'''
Created on 28. aug. 2015
@author: pab
'''
from __future__ import division, print_function
import numpy as np
from scipy import linalg
from scipy.ndimage.filters import convolve1d
import warnings
EPS = np.finfo(float).eps
_EPS = EPS
_TINY = np.finfo(float).tiny
[docs]def convolve(sequence, rule, **kwds):
'''Wrapper around scipy.ndimage.convolve1d that allows complex input
'''
if np.iscomplexobj(sequence):
return (convolve1d(sequence.real, rule, **kwds) + 1j *
convolve1d(sequence.imag, rule, **kwds))
return convolve1d(sequence, rule, **kwds)
[docs]class Dea(object):
'''
LIMEXP is the maximum number of elements the
epsilon table data can contain. The epsilon table
is stored in the first (LIMEXP+2) entries of EPSTAB.
LIST OF MAJOR VARIABLES
-----------------------
E0,E1,E2,E3 - DOUBLE PRECISION
The 4 elements on which the computation of
a new element in the epsilon table is based.
NRES - INTEGER
Number of extrapolation results actually
generated by the epsilon algorithm in prior
calls to the routine.
NEWELM - INTEGER
Number of elements to be computed in the
new diagonal of the epsilon table. The
condensed epsilon table is computed. Only
those elements needed for the computation of
the next diagonal are preserved.
RES - DOUBLE PREISION
New element in the new diagonal of the
epsilon table.
ERROR - DOUBLE PRECISION
An estimate of the absolute error of RES.
Routine decides whether RESULT=RES or
RESULT=SVALUE by comparing ERROR with
ABSERR from the previous call.
RES3LA - DOUBLE PREISION
Vector of DIMENSION 3 containing at most
the last 3 results.
'''
def __init__(self, limexp=3):
self.limexp = 2 * (limexp // 2) + 1
self.epstab = np.zeros(limexp+5)
self.ABSERR = 10.
self._n = 0
self._nres = 0
if (limexp < 3):
raise ValueError('LIMEXP IS LESS THAN 3')
def _compute_error(self, RES3LA, NRES, RES):
fact = [6.0, 2.0, 1.0][min(NRES-1, 2)]
error = fact * np.abs(RES - RES3LA[:NRES]).sum()
return error
def _shift_table(self, EPSTAB, N, NEWELM, NUM):
i_0 = 1 if ((NUM // 2) * 2 == NUM - 1) else 0
i_n = 2 * NEWELM + 2
EPSTAB[i_0:i_n:2] = EPSTAB[i_0 + 2:i_n + 2:2]
if (NUM != N):
i_n = NUM - N
EPSTAB[:N + 1] = EPSTAB[i_n:i_n + N + 1]
return EPSTAB
def _update_RES3LA(self, RES3LA, RESULT, NRES):
if NRES > 2:
RES3LA[:2] = RES3LA[1:]
RES3LA[2] = RESULT
else:
RES3LA[NRES] = RESULT
def __call__(self, SVALUE):
EPSTAB = self.epstab
RES3LA = EPSTAB[-3:]
RESULT = SVALUE
N = self._n
NRES = self._nres
EPSTAB[N] = SVALUE
if (N == 0):
ABSERR = abs(RESULT)
elif (N == 1):
ABSERR = 6.0 * abs(RESULT - EPSTAB[0])
else:
ABSERR = self.ABSERR
EPSTAB[N + 2] = EPSTAB[N]
NEWELM = N // 2
NUM = N
K1 = N
for I in range(NEWELM):
E0 = EPSTAB[K1 - 2]
E1 = EPSTAB[K1 - 1]
E2 = RES = EPSTAB[K1 + 2]
DELTA2, DELTA3 = E2 - E1, E1 - E0
ERR2, ERR3 = abs(DELTA2), abs(DELTA3)
TOL2 = max(abs(E2), abs(E1)) * _EPS
TOL3 = max(abs(E1), abs(E0)) * _EPS
converged = (ERR2 <= TOL2 and ERR3 <= TOL3)
