def main():
"""
Excerpt from the user manual of phcpy.
"""
from phcpy.phcpy2c3 import py2c_set_seed
py2c_set_seed(2019)
p = ['x^2 + y - 3;', 'x + 0.125*y^2 - 1.5;']
print('constructing a total degree start system ...')
from phcpy.solver import total_degree_start_system as tds
q, qsols = tds(p)
print('number of start solutions :', len(qsols))
from phcpy.trackers import initialize_standard_tracker
from phcpy.trackers import initialize_standard_solution
from phcpy.trackers import next_standard_solution
initialize_standard_tracker(p, q, False) # True)
# initialize_standard_tracker(p, q, True)
from phcpy.solutions import strsol2dict
import matplotlib.pyplot as plt
plt.ion()
fig = plt.figure()
for k in range(len(qsols)):
if(k == 0):
axs = fig.add_subplot(221)
elif(k == 1):
axs = fig.add_subplot(222)
elif(k == 2):
axs = fig.add_subplot(223)
elif(k == 3):
axs = fig.add_subplot(224)
startsol = qsols[k]
initialize_standard_solution(len(p),startsol)
dictsol = strsol2dict(startsol)
xpoints = [dictsol['x']]
ypoints = [dictsol['y']]
for k in range(300):
ns = next_standard_solution()
dictsol = strsol2dict(ns)
xpoints.append(dictsol['x'])
ypoints.append(dictsol['y'])
tval = dictsol['t'].real
# tval = eval(dictsol['t'].lstrip().split(' ')[0])
if(tval == 1.0):
break
print(ns)
xre = [point.real for point in xpoints]
yre = [point.real for point in ypoints]
axs.set_xlim(min(xre)-0.3, max(xre)+0.3)
axs.set_ylim(min(yre)-0.3, max(yre)+0.3)
dots, = axs.plot(xre,yre,'r-')
fig.canvas.draw()
fig.canvas.draw()
ans = input('hit return to exit')
def solve4circles(syst, verbose=True):
"""
Given in syst is a list of polynomial systems.
Returns a list of tuples. Each tuple in the list of return
consists of the coordinates of the center and the radius of
a circle touching the three given circles.
"""
from phcpy.solver import solve
from phcpy.solutions import strsol2dict, is_real
(circle, eqscnt) = (0, 0)
result = []
for eqs in syst:
eqscnt = eqscnt + 1
if verbose:
print('solving system', eqscnt, ':')
for pol in eqs:
print(pol)
sols = solve(eqs, silent=True)
if verbose:
print('system', eqscnt, 'has', len(sols), 'solutions')
for sol in sols:
if is_real(sol, 1.0e-8):
soldic = strsol2dict(sol)
if soldic['r'].real > 0:
circle = circle + 1
ctr = (soldic['x'].real, soldic['y'].real)
rad = soldic['r'].real
result.append((ctr, rad))
if verbose:
print('solution circle', circle)
print('center =', ctr)
print('radius =', rad)
return result
def straight_line(verbose=True):
"""
This function solves an instance where the five precision
points lie on a line. The coordinates are taken from Problem 7
of the paper by A.P. Morgan and C.W. Wampler.
Returns a list of solution dictionaries for the real solutions.
"""
from phcpy.solutions import strsol2dict, is_real
pt0 = Matrix([[ 0.50], [ 1.06]])
pt1 = Matrix([[-0.83], [-0.27]])
pt2 = Matrix([[-0.34], [ 0.22]])
pt3 = Matrix([[-0.13], [ 0.43]])
pt4 = Matrix([[ 0.22], [ 0.78]])
piv = Matrix([[1], [0]])
equ = polynomials(pt0,pt1,pt2,pt3,pt4,piv)
if verbose:
print 'the polynomial system :'
for pol in equ:
print pol
from phcpy.solver import solve
sols = solve(equ)
if verbose:
print 'the solutions :'
for sol in sols:
print sol
print 'computed', len(sols), 'solutions'
result = []
for sol in sols:
if is_real(sol, 1.0e-8):
soldic = strsol2dict(sol)
result.append(soldic)
return result
def eqs(ip):
#x1 -> omega_s
#x2 -> ns
#x3 -> np
eq1 = "{0:+.16f}*x1{1:+.16f}*x2{2:+.16f}*x3{3:+.16f};" .format((-gi - gs)/(gi*xs**2), -alpha**2 + 1, -2*alpha + 2, (-ei*gs + es*gi + 2*gs*omega_p_chosen)/(gi*xs**2))
