def TestStringRepresentations():
# Test string representations (note leading spaces)
Julian.day_offset = mpf("0")
Julian.interval_representation = "c"
j1, j2 = mpf("2451545.0"), mpf("2451545.5")
assert str(Julian(j1)) == " 1Jan2000:12:00"
assert str(Julian(mpi(j1, j2))) == " <<1Jan2000:12:00, 2Jan2000:00:00>>"
Julian.day_offset = mpf("0.5")
assert str(Julian(j1)) == " 1Jan2000:00:00"
assert str(Julian(mpi(j1, j2))) == " <<1Jan2000:00:00, 1Jan2000:12:00>>"
def check_spectral_norm(self, M, tolerance=1e-10):
"""
check iv_spectral_norm() and iv_spectral_norm_rough() against a numerical result from scipy.
@param tolerance: assumed guaranteed relative tolerance on scipy's result
"""
approx_spectral_norm = scipy.linalg.norm(
iv_matrix_to_numpy_ndarray(iv_matrix_mid_as_mp(M)), 2)
self.assertIn(approx_spectral_norm,
iv_spectral_norm(M) * (1 + tolerance * mp.mpi(-1, 1)))
self.assertIn(
approx_spectral_norm,
iv_spectral_norm_rough(M) * (1 + tolerance * mp.mpi(-1, 1)))
def is_in(region1, region2):
""" Returns True if the region1 is in the region2, returns False otherwise
Args:
region1 (list of pairs): (hyper)space defined by the regions
region2 (list of pairs): (hyper)space defined by the regions
"""
if len(region1) is not len(region2):
print("The intervals does not have the same size")
return False
for dimension in range(len(region1)):
if mpi(region1[dimension]) not in mpi(region2[dimension]):
return False
return True
def test_special_printers():
class IntervalPrinter(LambdaPrinter):
"""Use ``lambda`` printer but print numbers as ``mpi`` intervals. """
def _print_Integer(self, expr):
return "mpi('%s')" % super(IntervalPrinter,
self)._print_Integer(expr)
def _print_Rational(self, expr):
return "mpi('%s')" % super(IntervalPrinter,
self)._print_Rational(expr)
def intervalrepr(expr):
return IntervalPrinter().doprint(expr)
expr = sqrt(sqrt(2) + sqrt(3)) + S(1) / 2
func0 = lambdify((), expr, modules="mpmath", printer=intervalrepr)
func1 = lambdify((), expr, modules="mpmath", printer=IntervalPrinter)
func2 = lambdify((), expr, modules="mpmath", printer=IntervalPrinter())
mpi = type(mpmath.mpi(1, 2))
assert isinstance(func0(), mpi)
assert isinstance(func1(), mpi)
assert isinstance(func2(), mpi)
def test_special_printers():
from sympy.printing.lambdarepr import IntervalPrinter
def intervalrepr(expr):
return IntervalPrinter().doprint(expr)
expr = sqrt(sqrt(2) + sqrt(3)) + S.Half
func0 = lambdify((), expr, modules="mpmath", printer=intervalrepr)
func1 = lambdify((), expr, modules="mpmath", printer=IntervalPrinter)
func2 = lambdify((), expr, modules="mpmath", printer=IntervalPrinter())
mpi = type(mpmath.mpi(1, 2))
assert isinstance(func0(), mpi)
assert isinstance(func1(), mpi)
assert isinstance(func2(), mpi)
# To check Is lambdify loggamma works for mpmath or not
exp1 = lambdify(x, loggamma(x), 'mpmath')(5)
exp2 = lambdify(x, loggamma(x), 'mpmath')(1.8)
exp3 = lambdify(x, loggamma(x), 'mpmath')(15)
exp_ls = [exp1, exp2, exp3]
sol1 = mpmath.loggamma(5)
sol2 = mpmath.loggamma(1.8)
sol3 = mpmath.loggamma(15)
sol_ls = [sol1, sol2, sol3]
assert exp_ls == sol_ls
def iv_P_norm_expm(P_sqrt_T, M1, A, M2, tau):
"""
Bound on P-ellipsoid norm of (M1 (expm(A*t) - I) M2) for |t| < tau
using the theorem in arXiv:1911.02537, section "Norm bounding of summands"
@param P_sqrt_T: see iv_P_norm()
"""
P_sqrt_T = iv.matrix(P_sqrt_T)
M1 = iv.matrix(M1)
A = iv.matrix(A)
M2 = iv.matrix(M2)
# coerce tau to maximum
tau = abs(iv.mpf(tau)).b
# P-ellipsoid norms
M1_p = iv_P_norm(M=M1, P_sqrt_T=P_sqrt_T)
M2_p = iv_P_norm(M=M2, P_sqrt_T=P_sqrt_T)
A_p = iv_P_norm(M=A, P_sqrt_T=P_sqrt_T)
# A_pow[i] = A ** i
A_pow = _iv_matrix_powers(A)
# Work around bug in mpmath, see comment in iv_P_norm()
zero = iv.matrix(mp.zeros(len(A)))
M1 = zero + M1
# terms from [arXiv:1911.02537]
M1_Ai_M2_p = lambda i: iv_P_norm(M=M1 @ A_pow[i] @ M2, P_sqrt_T=P_sqrt_T)
gamma = lambda i: 1 / math.factorial(i) * (M1_Ai_M2_p(i) - M1_p * A_p ** i * M2_p)
max_norm = sum([gamma(i) * (tau ** i) for i in range(1, IV_NORM_EVAL_ORDER + 1)]) + M1_p * M2_p * (iv.exp(A_p * tau) - 1)
# the lower bound is always 0 (for t=0)
return mp.mpi([0, max_norm.b])
def LDL(mat):
"""
Algorithm for numeric LDL factization, exploiting sparse structure.
This function is a modification of scipy.sparse.SparseMatrix._LDL_sparse,
allowing mpmath.mpi interval arithmetic objects as entries.
L, D are SparseMatrix objects. However we assign values through _smat member
to avoid type conversions to Rational.
