def compute_diagonalisation(nf, lie_algebra, h_er_trunc, max_degree, tolerance,
system_prefix):
"""
Here, just go as far as we can with double precision, and dump the
output files.
"""
nf.set_tolerance(tolerance)
desired_grade = max_degree
nf.desired_grade = max_degree
logger.info('main computation started...')
#setup:
nf.h_er = nf.alg.isograde(nf.h_er, 0, desired_grade + 1)
nf.confirm_equilibrium_point()
#output max grade
print "Set up to degree", desired_grade,
#output h_er
print "with hamiltonian degree", nf.h_er.degree()
#diagonalize
nf.diagonalize()
#compute the complexification maps
eq_type = nf.eig.get_equilibrium_type()
nf.eq_type = eq_type
com = Complexifier(lie_algebra, eq_type)
nf.com = com
r_from_c = com.calc_sub_complex_into_real()
c_from_r = com.calc_sub_real_into_complex()
nf.r_in_terms_of_c = r_from_c
nf.c_in_terms_of_r = c_from_r
# convert to semi classical and then output files
if isinstance(nf, SemiclassicalNormalForm):
nf.quantize()
r_from_c = nf.r_in_terms_of_c
c_from_r = nf.c_in_terms_of_r
#make a polynomial ring io, poss SemiClassical
lie_algebra = nf.alg
ring_io = PolynomialRingIO(lie_algebra)
#dump the Taylor series
dump_isogrades(nf.alg, nf.h_er, max_degree, example.prefix, 'h-from-e')
#output vector of polynomials for er in terms of dr
er_from_dr = nf.er_in_terms_of_dr
file_name = '%s--e-from-d.vpol' % system_prefix
file_ostr = open(file_name, 'w')
ring_io.write_sexp_vector_of_polynomials(file_ostr, er_from_dr)
file_ostr.close()
#output equilibrium type
print nf.eig.get_equilibrium_type()
#output the complexification maps
file_name = '%s--r-from-c.vpol' % system_prefix
file_ostr = open(file_name, 'w')
ring_io.write_sexp_vector_of_polynomials(file_ostr, r_from_c)
file_ostr.close()
file_name = '%s--c-from-r.vpol' % system_prefix
file_ostr = open(file_name, 'w')
ring_io.write_sexp_vector_of_polynomials(file_ostr, c_from_r)
file_ostr.close()
def compute_diagonalisation(nf, lie_algebra, h_er_trunc, max_degree, tolerance, system_prefix):
"""
Here, just go as far as we can with double precision, and dump the
output files.
"""
nf.set_tolerance(tolerance)
desired_grade = max_degree
nf.desired_grade = max_degree
logger.info('main computation started...')
#setup:
nf.h_er = nf.alg.isograde(nf.h_er, 0, desired_grade+1)
nf.confirm_equilibrium_point()
#output max grade
print "Set up to degree", desired_grade,
#output h_er
print "with hamiltonian degree", nf.h_er.degree()
#diagonalize
nf.diagonalize()
#compute the complexification maps
eq_type = nf.eig.get_equilibrium_type()
nf.eq_type = eq_type
com = Complexifier(lie_algebra, eq_type)
nf.com = com
r_from_c = com.calc_sub_complex_into_real()
c_from_r = com.calc_sub_real_into_complex()
nf.r_in_terms_of_c = r_from_c
nf.c_in_terms_of_r = c_from_r
# convert to semi classical and then output files
if isinstance(nf, SemiclassicalNormalForm):
nf.quantize()
r_from_c = nf.r_in_terms_of_c
c_from_r = nf.c_in_terms_of_r
#make a polynomial ring io, poss SemiClassical
lie_algebra = nf.alg
ring_io = PolynomialRingIO(lie_algebra)
#dump the Taylor series
dump_isogrades(nf.alg, nf.h_er, max_degree, example.prefix, 'h-from-e')
#output vector of polynomials for er in terms of dr
er_from_dr = nf.er_in_terms_of_dr
file_name = '%s--e-from-d.vpol' % system_prefix
file_ostr = open(file_name, 'w')
ring_io.write_sexp_vector_of_polynomials(file_ostr, er_from_dr)
file_ostr.close()
#output equilibrium type
print nf.eig.get_equilibrium_type()
#output the complexification maps
file_name = '%s--r-from-c.vpol' % system_prefix
file_ostr = open(file_name, 'w')
ring_io.write_sexp_vector_of_polynomials(file_ostr, r_from_c)
file_ostr.close()
file_name = '%s--c-from-r.vpol' % system_prefix
file_ostr = open(file_name, 'w')
ring_io.write_sexp_vector_of_polynomials(file_ostr, c_from_r)
file_ostr.close()
def test_saddle(self):
lie = LieAlgebra(1)
diag = Diagonalizer(lie)
q = lie.q
p = lie.p
orig = q(0)*p(0)
eq_type = 's'
comp = Complexifier(lie, eq_type)
r2c = comp.calc_sub_complex_into_real()
c2r = comp.calc_sub_real_into_complex()
comp = orig.substitute(r2c)
real = comp.substitute(c2r)
diff = orig-real
for m, c in diff.powers_and_coefficients():
self.assert_(abs(c)<1.0e-15)
def test_saddle(self):
lie = LieAlgebra(1)
diag = Diagonalizer(lie)
q = lie.q
p = lie.p
orig = q(0) * p(0)
eq_type = 's'
comp = Complexifier(lie, eq_type)
r2c = comp.calc_sub_complex_into_real()
c2r = comp.calc_sub_real_into_complex()
comp = orig.substitute(r2c)
real = comp.substitute(c2r)
diff = orig - real
for m, c in diff.powers_and_coefficients():
self.assert_(abs(c) < 1.0e-15)
def test_mutual_inverses(self):
lie = LieAlgebra(6)
diag = Diagonalizer(lie)
pol = lie.zero()
for i in xrange(6):
pol += lie.q(i)
pol += lie.p(i)
eq_type = 'scccsc'
comp = Complexifier(lie, eq_type)
sub_c_into_r = comp.calc_sub_complex_into_real()
sub_r_into_c = comp.calc_sub_real_into_complex()
pol_c = pol.substitute(sub_c_into_r)
pol_r = pol_c.substitute(sub_r_into_c)
self.assert_(len(pol_r) == len(pol))
for i in xrange(6):
self.assert_((pol_r-pol).l_infinity_norm() < 1.0e-15)
self.assert_((pol_r-pol).l_infinity_norm() < 1.0e-15)
def test_mutual_inverses(self):
lie = LieAlgebra(6)
diag = Diagonalizer(lie)
pol = lie.zero()
for i in xrange(6):
pol += lie.q(i)
pol += lie.p(i)
eq_type = 'scccsc'
comp = Complexifier(lie, eq_type)
sub_c_into_r = comp.calc_sub_complex_into_real()
sub_r_into_c = comp.calc_sub_real_into_complex()
pol_c = pol.substitute(sub_c_into_r)
pol_r = pol_c.substitute(sub_r_into_c)
self.assert_(len(pol_r) == len(pol))
for i in xrange(6):
self.assert_((pol_r - pol).l_infinity_norm() < 1.0e-15)
self.assert_((pol_r - pol).l_infinity_norm() < 1.0e-15)
def test_centre_saddle(self):
lie = LieAlgebra(2)
diag = Diagonalizer(lie)
q = lie.q
p = lie.p
cent = (0.5*q(0)**2)+(0.5*p(0)**2)
sadd = (1.0*q(1))*(1.0*p(1))
orig = cent + sadd
self.assertEquals(len(orig), 3)
eq_type = 'cs'
comp = Complexifier(lie, eq_type)
r2c = comp.calc_sub_complex_into_real()
c2r = comp.calc_sub_real_into_complex()
comp = orig.substitute(r2c)
self.assertEquals(len(comp), 2)
real = comp.substitute(c2r)
diff = orig-real
for m, c in diff.powers_and_coefficients():
self.assert_(abs(c)<1.0e-15)
def test_centre_saddle(self):
lie = LieAlgebra(2)
diag = Diagonalizer(lie)
q = lie.q
p = lie.p
cent = (0.5 * q(0)**2) + (0.5 * p(0)**2)
sadd = (1.0 * q(1)) * (1.0 * p(1))
orig = cent + sadd
self.assertEquals(len(orig), 3)
eq_type = 'cs'
comp = Complexifier(lie, eq_type)
r2c = comp.calc_sub_complex_into_real()
c2r = comp.calc_sub_real_into_complex()
comp = orig.substitute(r2c)
self.assertEquals(len(comp), 2)
real = comp.substitute(c2r)
diff = orig - real
for m, c in diff.powers_and_coefficients():
self.assert_(abs(c) < 1.0e-15)
def compute_diagonalisation(lie_algebra, h_er_trunc, max_degree, tolerance, system_prefix):
"""
Here, just go as far as we can with double precision, and dump the
output files.
"""
nf = NormalForm(lie_algebra, h_er_trunc)
nf.set_tolerance(tolerance)
desired_grade = max_degree
nf.desired_grade = max_degree
logger.info('main computation started...')
