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#!/usr/bin/env python
"""
fitpack (dierckx in netlib) --- A Python-C wrapper to FITPACK (by P. Dierckx).
FITPACK is a collection of FORTRAN programs for curve and surface
fitting with splines and tensor product splines.
See
http://www.cs.kuleuven.ac.be/cwis/research/nalag/research/topics/fitpack.html
or
http://www.netlib.org/dierckx/index.html
Copyright 2002 Pearu Peterson all rights reserved,
Pearu Peterson <pearu@cens.ioc.ee>
Permission to use, modify, and distribute this software is given under the
terms of the SciPy (BSD style) license. See LICENSE.txt that came with
this distribution for specifics.
NO WARRANTY IS EXPRESSED OR IMPLIED. USE AT YOUR OWN RISK.
TODO: Make interfaces to the following fitpack functions:
For univariate splines: cocosp, concon, fourco, insert
For bivariate splines: profil, regrid, parsur, surev
"""
__all__ = ['splrep', 'splprep', 'splev', 'splint', 'sproot', 'spalde',
'bisplrep', 'bisplev', 'insert']
__version__ = "$Revision$"[10:-1]
import _fitpack
from numpy import atleast_1d, array, ones, zeros, sqrt, ravel, transpose, \
dot, sin, cos, pi, arange, empty, int32, asarray
myasarray = atleast_1d
# Try to replace _fitpack interface with
# f2py-generated version
import dfitpack
_iermess = {0:["""\
The spline has a residual sum of squares fp such that abs(fp-s)/s<=0.001""",None],
-1:["""\
The spline is an interpolating spline (fp=0)""",None],
-2:["""\
The spline is weighted least-squares polynomial of degree k.
fp gives the upper bound fp0 for the smoothing factor s""",None],
1:["""\
The required storage space exceeds the available storage space.
Probable causes: data (x,y) size is too small or smoothing parameter s is too small (fp>s).""",ValueError],
2:["""\
A theoretically impossible results when finding a smoothin spline
with fp = s. Probably causes: s too small. (abs(fp-s)/s>0.001)""",ValueError],
3:["""\
The maximal number of iterations (20) allowed for finding smoothing
spline with fp=s has been reached. Probably causes: s too small.
(abs(fp-s)/s>0.001)""",ValueError],
10:["""\
Error on input data""",ValueError],
'unknown':["""\
An error occurred""",TypeError]}
_iermess2 = {0:["""\
The spline has a residual sum of squares fp such that abs(fp-s)/s<=0.001""",None],
-1:["""\
The spline is an interpolating spline (fp=0)""",None],
-2:["""\
The spline is weighted least-squares polynomial of degree kx and ky.
fp gives the upper bound fp0 for the smoothing factor s""",None],
-3:["""\
Warning. The coefficients of the spline have been computed as the minimal
norm least-squares solution of a rank deficient system.""",None],
1:["""\
The required storage space exceeds the available storage space.
Probably causes: nxest or nyest too small or s is too small. (fp>s)""",ValueError],
2:["""\
A theoretically impossible results when finding a smoothin spline
with fp = s. Probably causes: s too small or badly chosen eps.
(abs(fp-s)/s>0.001)""",ValueError],
3:["""\
The maximal number of iterations (20) allowed for finding smoothing
spline with fp=s has been reached. Probably causes: s too small.
(abs(fp-s)/s>0.001)""",ValueError],
4:["""\
No more knots can be added because the number of B-spline coefficients
already exceeds the number of data points m. Probably causes: either
s or m too small. (fp>s)""",ValueError],
5:["""\
No more knots can be added because the additional knot would coincide
with an old one. Probably cause: s too small or too large a weight
to an inaccurate data point. (fp>s)""",ValueError],
10:["""\
Error on input data""",ValueError],
11:["""\
rwrk2 too small, i.e. there is not enough workspace for computing
the minimal least-squares solution of a rank deficient system of linear
equations.""",ValueError],
'unknown':["""\
An error occurred""",TypeError]}
_parcur_cache = {'t': array([],float), 'wrk': array([],float),
'iwrk':array([],int32), 'u': array([],float),'ub':0,'ue':1}
def splprep(x,w=None,u=None,ub=None,ue=None,k=3,task=0,s=None,t=None,
full_output=0,nest=None,per=0,quiet=1):
"""
Find the B-spline representation of an N-dimensional curve.