if converged:
ABSERR = ERR2 + ERR3
RESULT = RES
break
if (I != 0):
E3 = EPSTAB[K1]
DELTA1 = E1 - E3
ERR1 = abs(DELTA1)
TOL1 = max(abs(E1), abs(E3)) * _EPS
converged = (ERR1 <= TOL1 or ERR2 <= TOL2 or
ERR3 <= TOL3)
if not converged:
SS = 1.0 / DELTA1 + 1.0 / DELTA2 - 1.0 / DELTA3
else:
converged = (ERR2 <= TOL2 or ERR3 <= TOL3)
if not converged:
SS = 1.0 / DELTA2 - 1.0 / DELTA3
EPSTAB[K1] = E1
if (converged or abs(SS * E1) <= 1e-04):
N = 2 * I
if (NRES == 0):
ABSERR = ERR2 + ERR3
RESULT = RES
else:
RESULT = RES3LA[min(NRES-1, 2)]
break
RES = E1 + 1.0 / SS
EPSTAB[K1] = RES
K1 = K1 - 2
if (NRES == 0):
ABSERR = ERR2 + abs(RES - E2) + ERR3
RESULT = RES
continue
ERROR = self._compute_error(RES3LA, NRES, RES)
if (ERROR > 10.0 * ABSERR):
continue
ABSERR = ERROR
RESULT = RES
else:
ERROR = self._compute_error(RES3LA, NRES, RES)
# 50
if (N == self.limexp - 1):
N = 2 * (self.limexp // 2) - 1
EPSTAB = self._shift_table(EPSTAB, N, NEWELM, NUM)
self._update_RES3LA(RES3LA, RESULT, NRES)
ABSERR = max(ABSERR, 10.0*_EPS * abs(RESULT))
NRES = NRES + 1
N += 1
self._n = N
self._nres = NRES
# EPSTAB[-3:] = RES3LA
self.ABSERR = ABSERR
return RESULT, ABSERR
[docs]def test_dea():
def linfun(i):
return np.linspace(0, np.pi/2., 2**i+1)
dea = Dea(limexp=11)
print('NO. PANELS TRAP. APPROX APPROX W/EA ABSERR')
for k in np.arange(10):
x = linfun(k)
val = np.trapz(np.sin(x), x)
vale, err = dea(val)
print('%5d %20.8f %20.8f %20.8f' % (len(x)-1, val, vale, err))
[docs]def test_epsal():
HUGE = 1.E+60
TINY = 1.E-60
ZERO = 0.E0
ONE = 1.E0
true_vals = [0.78539816, 0.94805945, 0.99945672]
E = []
for N, SOFN in enumerate([0.78539816, 0.94805945, 0.98711580]):
E.append(SOFN)
if N == 0:
ESTLIM = SOFN
else:
AUX2 = ZERO
for J in range(N, 0, -1):
AUX1 = AUX2
AUX2 = E[J-1]
DIFF = E[J] - AUX2
if (abs(DIFF) <= TINY):
E[J-1] = HUGE
else:
E[J-1] = AUX1 + ONE/DIFF
if (N % 2) == 0:
ESTLIM = E[0]
else:
ESTLIM = E[1]
print(ESTLIM, true_vals[N])
[docs]def dea3(v0, v1, v2, symmetric=False):
"""
Extrapolate a slowly convergent sequence
Parameters
----------
v0, v1, v2 : array-like
3 values of a convergent sequence to extrapolate
Returns
-------
result : array-like
extrapolated value
abserr : array-like
absolute error estimate
Description
-----------
DEA3 attempts to extrapolate nonlinearly to a better estimate
of the sequence's limiting value, thus improving the rate of
convergence. The routine is based on the epsilon algorithm of
P. Wynn, see [1]_.
Example
-------
# integrate sin(x) from 0 to pi/2
>>> import numpy as np
>>> import numdifftools as nd
>>> Ei= np.zeros(3)
>>> linfun = lambda i : np.linspace(0, np.pi/2., 2**(i+5)+1)
>>> for k in np.arange(3):
... x = linfun(k)
... Ei[k] = np.trapz(np.sin(x),x)
>>> [En, err] = nd.dea3(Ei[0], Ei[1], Ei[2])
>>> truErr = Ei-1.
>>> (truErr, err, En)
(array([ -2.00805680e-04, -5.01999079e-05, -1.25498825e-05]),
array([ 0.00020081]), array([ 1.]))