"""
Lrowstruc = mat.row_structure_symbolic_cholesky()
print 'Number of entries in L: ', np.sum(map(len, Lrowstruc))
L = SparseMatrix(mat.rows, mat.rows,
dict([((i, i), mpi(0)) for i in range(mat.rows)]))
D = SparseMatrix(mat.rows, mat.cols, {})
for i in range(len(Lrowstruc)):
for j in Lrowstruc[i]:
if i != j:
L._smat[(i, j)] = mat._smat.get((i, j), mpi(0))
summ = 0
for p1 in Lrowstruc[i]:
if p1 < j:
for p2 in Lrowstruc[j]:
if p2 < j:
if p1 == p2:
summ += L[i, p1]*L[j, p1]*D[p1, p1]
else:
break
else:
break
L._smat[(i, j)] = L[i, j] - summ
L._smat[(i, j)] = L[i, j] / D[j, j]
elif i == j:
D._smat[(i, i)] = mat._smat.get((i, i), mpi(0))
summ = 0
for k in Lrowstruc[i]:
if k < i:
summ += L[i, k]**2*D[k, k]
else:
break
D._smat[(i, i)] -= summ
return L, D
def test_mpmath_intervals(self):
print(colored('mpi math sanity check', 'blue'))
## Check more here https://docs.sympy.org/0.6.7/modules/mpmath/basics.html
## Sanity check test
from mpmath import mpi ## Real intervals
my_interval = mpi(0, 5)
self.assertEqual(my_interval.a, 0)
self.assertEqual(my_interval.b, 5)
self.assertEqual(my_interval.mid, (5+0)/2)
self.assertEqual(my_interval.delta, abs(0-5))
def _iv_matrix_powers(A):
"""
return the first IV_NORM_EVAL_ORDER+1 powers of A:
[I, A, A**2, ..., A**(IV_NORM_EVAL_ORDER)]
"""
assert isinstance(A, iv.matrix)
A = iv.matrix(A) + mp.mpi(0,0) # workaround bug
A_pow = [iv.eye(len(A))]
for i in range(IV_NORM_EVAL_ORDER+1):
A_pow.append(A @ A_pow[-1])
return A_pow
def test_rel_tolerance(tol, matrixClass):
"""
test with given relative deviation
"""
lower = 1e3 * matrixClass.eye(4)
if matrixClass == generic_matrix.IntervalMatrix:
upper = lower * mp.mpi(1 - tol / 2, 1 + tol / 2)
else:
upper = lower * (1 + tol)
self.assertMatrixAlmostEqual(lower, upper)
def iv_matrix_mid_to_numpy_ndarray(M):
"""
convert midpoint of interval matrix to numpy ndarray
"""
if isinstance(M, numpy.ndarray):
return M
Y = numpy.zeros((M.rows, M.cols))
for i in range(M.rows):
for j in range(M.cols):
Y[i, j] = float(mp.mpf(mp.mpi(M[i, j]).mid))
return Y
def test_abs_tolerance(tol, matrixClass):
"""
test with given absolute deviation
"""
lower = 1e-3 * matrixClass.eye(4)
if matrixClass == generic_matrix.IntervalMatrix:
upper = lower + matrixClass.ones(4, 4) * mp.mpi(
-tol / 2, tol / 2)
else:
upper = lower + tol * matrixClass.ones(4, 4)
self.assertMatrixAlmostEqual(lower, upper)
def m1_a_m2_tau_uy(self, u_index, y_index):
"""
constituent parts of Delta_A_uy_i_j = M1 * (exp(A*t)-I) * M2, |t|<tau:
(M1, A, M2, tau, info_string) = m1_a_m2_tau_uy(i,j)
info_string is a textual identifier
"""
d = self.abstract_matrix_type
# tau = max(delta t_u - delta t_y) maximized over the range delta_t_u_min ... _max and delta_t_y_min ... _max
dtu = mp.mpi(self.sys.delta_t_u_min[u_index],
self.sys.delta_t_u_max[u_index])
dty = mp.mpi(self.sys.delta_t_y_min[y_index],
self.sys.delta_t_y_max[y_index])
tau = (
dty - dtu
).b # max(dty_j - dtu_i) for the given intervals of dty_j and dtu_i
tau = d.convert_scalar(tau) # convert to float if required
if tau < 0:
tau = 0
return (self.A_ctrl @ (self.A_y[y_index] - d.eye(self.n)), self.A_cont,
self.A_u[u_index] - d.eye(self.n), tau,
"u {} combined with y {}".format(u_index, y_index))
def v(self, s):
'''Interval numbers: allowed forms are
1. 'a +- b'
2. 'a (b%)' % sign is optional
3. '[a, b]'
In 1, a is the midpoint of the interval and b is the half-width.
In 2, a is the midpoint of the interval and b is the half-width.
In 3, the interval is indicated directly.
'''
e = ValueError("Improperly formed interval number '%s'" %s)
s = s.replace(" ", "")
if "+-" in s:
n = [mpf(strip(i)) for i in s.split("+-")]
return mpi(n[0] - n[1], n[0] + n[1])
elif "(" in s:
if s[0] == "(": # Don't confuse with a complex number (x,y)
return None
if ")" not in s:
raise e
s = s.replace(")", "")
percent = False
if "%" in s:
if s[-1] != "%":
raise e
percent = True
s = s.replace("%", "")
a, p = [mpf(strip(i)) for i in s.split("(")]
d = p
if percent:
d = a*p/mpf(100)
return mpi(a - d, a + d)
elif "," in s:
if "[" not in s: raise e
if "]" not in s: raise e
s = s.replace("[", "")
s = s.replace("]", "")
n = [mpf(strip(i)) for i in s.split(",")]
return mpi(n[0], n[1])
else:
return None
def mpiTests(): # Test with mpi's
onep2 = mpi("1", "2")
third = Rational(1,3)
# add
assert no_idiff(onep2 + third, onep2 + 1/mpi(3))
# radd
assert no_idiff(third + onep2, onep2 + 1/mpi(3))
# sub
assert no_idiff(onep2 - third, onep2 - 1/mpi(3))
# rsub
assert no_idiff(third - onep2, -onep2 + 1/mpi(3))
# mul
assert no_idiff(onep2 * third, onep2 * 1/mpi(3))
# rmul
assert no_idiff(third * onep2, onep2 * 1/mpi(3))
# div
assert no_idiff(onep2 / third, onep2 /(1/mpi(3)))
# rdiv
assert no_idiff(third / onep2, (1/mpi(3)) / onep2)
def iv_matrix_mid_as_mp(M):
"""
entrywise midpoint of interval matrix,
as (non-interval) mpmath matrix.
If the input is a mpmath matrix, it is returned unchanged.