eq2 = "{0:.16f}*x1^2{1:+.16f}*x1*x2{2:+.16f}*x1*x3{3:+.16f}*x1{4:+.16f}*x2^2{5:+.16f}*x2*x3{6:+.16f}*x2{7:+.16f}*x3^2{8:+.16f}*x3{9:+.16f};" .format(xs**(-4), (-4*alpha - 2)/xs**2, -4/xs**2, -2*es/xs**4, 4*alpha**2 + 4*alpha + 1, 8*alpha + 4, (4*alpha*es + 2*es)/xs**2, -alpha + 4, 4*es/xs**2, (4*es**2 + gs**2)/(4*xs**4))
eq3 = "{0:.16f}*x1^2*x2^2{1:+.16f}*x1*x2^3{2:+.16f}*x1*x2^2*x3{3:+.16f}*x1*x2^2{4:+.16f}*x1*x2*x3^2{5:+.16f}*x1*x2*x3{6:+.16f}*x2^4{7:+.16f}*x2^3*x3{8:+.16f}*x2^3{9:+.16f}*x2^2*x3^2{10:+.16f}*x2^2*x3{11:+.16f}*x2^2{12:+.16f}*x2*x3^3{13:+.16f}*x2*x3^2{14:+.16f}*x2*x3{15:+.16f}*x3^4{16:+.16f}*x3^3{17:+.16f}*x3^2{18:+.16f}*x3;" .format(4.0/xs**4, 1.0*(-16.0*alpha - 8.0)/xs**2, 1.0*(8.0*alpha - 8.0)/xs**2, -8.0*es/xs**4, 4.0/xs**2, 1.0*(4.0*ep - 4.0*omega_p_chosen)/(xp**2*xs**2), 16.0*alpha**2 + 16.0*alpha + 4.0, -16.0*alpha**2 + 8.0*alpha + 8.0, 1.0*(16.0*alpha*es + 8.0*es)/xs**2, 4.0*alpha**2 - 16.0*alpha, 1.0*(-8.0*alpha*ep*xs**2 - 8.0*alpha*es*xp**2 + 8.0*alpha*omega_p_chosen*xs**2 - 4.0*ep*xs**2 + 8.0*es*xp**2 + 4.0*omega_p_chosen*xs**2)/(xp**2*xs**2), 1.0*(4.0*es**2 + 1.0*gs**2)/xs**4, 4.0*alpha - 4.0, 1.0*(4.0*alpha*ep*xs**2 - 4.0*alpha*omega_p_chosen*xs**2 - 4.0*ep*xs**2 - 4.0*es*xp**2 + 4.0*omega_p_chosen*xs**2)/(xp**2*xs**2), 1.0*(-4.0*ep*es + 4.0*es*omega_p_chosen + 1.0*gs)/(xp**2*xs**2), 1.00000000000000, 1.0*(2.0*ep - 2.0*omega_p_chosen)/xp**2, 1.0*(1.0*ep**2 - 2.0*ep*omega_p_chosen + 1.0*omega_p_chosen**2 + 0.25)/xp**4, -1.0*ip)
f = [eq1, eq2, eq3]
#seeding random number generator for reproducible results
py2c_set_seed(130683)
s = solve(f,silent=True)
ns = len(s) #number of solutions
R = []
for k in range(ns):
d = strsol2dict(s[k])
x123 = np.array([d['x1'], d['x2'], d['x3']])
real = np.real(x123)
imag = np.fabs(np.imag(x123))
sol = np.empty((real.size + imag.size,), dtype=real.dtype)
sol[0::2] = real
sol[1::2] = imag
if np.allclose(imag, np.zeros(3)):
if real[1] > 1e-7:
R.append((sol[0], sol[2], sol[4], ip))
return R
def phcpy(line, cell=None):
"Solve a system of polynomials formatted for phcpy."
# TODO: progress bar
from phcpy.solver import solve
from phcpy.solutions import strsol2dict
if cell is None:
with open(line, 'r') as poly:
sys = ' '.join([f.rstrip() for f in poly])
else:
# jupyter uses unix newlines
sys = ' '.join([f.rstrip() for f in cell.split('\n')])
# cell.replace('\n',' ').lstrip()
# discard equation count (and undeterminate count) if present
while sys.split(' ')[0].isdigit():
sys = sys.split(' ', 1)[1]
# convert to list with delimiters intact
sys = [f+';' for f in sys.split(';')[:-1]]
# for l in sys: print(l)
ret = solve(sys)
return [strsol2dict(s) for s in ret]
def findEmbeddings_dist(syst, interval):
global prevSystem
global usePrev
global prevSolutions
# i = 0
y1_left, y1_right, y4_left, y4_right = interval
# print fileNamePref
while True:
if prevSystem and usePrev:
sols = track(syst, prevSystem, prevSolutions, tasks=tasks_num)
else:
sols = solve(syst, verbose=0, tasks=tasks_num)
result_real = []
for sol in sols:
soldic = strsol2dict(sol)
if is_real(sol, tolerance) and soldic['y1'].real>=y1_left and soldic['y1'].real<=y1_right and soldic['y4'].real>=y4_left and soldic['y4'].real<=y4_right:
result_real.append(soldic)
num_real = len(result_real)
if num_real%2==0 and len(sols)==numAll:
prevSystem = syst
prevSolutions = sols
return num_real
def coordinates_and_slopes(sol):
"""
Given a solution, return the 3-tuple with the x and y coordinates
of the tangent point and the slope of the tangent line.