Given a list of N rank-1 arrays, x, which represent a curve in
N-dimensional space parametrized by u, find a smooth approximating
spline curve g(u). Uses the FORTRAN routine parcur from FITPACK.
Parameters
----------
x : array_like
A list of sample vector arrays representing the curve.
w : array_like
Strictly positive rank-1 array of weights the same length as x[0].
The weights are used in computing the weighted least-squares spline
fit. If the errors in the x values have standard-deviation given by the
vector d, then w should be 1/d. Default is ones(len(x[0])).
u : array_like, optional
An array of parameter values. If not given, these values are
calculated automatically as ``M = len(x[0])``::
v[0] = 0
v[i] = v[i-1] + distance(x[i],x[i-1])
u[i] = v[i] / v[M-1]
ub, ue : int, optional
The end-points of the parameters interval. Defaults to
u[0] and u[-1].
k : int, optional
Degree of the spline. Cubic splines are recommended.
Even values of `k` should be avoided especially with a small s-value.
``1 <= k <= 5``, default is 3.
task : int, optional
If task==0 (default), find t and c for a given smoothing factor, s.
If task==1, find t and c for another value of the smoothing factor, s.
There must have been a previous call with task=0 or task=1
for the same set of data.
If task=-1 find the weighted least square spline for a given set of
knots, t.
s : float, optional
A smoothing condition.
The amount of smoothness is determined by
satisfying the conditions: ``sum((w * (y - g))**2,axis=0) <= s``,
where g(x) is the smoothed interpolation of (x,y). The user can
use `s` to control the trade-off between closeness and smoothness
of fit. Larger `s` means more smoothing while smaller values of `s`
indicate less smoothing. Recommended values of `s` depend on the
weights, w. If the weights represent the inverse of the
standard-deviation of y, then a good `s` value should be found in
the range ``(m-sqrt(2*m),m+sqrt(2*m))``, where m is the number of
data points in x, y, and w.
t : int, optional
The knots needed for task=-1.
full_output : int, optional
If non-zero, then return optional outputs.
nest : int, optional
An over-estimate of the total number of knots of the spline to
help in determining the storage space. By default nest=m/2.
Always large enough is nest=m+k+1.
per : int, optional
If non-zero, data points are considered periodic with period
``x[m-1] - x[0]`` and a smooth periodic spline approximation is
returned. Values of ``y[m-1]`` and ``w[m-1]`` are not used.
quiet : int, optional
Non-zero to suppress messages.
Returns
-------
tck : tuple
A tuple (t,c,k) containing the vector of knots, the B-spline
coefficients, and the degree of the spline.
u : array
An array of the values of the parameter.
fp : float
The weighted sum of squared residuals of the spline approximation.
ier : int
An integer flag about splrep success. Success is indicated
if ier<=0. If ier in [1,2,3] an error occurred but was not raised.
Otherwise an error is raised.
msg : str
A message corresponding to the integer flag, ier.
See Also
--------
splrep, splev, sproot, spalde, splint,
bisplrep, bisplev
UnivariateSpline, BivariateSpline
Notes
-----
See `splev` for evaluation of the spline and its derivatives.
References
----------
.. [1] P. Dierckx, "Algorithms for smoothing data with periodic and
parametric splines, Computer Graphics and Image Processing",
20 (1982) 171-184.
.. [2] P. Dierckx, "Algorithms for smoothing data with periodic and
parametric splines", report tw55, Dept. Computer Science,
K.U.Leuven, 1981.
.. [3] P. Dierckx, "Curve and surface fitting with splines", Monographs on
Numerical Analysis, Oxford University Press, 1993.
