
The Littlewood-Richardson homotopies compute k-planes that
meet the flags at spaces of dimensions prescribed by the brackets,
in standard double precision.
On return is a 4-tuple. The first item of the tuple is the
formal root count, sharp for general flags, then as second
item the coordinates of the flags. The coordinates of the
flags are stored row wise in a list of real and imaginary parts.
The third and fourth item of the tuple on return are respectively
the polynomial system that has been solved and its solutions.
The length of the list of solution should match the root count.
"""
from phcpy.phcpy2c2 import py2c_solcon_clear_standard_solutions
from phcpy.phcpy2c2 \
import py2c_schubert_standard_littlewood_richardson_homotopies \
as stlrhom
from phcpy.interface import load_standard_solutions, load_standard_system
py2c_solcon_clear_standard_solutions()
nbc = len(brackets)
cds = ''
for bracket in brackets:
for num in bracket:
cds = cds + ' ' + str(num)
# print 'the condition string :', cds
(roco, sflags) = stlrhom(ndim, kdim, nbc, len(cds), cds, int(verbose), \
int(vrfcnd), len(outputfilename), outputfilename)
rflags = eval(sflags)
flgs = []
for k in range(len(rflags)/2):
flgs.append(complex(rflags[2*k], rflags[2*k+1]))
fsys = load_standard_system()
sols = load_standard_solutions()
return (roco, flgs, fsys, sols)
def standard_random_coefficient_system(silent=False):
"""
Runs the polyhedral homotopies and returns a random coefficient
system based on the contents of the cell container,
in standard double precision arithmetic.
For this to work, the mixed_volume function must be called first.
"""
from phcpy.phcpy2c2 import py2c_celcon_standard_random_coefficient_system
from phcpy.phcpy2c2 import py2c_celcon_copy_into_standard_systems_container
from phcpy.phcpy2c2 import py2c_celcon_standard_polyhedral_homotopy
from phcpy.phcpy2c2 import py2c_celcon_number_of_cells
from phcpy.phcpy2c2 import py2c_solcon_clear_standard_solutions
from phcpy.phcpy2c2 import py2c_celcon_solve_standard_start_system
from phcpy.phcpy2c2 import py2c_celcon_track_standard_solution_path
from phcpy.phcpy2c2 \
import py2c_celcon_copy_target_standard_solution_to_container
py2c_celcon_standard_random_coefficient_system()
py2c_celcon_copy_into_standard_systems_container()
from phcpy.interface import load_standard_system, load_standard_solutions
# py2c_syscon_write_system()
result = load_standard_system()
# print result
py2c_celcon_standard_polyhedral_homotopy()
nbcells = py2c_celcon_number_of_cells()
py2c_solcon_clear_standard_solutions()
for cell in range(1, nbcells+1):
mixvol = py2c_celcon_solve_standard_start_system(cell)
if not silent:
print 'system %d has %d solutions' % (cell, mixvol)
for j in range(1, mixvol+1):
if not silent:
print '-> tracking path %d out of %d' % (j, mixvol)
py2c_celcon_track_standard_solution_path(cell, j, 0)
py2c_celcon_copy_target_standard_solution_to_container(cell, j)
sols = load_standard_solutions()
# print sols
# newton_step(result, sols)
return (result, sols)
def standard_monodromy_breakup(embsys, esols, dim, \
islaurent=0, verbose=True, nbloops=0):
r"""
Applies the monodromy breakup algorithm in standard 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.phcpy2c2 import py2c_factor_set_standard_to_mute
from phcpy.phcpy2c2 import py2c_factor_set_standard_to_verbose
from phcpy.phcpy2c2 import py2c_factor_standard_assign_labels
from phcpy.phcpy2c2 import py2c_factor_initialize_standard_monodromy
from phcpy.phcpy2c2 import py2c_factor_initialize_standard_sampler
from phcpy.phcpy2c2 import py2c_factor_initialize_standard_Laurent_sampler
from phcpy.phcpy2c2 import py2c_factor_standard_trace_grid_diagnostics
from phcpy.phcpy2c2 import py2c_factor_set_standard_trace_slice
from phcpy.phcpy2c2 import py2c_factor_store_standard_gammas
from phcpy.phcpy2c2 import py2c_factor_standard_track_paths
from phcpy.phcpy2c2 import py2c_factor_store_standard_solutions
from phcpy.phcpy2c2 import py2c_factor_restore_standard_solutions
from phcpy.phcpy2c2 import py2c_factor_new_standard_slices
from phcpy.phcpy2c2 import py2c_factor_swap_standard_slices
from phcpy.phcpy2c2 import py2c_factor_permutation_after_standard_loop
from phcpy.phcpy2c2 import py2c_factor_number_of_standard_components
from phcpy.phcpy2c2 import py2c_factor_update_standard_decomposition
from phcpy.phcpy2c2 import py2c_solcon_write_standard_solutions
from phcpy.phcpy2c2 import py2c_solcon_clear_standard_solutions
from phcpy.interface import store_standard_witness_set
from phcpy.interface import store_standard_laurent_witness_set
if(verbose):
print('... applying monodromy factorization with standard doubles ...')
py2c_factor_set_standard_to_verbose()
else:
py2c_factor_set_standard_to_mute()
deg = len(esols)
nvar = len(embsys)
if(verbose):
print('dim =', dim)
if(islaurent == 1):
store_standard_laurent_witness_set(nvar, dim, embsys, esols)
py2c_factor_standard_assign_labels(nvar, deg)
py2c_factor_initialize_standard_Laurent_sampler(dim)
else:
store_standard_witness_set(nvar, dim, embsys, esols)
py2c_factor_standard_assign_labels(nvar, deg)
py2c_factor_initialize_standard_sampler(dim)
if(verbose):
py2c_solcon_write_standard_solutions()
if(nbloops == 0):
strnbloops = input('give the maximum number of loops : ')
nbloops = int(strnbloops)
py2c_factor_initialize_standard_monodromy(nbloops, deg, dim)
py2c_factor_store_standard_solutions()
if(verbose):
print('... initializing the grid in standard double precision ...')
for i in range(1, 3):
py2c_factor_set_standard_trace_slice(i)
py2c_factor_store_standard_gammas(nvar)
py2c_factor_standard_track_paths(islaurent)
py2c_factor_store_standard_solutions()
py2c_factor_restore_standard_solutions()
py2c_factor_swap_standard_slices()
(err, dis) = py2c_factor_standard_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_standard_slices(dim, nvar)
py2c_factor_store_standard_gammas(nvar)
py2c_factor_standard_track_paths(islaurent)
py2c_solcon_clear_standard_solutions()
py2c_factor_store_standard_gammas(nvar)
py2c_factor_standard_track_paths(islaurent)
py2c_factor_store_standard_solutions()
sprm = py2c_factor_permutation_after_standard_loop(deg)
if(verbose):
perm = eval(sprm)
print('the permutation :', perm)
nb0 = py2c_factor_number_of_standard_components()
done = py2c_factor_update_standard_decomposition(deg, len(sprm), sprm)
nb1 = py2c_factor_number_of_standard_components()
if(verbose):
print('number of factors : %d -> %d' % (nb0, nb1))
deco = decomposition(deg)
print('decomposition :', deco)
if(done == 1):
break
py2c_factor_restore_standard_solutions()