If the input is a 2-dimensional numpy.ndarray, it is converted to mpmath.
"""
if isinstance(M, numpy.ndarray):
return mp.matrix(M)
Y = mp.matrix(M.rows, M.cols)
for i in range(M.rows):
for j in range(M.cols):
Y[i, j] = mp.mpi(M[i, j]).mid
return Y
def ParseRawData(show=False):
'''Set show to True to have the names printed to stdout.
'''
def Compact(s):
return s.replace(" ", "")
# Locations of fields in data
locations = {
"name": (0, 55),
"value": (55, 77),
"uncertainty": (77, 98),
}
s = StringIO(raw_data)
lines = s.readlines()
constants = {}
fix = 1
for line in lines:
line = line.strip()
if not line:
continue
a, b = locations["name"]
name = line[a:b].strip()
a, b = locations["value"]
value = Compact(line[a:b])
a, b = locations["uncertainty"]
uncertainty = Compact(line[a:b])
if fix:
if uncertainty == "(exact)":
uncertainty = 0
if "..." in value:
value = value.replace("...", "")
try:
x = mpf(value)
dx = mpf(uncertainty)
if dx == 0:
num = x
else:
num = mpi(x - dx, x + dx)
constants[name] = num
except Exception as e:
print(name)
print(" ", str(e))
names = list(constants.keys())
names.sort()
if show:
for name in names:
print(name)
return constants
def test_special_printers():
from sympy.polys.numberfields import IntervalPrinter
def intervalrepr(expr):
return IntervalPrinter().doprint(expr)
expr = sqrt(sqrt(2) + sqrt(3)) + S(1)/2
func0 = lambdify((), expr, modules="mpmath", printer=intervalrepr)
func1 = lambdify((), expr, modules="mpmath", printer=IntervalPrinter)
func2 = lambdify((), expr, modules="mpmath", printer=IntervalPrinter())
mpi = type(mpmath.mpi(1, 2))
assert isinstance(func0(), mpi)
assert isinstance(func1(), mpi)
assert isinstance(func2(), mpi)
def ParseRawData(show=False):
'''Set show to True to have the names printed to stdout.
'''
def Compact(s):
return s.replace(" ", "")
# Locations of fields in data
locations = {
"name" : (0, 55),
"value" : (55, 77),
"uncertainty" : (77, 98),
}
s = StringIO(raw_data)
lines = s.readlines()
constants = {}
fix = 1
for line in lines:
line = strip(line)
if not line:
continue
a, b = locations["name"]
name = strip(line[a:b])
a, b = locations["value"]
value = Compact(line[a:b])
a, b = locations["uncertainty"]
uncertainty = Compact(line[a:b])
if fix:
if uncertainty == "(exact)":
uncertainty = 0
if "..." in value:
value = value.replace("...", "")
try:
x = mpf(value)
dx = mpf(uncertainty)
if dx == 0:
num = x
else:
num = mpi(x-dx, x+dx)
constants[name] = num
except Exception, e:
print name
print " ", str(e)
def compute_strategy(self):
if self.tau() > 1:
# self.repetition -= 1
# if self.repetition:
# return self.last_strategy
# else:
# self.repetition = self.n_rep
p = 0.5 / (self.tau() ** self.power)
b = (self.norm_const *
sqrt(-log(p) / (self.tau())))
interval = mpmath.mpi(self.avg_loss - b,
self.avg_loss + b)
for p in sorted(self.profiles, key=lambda x: self.exp_adv_loss[x]):
if self.exp_adv_loss[p] in interval:
# print("exp_loss", self.exp_adv_loss[p], "is in", interval)
# print("with b", b, "and avg_loss", self.avg_loss)
return p.get_best_responder().compute_strategy()
# print("unknown")
# MODIFY to take into account the not-Unknown case
# if I have not returned yet, then I must use fpl
return super().compute_strategy()
def check_qlf_bounds(self, P_sqrt_T, M, abstractmatrix: AbstractMatrix):
"""
test qlf_upper_bound, qlf_lower_bound for given matrices P_sqrt_T, M
with random vectors x.
"""
c1 = abstractmatrix.qlf_upper_bound(P_sqrt_T,
M) # c1 sqrt(V(Mx)) <= |x|
c2 = abstractmatrix.qlf_lower_bound(P_sqrt_T,
M) # |Mx| <= c2 sqrt(V(x))
def sqrt_V(x):
"""
V(x) = x.T P_sqrt_T.T P_sqrt_T x.
"""
return np.sqrt(x.T @ P_sqrt_T.T @ P_sqrt_T @ x)
def mag(x):
"""
|x|
"""
return np.sqrt(np.sum(x**2))
M_actual = M
if M_actual is None:
M_actual = np.eye(len(P_sqrt_T))
RELTOL = 1e-10
for i in range(100):
x = np.random.uniform(low=-42, high=+42, size=len(P_sqrt_T))
# c1 sqrt(V(Mx)) <= |x|
if not mp.iv.isnan(c1) and c1 != mp.mpi(
'-inf', '+inf') and not mp.iv.isinf(c1):
self.assertLessEqual(c1 * sqrt_V(M_actual @ x),
mag(x) * (1 + RELTOL))
# |Mx| <= c2 sqrt(V(x))
self.assertLessEqual(mag(M_actual @ x),
c2 * sqrt_V(x) * (1 + RELTOL))
def test_special_printers():
class IntervalPrinter(LambdaPrinter):
"""Use ``lambda`` printer but print numbers as ``mpi`` intervals."""