See also
--------
dea
Reference
---------
.. [1] C. Brezinski (1977)
"Acceleration de la convergence en analyse numerique",
"Lecture Notes in Math.", vol. 584,
Springer-Verlag, New York, 1977.
"""
E0, E1, E2 = np.atleast_1d(v0, v1, v2)
abs, max = np.abs, np.maximum # @ReservedAssignment
with warnings.catch_warnings():
warnings.simplefilter("ignore") # ignore division by zero and overflow
delta2, delta1 = E2 - E1, E1 - E0
err2, err1 = abs(delta2), abs(delta1)
tol2, tol1 = max(abs(E2), abs(E1)) * _EPS, max(abs(E1), abs(E0)) * _EPS
delta1[err1 < _TINY] = _TINY
delta2[err2 < _TINY] = _TINY # avoid division by zero and overflow
ss = 1.0 / delta2 - 1.0 / delta1 + _TINY
smalle2 = (abs(ss * E1) <= 1.0e-3)
converged = (err1 <= tol1) & (err2 <= tol2) | smalle2
result = np.where(converged, E2 * 1.0, E1 + 1.0 / ss)
abserr = err1 + err2 + np.where(converged, tol2 * 10, abs(result-E2))
if symmetric and len(result) > 1:
return result[:-1], abserr[1:]