The real parts of the coordinates are selected.
"""
from phcpy.solutions import strsol2dict
sdc = strsol2dict(sol)
return (sdc['x'].real, sdc['y'].real, sdc['s'].real)
def run_tracker(start, sols, target, sol_num, gamma, min_step, max_step,
num_steps):
set_homotopy_continuation_gamma(gamma.real, gamma.imag)
set_homotopy_continuation_parameter(5, min_step)
set_homotopy_continuation_parameter(4, max_step)
set_homotopy_continuation_parameter(12, num_steps)
dim = number_of_symbols(start)
result = [[], [], [], [], [], [], []]
for i in range(dim):
result[2].append([])
standard_set_homotopy(target, start, False)
idx = 0
choice_sol = sol_num
solution = sols[choice_sol - 1]
idx = idx + 1
standard_set_solution(dim, solution, False)
solution = standard_get_solution(False)
solution_dict = strsol2dict(solution)
var_names = variables(solution_dict)
var_names.sort()
result[0].append(standard_t_value())
result[1].append(standard_step_size())
for i in range(len(var_names)):
result[2][i].append(solution_dict[var_names[i]])
result[3].append(standard_closest_pole())
result[4].append(standard_pole_radius())
result[5].append(standard_series_coefficients(dim))
result[6].append(standard_pade_vector(dim))
for i in range(num_steps):
standard_predict_correct(False)
solution = standard_get_solution(False)
solution_dict = strsol2dict(solution)
result[0].append(standard_t_value())
result[1].append(standard_step_size())
for j in range(len(var_names)):
result[2][j].append(solution_dict[var_names[j]])
result[3].append(standard_closest_pole())
result[4].append(standard_pole_radius())
result[5].append(standard_series_coefficients(dim))
result[6].append(standard_pade_vector(dim))
for i in range(len(result[2])):
for j in range(len(result[2][i])):
result[2][i][j] = (result[2][i][j].real, result[2][i][j].imag)
return result
def verify(pols, sols):
"""
Verifies whether the solutions in sols
satisfy the polynomials of the system in pols.
"""
from phcpy.solutions import strsol2dict, evaluate
dictsols = [strsol2dict(sol) for sol in sols]
checksum = 0
for sol in dictsols:
sumeval = sum(evaluate(pols, sol))
print(sumeval)
checksum = checksum + sumeval
print('the total check sum :', checksum)
def verify(pols, sols):
"""
Verifies whether the solutions in sols
satisfy the polynomials of the system in pols.
"""
from phcpy.solutions import strsol2dict, evaluate
dictsols = [strsol2dict(sol) for sol in sols]
checksum = 0
for sol in dictsols:
sumeval = sum(evaluate(pols, sol))
print(sumeval)
checksum = checksum + sumeval
print('the total check sum :', checksum)
def find_more_points(self, num_points, return_complex=False):
"""
We find additional points on the variety.
"""
points = []
failure = 0
while (len(points) < num_points):
phcsystem = self._system()
phcsolutions = track(phcsystem, self._startsystem, self._startsol)
# Parsing the output of solutions
for sol in phcsolutions:
solutiondict = strsol2dict(sol)
point = [solutiondict[variable] for variable in self.varlist]
if return_complex:
points.append(tuple(point))
else:
closeness = True
for component in point:
# choses the points we want
if self._is_close(component.imag) \
and self._in_bounds(component.real):
# sometimes phcpy gives more points than we
# ask, thus the additional check
if component == point[-1] \
and closeness \
and len(points) < num_points:
points.append(
tuple([
component.real for component in point
]))
assert len(points) <= num_points
failure = 0
else:
closeness = False
failure = failure + 1
if self._failure <= failure:
raise RuntimeError(
"equation has too many complex solutions in a row")
self.points = points + self.points
def extend_solutions(sols, dim, pivots):
"""
Adds one extra zero coordinate to all solutions in sols,
corresponding to the rightmost nonfull pivot.