def _print_Integer(self, expr):
return f"mpi('{super()._print_Integer(expr)}')"
def _print_Rational(self, expr):
return f"mpi('{super()._print_Rational(expr)}')"
def intervalrepr(expr):
return IntervalPrinter().doprint(expr)
expr = diofant.sqrt(diofant.sqrt(2) + diofant.sqrt(3)) + diofant.Rational(
1, 2)
func0 = lambdify((), expr, modules='mpmath', printer=intervalrepr)
func1 = lambdify((), expr, modules='mpmath', printer=IntervalPrinter)
func2 = lambdify((), expr, modules='mpmath', printer=IntervalPrinter())
mpi = type(mpmath.mpi(1, 2))
assert isinstance(func0(), mpi)
assert isinstance(func1(), mpi)
assert isinstance(func2(), mpi)
def test_special_printers():
class IntervalPrinter(LambdaPrinter):
"""Use ``lambda`` printer but print numbers as ``mpi`` intervals. """
def _print_Integer(self, expr):
return "mpi('%s')" % super(IntervalPrinter, self)._print_Integer(expr)
def _print_Rational(self, expr):
return "mpi('%s')" % super(IntervalPrinter, self)._print_Rational(expr)
def intervalrepr(expr):
return IntervalPrinter().doprint(expr)
expr = sympy.sqrt(sympy.sqrt(2) + sympy.sqrt(3)) + sympy.S(1)/2
func0 = lambdify((), expr, modules="mpmath", printer=intervalrepr)
func1 = lambdify((), expr, modules="mpmath", printer=IntervalPrinter)
func2 = lambdify((), expr, modules="mpmath", printer=IntervalPrinter())
mpi = type(mpmath.mpi(1, 2))
assert isinstance(func0(), mpi)
assert isinstance(func1(), mpi)
assert isinstance(func2(), mpi)
def list_of_basis(N,weight,prec=501,tol=1e-20,sv_min=1E-1,sv_max=1E15,set_dim=None):
r""" Returns a list of pairs (r,D) forming a basis
"""
# First we find the smallest Discriminant for each of the components
if(set_dim<>None and set_dim >0):
dim=set_dim
else:
dim=dimension_jac_cusp_forms(int(weight+0.5),N,-1)
basislist=dict()
num_gotten=0
co_tmp=dict()
num_gotten=0
C0=1
RF=RealField(prec)
if(silent>1):
print "N=",N
print "dim=",dim
print "sv_min=",sv_min
print "sv_max=",sv_max
Aold=Matrix(RF,1)
tol0=1E-20 #tol
# we start with the first discriminant, then the second etc.
Z2N=IntegerModRing(2*N)
ZZ4N=IntegerModRing(4*N)
for Dp in [1..max(1000,100*dim)]:
D=-Dp # we use the dual of the Weil representation
D4N=ZZ4N(D)
if(not(is_square(D4N))):
continue
for r in my_modsqrt(D4N,N):
# I want to make sure that P_{(D,r)} is independent from the previously computed functions
# The only sure way to do this is to compute all submatrices (to a much smaller precision than what we want at the end)
# The candidate is [D,r] and we need to compute the vector of [D,r,D',r']
# for all D',r' already in the list
ltmp1=dict()
ltmp2=dict()
j=0
for [Dp,rp] in basislist.values():
ltmp1[j]=[D,r,Dp,rp]
ltmp2[j]=[Dp,rp,D,r]
j=j+1
ltmp1[j]=[D,r,D,r]
#print "Checking: D,r,D,r=",ltmp1
ctmp1=ps_coefficients_holomorphic_vec(N,weight,ltmp1,tol0)
# print "ctmp1=",ctmp1
if(j >0):
#print "Checking: D,r,Dp,rp=",ltmp2 # Data is ok?: {0: True}
ctmp2=ps_coefficients_holomorphic_vec(N,weight,ltmp2,tol0)
# print "ctmp2=",ctmp2
#print "num_gotten=",num_gotten
A=matrix(RF,num_gotten+1)
# The old matrixc with the elements that are already added to the basis
# print "Aold=\n",A,"\n"
# print "num_gotten=",num_gotten
# print "Aold=\n",Aold,"\n"
for k in range(Aold.nrows()):
for l in range(Aold.ncols()):
A[k,l]=Aold[k,l]
# endfor
# print "A set by old=\n",A,"\n"
# Add the (D',r',D,r) for each D',r' in the list
tmp=RF(1.0)
for l in range(num_gotten):
# we do not use the scaling factor when
# determining linear independence
# mm=RF(abs(ltmp2[l][0]))/N4
# tmp=RF(mm**(weight-one))
A[num_gotten,l]=ctmp2['data'][l]*tmp
# Add the (D,r,D',r') for each D',r' in the list
# print "ctmp1.keys()=",ctmp1.keys()
for l in range(num_gotten+1):
#mm=RF(abs(ltmp1[l][2]))/4N
#tmp=RF(mm**(weight-one))
# print "scaled with=",tmp.n(200)
A[l,num_gotten]=ctmp1['data'][l]*tmp
#[d,B]=mat_inverse(A) # d=det(A)
#if(silent>1):
#d=det(A)
#print "det A = ",d
# Now we have to determine whether we have a linearly independent set or not
dold=mpmath.mp.dps
mpmath.mp.dps=int(prec/3.3)
AInt=mpmath.matrix(int(A.nrows()),int(A.ncols()))
AMp=mpmath.matrix(int(A.nrows()),int(A.ncols()))
if(silent>0):
print "tol0=",tol0
for ir in range(A.nrows()):
for ik in range(A.ncols()):
AInt[ir,ik]=mpmath.mp.mpi(A[ir,ik]-tol0,A[ir,ik]+tol0)
AMp[ir,ik]=mpmath.mpf(A[ir,ik])
d=mpmath.det(AMp)
di=mpmath.mp.mpi(mpmath.mp.det(AInt))
#for ir in range(A.nrows()):
# for ik in range(A.ncols()):
# #print "A.d=",AInt[ir,ik].delta
if(silent>0):
print "mpmath.mp.dps=",mpmath.mp.dps
print "det(A)=",d
print "det(A-as-interval)=",di
print "d.delta=",di.delta
#if(not mpmath.mpi(d) in di):
# raise ArithmeticError," Interval determinant not ok?"
#ANP=A.numpy()
#try:
# u,s,vnp=svd(ANP) # s are the singular values
# sl=s.tolist()
# mins=min(sl) # the smallest singular value
# maxs=max(sl)
# if(silent>1):
# print "singular values = ",s
#except LinAlgError:
# if(silent>0):
# print "could not compute SVD!"
# print "using abs(det) instead"
# mins=abs(d)
# maxs=abs(d)
#if((mins>sv_min and maxs< sv_max)):
zero=mpmath.mpi(0)
if(zero not in di):
if(silent>1):
print "Adding D,r=",D,r
basislist[num_gotten]=[D,r]
num_gotten=num_gotten+1
if(num_gotten>=dim):
return basislist
else:
#print "setting Aold to A"
Aold=A
else:
if(silent>1):
print " do not use D,r=",D,r
# endif
mpmath.mp.dps=dold
# endfor
if(num_gotten < dim):
raise ValueError," did not find enough good elements for a basis list!"