return result, abserr
# class Richardson(object):
# '''
# Extrapolates as sequence with Richardsons method
#
# Notes
# -----
# Suppose you have series expansion that goes like this
#
# L = f(h) + a0 * h^p_0 + a1 * h^p_1+ a2 * h^p_2 + ...
#
# where p_i = order + step * i and f(h) -> L as h -> 0, but f(0) != L.
#
# If we evaluate the right hand side for different stepsizes h
# we can fit a polynomial to that sequence of approximations.
# This is exactly what this class does.
#
# Example
# -------
# >>> import numpy as np
# >>> import numdifftools as nd
# >>> n = 3
# >>> Ei = np.zeros((n,1))
# >>> h = np.zeros((n,1))
# >>> linfun = lambda i : np.linspace(0, np.pi/2., 2**(i+5)+1)
# >>> for k in np.arange(n):
# ... x = linfun(k)
# ... h[k] = x[1]
# ... Ei[k] = np.trapz(np.sin(x),x)
# >>> En, err, step = nd.Richardson(step=1, order=1)(Ei, h)
# >>> truErr = Ei-1.
# >>> (truErr, err, En)
# (array([[ -2.00805680e-04],
# [ -5.01999079e-05],
# [ -1.25498825e-05]]), array([[ 0.00320501]]), array([[ 1.]]))
#
# '''
# def __init__(self, step_ratio=2.0, step=1, order=1, num_terms=2):
# self.num_terms = num_terms
# self.order = order
# self.step = step
# self.step_ratio = step_ratio
#
# def _r_matrix(self, num_terms):
# step = self.step
# i, j = np.ogrid[0:num_terms+1, 0:num_terms]
# r_mat = np.ones((num_terms + 1, num_terms + 1))
# r_mat[:, 1:] = (1.0 / self.step_ratio) ** (i*(step*j + self.order))
# return r_mat
#
# def _get_richardson_rule(self, sequence_length=None):
# if sequence_length is None:
# sequence_length = self.num_terms + 1
# num_terms = min(self.num_terms, sequence_length - 1)
# if num_terms > 0:
# r_mat = self._r_matrix(num_terms)
# return linalg.pinv(r_mat)[0]
# return np.ones((1,))
#
# def _estimate_error(self, new_sequence, old_sequence, steps, rule):
# m, _n = new_sequence.shape
#
# if m < 2:
# return (np.abs(new_sequence) * EPS + steps) * 10.0
# cov1 = np.sum(rule**2) # 1 spare dof
# fact = np.maximum(12.7062047361747 * np.sqrt(cov1), EPS * 10.)
# err = np.abs(np.diff(new_sequence, axis=0)) * fact
# tol = np.maximum(np.abs(new_sequence[1:]),
# np.abs(new_sequence[:-1])) * EPS * fact
# converged = err <= tol
# abserr = err + np.where(converged, tol * 10,
# abs(new_sequence[:-1]-old_sequence[1:])*fact)
# # abserr = err1 + err2 + np.where(converged, tol2 * 10, abs(result-E2))
# # abserr = s * fact + np.abs(new_sequence) * EPS * 10.0
# return abserr
#
# def extrapolate(self, sequence, steps):
# return self.__call__(sequence, steps)
#
# def __call__(self, sequence, steps):
# ne = sequence.shape[0]
# rule = self._get_richardson_rule(ne)
# nr = rule.size - 1
# m = ne - nr
# new_sequence = convolve1d(sequence, rule[::-1], axis=0,
# origin=(nr//2))
# if np.any(sequence.imag()):
# new_sequence = new_sequence + 1j*convolve1d(np.imag(sequence),
# rule[::-1], axis=0,
# origin=(nr//2))
# abserr = self._estimate_error(new_sequence, sequence, steps, rule)
# return new_sequence[:m], abserr[:m], steps[:m]
[docs]class Richardson(object):
'''
Extrapolates as sequence with Richardsons method
Notes
-----
Suppose you have series expansion that goes like this
L = f(h) + a0 * h^p_0 + a1 * h^p_1+ a2 * h^p_2 + ...
where p_i = order + step * i and f(h) -> L as h -> 0, but f(0) != L.
If we evaluate the right hand side for different stepsizes h
we can fit a polynomial to that sequence of approximations.
This is exactly what this class does.
Example
-------
>>> import numpy as np
>>> import numdifftools as nd
>>> n = 3
>>> Ei = np.zeros((n,1))
>>> h = np.zeros((n,1))
>>> linfun = lambda i : np.linspace(0, np.pi/2., 2**(i+5)+1)
>>> for k in np.arange(n):
... x = linfun(k)
... h[k] = x[1]
... Ei[k] = np.trapz(np.sin(x),x)
>>> En, err, step = nd.Richardson(step=1, order=1)(Ei, h)
>>> truErr = Ei-1.
>>> (truErr, err, En)
(array([[ -2.00805680e-04],
[ -5.01999079e-05],
[ -1.25498825e-05]]), array([[ 0.00320501]]), array([[ 1.]]))
'''
def __init__(self, step_ratio=2.0, step=1, order=1, num_terms=2):
self.num_terms = num_terms
self.order = order
self.step = step
self.step_ratio = step_ratio
def _r_matrix(self, num_terms):
step = self.step
i, j = np.ogrid[0:num_terms+1, 0:num_terms]
r_mat = np.ones((num_terms + 1, num_terms + 1))
r_mat[:, 1:] = (1.0 / self.step_ratio) ** (i*(step*j + self.order))
return r_mat
def _get_richardson_rule(self, sequence_length=None):
if sequence_length is None:
sequence_length = self.num_terms + 1
num_terms = min(self.num_terms, sequence_length - 1)
if num_terms > 0:
r_mat = self._r_matrix(num_terms)
return linalg.pinv(r_mat)[0]
return np.ones((1,))
def _estimate_error(self, new_sequence, old_sequence, steps, rule):
m, _n = new_sequence.shape
if m < 2:
return (np.abs(new_sequence) * EPS + steps) * 10.0
cov1 = np.sum(rule**2) # 1 spare dof
fact = np.maximum(12.7062047361747 * np.sqrt(cov1), EPS * 10.)
err = np.abs(np.diff(new_sequence, axis=0)) * fact
tol = np.maximum(np.abs(new_sequence[1:]),
np.abs(new_sequence[:-1])) * EPS * fact
converged = err <= tol
abserr = err + np.where(converged, tol * 10,
abs(new_sequence[:-1]-old_sequence[1:])*fact)
# abserr = err1 + err2 + np.where(converged, tol2 * 10, abs(result-E2))
# abserr = s * fact + np.abs(new_sequence) * EPS * 10.0
return abserr
def __call__(self, sequence, steps):
ne = sequence.shape[0]
rule = self._get_richardson_rule(ne)
nr = rule.size - 1
m = ne - nr
new_sequence = convolve(sequence, rule[::-1], axis=0, origin=(nr // 2))
abserr = self._estimate_error(new_sequence, sequence, steps, rule)
return new_sequence[:m], abserr[:m], steps[:m]
if __name__ == '__main__':
pass