#setup:
nf.h_er = nf.alg.isograde(nf.h_er, 0, desired_grade+1)
nf.confirm_equilibrium_point()
#output max grade
print desired_grade
#output h_er
print nf.h_er
#diagonalize
nf.diagonalize()
#make a polynomial ring io
ring_io = PolynomialRingIO(lie_algebra)
#output vector of polynomials for er in terms of dr
er_from_dr = nf.er_in_terms_of_dr
file_name = '%s--e-from-d.vpol' % system_prefix
file_ostr = open(file_name, 'w')
ring_io.write_sexp_vector_of_polynomials(file_ostr, er_from_dr)
file_ostr.close()
#output equilibrium type
print nf.eig.get_equilibrium_type()
#compute the complexification maps
eq_type = nf.eig.get_equilibrium_type()
com = Complexifier(lie_algebra, eq_type)
r_from_c = com.calc_sub_complex_into_real()
c_from_r = com.calc_sub_real_into_complex()
#output the complexification maps
file_name = '%s--r-from-c.vpol' % system_prefix
file_ostr = open(file_name, 'w')
ring_io.write_sexp_vector_of_polynomials(file_ostr, r_from_c)
file_ostr.close()
def test_centre(self):
lie = LieAlgebra(1)
diag = Diagonalizer(lie)
q = lie.q
p = lie.p
orig = (0.5 * q(0)**2) + (0.5 * p(0)**2)
self.assertEquals(len(orig), 2)
eq_type = 'c'
comp = Complexifier(lie, eq_type)
r2c = comp.calc_sub_complex_into_real()
c2r = comp.calc_sub_real_into_complex()
comp = orig.substitute(r2c)
self.assertEquals(len(comp), 1)
for m, c in comp.powers_and_coefficients():
self.assert_(c.real == 0.0)
self.assert_(m[0] == 1)
self.assert_(m[1] == 1)
real = comp.substitute(c2r)
diff = orig - real
for m, c in diff.powers_and_coefficients():
self.assert_(abs(c) < 1.0e-15)
def test_centre(self):
lie = LieAlgebra(1)
diag = Diagonalizer(lie)
q = lie.q
p = lie.p
orig = (0.5*q(0)**2)+(0.5*p(0)**2)
self.assertEquals(len(orig), 2)
eq_type = 'c'
comp = Complexifier(lie, eq_type)
r2c = comp.calc_sub_complex_into_real()
c2r = comp.calc_sub_real_into_complex()
comp = orig.substitute(r2c)
self.assertEquals(len(comp), 1)
for m, c in comp.powers_and_coefficients():
self.assert_(c.real == 0.0)
self.assert_(m[0] == 1)
self.assert_(m[1] == 1)
real = comp.substitute(c2r)
diff = orig-real
for m, c in diff.powers_and_coefficients():
self.assert_(abs(c)<1.0e-15)
def diagonalize(self, dof, h):
tolerance = 5.0e-15
lie = self.alg
diag = Diagonalizer(lie)
eig = diag.compute_eigen_system(h, tolerance=tolerance)
diag.compute_diagonal_change()
mat = diag.get_matrix_diag_to_equi()
assert diag.matrix_is_symplectic(mat)
sub_diag_into_equi = diag.matrix_as_vector_of_row_polynomials(mat)
h_diag = h.substitute(sub_diag_into_equi)
eq_type = eig.get_equilibrium_type()
comp = Complexifier(lie, eq_type)
self.sub_c_into_r = comp.calc_sub_complex_into_real()
self.sub_r_into_c = comp.calc_sub_real_into_complex()
h_comp = h_diag.substitute(self.sub_c_into_r)
h_comp = h_comp.with_small_coeffs_removed(tolerance)
self.assert_(lie.is_diagonal_polynomial(h_comp.homogeneous(2)))
return h_comp
class SemiclassicalNormalForm(NormalForm):
def __init__(self, classical_lie_algebra, h_er):
NormalForm.__init__(self, classical_lie_algebra, h_er)
def quantize(self):
cla_to_sem = ClassicalToSemiclassical(self.get_lie_algebra())
#keep classical versions
self.alg_classical = self.alg
self.com_classical = self.com
self.c_in_terms_of_r_classical = self.c_in_terms_of_r
self.r_in_terms_of_c_classical = self.r_in_terms_of_c
# if we have h_dc convert it
if hasattr(self,"h_dc"):
self.h_dc_classical = self.h_dc
self.h_dc = cla_to_sem(self.h_dc)
#make semiclassical versions
self.alg = cla_to_sem.semi_classical_algebra()
n_vars = self.alg.n_vars()
self.com = Complexifier(self.alg, self.eq_type)
self.r_in_terms_of_c = self.com.calc_sub_complex_into_real()
self.c_in_terms_of_r = self.com.calc_sub_real_into_complex()
#make polys for use by cplusplus
self.h_er_classical = self.h_er
self.h_er = cla_to_sem(self.h_er)
self.er_in_terms_of_dr_classical = self.er_in_terms_of_dr
er_in_terms_of_dr = []
for i in xrange(len(self.er_in_terms_of_dr)):
er_in_terms_of_dr.append(cla_to_sem(self.er_in_terms_of_dr[i]))
er_in_terms_of_dr.append(Polynomial.Polynomial.CoordinateMonomial(n_vars,n_vars-1))
self.er_in_terms_of_dr = er_in_terms_of_dr
def perform_all_computations(self, desired_grade):
"""
Overrides classical normal form.
"""
logger.info('main computation started...')
#flags for extra computations:
do_check_generating_function_invariant = False
do_check_diag_transforms_to_norm = False
do_compute_complex_diag_vs_norm = False
do_check_diag_vs_norm_using_conversion = False
#setup:
self.desired_grade = desired_grade
self.h_er = self.alg.isograde(self.h_er, 0, desired_grade+1)
self.confirm_equilibrium_point()
#main steps:
self.diagonalize()
self.complexify()
self.quantize()
self.normalize()
#the following need semiclassical real-complex relations...
self.normalize_real()
#self.extract_integrals()
#optional computations:
#if do_check_generating_function_invariant:
# self.check_generating_function_invariant()
#if do_check_diag_transforms_to_norm:
# self.check_diag_transforms_to_norm()
#if do_compute_complex_diag_vs_norm:
# self.compute_diag_to_norm()
# self.compute_norm_to_diag()
#coordinate change maps:
#self.compute_diag_to_norm_real_using_w_real()
#self.compute_norm_to_diag_real_using_w_real()
#optional computations:
#if do_check_diag_vs_norm_using_conversion:
# self.compute_diag_to_norm_real_using_conversion()
# self.compute_norm_to_diag_real_using_conversion()
logger.info('...main computation done.')
class NormalForm:
def __init__(self, lie_algebra, h_er):
logger.info('Initializing...')
self.alg = lie_algebra
self.h_er = h_er
logger.info('...done initializing.')
def get_lie_algebra(self):
return self.alg
def set_tolerance(self, tolerance):
self.tolerance = tolerance
def confirm_equilibrium_point(self):
logger.info('Confirming equilibrium point...')
alg = self.alg
#logger.info('Real Equilibrium Hamiltonian (terms up to grade')
#logger.info(truncated_poly(self.alg, self.h_er))
logger.info('Equilibrium Hamiltonian has %d terms'%self.h_er.n_terms())
gra = alg.grade(self.h_er)
if alg.isograde(self.h_er, 0):
logger.error('WARNING! constant term on input Hamiltonian removed')
logger.error('size %s', alg.isograde(self.h_er, 0).l_infinity_norm())
self.h_er = alg.isograde(self.h_er, 1, alg.grade(self.h_er)+1)
if alg.isograde(self.h_er, 1):
logger.error('WARNING! linear term on input Hamiltonian removed')
logger.error('size %s', alg.isograde(self.h_er, 1).l_infinity_norm())
self.h_er = alg.isograde(self.h_er, 2, alg.grade(self.h_er)+1)
assert not alg.isograde(self.h_er, 0, 2)
assert gra == alg.grade(self.h_er)
logger.info('WARNING! Removing small coefficients of size <=%s'%tol_equi_real_coeffs)
self.h_er = self.h_er.with_small_coeffs_removed(tol_equi_real_coeffs)
logger.info('Real Equilibrium Hamiltonian has %d terms'%self.h_er.n_terms())
logger.info('...done confirming equilibrium point.')
def diagonalize(self):
logger.info('Diagonalizing...')
self.dia = Diagonalizer(self.alg)
logger.info('Equilibrium Hamiltonian has %d terms'%self.h_er.n_terms())
logger.info('Computing the eigensystem...')
self.eig = self.dia.compute_eigen_system(self.h_er, self.tolerance)
logger.info('...done computing the eigensystem.')
logger.info('Computing the diagonal change...')
self.dia.compute_diagonal_change()
logger.info('...done computing the diagonal change.')
logger.info('Measuring error in symplectic identities...')
mat = self.dia.get_matrix_diag_to_equi()
assert self.dia.matrix_is_symplectic(mat)
logger.info('...done measuring error in symplectic identities.')
logger.info('Converting the matrix into vec<poly> representation...')
er_in_terms_of_dr = self.dia.matrix_as_vector_of_row_polynomials(mat)
self.er_in_terms_of_dr = er_in_terms_of_dr
logger.info('... done converting the matrix into vec<poly> repn.')
logger.info('Applying diagonalizing change to equilibrium H...')