"""
if task<=0:
_parcur_cache = {'t': array([],float), 'wrk': array([],float),
'iwrk':array([],int32),'u': array([],float),
'ub':0,'ue':1}
x=myasarray(x)
idim,m=x.shape
if per:
for i in range(idim):
if x[i][0]!=x[i][-1]:
if quiet<2:print 'Warning: Setting x[%d][%d]=x[%d][0]'%(i,m,i)
x[i][-1]=x[i][0]
if not 0 < idim < 11:
raise TypeError('0 < idim < 11 must hold')
if w is None:
w = ones(m, float)
else:
w = myasarray(w)
ipar = (u is not None)
if ipar:
_parcur_cache['u']=u
if ub is None: _parcur_cache['ub']=u[0]
else: _parcur_cache['ub']=ub
if ue is None: _parcur_cache['ue']=u[-1]
else: _parcur_cache['ue']=ue
else: _parcur_cache['u']=zeros(m,float)
if not (1 <= k <= 5):
raise TypeError('1 <= k= %d <=5 must hold' % k)
if not (-1 <= task <=1):
raise TypeError('task must be -1, 0 or 1')
if (not len(w)==m) or (ipar==1 and (not len(u)==m)):
raise TypeError('Mismatch of input dimensions')
if s is None: s=m-sqrt(2*m)
if t is None and task == -1:
raise TypeError('Knots must be given for task=-1')
if t is not None:
_parcur_cache['t']=myasarray(t)
n=len(_parcur_cache['t'])
if task==-1 and n<2*k+2:
raise TypeError('There must be at least 2*k+2 knots for task=-1')
if m <= k:
raise TypeError('m > k must hold')
if nest is None: nest=m+2*k
if (task>=0 and s==0) or (nest<0):
if per: nest=m+2*k
else: nest=m+k+1
nest=max(nest,2*k+3)
u=_parcur_cache['u']
ub=_parcur_cache['ub']
ue=_parcur_cache['ue']
t=_parcur_cache['t']
wrk=_parcur_cache['wrk']
iwrk=_parcur_cache['iwrk']
t,c,o=_fitpack._parcur(ravel(transpose(x)),w,u,ub,ue,k,task,ipar,s,t,
nest,wrk,iwrk,per)
_parcur_cache['u']=o['u']
_parcur_cache['ub']=o['ub']
_parcur_cache['ue']=o['ue']
_parcur_cache['t']=t
_parcur_cache['wrk']=o['wrk']
_parcur_cache['iwrk']=o['iwrk']
ier,fp,n=o['ier'],o['fp'],len(t)
u=o['u']
c.shape=idim,n-k-1
tcku = [t,list(c),k],u
if ier<=0 and not quiet:
print _iermess[ier][0]
print "\tk=%d n=%d m=%d fp=%f s=%f"%(k,len(t),m,fp,s)
if ier>0 and not full_output:
if ier in [1,2,3]:
print "Warning: "+_iermess[ier][0]
else:
try:
raise _iermess[ier][1](_iermess[ier][0])
except KeyError:
raise _iermess['unknown'][1](_iermess['unknown'][0])
if full_output:
try:
return tcku,fp,ier,_iermess[ier][0]
except KeyError:
return tcku,fp,ier,_iermess['unknown'][0]
else:
return tcku
_curfit_cache = {'t': array([],float), 'wrk': array([],float),
'iwrk':array([],int32)}
def splrep(x,y,w=None,xb=None,xe=None,k=3,task=0,s=None,t=None,
full_output=0,per=0,quiet=1):
"""
Find the B-spline representation of 1-D curve.
Given the set of data points (x[i], y[i]) determine a smooth spline
approximation of degree k on the interval xb <= x <= xe. The coefficients,
c, and the knot points, t, are returned. Uses the FORTRAN routine
curfit from FITPACK.
Parameters
----------
x, y : array_like
The data points defining a curve y = f(x).
w : array_like
Strictly positive rank-1 array of weights the same length as x and y.
The weights are used in computing the weighted least-squares spline
fit. If the errors in the y values have standard-deviation given by the
vector d, then w should be 1/d. Default is ones(len(x)).
xb, xe : float
The interval to fit. If None, these default to x[0] and x[-1]
respectively.
k : int
The order of the spline fit. It is recommended to use cubic splines.
Even order splines should be avoided especially with small s values.
