def __init__(self,f,fi,exponent,fps=None,fixpoint=0,z0=0,u=None,prec=53,iprec=None,N=5,direction=-1,debug=0):
self.attracting = False
self.fp = fixpoint
self.prec = prec
self.f = f
self.fi = fi
self.N = N
if iprec == None:
iprec = prec + 10
self.iprec = iprec
R = RealField(iprec)
F = FormalPowerSeriesRing(R)
if fps == None:
p = F(f)
else:
p = F.by_formal_powerseries(fps)
pit = p.it(exponent)
self.iterate_poly = pit.polynomial(N)
self.iterate_raw0 = lambda z: self.fp + self.iterate_poly(z-self.fp)
def __init__(self,b,N,iprec=512,u=None,x0=0):
"""
x0 is the development point for the Carleman matrix for the slog
u is the initial value such that slog(u)=0 or equivalently sexp(0)=u
if no u is specified we have slog(x0)=0
"""
bsym = b
self.bsym = bsym
self.N = N
self.iprec = iprec
x0sym = x0
self.x0sym = x0sym
self.prec = None
bname = repr(bsym).strip('0').replace('.',',')
if bsym == sqrt(2):
bname = "sqrt2"
if bsym == e**(1/e):
bname = "eta"
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
self.path = "savings/itet_%s"%bname + "_N%04d"%N + "_iprec%05d"%iprec + "_a%s"%x0name
if iprec != None:
b = num(bsym,iprec)
self.b = b
x0 = num(x0sym,iprec)
if x0.is_real():
R = RealField(iprec)
else:
R = ComplexField(iprec)
self.x0 = x0
else:
if b == e and x0 == 0:
R = QQ
else:
R = SR
self.R = R
#Carleman matrix
if x0 == 0:
#C = Matrix([[ m**n*log(b)**n/factorial(n) for n in range(N)] for m in range(N)])
coeffs = [ln(b)**n/factorial(n) for n in xrange(N)]
else:
#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)])
coeffs = [b**x0-x0]+[b**x0*ln(b)**n/factorial(n) for n in xrange(1,N)]
def psmul(A,B):
N = len(B)
return [sum([A[k]*B[n-k] for k in xrange(n+1)]) for n in xrange(N)]
C = Matrix(R,N)
row = vector(R,[1]+(N-1)*[0])
C[0] = row
for m in xrange(1,N):
row = psmul(row,coeffs)
C[m] = row
A = (C - identity_matrix(N)).submatrix(1,0,N-1,N-1)
self.A = A
print "A computed."
if iprec != None:
A = num(A,iprec)
row = A.solve_left(vector([1] + (N-2)*[0]))
print "A solved."
self.slog0coeffs = [0]+[row[n] for n in range(N-1)]
self.slog0poly = PolynomialRing(R,'x')(self.slog0coeffs[:int(N)/2])
slog0ps = FormalPowerSeriesRing(R)(self.slog0coeffs)
sexp0ps = slog0ps.inv()
#print self.slog0ps | self.sexp0ps
self.sexp0coeffs = sexp0ps[:N]
self.sexp0poly = PolynomialRing(R,'x')(self.sexp0coeffs[:int(N)/2])
self.slog_raw0 = lambda z: self.slog0poly(z-self.x0)
print "slog reversed."
#the primary or the upper fixed point
pfp = exp_fixpoint(b,1,prec=iprec)
self.pfp = pfp
r = abs(x0-pfp)
#lower fixed point
lfp = None
if b <= R(e**(1/e)):
lfp = exp_fixpoint(b,0,prec=iprec)
r = min(r,abs(x0-lfp))
self.lfp = lfp
self.r = r
self.c = 0
if not u == None:
self.c = - self.slog(u)