self.h_dr = self.h_er.substitute(er_in_terms_of_dr)
logger.info('...done applying diagonalizing change to equilibrium H.')
del er_in_terms_of_dr
logger.info('Real diagonal Hamiltonian has %d terms'%self.h_dr.n_terms())
err = self.h_dr.imag().l_infinity_norm()
logger.info('Size of imaginary part: %s', err)
assert err < tol_diag_real_imag_part
logger.info('WARNING! Removing imaginary terms')
self.h_dr = self.h_dr.real()
logger.info('Real diagonal Hamiltonian has %d terms'%self.h_dr.n_terms())
logger.info('WARNING! Removing small coefficients of size <=%s'%tol_diag_real_coeffs)
self.h_dr = self.h_dr.with_small_coeffs_removed(tol_diag_real_coeffs)
logger.info('Real diagonal Hamiltonian has %d terms'%self.h_dr.n_terms())
#logger.info('Real Diagonal Hamiltonian (terms up to gra 3)')
#logger.info(truncated_poly(self.alg, self.h_dr))
logger.info('Quadratic part')
logger.info(truncated_poly(self.alg, self.alg.isograde(self.h_dr, 2)))
logger.info('Terms in quadratic part %d'%(self.alg.isograde(self.h_dr, 2).n_terms()))
logger.info('...done diagonalizing.')
def complexify(self):
logger.info('Complexifying...')
self.eq_type = self.eig.get_equilibrium_type()
logger.info('Equilibrium type %s'%self.eq_type)
logger.info('Terms in real diagonal %d'%self.h_dr.n_terms())
self.com = Complexifier(self.alg, self.eq_type)
self.r_in_terms_of_c = self.com.calc_sub_complex_into_real()
self.c_in_terms_of_r = self.com.calc_sub_real_into_complex()
logger.info('Performing complex substitution...')
self.h_dc = self.h_dr.substitute(self.r_in_terms_of_c)
logger.info('Terms in complex diagonal %d'%self.h_dc.n_terms())
logger.info('Removing small coefficients...')
self.h_dc = self.h_dc.with_small_coeffs_removed(self.tolerance)
logger.info('Terms in complex diagonal %d'%self.h_dc.n_terms())
h_dc_2 = self.alg.isograde(self.h_dc, 2)
off_diag = h_dc_2 - self.alg.diagonal_part_of_polynomial(h_dc_2)
err = off_diag.l_infinity_norm()
logger.info('Size of off-diagonal part of quadratic part of diagonal h: %s', err)
assert err < tol_diag_comp_off_diag_part
logger.info('Removing off-diagonal part')
self.h_dc -= off_diag
del off_diag
logger.info('Terms in complex diagonal %d'%self.h_dc.n_terms())
h_dc_2 = self.alg.isograde(self.h_dc, 2)
logger.info('Quadratic part')
logger.info(self.alg.display(h_dc_2))
assert (self.alg.is_diagonal_polynomial(h_dc_2))
del h_dc_2
self.eig_vals = self.eig.get_eigen_values_in_plus_minus_pairs()
logger.info('Complex diagonal Hamiltonian has %d terms'%self.h_dc.n_terms())
logger.info('...done complexifying.')
def cvec2r_to_rvec2r(self, p):
"""
@param p: a polynomial map from a vector of complex
coordinates to a real scalar.
@return: a polynomial map from the real version of the
coordinates to the real scalar.
"""
return p.substitute(self.c_in_terms_of_r)
def cvec2cvec_to_rvec2rvec(self, vp):
"""
@param vp: vector of polynomial maps from vector of complex
coordinates to a complex coordinate.
@return: vector of polynomial maps from vector of reals to
real.
"""
cvec_in_rvec = []
for i in xrange(len(vp)):
cvec_in_rvec.append(vp[i].substitute(self.c_in_terms_of_r))
cvec_in_cvec = []
for r_in_c_i in self.r_in_terms_of_c:
cvec_in_cvec.append(r_in_c_i.substitute(cvec_in_rvec))
return cvec_in_cvec
def normalize(self):
logger.info('Normalizing...')
steps = self.desired_grade-1 #1+h_dc.grade()-2
self.w_c_list = []
self.h_nc_list = []
self.h_nc_dict = {}
self.h_nc = self.alg.zero()
self.w_c = self.alg.zero()
h_dc_2 = self.alg.isograde(self.h_dc, 2)
state = (self.h_nc_dict, self.w_c_list, -1)
tri = LieTriangle.LieTriangle(self.alg, h_dc_2, state)
#
# note that, in the following, h_term is _NOT_ h_i for the
# Deprit triangle; it is (h_i)/factorial(i) and this is
# taken care of _WITHIN_ the triangle procedure.
#
for step in xrange(0, steps + 1):
h_term = self.alg.isograde(self.h_dc, step + 2)
k_term, w_term = tri.compute_normal_form_and_generating_function(h_term)
logger.info('Resulting normal form terms:')
logger.info(truncated_poly(self.alg, k_term))
self.h_nc += k_term
self.w_c += w_term
logger.info('Complex normal form:')
logger.info(truncated_poly(self.alg, self.h_nc))
self.iso_nf = tri.get_isograde_list_handler()
#h_nc_new = tri.compute_lie_transform_given_generating_function([], steps)
#logger.info('Normalized using computed generating function:')
#logger.info(self.alg.display(h_nc_new))
def check_generating_function_invariant(self):
logger.info('confirming generating function invariant...')
grade = self.alg.grade
#1+self.h_dc.grade()-3 #check
steps = self.desired_grade-2
logger.info('surely we can use independent no. steps in coord change,')
logger.info('than we did in the nf computation? CARE!!!')
logger.info('the generating function is only invariant with correct steps.')
cc = CoordinateChange(self.alg, self.w_c_list)
res1 = []
cc.express_norm_in_diag(self.w_c, res1, {}, steps)
res2 = []
cc.express_diag_in_norm(self.w_c, res2, {}, steps)
iso_cc = cc.get_isograde_list_handler()
for res in [res1, res2]:
w_c_new = iso_cc.list_to_poly(res)
logger.info('Grade %d to %d', grade(self.w_c), grade(w_c_new))
gra = min((grade(self.w_c), grade(w_c_new)))
err = self.alg.isograde((w_c_new-self.w_c), 0, gra+1).l_infinity_norm()
logger.info('Change in like-grade part of generating function: %s', err)
assert err < tol_comp_generator_through_lie_triangle
del cc
del res1
del res2
del iso_cc
del w_c_new
def check_diag_transforms_to_norm(self):
logger.info('confirming that diagonal Hamiltonian transforms to normal...')
grade = self.alg.grade
steps = self.desired_grade-2 #1+self.h_dc.grade()-3
cc = CoordinateChange(self.alg, self.w_c_list)
iso_cc = cc.get_isograde_list_handler()
res1 = []
cc.express_norm_in_diag(self.h_nc, res1, {}, steps)
h_dc_new = iso_cc.list_to_poly(res1)
gra = min((grade(self.h_dc), grade(h_dc_new)))
err = self.alg.isograde(self.h_dc-h_dc_new, 0, gra+1).l_infinity_norm()
logger.info('Error expressing H_{nc} in x_{dc}')
logger.info('Grade %d to %d',
grade(self.h_dc),
grade(h_dc_new))
logger.info(err)
logger.info('confirming that normal Hamiltonian transforms to diagonal...')
res2 = []
cc.express_diag_in_norm(self.h_dc, res2, {}, steps)
h_nc_new = iso_cc.list_to_poly(res2)
gra = min((grade(self.h_nc), grade(h_nc_new)))
err = self.alg.isograde(self.h_nc-h_nc_new, 0, gra+1).l_infinity_norm()
logger.info('Error expressing H_{dc} in x_{nc}')
logger.info('Grade %d to %d', grade(self.h_nc), grade(h_nc_new))
logger.info(err)
del cc
del iso_cc
del h_dc_new
del h_nc_new
del res1
del res2
def normalize_real(self):
logger.info('computing real normalization...')
self.h_nr = self.h_nc.substitute(self.c_in_terms_of_r)
err = self.h_nr.imag().l_infinity_norm()
tol_norm_real_imag_part = (1.0 + self.h_nr.l_infinity_norm()) * 1.0e-10
logger.info('l infinity norm of real normal form imaginary part: %s', err)
assert err < tol_norm_real_imag_part
self.h_nr = self.h_nr.real()
logger.info('Real normal form:')
logger.info(truncated_poly(self.alg, self.h_nr))
self.w_r_list = [p.substitute(self.c_in_terms_of_r) for p in self.w_c_list]
#self.w_r = self.iso_nf.list_to_poly(self.w_r_list)
def extract_integrals(self):
logger.info('extracting the integrals...')
