def test_geometric(self):
G = (1 - x) * y - 1
S = newton_iteration(G, 9).truncate(x, 10)
z = var('z')
series = taylor(1 / (1 - z), z, 0, 9)
series = R(series.subs({z: x}))
self.assertEqual(S, series)
def test_geometric(self):
G = (1-x)*y - 1
S = newton_iteration(G,9).truncate(x,10)
z = var('z')
series = taylor(1/(1-z),z,0,9)
series = R(series.subs({z:x}))
self.assertEqual(S,series)
def test_cuberoot(self):
# recenter cuberoot(x) at x+1
G = (y + 1)**3 - (x + 1)
S = newton_iteration(G, 9).truncate(x, 10) + 1
z = var('z')
series = taylor(z**(QQ(1) / QQ(3)), z, 1, 9)
series = R(series.subs({z: x + 1}))
self.assertEqual(S, series)
def test_sqrt(self):
# recenter sqrt(x) at x+1
G = (y + 1)**2 - (x + 1)
S = newton_iteration(G, 9).truncate(x, 10) + 1
z = var('z')
series = taylor(sqrt(z), z, 1, 9)
series = R(series.subs({z: x + 1}))
self.assertEqual(S, series)
def test_cuberoot(self):
# recenter cuberoot(x) at x+1
G = (y+1)**3 - (x+1)
S = newton_iteration(G,9).truncate(x,10) + 1
z = var('z')
series = taylor(z**(QQ(1)/QQ(3)),z,1,9)
series = R(series.subs({z:x+1}))
self.assertEqual(S,series)
def test_sqrt(self):
# recenter sqrt(x) at x+1
G = (y+1)**2 - (x+1)
S = newton_iteration(G,9).truncate(x,10) + 1
z = var('z')
series = taylor(sqrt(z),z,1,9)
series = R(series.subs({z:x+1}))
self.assertEqual(S,series)
def _extend1(self):
"Increases the matrix size by 1"
#save current self
self.iv0 = copy(self)
self.iv0.abel_raw0 = lambda z: self.iv0.abel0poly(z-self.iv0.x0)
N = self.N
A = self.A
#Carleman matrix without 0-th row:
Ct = A + identity_matrix(self.R,N).submatrix(1,0,N-1,N-1)
AI = self.AI
#assert AI*self.A == identity_matrix(N-1)
if isinstance(self.f,FormalPowerSeries):
coeffs = [ self.f[n] for n in xrange(0,N) ]
else:
x = self.f.args()[0]
coeffs = taylor(self.f.substitute({x:x+self.x0sym})-self.x0sym,x,0,N).polynomial(self.R)
self.Ct = matrix(self.R,N,N)
self.Ct.set_block(0,0,Ct)
self.Ct[0,N-1] = coeffs[N-1]
for m in range(1,N-1):
self.Ct[m,N-1] = psmul_at(self.Ct[0],self.Ct[m-1],N-1)
self.Ct[N-1] = psmul(self.Ct[0],self.Ct[N-2])
print "C extended"
self.A = self.Ct - identity_matrix(self.R,N+1).submatrix(1,0,N,N)
av=self.A.column(N-1)[:N-1]
ah=self.A.row(N-1)[:N-1]
a_n=self.A[N-1,N-1]
# print 'A:';print A
# print 'A\''; print self.A
# print 'ah:',ah
# print 'av:',av
# print 'a_n:',a_n
AI0 = matrix(self.R,N,N)
AI0.set_block(0,0,self.AI)
horiz = matrix(self.R,1,N)
horiz.set_block(0,0,(ah*AI).transpose().transpose())
horiz[0,N-1] = -1
vert = matrix(self.R,N,1)
vert.set_block(0,0,(AI*av).transpose())
vert[N-1,0] = -1
self.N += 1
self.AI = AI0 + vert*horiz/(a_n-ah*AI*av)
def __init__(self,f,N,iprec=512,u=None,x0=0,fname=None,extendable=True):
"""
x0 is the development point for the Carleman matrix for the abel function
u is the initial value such that abel(u)=0 or equivalently super(0)=u
if no u is specified we have abel(x0)=0
iprec=None means try to work with the rational numbers
if it is extendable then you can increase N without recomputing everything.
"""
self.fps = None
if isinstance(f,sage.symbolic.expression.Expression):
x = f.variables()[0]
f = f.function(x)
if isinstance(f,FormalPowerSeries):
self.fps = f
f = f.polynomial(N-1)
self.f = f
self.N = N
self.iprec = iprec
x0sym = x0
self.x0sym = x0sym
self.prec = None
self.extendable = extendable
self.u = u
x0name = repr(x0sym)
if x0name.find('.') > -1:
if x0.is_real():
x0name = repr(float(x0sym)).strip('0').replace('.',',')
else:
x0name = repr(complex(x0sym)).strip('0').replace('.',',')
# by some reason save does not work with additional . inside the path
if iprec==None:
iprec_part = 'QQ'
else:
iprec_part = '%05d'%iprec
self.path = "savings/iabel_%s"%fname + "_N%04d"%N + "_iprec" + iprec_part + "_a%s"%x0name
if iprec != None:
x0 = num(x0sym,iprec)
if x0.is_real():
R = RealField(iprec)
else:
R = ComplexField(iprec)
else:
R = QQ
self.x0 = x0
self.R = R
#Carleman matrix
#too slow
#C = Matrix([ [ln(b)**n/factorial(n)*sum([binomial(m,k)*k**n*(b**x0)**k*(-x0)**(m-k) for k in range(m+1)]) for n in range(N)] for m in range(N)])
if self.fps == None:
x = self.f.args()[0]
coeffs = taylor(self.f.substitute({x:x+x0sym})-x0sym,x,0,N-1).polynomial(self.R)
coeffs = [coeffs[n] for n in xrange(N-1)]
else:
coeffs = [ self.fps[n] for n in xrange(0,N) ]
print "taylor computed"
C = self.fast_carleman_matrix(coeffs)
self.A = C.submatrix(1,0,N-1,N-1) - identity_matrix(R,N).submatrix(1,0,N-1,N-1)
print "A computed."
self._init_abel()