def test_interval_to_mpi():
assert Interval(0, 1).to_mpi() == mpi(0, 1)
assert Interval(0, 1, True, False).to_mpi() == mpi(0, 1)
assert type(Interval(0, 1).to_mpi()) == type(mpi(0, 1))
# Rational numbers
Rational(3, 8) : (
"3/8", "6/16", "0 12/32", "0-15/40", "0+18/48",
),
Rational(-3, 7) : (
"-3/7", "-6/14", "-0 12/28", "-0-15/35", "-0+18/42",
),
Rational(3, -7) : (
"-3/7", "-6/14", "-0 12/28", "-0-15/35", "-0+18/42",
),
}
# Because of a bug in mpi == and != tests, we have to test them
# differently.
mpi_tests = {
# Interval numbers
mpi(1, 3) : (
"[1, 3]", "[1.0, 3]", "[1, 3.0]", "[1.0, 3.0]",
"[1,3]", "[1.0,3]", "[1,3.0]", "[1.0,3.0]",
"[ 1, 3]", "[ 1.0,3]", "[ 1, 3.0]",
"1.5 +- 0.5", "1.5+-0.5", "1.5+- 0.5", "1.5 +-0.5",
"1.5 +- 0.5", "15e-1 +- 500e-3",
"1.5(33.33333333333333333333%)", "1.5 (33.33333333333333333333%)",
"1.5 ( 33.33333333333333333333%)",
"1.5 ( 33.33333333333333333333 % )",
"1.5( 33.33333333333333333333 % )",
"1.5(33.33333333333333333333 % )",
),
}
n = Number()
status = 0
def test_Interval(self):
print(colored("Interval test here", 'blue'))
self.assertEqual(1.0 in mpi(1, 1), True)
self.assertEqual(1.0 in mpi(1, 2), True)
self.assertEqual(1.0 in mpi(0, 1), True)
self.assertEqual(1.0 in mpi(0, 2), True)
self.assertEqual(5.0 not in mpi(1, 1), True)
self.assertEqual(5.0 not in mpi(1, 2), True)
self.assertEqual(5.0 not in mpi(0, 1), True)
self.assertEqual(5.0 not in mpi(0, 2), True)
self.assertEqual(mpi(1, 1) in mpi(1, 1), True)
self.assertEqual(mpi(1, 1) in mpi(1, 2), True)
self.assertEqual(mpi(1, 1) in mpi(0, 1), True)
self.assertEqual(mpi(1, 1) in mpi(0, 2), True)
self.assertEqual(mpi(1, 2) in mpi(1, 3), True)
self.assertEqual(mpi(1, 2) in mpi(0, 2), True)
self.assertEqual(mpi(1, 2) in mpi(0, 3), True)
self.assertEqual(mpi(1, 2) not in mpi(0, 1), True)
self.assertEqual(mpi(1, 2) not in mpi(2, 3), True)
self.assertEqual(mpi(1, 2) not in mpi(1.5, 2), True)
def test_interval_to_mpi():
assert Interval(0, 1).to_mpi() == mpi(0, 1)
assert Interval(0, 1, True, False).to_mpi() == mpi(0, 1)
assert isinstance(Interval(0, 1).to_mpi(), type(mpi(0, 1)))
def list_of_basis(N,
weight,
prec=501,
tol=1e-20,
sv_min=1E-1,
sv_max=1E15,
set_dim=None):
r""" Returns a list of pairs (r,D) forming a basis
"""
# First we find the smallest Discriminant for each of the components
if (set_dim <> None and set_dim > 0):
dim = set_dim
else:
dim = dimension_jac_cusp_forms(int(weight + 0.5), N, -1)
basislist = dict()
num_gotten = 0
co_tmp = dict()
num_gotten = 0
C0 = 1
RF = RealField(prec)
if (silent > 1):
print "N=", N
print "dim=", dim
print "sv_min=", sv_min
print "sv_max=", sv_max
Aold = Matrix(RF, 1)
tol0 = 1E-20 #tol
# we start with the first discriminant, then the second etc.
Z2N = IntegerModRing(2 * N)
ZZ4N = IntegerModRing(4 * N)
for Dp in [1..max(1000, 100 * dim)]:
D = -Dp # we use the dual of the Weil representation
D4N = ZZ4N(D)
if (not (is_square(D4N))):
continue
for r in my_modsqrt(D4N, N):
# I want to make sure that P_{(D,r)} is independent from the previously computed functions
# The only sure way to do this is to compute all submatrices (to a much smaller precision than what we want at the end)
# The candidate is [D,r] and we need to compute the vector of [D,r,D',r']
# for all D',r' already in the list
ltmp1 = dict()
ltmp2 = dict()
j = 0
for [Dp, rp] in basislist.values():
ltmp1[j] = [D, r, Dp, rp]
ltmp2[j] = [Dp, rp, D, r]
j = j + 1
ltmp1[j] = [D, r, D, r]
#print "Checking: D,r,D,r=",ltmp1
ctmp1 = ps_coefficients_holomorphic_vec(N, weight, ltmp1, tol0)
# print "ctmp1=",ctmp1
if (j > 0):
#print "Checking: D,r,Dp,rp=",ltmp2 # Data is ok?: {0: True}
ctmp2 = ps_coefficients_holomorphic_vec(N, weight, ltmp2, tol0)
# print "ctmp2=",ctmp2
#print "num_gotten=",num_gotten
A = matrix(RF, num_gotten + 1)
# The old matrixc with the elements that are already added to the basis
# print "Aold=\n",A,"\n"
# print "num_gotten=",num_gotten
# print "Aold=\n",Aold,"\n"
for k in range(Aold.nrows()):
for l in range(Aold.ncols()):
A[k, l] = Aold[k, l]
# endfor
# print "A set by old=\n",A,"\n"
# Add the (D',r',D,r) for each D',r' in the list
tmp = RF(1.0)
for l in range(num_gotten):
# we do not use the scaling factor when
# determining linear independence
# mm=RF(abs(ltmp2[l][0]))/N4
# tmp=RF(mm**(weight-one))
A[num_gotten, l] = ctmp2['data'][l] * tmp
# Add the (D,r,D',r') for each D',r' in the list
# print "ctmp1.keys()=",ctmp1.keys()
for l in range(num_gotten + 1):
#mm=RF(abs(ltmp1[l][2]))/4N
#tmp=RF(mm**(weight-one))
# print "scaled with=",tmp.n(200)
A[l, num_gotten] = ctmp1['data'][l] * tmp
#[d,B]=mat_inverse(A) # d=det(A)
#if(silent>1):
#d=det(A)
#print "det A = ",d
# Now we have to determine whether we have a linearly independent set or not
dold = mpmath.mp.dps
mpmath.mp.dps = int(prec / 3.3)
AInt = mpmath.matrix(int(A.nrows()), int(A.ncols()))
AMp = mpmath.matrix(int(A.nrows()), int(A.ncols()))
if (silent > 0):
print "tol0=", tol0
for ir in range(A.nrows()):
for ik in range(A.ncols()):
AInt[ir, ik] = mpmath.mp.mpi(A[ir, ik] - tol0,
A[ir, ik] + tol0)
AMp[ir, ik] = mpmath.mpf(A[ir, ik])
d = mpmath.det(AMp)
di = mpmath.mp.mpi(mpmath.mp.det(AInt))
#for ir in range(A.nrows()):
# for ik in range(A.ncols()):
# #print "A.d=",AInt[ir,ik].delta
if (silent > 0):
print "mpmath.mp.dps=", mpmath.mp.dps
print "det(A)=", d
print "det(A-as-interval)=", di
print "d.delta=", di.delta
#if(not mpmath.mpi(d) in di):