int_ext = IntegralExtractor(self.alg)
int_ext.set_complex_normal_hamiltonian(self.h_nc)
int_ext.find_lost_simple_integrals_and_non_simple_integral()
logger.info('Lost integrals:')
logger.info(int_ext._lost_integrals)
assert ((not int_ext._lost_integrals), int_ext._lost_integrals)
logger.info('Real simple part:')
logger.info((self.cvec2r_to_rvec2r(int_ext._simple_part)))
logger.info('Real non-simple part:')
logger.info((self.cvec2r_to_rvec2r(int_ext._non_simple_integral)))
int_ext.list_the_simple_integrals()
logger.info('Kept integrals:')
logger.info(int_ext._kept_integrals)
int_ext.express_simple_part_as_polynomial_over_all_integrals()
logger.info('Simple part in terms of integrals:')
logger.info((int_ext._simple_in_integrals))
int_ext.express_both_parts_as_polynomial_over_all_integrals()
logger.info('Total in terms of integrals')
logger.info((int_ext._total_in_integrals))
int_ext.express_all_integrals_in_normal_form_coords()
logger.info('Complex Integrals in complex normal form coords:')
logger.info((int_ext._integrals_in_coords))
logger.info('Complex Integrals in real normal form coords:')
ic_in_nr = [self.cvec2r_to_rvec2r(int_i) for int_i in int_ext._integrals_in_coords]
logger.info(ic_in_nr)
#convert simple part in terms of complex integrals,
#to simple part in terms of real integrals.
def simple_int_is_saddle(real_integral):
return len(real_integral) == 1
def simple_int_is_centre(real_integral):
return len(real_integral) == 2
comp_ints_in_real_ints = []
real_integrals = []
n_integrals = len(int_ext._integrals_in_coords)
ring_ints = PolynomialRing(n_integrals)
for i, integral in zip(int_ext._kept_integrals,
int_ext._integrals_in_coords):
# This is the complex integral in real coordinates
real_integral = integral.substitute(self.c_in_terms_of_r)
mon = ring_ints.coordinate_monomial(i)
if simple_int_is_centre(real_integral):
# Note that a centre real integral = i * complex integral
comp_ints_in_real_ints.append((-1.0J)*mon)
real_integrals.append((1.0J)*real_integral)
logger.info('centre')
elif simple_int_is_saddle(real_integral):
comp_ints_in_real_ints.append(mon)
real_integrals.append(real_integral)
logger.info('saddle')
else:
assert 0, 'Simple integral, neither saddle nor centre!'
h_ir = int_ext._simple_in_integrals.substitute(comp_ints_in_real_ints)
logger.info('Simple part in terms of the real simple integrals:')
logger.info(h_ir)
logger.info('Real integrals in real nf coords:')
logger.info(real_integrals)
#we should check this by converting the simple part to real
h_ir_check = self.cvec2r_to_rvec2r(int_ext._simple_part)
err = (h_ir_check-h_ir.substitute(real_integrals)).l_infinity_norm()
logger.info('Error in composite of real integrals and real simple part')
logger.info(err)
self.h_ir = h_ir
self.real_integrals = real_integrals
def compute_diag_to_norm(self):
logger.info('computing coordinate changes...')
steps = self.desired_grade-2 #1+self.h_dc.grade()-3 #check!
res = []
for s in xrange(self.alg.n_vars()):
#do we request a coordinate change in h-bar???
logger.info('complex coord change, component %d of %d', s, self.alg.n_vars())
xs = self.alg.coordinate_monomial(s)
sp = [0,]*(self.alg.n_vars())
sp[s] = 1
#use complex nf
logger.info('COMPLEX component of x_{nc} expressed IN x_{dc} (diag TO norm):')
x_i_list = []
x_ij_dict = {}
cc = CoordinateChange(self.alg, self.w_c_list)
cc.express_norm_in_diag(xs, x_i_list, x_ij_dict, steps)
#near-identity?
assert (len(x_i_list[0]) == 1)
assert (x_i_list[0][Powers(tuple(sp))] == 1.0)
for gra_minus_1, pol in enumerate(x_i_list):
gra = self.alg.grade(pol)
assert ((gra == 0) or (gra == gra_minus_1+1))
iso_cc = cc.get_isograde_list_handler()
#for i, polc in enumerate(x_i_list):
#logger.info(truncated_poly(self.alg, iso_cc.inner_taylor_to_poly(polc, i)))
poly = iso_cc.list_to_poly(x_i_list)
res.append(poly)
self.nc_in_dc = res
def compute_diag_to_norm_real_using_w_real(self):
logger.info('computing real coordinate changes using real generator...')
steps = self.desired_grade-2 #self.h_dc.grade()-2 #check!
res = []
for s in xrange(self.alg.n_vars()):
logger.info('complex coord change, component %d of %d', s, self.alg.n_vars())
xs = self.alg.coordinate_monomial(s)
sp = [0,]*(self.alg.n_vars())
sp[s] = 1
#use real nf
logger.info('REAL component of x_{nr} expressed IN x_{dc} (diag TO norm):')
x_i_list = []
x_ij_dict = {}
cc = CoordinateChange(self.alg, self.w_r_list)
cc.express_norm_in_diag(xs, x_i_list, x_ij_dict, steps)
#near-identity?
assert (len(x_i_list[0]) == 1)
assert (x_i_list[0][Powers(tuple(sp))] == 1.0)
for gra_minus_1, pol in enumerate(x_i_list):
gra = self.alg.grade(pol)
assert ((gra == 0) or (gra == gra_minus_1+1))
iso_cc = cc.get_isograde_list_handler()
for i, polc in enumerate(x_i_list):
err = polc.imag().l_infinity_norm()
logger.info('l_infinity_norm of imaginary part: %s', err)
tol_norm_real_imag_part = (1.0 + self.w_r_list[i].l_infinity_norm()) * 1.0e-10
assert err < tol_norm_real_imag_part
#logger.info(truncated_poly(self.alg, iso_cc.inner_taylor_to_poly(polc.real(), i)))
poly = iso_cc.list_to_poly(x_i_list)
res.append(poly.real())
self.nr_in_dr_via_wr = res
def compute_diag_to_norm_real_using_conversion(self):
logger.info('computing real coordinate changes using real conversion...')
res = self.cvec2cvec_to_rvec2rvec(self.nc_in_dc)
self.nr_in_dr_via_nc_in_dc = res
logger.info('ERRORS in real generator vs. conversion:')
for m0, m1 in zip(self.nr_in_dr_via_nc_in_dc, self.nr_in_dr_via_wr):
err = (m1-m0).l_infinity_norm()
logger.info(err)
def compute_norm_to_diag(self):
logger.info('computing inverse coordinate changes...')
steps = self.desired_grade-2 #1+self.h_dc.grade()-3 #check!
res = []
for s in xrange(self.alg.n_vars()):
logger.info('complex coord change, component %d of %d', s, self.alg.n_vars())
xs = self.alg.coordinate_monomial(s)
sp = [0,]*(self.alg.n_vars())
sp[s] = 1
#use complex nf
logger.info('COMPLEX component of x_{dc} expressed IN x_{nc} (norm TO diag):')
x_i_list = []
x_ij_dict = {}
cc = CoordinateChange(self.alg, self.w_c_list)
cc.express_diag_in_norm(xs, x_i_list, x_ij_dict, steps)
#near-identity?
assert (len(x_i_list[0]) == 1)
assert (x_i_list[0][Powers(tuple(sp))] == 1.0)
for gra_minus_1, pol in enumerate(x_i_list):
gra = self.alg.grade(pol)
assert ((gra == 0) or (gra == gra_minus_1+1))
iso_cc = cc.get_isograde_list_handler()
#for i, polc in enumerate(x_i_list):
#logger.info(truncated_poly(self.alg, iso_cc.inner_taylor_to_poly(polc, i)))
poly = iso_cc.list_to_poly(x_i_list)
res.append(poly)
self.dc_in_nc = res
def compute_norm_to_diag_real_using_w_real(self):
logger.info('computing real inverse coordinate change using real generator...')
steps = self.desired_grade-2 #self.h_dc.grade()-2 #check!
res = []
for s in xrange(self.alg.n_vars()):
logger.info('complex coord change, component %d of %d', s, self.alg.n_vars())
xs = self.alg.coordinate_monomial(s)
sp = [0,]*(self.alg.n_vars())
sp[s] = 1
#use real nf
logger.info('REAL component of x_{dc} expressed IN x_{nr} (norm TO diag):')
x_i_list = []
x_ij_dict = {}
cc = CoordinateChange(self.alg, self.w_r_list)
cc.express_diag_in_norm(xs, x_i_list, x_ij_dict, steps)
#near-identity?
assert (len(x_i_list[0]) == 1)
assert (x_i_list[0][Powers(tuple(sp))] == 1.0)
for gra_minus_1, pol in enumerate(x_i_list):
gra = self.alg.grade(pol)
assert ((gra == 0) or (gra == gra_minus_1+1))
iso_cc = cc.get_isograde_list_handler()
for i, polc in enumerate(x_i_list):
err = polc.imag().l_infinity_norm()
logger.info('l_infinity_norm of imaginary part: %s', err)
tol_norm_real_imag_part = (1.0 + self.w_r_list[i].l_infinity_norm()) * 1.0e-10
assert err < tol_norm_real_imag_part
poly = iso_cc.list_to_poly(x_i_list)
res.append(poly.real())
self.dr_in_nr_via_wr = res
def compute_norm_to_diag_real_using_conversion(self):
logger.info('computing real inverse coordinate changes using real conversion...')
res = self.cvec2cvec_to_rvec2rvec(self.dc_in_nc)
self.nr_in_dr_via_nc_in_dc = res
logger.info('ERRORS in real generator vs. conversion:')
for m0, m1 in zip(self.nr_in_dr_via_nc_in_dc, self.dr_in_nr_via_wr):
err = (m1-m0).l_infinity_norm()
logger.info(err)
def perform_all_computations(self, desired_grade):
logger.info('main computation started...')