"""
from phcpy.solutions import strsol2dict, variables, make_solution
piv = rightmost_nonfull_pivot(dim, pivots)
print '-> in extend_solutions, piv =', piv
if (piv < 0):
result = sols
else:
rown = pivots[piv] + 1 # row number of new variable
name = 'x_' + str(rown + 1) + '_' + str(piv + 1)
print '-> the name of the new variable :', name
result = []
for sol in sols:
dicsol = strsol2dict(sol)
solvar = variables(dicsol)
solval = [dicsol[var] for var in solvar]
solvar.append(name)
solval.append(complex(0.0))
result.append(make_solution(solvar, solval))
return result
def extend_solutions(sols, dim, pivots):
"""
Adds one extra zero coordinate to all solutions in sols,
corresponding to the rightmost nonfull pivot.
"""
from phcpy.solutions import strsol2dict, variables, make_solution
piv = rightmost_nonfull_pivot(dim, pivots)
print '-> in extend_solutions, piv =', piv
if(piv < 0):
result = sols
else:
rown = pivots[piv] + 1 # row number of new variable
name = 'x_' + str(rown+1) + '_' + str(piv+1)
print '-> the name of the new variable :', name
result = []
for sol in sols:
dicsol = strsol2dict(sol)
solvar = variables(dicsol)
solval = [dicsol[var] for var in solvar]
solvar.append(name)
solval.append(complex(0.0))
result.append(make_solution(solvar, solval))
return result
def findEmbeddings(syst):
global prevSystem
global usePrev
global prevSolutions
i = 0
while True:
if prevSystem and usePrev:
sols = track(syst, prevSystem, prevSolutions, tasks=tasks_num)
else:
sols = solve(syst, verbose=0, tasks=tasks_num)
result_real = []
for sol in sols:
soldic = strsol2dict(sol)
if is_real(sol, tolerance):
result_real.append(soldic)
num_real = len(result_real)
if num_real%4==0 and len(sols)==numAll:
prevSystem = syst
prevSolutions = sols
return num_real
elif numAllowedMissing and len(sols)==numAll:
print 'The number of real embeddings was not divisible by 4. ',
return num_real
elif numAllowedMissing and len(sols)>=numAll-numAllowedMissing:
print 'Continuing although there were '+str(numAll-len(sols))+' solutions missing. ',
return num_real
else:
usePrev = False
i += 1
# print 'PHC failed, trying again: '+str(i)
if i>=3:
print 'PHC failed 3 times',
return -1
"""
Get the x and y coordinates of a solution path.
First we take two quadrics and construct a start system
based on the total degree applying the theorem of Bezout.
To extract the coordinates of the points on a solution path,
we convert the string representation into a dictionary.
"""
f = ['x*y + y^2 - 8;', 'x^2 - 3*y + 4;']
print 'the polynomials in the target system :'
for pol in f:
print pol
from phcpy.solver import total_degree_start_system as tds
(g, gsols) = tds(f)
print 'the polynomials in the start system :'
for pol in g:
print pol
print 'number of start solutions :', len(gsols)
from phcpy.trackers import initialize_standard_tracker
from phcpy.trackers import initialize_standard_solution
from phcpy.trackers import next_standard_solution as nxtsol
initialize_standard_tracker(f, g)
initialize_standard_solution(len(g), gsols[0])
points = [nxtsol() for k in range(10)]
from phcpy.solutions import strsol2dict
dicpts = [strsol2dict(sol) for sol in points]
coords = [(sol['x'].real, sol['y'].real) for sol in dicpts]
print 'the real parts of the first ten points on the path :'
for point in coords:
print point
Get the x and y coordinates of a solution path.
First we take two quadrics and construct a start system
based on the total degree applying the theorem of Bezout.
To extract the coordinates of the points on a solution path,
we convert the string representation into a dictionary.
"""
f = ['x*y + y^2 - 8;', 'x^2 - 3*y + 4;']
print 'the polynomials in the target system :'
for pol in f:
print pol
from phcpy.solver import total_degree_start_system as tds
(g, gsols) = tds(f)
print 'the polynomials in the start system :'
for pol in g:
print pol
print 'number of start solutions :', len(gsols)
from phcpy.trackers import initialize_standard_tracker
from phcpy.trackers import initialize_standard_solution
from phcpy.trackers import next_standard_solution as nxtsol
initialize_standard_tracker(f, g)
initialize_standard_solution(len(g), gsols[0])
points = [nxtsol() for k in range(10)]
from phcpy.solutions import strsol2dict
dicpts = [strsol2dict(sol) for sol in points]
coords = [(sol['x'].real, sol['y'].real) for sol in dicpts]
print 'the real parts of the first ten points on the path :'
for point in coords:
print point