# raise ArithmeticError," Interval determinant not ok?"
#ANP=A.numpy()
#try:
# u,s,vnp=svd(ANP) # s are the singular values
# sl=s.tolist()
# mins=min(sl) # the smallest singular value
# maxs=max(sl)
# if(silent>1):
# print "singular values = ",s
#except LinAlgError:
# if(silent>0):
# print "could not compute SVD!"
# print "using abs(det) instead"
# mins=abs(d)
# maxs=abs(d)
#if((mins>sv_min and maxs< sv_max)):
zero = mpmath.mpi(0)
if (zero not in di):
if (silent > 1):
print "Adding D,r=", D, r
basislist[num_gotten] = [D, r]
num_gotten = num_gotten + 1
if (num_gotten >= dim):
return basislist
else:
#print "setting Aold to A"
Aold = A
else:
if (silent > 1):
print " do not use D,r=", D, r
# endif
mpmath.mp.dps = dold
# endfor
if (num_gotten < dim):
raise ValueError, " did not find enough good elements for a basis list!"
def gram_matrix(N,weight,prec=501,tol=1E-40,sv_min=1E-1,sv_max=1E15,bl=None,set_dim=None,force_prec=False):
r""" Computes a matrix of p_{r,D}(r',D')
for a basis of P_{r,D}, i.e. dim linearly independent P's
INPUT: N = Integer
weight = Real
OPTIONAL:
tol = error bound for the Poincaré series
sv_min = minimal allowed singular value when determining whether a given set is linarly independent or not.
sv_max = maximally allowed singular value
bl = list of pairs (D_i,r_i) from which we compute a matrix of coeffficients p_{D_i,r_i}(D_j,r_j)
"""
# If we have supplied a list of D's and r's we make a gram matrix relative to these
# otherwise we find a basis, i.e. linearly independent forms with correct dimension
# find the dimension
wt='%.4f'% weight
if(N<10):
stN="0"+str(N)
else:
stN=str(N)
v=dict()
filename_work="__N"+stN+"-"+wt+"--finding basis.txt"
fp=open(filename_work,"write")
fp.write("starting to find basis")
fp.close()
if(silent>0):
print "Forcing precision:",force_prec
set_silence_level(0)
if(bl<>None):
dim=len(bl)
l=bl
else:
if(set_dim<>None and set_dim >0):
dim=set_dim
else:
dim=dimension_jac_cusp_forms(int(weight+0.5),N,-1)
l=list_of_basis(N,weight,prec,tol,sv_min,sv_max,set_dim=dim)
j=0
for [D,r] in l.values():
for [Dp,rp] in l.values():
# Recall that the gram matrix is symmetric. We need only compute the upper diagonal
if(v.values().count([Dp,rp,D,r])==0):
v[j]=[D,r,Dp,rp]
j=j+1
# now v is a list we can get into computing coefficients
# first we print the "gram data" (list of indices) to the file
s=str(N)+": (AI["+str(N)+"],["
indices=dict()
for j in range(len(l)):
Delta=l[j][0]
r=l[j][1]
diff=(r*r-Delta) % (4*N)
if(diff<>0):
raise ValueError, "ERROR r^2=%s not congruent to Delta=%s mod %s!" %(r*r, Delta, 4*N)
s=s+"("+str(Delta)+","+str(r)+")"
indices[j]=[Delta,r]
if(j<len(l)-1):
s=s+","
else:
s=s+"]),"
s=s+"\n"
if(silent>0):
print s+"\n"
filename2="PS_Gramdata"+stN+"-"+wt+".txt"
fp=open(filename2,"write")
fp.write(s)
fp.close()
try:
os.remove(filename_work)
except os.error:
print "Could not remove file:",filename_work
pass
filename_work="__N"+stN+"-"+wt+"--computing_gram_matrix.txt"
fp=open(filename_work,"write")
fp.write("")
fp.close()
#print "tol=",tol
#set_silence_level(2)
#print "force_prec(gram_mat)=",force_prec
res=ps_coefficients_holomorphic_vec(N,weight,v,tol,prec,force_prec=force_prec)
set_silence_level(0)
res['indices']=indices
maxerr=0.0
for j in res['errs'].keys():
tmperr=abs(res['errs'][j])
#print "err(",j,")=",tmperr
if(tmperr>maxerr):
maxerr=tmperr
# switch format for easier vewing
res['errs'][j]=RR(tmperr)
if(silent>0):
print "maxerr=",RR(maxerr)
res['maxerr']=maxerr
wt_phalf='%.4f'% (weight+0.5)
filename3="PS_Gramerr"+stN+"-"+wt+".txt"
fp=open(filename3,"write")
wt
s="MAXERR["+wt_phalf+"]["+stN+"]="+str(RR(maxerr))
fp.write(s)
fp.close()
if(res['ok']):
Cps=res['data']
else:
print "Failed to compute Fourier coefficients!"