#flags for extra computations:
do_check_generating_function_invariant = False
do_check_diag_transforms_to_norm = False
do_compute_complex_diag_vs_norm = False
do_check_diag_vs_norm_using_conversion = False
#setup:
self.desired_grade = desired_grade
self.h_er = self.alg.isograde(self.h_er, 0, desired_grade+1)
self.confirm_equilibrium_point()
#main steps:
self.diagonalize()
self.complexify()
self.normalize()
self.normalize_real()
self.extract_integrals()
#optional computations:
if do_check_generating_function_invariant:
self.check_generating_function_invariant()
if do_check_diag_transforms_to_norm:
self.check_diag_transforms_to_norm()
if do_compute_complex_diag_vs_norm:
self.compute_diag_to_norm()
self.compute_norm_to_diag()
#coordinate change maps:
self.compute_diag_to_norm_real_using_w_real()
self.compute_norm_to_diag_real_using_w_real()
#optional computations:
if do_check_diag_vs_norm_using_conversion:
self.compute_diag_to_norm_real_using_conversion()
self.compute_norm_to_diag_real_using_conversion()
logger.info('...main computation done.')
#h_nr_check = self.h_dr.substitute(self.dr_in_nr_via_wr)
#print "(self.h_dr.substitute(self.dr_in_nr_via_wr)-self.h_nr).l_infinity_norm()\n", h_nr_check.l_infinity_norm(), self.h_nr.l_infinity_norm(), (h_nr_check-self.h_nr).l_infinity_norm(), "\n", h_nr_check.l1_norm(), self.h_nr.l1_norm(), (h_nr_check-self.h_nr).l1_norm()
#h_dr_check = self.h_nr.substitute(self.nr_in_dr_via_wr)
#print "(self.h_nr.substitute(self.nr_in_dr_via_wr)-self.h_dr).l_infinity_norm()\n", h_dr_check.l_infinity_norm(), self.h_dr.l_infinity_norm(), (h_dr_check-self.h_dr).l_infinity_norm(), "\n", h_dr_check.l1_norm(), self.h_dr.l1_norm(), (h_dr_check-self.h_dr).l1_norm()
#print "truncated to degree 5"
#print self.alg.isograde(h_dr_check-self.h_dr, 1, up_to=5).l1_norm()
#print self.alg.isograde(h_nr_check-self.h_nr, 1, up_to=5).l1_norm()
self.write_out_integrals()
self.write_out_transforms()
def write_out_transforms(self):
#write out in Mathematica xxpp format
is_xxpp_format = True
write_file_mat("diag_to_equi.mat", self.dia.matrix_diag_to_equi)
write_file_mat("equi_to_diag.mat",
self.dia.get_matrix_equi_to_diag())
write_file_vec_polynomials("norm_to_diag.vec",
self.dr_in_nr_via_wr, is_xxpp_format)
write_file_vec_polynomials("diag_to_norm.vec",
self.nr_in_dr_via_wr, is_xxpp_format)
write_file_vec_polynomials("norm_to_ints.vec",
self.real_integrals, is_xxpp_format)
write_file_polynomial("equi_to_tham.pol",
self.h_er, is_xxpp_format)
def write_out_integrals(self):
#write out in Mathematica xxpp format
is_xxpp_format = True
# equi_to_tvec is J.Hessian
hess = []
for i in xrange(self.h_er.n_vars()):
hess.append(self.h_er.diff(i))
e2tv = [ poly.substitute(hess) for poly in self.dia.matrix_as_vector_of_row_polynomials(self.dia.skew_symmetric_matrix())]
write_file_vec_polynomials("equi_to_tvec.vec",e2tv,is_xxpp_format)
#These are dim dof, is_xxpp_format doesn't apply
i2f = []
for i in xrange(self.h_ir.n_vars()):
i2f.append(self.h_ir.diff(i))
write_file_vec_polynomials("ints_to_freq.vec", i2f, False)
file_ostr = open("ints_to_tham.pol", 'w')
write_ascii_polynomial(file_ostr, self.h_ir.real(), False)
file_ostr.close()
logger.info('...done')
class NormalForm:
def __init__(self, lie_algebra, h_er):
logger.info('Initializing...')
self.alg = lie_algebra
self.h_er = h_er
logger.info('...done initializing.')
def get_lie_algebra(self):
return self.alg
def set_tolerance(self, tolerance):
self.tolerance = tolerance
def confirm_equilibrium_point(self):
logger.info('Confirming equilibrium point...')
alg = self.alg
#logger.info('Real Equilibrium Hamiltonian (terms up to grade')
#logger.info(truncated_poly(self.alg, self.h_er))
logger.info('Equilibrium Hamiltonian has %d terms' %
self.h_er.n_terms())
gra = alg.grade(self.h_er)
if alg.isograde(self.h_er, 0):
logger.error('WARNING! constant term on input Hamiltonian removed')
logger.error('size %s',
alg.isograde(self.h_er, 0).l_infinity_norm())
self.h_er = alg.isograde(self.h_er, 1, alg.grade(self.h_er) + 1)
if alg.isograde(self.h_er, 1):
logger.error('WARNING! linear term on input Hamiltonian removed')
logger.error('size %s',
alg.isograde(self.h_er, 1).l_infinity_norm())
self.h_er = alg.isograde(self.h_er, 2, alg.grade(self.h_er) + 1)
assert not alg.isograde(self.h_er, 0, 2)
assert gra == alg.grade(self.h_er)
logger.info('WARNING! Removing small coefficients of size <=%s' %
tol_equi_real_coeffs)
self.h_er = self.h_er.with_small_coeffs_removed(tol_equi_real_coeffs)
logger.info('Real Equilibrium Hamiltonian has %d terms' %
self.h_er.n_terms())
logger.info('...done confirming equilibrium point.')
def diagonalize(self):
logger.info('Diagonalizing...')
self.dia = Diagonalizer(self.alg)
logger.info('Equilibrium Hamiltonian has %d terms' %
self.h_er.n_terms())
logger.info('Computing the eigensystem...')
self.eig = self.dia.compute_eigen_system(self.h_er, self.tolerance)
logger.info('...done computing the eigensystem.')
logger.info('Computing the diagonal change...')
self.dia.compute_diagonal_change()
logger.info('...done computing the diagonal change.')
logger.info('Measuring error in symplectic identities...')
mat = self.dia.get_matrix_diag_to_equi()
assert self.dia.matrix_is_symplectic(mat)
logger.info('...done measuring error in symplectic identities.')
logger.info('Converting the matrix into vec<poly> representation...')
er_in_terms_of_dr = self.dia.matrix_as_vector_of_row_polynomials(mat)
self.er_in_terms_of_dr = er_in_terms_of_dr
logger.info('... done converting the matrix into vec<poly> repn.')
logger.info('Applying diagonalizing change to equilibrium H...')
self.h_dr = self.h_er.substitute(er_in_terms_of_dr)
logger.info('...done applying diagonalizing change to equilibrium H.')
del er_in_terms_of_dr
logger.info('Real diagonal Hamiltonian has %d terms' %
self.h_dr.n_terms())
err = self.h_dr.imag().l_infinity_norm()
logger.info('Size of imaginary part: %s', err)
assert err < tol_diag_real_imag_part
logger.info('WARNING! Removing imaginary terms')
self.h_dr = self.h_dr.real()
logger.info('Real diagonal Hamiltonian has %d terms' %
self.h_dr.n_terms())
logger.info('WARNING! Removing small coefficients of size <=%s' %
tol_diag_real_coeffs)
self.h_dr = self.h_dr.with_small_coeffs_removed(tol_diag_real_coeffs)
logger.info('Real diagonal Hamiltonian has %d terms' %
self.h_dr.n_terms())
#logger.info('Real Diagonal Hamiltonian (terms up to gra 3)')
#logger.info(truncated_poly(self.alg, self.h_dr))
logger.info('Quadratic part')
logger.info(truncated_poly(self.alg, self.alg.isograde(self.h_dr, 2)))
logger.info('Terms in quadratic part %d' %
(self.alg.isograde(self.h_dr, 2).n_terms()))
logger.info('...done diagonalizing.')
def complexify(self):
logger.info('Complexifying...')
self.eq_type = self.eig.get_equilibrium_type()
logger.info('Equilibrium type %s' % self.eq_type)
logger.info('Terms in real diagonal %d' % self.h_dr.n_terms())
self.com = Complexifier(self.alg, self.eq_type)
self.r_in_terms_of_c = self.com.calc_sub_complex_into_real()
self.c_in_terms_of_r = self.com.calc_sub_real_into_complex()
logger.info('Performing complex substitution...')
self.h_dc = self.h_dr.substitute(self.r_in_terms_of_c)
logger.info('Terms in complex diagonal %d' % self.h_dc.n_terms())
logger.info('Removing small coefficients...')
self.h_dc = self.h_dc.with_small_coeffs_removed(self.tolerance)
logger.info('Terms in complex diagonal %d' % self.h_dc.n_terms())
h_dc_2 = self.alg.isograde(self.h_dc, 2)
off_diag = h_dc_2 - self.alg.diagonal_part_of_polynomial(h_dc_2)
err = off_diag.l_infinity_norm()
logger.info(
'Size of off-diagonal part of quadratic part of diagonal h: %s',
err)
assert err < tol_diag_comp_off_diag_part
logger.info('Removing off-diagonal part')
self.h_dc -= off_diag
del off_diag
logger.info('Terms in complex diagonal %d' % self.h_dc.n_terms())
h_dc_2 = self.alg.isograde(self.h_dc, 2)
logger.info('Quadratic part')
logger.info(self.alg.display(h_dc_2))
assert (self.alg.is_diagonal_polynomial(h_dc_2))
del h_dc_2
self.eig_vals = self.eig.get_eigen_values_in_plus_minus_pairs()
logger.info('Complex diagonal Hamiltonian has %d terms' %
self.h_dc.n_terms())
logger.info('...done complexifying.')
def cvec2r_to_rvec2r(self, p):
"""
@param p: a polynomial map from a vector of complex
coordinates to a real scalar.
@return: a polynomial map from the real version of the
coordinates to the real scalar.
"""
return p.substitute(self.c_in_terms_of_r)
def cvec2cvec_to_rvec2rvec(self, vp):
"""
@param vp: vector of polynomial maps from vector of complex
coordinates to a complex coordinate.
@return: vector of polynomial maps from vector of reals to
real.
"""
cvec_in_rvec = []
for i in xrange(len(vp)):
cvec_in_rvec.append(vp[i].substitute(self.c_in_terms_of_r))
cvec_in_cvec = []
for r_in_c_i in self.r_in_terms_of_c:
cvec_in_cvec.append(r_in_c_i.substitute(cvec_in_rvec))
return cvec_in_cvec
def normalize(self):
logger.info('Normalizing...')
steps = self.desired_grade - 1 #1+h_dc.grade()-2
self.w_c_list = []
self.h_nc_list = []
self.h_nc_dict = {}
self.h_nc = self.alg.zero()
self.w_c = self.alg.zero()
h_dc_2 = self.alg.isograde(self.h_dc, 2)
state = (self.h_nc_dict, self.w_c_list, -1)
tri = LieTriangle.LieTriangle(self.alg, h_dc_2, state)
#
# note that, in the following, h_term is _NOT_ h_i for the
# Deprit triangle; it is (h_i)/factorial(i) and this is
# taken care of _WITHIN_ the triangle procedure.
#
for step in xrange(0, steps + 1):
h_term = self.alg.isograde(self.h_dc, step + 2)
k_term, w_term = tri.compute_normal_form_and_generating_function(
h_term)
logger.info('Resulting normal form terms:')
logger.info(truncated_poly(self.alg, k_term))
self.h_nc += k_term
self.w_c += w_term
logger.info('Complex normal form:')
logger.info(truncated_poly(self.alg, self.h_nc))
self.iso_nf = tri.get_isograde_list_handler()
#h_nc_new = tri.compute_lie_transform_given_generating_function([], steps)
#logger.info('Normalized using computed generating function:')
#logger.info(self.alg.display(h_nc_new))
def check_generating_function_invariant(self):
logger.info('confirming generating function invariant...')
grade = self.alg.grade
#1+self.h_dc.grade()-3 #check
steps = self.desired_grade - 2
logger.info('surely we can use independent no. steps in coord change,')
logger.info('than we did in the nf computation? CARE!!!')
logger.info(
'the generating function is only invariant with correct steps.')
cc = CoordinateChange(self.alg, self.w_c_list)
res1 = []
cc.express_norm_in_diag(self.w_c, res1, {}, steps)
res2 = []
cc.express_diag_in_norm(self.w_c, res2, {}, steps)
iso_cc = cc.get_isograde_list_handler()
for res in [res1, res2]:
w_c_new = iso_cc.list_to_poly(res)
logger.info('Grade %d to %d', grade(self.w_c), grade(w_c_new))
gra = min((grade(self.w_c), grade(w_c_new)))
err = self.alg.isograde((w_c_new - self.w_c), 0,
gra + 1).l_infinity_norm()
logger.info('Change in like-grade part of generating function: %s',
err)
assert err < tol_comp_generator_through_lie_triangle
del cc
del res1
del res2
del iso_cc
del w_c_new
def check_diag_transforms_to_norm(self):
logger.info(
'confirming that diagonal Hamiltonian transforms to normal...')
grade = self.alg.grade
steps = self.desired_grade - 2 #1+self.h_dc.grade()-3
cc = CoordinateChange(self.alg, self.w_c_list)
iso_cc = cc.get_isograde_list_handler()
res1 = []
cc.express_norm_in_diag(self.h_nc, res1, {}, steps)
h_dc_new = iso_cc.list_to_poly(res1)
gra = min((grade(self.h_dc), grade(h_dc_new)))
err = self.alg.isograde(self.h_dc - h_dc_new, 0,
gra + 1).l_infinity_norm()
logger.info('Error expressing H_{nc} in x_{dc}')
logger.info('Grade %d to %d', grade(self.h_dc), grade(h_dc_new))
logger.info(err)
logger.info(
'confirming that normal Hamiltonian transforms to diagonal...')
res2 = []
cc.express_diag_in_norm(self.h_dc, res2, {}, steps)
h_nc_new = iso_cc.list_to_poly(res2)
gra = min((grade(self.h_nc), grade(h_nc_new)))
err = self.alg.isograde(self.h_nc - h_nc_new, 0,
gra + 1).l_infinity_norm()
logger.info('Error expressing H_{dc} in x_{nc}')
logger.info('Grade %d to %d', grade(self.h_nc), grade(h_nc_new))
logger.info(err)
del cc
del iso_cc
del h_dc_new
del h_nc_new
del res1
del res2
def normalize_real(self):
logger.info('computing real normalization...')
self.h_nr = self.h_nc.substitute(self.c_in_terms_of_r)
err = self.h_nr.imag().l_infinity_norm()
tol_norm_real_imag_part = (1.0 + self.h_nr.l_infinity_norm()) * 1.0e-10
logger.info('l infinity norm of real normal form imaginary part: %s',
err)
assert err < tol_norm_real_imag_part
self.h_nr = self.h_nr.real()
logger.info('Real normal form:')
logger.info(truncated_poly(self.alg, self.h_nr))
self.w_r_list = [
p.substitute(self.c_in_terms_of_r) for p in self.w_c_list
]
#self.w_r = self.iso_nf.list_to_poly(self.w_r_list)
def extract_integrals(self):
logger.info('extracting the integrals...')
int_ext = IntegralExtractor(self.alg)
int_ext.set_complex_normal_hamiltonian(self.h_nc)
int_ext.find_lost_simple_integrals_and_non_simple_integral()
logger.info('Lost integrals:')
logger.info(int_ext._lost_integrals)
assert ((not int_ext._lost_integrals), int_ext._lost_integrals)
logger.info('Real simple part:')
logger.info((self.cvec2r_to_rvec2r(int_ext._simple_part)))
logger.info('Real non-simple part:')
logger.info((self.cvec2r_to_rvec2r(int_ext._non_simple_integral)))
int_ext.list_the_simple_integrals()
logger.info('Kept integrals:')
logger.info(int_ext._kept_integrals)
int_ext.express_simple_part_as_polynomial_over_all_integrals()
logger.info('Simple part in terms of integrals:')
logger.info((int_ext._simple_in_integrals))
int_ext.express_both_parts_as_polynomial_over_all_integrals()
logger.info('Total in terms of integrals')
logger.info((int_ext._total_in_integrals))
int_ext.express_all_integrals_in_normal_form_coords()
logger.info('Complex Integrals in complex normal form coords:')
logger.info((int_ext._integrals_in_coords))
logger.info('Complex Integrals in real normal form coords:')
ic_in_nr = [
self.cvec2r_to_rvec2r(int_i)
for int_i in int_ext._integrals_in_coords
]
logger.info(ic_in_nr)
#convert simple part in terms of complex integrals,
#to simple part in terms of real integrals.
def simple_int_is_saddle(real_integral):
return len(real_integral) == 1
def simple_int_is_centre(real_integral):
return len(real_integral) == 2
comp_ints_in_real_ints = []
real_integrals = []
n_integrals = len(int_ext._integrals_in_coords)
ring_ints = PolynomialRing(n_integrals)
for i, integral in zip(int_ext._kept_integrals,
int_ext._integrals_in_coords):
# This is the complex integral in real coordinates
real_integral = integral.substitute(self.c_in_terms_of_r)
mon = ring_ints.coordinate_monomial(i)
if simple_int_is_centre(real_integral):
# Note that a centre real integral = i * complex integral
comp_ints_in_real_ints.append((-1.0J) * mon)
real_integrals.append((1.0J) * real_integral)
logger.info('centre')
elif simple_int_is_saddle(real_integral):
comp_ints_in_real_ints.append(mon)
real_integrals.append(real_integral)
logger.info('saddle')
else:
assert 0, 'Simple integral, neither saddle nor centre!'