return 0
RF=RealField(prec)
A=matrix(RF,dim)
kappa=weight
fourpi=RF(4.0)*pi.n(prec)
one=RF(1.0)
N4=RF(4*N)
C=dict()
if(silent>1):
print "v=",v
print "dim=",dim
lastix=0
# First set the upper right part of A
for j in range(dim):
ddim=dim-j
if(silent>1):
print "j=",j,"ddim=",ddim," lastix=",lastix
for k in range(0,ddim):
# need to scale with |D|^(k+0.5)
if(silent>1):
print "k=",k
print "lastix+k=",lastix+k
mm=RF(abs(v[lastix+k][0]))/N4
tmp=RF(mm**(weight-one))
if(silent>1):
print "ddim+k=",ddim+k
A[j,j+k]=Cps[lastix+k]*tmp
C[v[lastix+k][0],v[lastix+k][1]]=Cps[lastix+k]
lastix=lastix+k+1
# And add the lower triangular part to mak the matrix symmetric
for j in range(dim):
for k in range(0,j):
A[j,k]=A[k,j]
# And print the gram matrix
res['matrix']=A
dold=mpmath.mp.dps
mpmath.mp.dps=int(prec/3.3)
AInt=mpmath.matrix(int(A.nrows()),int(A.ncols()))
AMp=mpmath.matrix(int(A.nrows()),int(A.ncols()))
for ir in range(A.nrows()):
for ik in range(A.ncols()):
AInt[ir,ik]=mpmath.mpi(A[ir,ik]-tol,A[ir,ik]+tol)
AMp[ir,ik]=mpmath.mpf(A[ir,ik])
d=mpmath.det(AMp)
if(silent>1):
print "det(A-as-mpmath)=",d
di=mpmath.det(AInt)
if(silent>1):
print "det(A-as-interval)=",di
res['det']=(RF(di.a),RF(di.b))
filename="PS_Gram"+stN+"-"+wt+".txt"
if(silent>1):
print "printing to file: "+filename
print_matrix_to_file(A,filename,'A['+str(N)+']')
if(silent>1):
print "A-A.transpose()=",norm(A-A.transpose())
B=A^-1
#[d,B]=mat_inverse(A)
if(silent>1):
print "A=",A.n(100)
print "det(A)=",di
print "Done making inverse!"
#res['det']=d
res['inv']=B
mpmath.mp.dps=dold
filename="PS_Gram-inv"+stN+"-"+wt+".txt"
print_matrix_to_file(B,filename,' AI['+str(N)+']')
# first make the filename
s='%.1e'%tol
filename3="PS_Coeffs"+stN+"-"+wt+"-"+s+".sobj"
# If the file already exist we load it and append the new data
if(silent>0):
print "saving data to ",filename3
try:
f=open(filename3,"read")
except IOError:
if(silent>0):
print "no file before!"
# do nothing
else:
if(silent>0):
print "file: "+filename3+" exists!"
f.close()
Cold=load(filename3)
for key in Cold.keys():
# print"key:",key
if(not C.has_key(key)): # then we addd it
print"key:",key," does not exist in the new version!"
C[key]=Cold[key]
save(C,filename3)
## Save the whole thing
filename="PS_all_gram"+stN+"-"+wt+".sobj"
save(res,filename)
## our work is comleted and we can remove the file
try:
os.remove(filename_work)
except os.error:
print "Could not remove file:",filename_work
pass
return res
Entries assumed to be integers.
"""
f = open(filename)
v = np.array(eval(f.readline()), dtype=int)
f.close()
ind = np.array(v, dtype=int)[:, :2]
val = np.array(v)[:, 2]
return ss.csc_matrix((val, (ind[:, 0], ind[:, 1])))
text = open('results/ldl_mpi.txt', 'w')
for number in range(1, 21):
positive = get_matrix('./results/CHOLMOD_permuted/permuted{}.txt'.format(number))
size = max(positive.indices)+1
print >>text, 'Domain {}:'.format(number)
print >>text, 'Number of entries: ', len(positive.data)
# change integers into mpi intervals
print >>text, 'Entries :', np.unique(positive.data)
positive.data = map(lambda x: mpi(x), positive.data)
positive = positive.todok()
sympy_positive = SparseMatrix(size, size, positive)
L, D = LDL(sympy_positive)
D = D._smat.values()
delta = [x.delta/x.mid for x in D]
print >>text, "The smallest diagonal element in LDL': ", min(D)
print >>text, 'Ratio largest/smallest : ', max(D)/min(D)
print >>text, "Maximal relative delta around diagonal elements: ", max(delta)
print >>text, '\n'
text.close()
def test_interval_to_mpi():
assert Interval(0, 1).to_mpi() == mpi(0, 1)
assert Interval(0, 1, True, False).to_mpi() == mpi(0, 1)
assert isinstance(Interval(0, 1).to_mpi(), type(mpi(0, 1)))
def gram_matrix(N,
weight,
prec=501,
tol=1E-40,
sv_min=1E-1,
sv_max=1E15,
bl=None,
set_dim=None,
force_prec=False):
r""" Computes a matrix of p_{r,D}(r',D')
for a basis of P_{r,D}, i.e. dim linearly independent P's
INPUT: N = Integer
weight = Real
OPTIONAL:
tol = error bound for the Poincaré series
sv_min = minimal allowed singular value when determining whether a given set is linarly independent or not.