h_ir = int_ext._simple_in_integrals.substitute(comp_ints_in_real_ints)
logger.info('Simple part in terms of the real simple integrals:')
logger.info(h_ir)
logger.info('Real integrals in real nf coords:')
logger.info(real_integrals)
#we should check this by converting the simple part to real
h_ir_check = self.cvec2r_to_rvec2r(int_ext._simple_part)
err = (h_ir_check - h_ir.substitute(real_integrals)).l_infinity_norm()
logger.info(
'Error in composite of real integrals and real simple part')
logger.info(err)
self.h_ir = h_ir
self.real_integrals = real_integrals
def compute_diag_to_norm(self):
logger.info('computing coordinate changes...')
steps = self.desired_grade - 2 #1+self.h_dc.grade()-3 #check!
res = []
for s in xrange(self.alg.n_vars()):
#do we request a coordinate change in h-bar???
logger.info('complex coord change, component %d of %d', s,
self.alg.n_vars())
xs = self.alg.coordinate_monomial(s)
sp = [
0,
] * (self.alg.n_vars())
sp[s] = 1
#use complex nf
logger.info(
'COMPLEX component of x_{nc} expressed IN x_{dc} (diag TO norm):'
)
x_i_list = []
x_ij_dict = {}
cc = CoordinateChange(self.alg, self.w_c_list)
cc.express_norm_in_diag(xs, x_i_list, x_ij_dict, steps)
#near-identity?
assert (len(x_i_list[0]) == 1)
assert (x_i_list[0][Powers(tuple(sp))] == 1.0)
for gra_minus_1, pol in enumerate(x_i_list):
gra = self.alg.grade(pol)
assert ((gra == 0) or (gra == gra_minus_1 + 1))
iso_cc = cc.get_isograde_list_handler()
#for i, polc in enumerate(x_i_list):
#logger.info(truncated_poly(self.alg, iso_cc.inner_taylor_to_poly(polc, i)))
poly = iso_cc.list_to_poly(x_i_list)
res.append(poly)
self.nc_in_dc = res
def compute_diag_to_norm_real_using_w_real(self):
logger.info(
'computing real coordinate changes using real generator...')
steps = self.desired_grade - 2 #self.h_dc.grade()-2 #check!
res = []
for s in xrange(self.alg.n_vars()):
logger.info('complex coord change, component %d of %d', s,
self.alg.n_vars())
xs = self.alg.coordinate_monomial(s)
sp = [
0,
] * (self.alg.n_vars())
sp[s] = 1
#use real nf
logger.info(
'REAL component of x_{nr} expressed IN x_{dc} (diag TO norm):')
x_i_list = []
x_ij_dict = {}
cc = CoordinateChange(self.alg, self.w_r_list)
cc.express_norm_in_diag(xs, x_i_list, x_ij_dict, steps)
#near-identity?
assert (len(x_i_list[0]) == 1)
assert (x_i_list[0][Powers(tuple(sp))] == 1.0)
for gra_minus_1, pol in enumerate(x_i_list):
gra = self.alg.grade(pol)
assert ((gra == 0) or (gra == gra_minus_1 + 1))
iso_cc = cc.get_isograde_list_handler()
for i, polc in enumerate(x_i_list):
err = polc.imag().l_infinity_norm()
logger.info('l_infinity_norm of imaginary part: %s', err)
tol_norm_real_imag_part = (
1.0 + self.w_r_list[i].l_infinity_norm()) * 1.0e-10
assert err < tol_norm_real_imag_part
#logger.info(truncated_poly(self.alg, iso_cc.inner_taylor_to_poly(polc.real(), i)))
poly = iso_cc.list_to_poly(x_i_list)
res.append(poly.real())
self.nr_in_dr_via_wr = res
def compute_diag_to_norm_real_using_conversion(self):
logger.info(
'computing real coordinate changes using real conversion...')
res = self.cvec2cvec_to_rvec2rvec(self.nc_in_dc)
self.nr_in_dr_via_nc_in_dc = res
logger.info('ERRORS in real generator vs. conversion:')
for m0, m1 in zip(self.nr_in_dr_via_nc_in_dc, self.nr_in_dr_via_wr):
err = (m1 - m0).l_infinity_norm()
logger.info(err)
def compute_norm_to_diag(self):
logger.info('computing inverse coordinate changes...')
steps = self.desired_grade - 2 #1+self.h_dc.grade()-3 #check!
res = []
for s in xrange(self.alg.n_vars()):
logger.info('complex coord change, component %d of %d', s,
self.alg.n_vars())
xs = self.alg.coordinate_monomial(s)
sp = [
0,
] * (self.alg.n_vars())
sp[s] = 1
#use complex nf
logger.info(
'COMPLEX component of x_{dc} expressed IN x_{nc} (norm TO diag):'
)
x_i_list = []
x_ij_dict = {}
cc = CoordinateChange(self.alg, self.w_c_list)
cc.express_diag_in_norm(xs, x_i_list, x_ij_dict, steps)
#near-identity?
assert (len(x_i_list[0]) == 1)
assert (x_i_list[0][Powers(tuple(sp))] == 1.0)
for gra_minus_1, pol in enumerate(x_i_list):
gra = self.alg.grade(pol)
assert ((gra == 0) or (gra == gra_minus_1 + 1))
iso_cc = cc.get_isograde_list_handler()
#for i, polc in enumerate(x_i_list):
#logger.info(truncated_poly(self.alg, iso_cc.inner_taylor_to_poly(polc, i)))
poly = iso_cc.list_to_poly(x_i_list)
res.append(poly)
self.dc_in_nc = res
def compute_norm_to_diag_real_using_w_real(self):
logger.info(
'computing real inverse coordinate change using real generator...')
steps = self.desired_grade - 2 #self.h_dc.grade()-2 #check!
res = []
for s in xrange(self.alg.n_vars()):
logger.info('complex coord change, component %d of %d', s,
self.alg.n_vars())
xs = self.alg.coordinate_monomial(s)
sp = [
0,
] * (self.alg.n_vars())
sp[s] = 1
#use real nf
logger.info(
'REAL component of x_{dc} expressed IN x_{nr} (norm TO diag):')
x_i_list = []
x_ij_dict = {}
cc = CoordinateChange(self.alg, self.w_r_list)
cc.express_diag_in_norm(xs, x_i_list, x_ij_dict, steps)
#near-identity?
assert (len(x_i_list[0]) == 1)
assert (x_i_list[0][Powers(tuple(sp))] == 1.0)
for gra_minus_1, pol in enumerate(x_i_list):
gra = self.alg.grade(pol)
assert ((gra == 0) or (gra == gra_minus_1 + 1))
iso_cc = cc.get_isograde_list_handler()
for i, polc in enumerate(x_i_list):
err = polc.imag().l_infinity_norm()
logger.info('l_infinity_norm of imaginary part: %s', err)
tol_norm_real_imag_part = (
1.0 + self.w_r_list[i].l_infinity_norm()) * 1.0e-10
assert err < tol_norm_real_imag_part
poly = iso_cc.list_to_poly(x_i_list)
res.append(poly.real())
self.dr_in_nr_via_wr = res
def compute_norm_to_diag_real_using_conversion(self):
logger.info(
'computing real inverse coordinate changes using real conversion...'
)
res = self.cvec2cvec_to_rvec2rvec(self.dc_in_nc)
self.nr_in_dr_via_nc_in_dc = res
logger.info('ERRORS in real generator vs. conversion:')
for m0, m1 in zip(self.nr_in_dr_via_nc_in_dc, self.dr_in_nr_via_wr):
err = (m1 - m0).l_infinity_norm()
logger.info(err)
def perform_all_computations(self, desired_grade):
logger.info('main computation started...')
#flags for extra computations:
do_check_generating_function_invariant = False
do_check_diag_transforms_to_norm = False
do_compute_complex_diag_vs_norm = False
do_check_diag_vs_norm_using_conversion = False
#setup:
self.desired_grade = desired_grade
self.h_er = self.alg.isograde(self.h_er, 0, desired_grade + 1)
self.confirm_equilibrium_point()
#main steps:
self.diagonalize()
self.complexify()
self.normalize()
self.normalize_real()
self.extract_integrals()
#optional computations:
if do_check_generating_function_invariant:
self.check_generating_function_invariant()
if do_check_diag_transforms_to_norm:
self.check_diag_transforms_to_norm()
if do_compute_complex_diag_vs_norm:
self.compute_diag_to_norm()
self.compute_norm_to_diag()
#coordinate change maps:
self.compute_diag_to_norm_real_using_w_real()
self.compute_norm_to_diag_real_using_w_real()
#optional computations:
if do_check_diag_vs_norm_using_conversion:
self.compute_diag_to_norm_real_using_conversion()
self.compute_norm_to_diag_real_using_conversion()
logger.info('...main computation done.')