sv_max = maximally allowed singular value
bl = list of pairs (D_i,r_i) from which we compute a matrix of coeffficients p_{D_i,r_i}(D_j,r_j)
"""
# If we have supplied a list of D's and r's we make a gram matrix relative to these
# otherwise we find a basis, i.e. linearly independent forms with correct dimension
# find the dimension
wt = '%.4f' % weight
if (N < 10):
stN = "0" + str(N)
else:
stN = str(N)
v = dict()
filename_work = "__N" + stN + "-" + wt + "--finding basis.txt"
fp = open(filename_work, "write")
fp.write("starting to find basis")
fp.close()
if (silent > 0):
print "Forcing precision:", force_prec
set_silence_level(0)
if (bl <> None):
dim = len(bl)
l = bl
else:
if (set_dim <> None and set_dim > 0):
dim = set_dim
else:
dim = dimension_jac_cusp_forms(int(weight + 0.5), N, -1)
l = list_of_basis(N, weight, prec, tol, sv_min, sv_max, set_dim=dim)
j = 0
for [D, r] in l.values():
for [Dp, rp] in l.values():
# Recall that the gram matrix is symmetric. We need only compute the upper diagonal
if (v.values().count([Dp, rp, D, r]) == 0):
v[j] = [D, r, Dp, rp]
j = j + 1
# now v is a list we can get into computing coefficients
# first we print the "gram data" (list of indices) to the file
s = str(N) + ": (AI[" + str(N) + "],["
indices = dict()
for j in range(len(l)):
Delta = l[j][0]
r = l[j][1]
diff = (r * r - Delta) % (4 * N)
if (diff <> 0):
raise ValueError, "ERROR r^2=%s not congruent to Delta=%s mod %s!" % (
r * r, Delta, 4 * N)
s = s + "(" + str(Delta) + "," + str(r) + ")"
indices[j] = [Delta, r]
if (j < len(l) - 1):
s = s + ","
else:
s = s + "]),"
s = s + "\n"
if (silent > 0):
print s + "\n"
filename2 = "PS_Gramdata" + stN + "-" + wt + ".txt"
fp = open(filename2, "write")
fp.write(s)
fp.close()
try:
os.remove(filename_work)
except os.error:
print "Could not remove file:", filename_work
pass
filename_work = "__N" + stN + "-" + wt + "--computing_gram_matrix.txt"
fp = open(filename_work, "write")
fp.write("")
fp.close()
#print "tol=",tol
#set_silence_level(2)
#print "force_prec(gram_mat)=",force_prec
res = ps_coefficients_holomorphic_vec(N,
weight,
v,
tol,
prec,
force_prec=force_prec)
set_silence_level(0)
res['indices'] = indices
maxerr = 0.0
for j in res['errs'].keys():
tmperr = abs(res['errs'][j])
#print "err(",j,")=",tmperr
if (tmperr > maxerr):
maxerr = tmperr
# switch format for easier vewing
res['errs'][j] = RR(tmperr)
if (silent > 0):
print "maxerr=", RR(maxerr)
res['maxerr'] = maxerr
wt_phalf = '%.4f' % (weight + 0.5)
filename3 = "PS_Gramerr" + stN + "-" + wt + ".txt"
fp = open(filename3, "write")
wt
s = "MAXERR[" + wt_phalf + "][" + stN + "]=" + str(RR(maxerr))
fp.write(s)
fp.close()
if (res['ok']):
Cps = res['data']
else:
print "Failed to compute Fourier coefficients!"
return 0
RF = RealField(prec)
A = matrix(RF, dim)
kappa = weight
fourpi = RF(4.0) * pi.n(prec)
one = RF(1.0)
N4 = RF(4 * N)
C = dict()
if (silent > 1):
print "v=", v
print "dim=", dim
lastix = 0
# First set the upper right part of A
for j in range(dim):
ddim = dim - j
if (silent > 1):
print "j=", j, "ddim=", ddim, " lastix=", lastix
for k in range(0, ddim):
# need to scale with |D|^(k+0.5)
if (silent > 1):
print "k=", k
print "lastix+k=", lastix + k
mm = RF(abs(v[lastix + k][0])) / N4
tmp = RF(mm**(weight - one))
if (silent > 1):
print "ddim+k=", ddim + k
A[j, j + k] = Cps[lastix + k] * tmp
C[v[lastix + k][0], v[lastix + k][1]] = Cps[lastix + k]
lastix = lastix + k + 1
# And add the lower triangular part to mak the matrix symmetric
for j in range(dim):
for k in range(0, j):
A[j, k] = A[k, j]
# And print the gram matrix
res['matrix'] = A
dold = mpmath.mp.dps
mpmath.mp.dps = int(prec / 3.3)
AInt = mpmath.matrix(int(A.nrows()), int(A.ncols()))
AMp = mpmath.matrix(int(A.nrows()), int(A.ncols()))
for ir in range(A.nrows()):
for ik in range(A.ncols()):
AInt[ir, ik] = mpmath.mpi(A[ir, ik] - tol, A[ir, ik] + tol)
AMp[ir, ik] = mpmath.mpf(A[ir, ik])
d = mpmath.det(AMp)
if (silent > 1):
print "det(A-as-mpmath)=", d
di = mpmath.det(AInt)
if (silent > 1):
print "det(A-as-interval)=", di
res['det'] = (RF(di.a), RF(di.b))
filename = "PS_Gram" + stN + "-" + wt + ".txt"
if (silent > 1):
print "printing to file: " + filename
print_matrix_to_file(A, filename, 'A[' + str(N) + ']')
if (silent > 1):
print "A-A.transpose()=", norm(A - A.transpose())
B = A ^ -1
#[d,B]=mat_inverse(A)
if (silent > 1):
print "A=", A.n(100)
print "det(A)=", di
print "Done making inverse!"
#res['det']=d
res['inv'] = B
mpmath.mp.dps = dold
filename = "PS_Gram-inv" + stN + "-" + wt + ".txt"
print_matrix_to_file(B, filename, ' AI[' + str(N) + ']')
# first make the filename
s = '%.1e' % tol
filename3 = "PS_Coeffs" + stN + "-" + wt + "-" + s + ".sobj"
# If the file already exist we load it and append the new data
if (silent > 0):
print "saving data to ", filename3
try:
f = open(filename3, "read")
except IOError:
if (silent > 0):
print "no file before!"
# do nothing
else:
if (silent > 0):
print "file: " + filename3 + " exists!"
f.close()
Cold = load(filename3)
for key in Cold.keys():
# print"key:",key
if (not C.has_key(key)): # then we addd it
print "key:", key, " does not exist in the new version!"
C[key] = Cold[key]
save(C, filename3)
## Save the whole thing
filename = "PS_all_gram" + stN + "-" + wt + ".sobj"
save(res, filename)
## our work is comleted and we can remove the file
try:
os.remove(filename_work)
except os.error:
print "Could not remove file:", filename_work
pass
return res
def mpi(self):
n = mpf(self.n)/mpf(self.d)
return mpi(n, n)
# coding: utf-8
# In[2]:
from mpmath import mpi
# In[3]:
a = mpi([0, 1])
# In[4]:
def f0(x):
return x**2 - x
# In[5]:
def f1(x):
return x * (x - 1)
# In[6]:
def f2(x):
return 1 / 4 - (x - 1 / 2)**2