#h_nr_check = self.h_dr.substitute(self.dr_in_nr_via_wr)
#print "(self.h_dr.substitute(self.dr_in_nr_via_wr)-self.h_nr).l_infinity_norm()\n", h_nr_check.l_infinity_norm(), self.h_nr.l_infinity_norm(), (h_nr_check-self.h_nr).l_infinity_norm(), "\n", h_nr_check.l1_norm(), self.h_nr.l1_norm(), (h_nr_check-self.h_nr).l1_norm()
#h_dr_check = self.h_nr.substitute(self.nr_in_dr_via_wr)
#print "(self.h_nr.substitute(self.nr_in_dr_via_wr)-self.h_dr).l_infinity_norm()\n", h_dr_check.l_infinity_norm(), self.h_dr.l_infinity_norm(), (h_dr_check-self.h_dr).l_infinity_norm(), "\n", h_dr_check.l1_norm(), self.h_dr.l1_norm(), (h_dr_check-self.h_dr).l1_norm()
#print "truncated to degree 5"
#print self.alg.isograde(h_dr_check-self.h_dr, 1, up_to=5).l1_norm()
#print self.alg.isograde(h_nr_check-self.h_nr, 1, up_to=5).l1_norm()
self.write_out_integrals()
self.write_out_transforms()
def write_out_transforms(self):
#write out in Mathematica xxpp format
is_xxpp_format = True
write_file_mat("diag_to_equi.mat", self.dia.matrix_diag_to_equi)
write_file_mat("equi_to_diag.mat", self.dia.get_matrix_equi_to_diag())
write_file_vec_polynomials("norm_to_diag.vec", self.dr_in_nr_via_wr,
is_xxpp_format)
write_file_vec_polynomials("diag_to_norm.vec", self.nr_in_dr_via_wr,
is_xxpp_format)
write_file_vec_polynomials("norm_to_ints.vec", self.real_integrals,
is_xxpp_format)
write_file_polynomial("equi_to_tham.pol", self.h_er, is_xxpp_format)
def write_out_integrals(self):
#write out in Mathematica xxpp format
is_xxpp_format = True
# equi_to_tvec is J.Hessian
hess = []
for i in xrange(self.h_er.n_vars()):
hess.append(self.h_er.diff(i))
e2tv = [
poly.substitute(hess)
for poly in self.dia.matrix_as_vector_of_row_polynomials(
self.dia.skew_symmetric_matrix())
]
write_file_vec_polynomials("equi_to_tvec.vec", e2tv, is_xxpp_format)
#These are dim dof, is_xxpp_format doesn't apply
i2f = []
for i in xrange(self.h_ir.n_vars()):
i2f.append(self.h_ir.diff(i))
write_file_vec_polynomials("ints_to_freq.vec", i2f, False)
file_ostr = open("ints_to_tham.pol", 'w')
write_ascii_polynomial(file_ostr, self.h_ir.real(), False)
file_ostr.close()
logger.info('...done')
class Hill:
def __init__(self, max_grade):
dof = 3
self.alg = LieAlgebra(dof)
self.h = self.alg.isograde(self.hill_about_l1(), 0, max_grade + 1)
self.h_dc = self.diagonalize(self.alg, self.h)
self.h_dr = self.h_dc.substitute(self.sub_r_into_c)
self.k_dc = self.normalize_to_grade(max_grade)
self.k_dr = self.k_dc.substitute(self.sub_r_into_c)
def hill_about_l1(self):
#print 'reading hill hamiltonian for test...'
in_name = '../../test/ma-files/hill_l1_18_equi_to_tham.pol'
in_file = open(in_name, 'r')
h = read_ascii_polynomial(in_file, is_xxpp_format=1)
in_file.close()
#print 'done'
terms = {
Powers((2, 0, 0, 0, 0, 0)): -4.0, #x^2
Powers((0, 0, 2, 0, 0, 0)): +2.0, #y^2
Powers((0, 0, 0, 0, 2, 0)): +2.0, #z^2
Powers((0, 2, 0, 0, 0, 0)): +0.5, #px^2
Powers((0, 0, 0, 2, 0, 0)): +0.5, #py^2
Powers((0, 0, 0, 0, 0, 2)): +0.5, #pz^2
Powers((1, 0, 0, 1, 0, 0)): -1.0, #xpy
Powers((0, 1, 1, 0, 0, 0)): +1.0
} #ypx
assert len(terms) == 8
h_2 = self.alg.polynomial(terms)
assert (h_2 == self.alg.isograde(h, 2))
return h
def diagonalize(self, dof, h):
tolerance = 5.0e-15
self.diag = Diagonalizer(self.alg)
self.eig = self.diag.compute_eigen_system(h, tolerance)
self.diag.compute_diagonal_change()
mat = self.diag.get_matrix_diag_to_equi()
assert self.diag.matrix_is_symplectic(mat)
sub_diag_into_equi = self.diag.matrix_as_vector_of_row_polynomials(mat)
h_diag = h.substitute(sub_diag_into_equi)
self.eq_type = self.eig.get_equilibrium_type()
self.comp = Complexifier(self.alg, self.eq_type)
self.sub_c_into_r = self.comp.calc_sub_complex_into_real()
self.sub_r_into_c = self.comp.calc_sub_real_into_complex()
h_comp = h_diag.substitute(self.sub_c_into_r)
h_comp = h_comp.with_small_coeffs_removed(tolerance)
assert (self.alg.is_diagonal_polynomial(self.alg.isograde(h_comp, 2)))
return h_comp
def normalize_to_grade(self, max_grade):
steps = max_grade
h_dc = self.alg.isograde(self.h_dc, 2, max_grade + 1)
h_dc_2 = self.alg.isograde(h_dc, 2)
k_comp = self.alg.zero()
w_comp = self.alg.zero()
#the following is not very efficient for separating grades!
h_list = [self.alg.isograde(h_dc, i + 2) for i in xrange(0, steps + 1)]
nf = LieTriangle(self.alg, h_dc_2)
for i in xrange(0, steps + 1):
h_term = h_list[i]
k_term, w_term = nf.compute_normal_form_and_generating_function(
h_term)
k_comp += k_term
w_comp += w_term
#print k_comp
k_real = k_comp.substitute(self.sub_r_into_c)
assert (k_real.imag().l_infinity_norm() < 1.0e-14)
k_real = k_real.real()
return k_comp
class Hill:
def __init__(self, max_grade):
dof = 3
self.alg = LieAlgebra(dof)
self.h = self.alg.isograde(self.hill_about_l1(), 0, max_grade+1)
self.h_dc = self.diagonalize(self.alg, self.h)
self.h_dr = self.h_dc.substitute(self.sub_r_into_c)
self.k_dc = self.normalize_to_grade(max_grade)
self.k_dr = self.k_dc.substitute(self.sub_r_into_c)
def hill_about_l1(self):
#print 'reading hill hamiltonian for test...'
in_name = '../../test/ma-files/hill_l1_18_equi_to_tham.pol'
in_file = open(in_name, 'r')
h = read_ascii_polynomial(in_file, is_xxpp_format=1)
in_file.close()
#print 'done'
terms = {Powers((2, 0, 0, 0, 0, 0)): -4.0, #x^2
Powers((0, 0, 2, 0, 0, 0)): +2.0, #y^2
Powers((0, 0, 0, 0, 2, 0)): +2.0, #z^2
Powers((0, 2, 0, 0, 0, 0)): +0.5, #px^2
Powers((0, 0, 0, 2, 0, 0)): +0.5, #py^2
Powers((0, 0, 0, 0, 0, 2)): +0.5, #pz^2
Powers((1, 0, 0, 1, 0, 0)): -1.0, #xpy
Powers((0, 1, 1, 0, 0, 0)): +1.0} #ypx
assert len(terms) == 8
h_2 = self.alg.polynomial(terms)
assert (h_2 == self.alg.isograde(h, 2))
return h
def diagonalize(self, dof, h):
tolerance = 5.0e-15
self.diag = Diagonalizer(self.alg)
self.eig = self.diag.compute_eigen_system(h, tolerance)
self.diag.compute_diagonal_change()
mat = self.diag.get_matrix_diag_to_equi()
assert self.diag.matrix_is_symplectic(mat)
sub_diag_into_equi = self.diag.matrix_as_vector_of_row_polynomials(mat)
h_diag = h.substitute(sub_diag_into_equi)
self.eq_type = self.eig.get_equilibrium_type()
self.comp = Complexifier(self.alg, self.eq_type)
self.sub_c_into_r = self.comp.calc_sub_complex_into_real()
self.sub_r_into_c = self.comp.calc_sub_real_into_complex()
h_comp = h_diag.substitute(self.sub_c_into_r)
h_comp = h_comp.with_small_coeffs_removed(tolerance)
assert (self.alg.is_diagonal_polynomial(self.alg.isograde(h_comp, 2)))
return h_comp
def normalize_to_grade(self, max_grade):
steps = max_grade
h_dc = self.alg.isograde(self.h_dc, 2, max_grade+1)
h_dc_2 = self.alg.isograde(h_dc, 2)
k_comp = self.alg.zero()
w_comp = self.alg.zero()
#the following is not very efficient for separating grades!
h_list = [self.alg.isograde(h_dc, i + 2) for i in xrange(0, steps + 1)]
nf = LieTriangle(self.alg, h_dc_2)
for i in xrange(0, steps + 1):
h_term = h_list[i]
k_term, w_term = nf.compute_normal_form_and_generating_function(h_term)
k_comp += k_term
w_comp += w_term
#print k_comp
k_real = k_comp.substitute(self.sub_r_into_c)
assert (k_real.imag().l_infinity_norm() < 1.0e-14)
k_real = k_real.real()
return k_comp