1 <= k <= 5
task : {1, 0, -1}
If task==0 find t and c for a given smoothing factor, s.
If task==1 find t and c for another value of the smoothing factor, s.
There must have been a previous call with task=0 or task=1 for the same
set of data (t will be stored an used internally)
If task=-1 find the weighted least square spline for a given set of
knots, t. These should be interior knots as knots on the ends will be
added automatically.
s : float
A smoothing condition. The amount of smoothness is determined by
satisfying the conditions: sum((w * (y - g))**2,axis=0) <= s where g(x)
is the smoothed interpolation of (x,y). The user can use s to control
the tradeoff between closeness and smoothness of fit. Larger s means
more smoothing while smaller values of s indicate less smoothing.
Recommended values of s depend on the weights, w. If the weights
represent the inverse of the standard-deviation of y, then a good s
value should be found in the range (m-sqrt(2*m),m+sqrt(2*m)) where m is
the number of datapoints in x, y, and w. default : s=m-sqrt(2*m) if
weights are supplied. s = 0.0 (interpolating) if no weights are
supplied.
t : int
The knots needed for task=-1. If given then task is automatically set
to -1.
full_output : bool
If non-zero, then return optional outputs.
per : bool
If non-zero, data points are considered periodic with period x[m-1] -
x[0] and a smooth periodic spline approximation is returned. Values of
y[m-1] and w[m-1] are not used.
quiet : bool
Non-zero to suppress messages.
Returns
-------
tck : tuple
(t,c,k) a tuple containing the vector of knots, the B-spline
coefficients, and the degree of the spline.
fp : array, optional
The weighted sum of squared residuals of the spline approximation.
ier : int, optional
An integer flag about splrep success. Success is indicated if ier<=0.
If ier in [1,2,3] an error occurred but was not raised. Otherwise an
error is raised.
msg : str, optional
A message corresponding to the integer flag, ier.
Notes
-----
See splev for evaluation of the spline and its derivatives.
See Also
--------
UnivariateSpline, BivariateSpline
splprep, splev, sproot, spalde, splint
bisplrep, bisplev
References
----------
Based on algorithms described in [1], [2], [3], and [4]:
.. [1] P. Dierckx, "An algorithm for smoothing, differentiation and
integration of experimental data using spline functions",
J.Comp.Appl.Maths 1 (1975) 165-184.
.. [2] P. Dierckx, "A fast algorithm for smoothing data on a rectangular
grid while using spline functions", SIAM J.Numer.Anal. 19 (1982)
1286-1304.
.. [3] P. Dierckx, "An improved algorithm for curve fitting with spline
functions", report tw54, Dept. Computer Science,K.U. Leuven, 1981.
.. [4] P. Dierckx, "Curve and surface fitting with splines", Monographs on
Numerical Analysis, Oxford University Press, 1993.
Examples
--------
>>> x = linspace(0, 10, 10)
>>> y = sin(x)
>>> tck = splrep(x, y)
>>> x2 = linspace(0, 10, 200)
>>> y2 = splev(x2, tck)
>>> plot(x, y, 'o', x2, y2)
"""
if task<=0:
_curfit_cache = {}
x,y=map(myasarray,[x,y])
m=len(x)
if w is None:
w=ones(m,float)
if s is None: s = 0.0
else:
w=myasarray(w)
if s is None: s = m-sqrt(2*m)
if not len(w) == m:
raise TypeError('len(w)=%d is not equal to m=%d' % (len(w),m))
if (m != len(y)) or (m != len(w)):
raise TypeError('Lengths of the first three arguments (x,y,w) must be equal')
if not (1 <= k <= 5):
raise TypeError('Given degree of the spline (k=%d) is not supported. (1<=k<=5)' % k)
if m <= k:
raise TypeError('m > k must hold')
if xb is None: xb=x[0]
if xe is None: xe=x[-1]
if not (-1 <= task <= 1):
raise TypeError('task must be -1, 0 or 1')
if t is not None:
task = -1
if task == -1:
if t is None:
raise TypeError('Knots must be given for task=-1')
numknots = len(t)
_curfit_cache['t'] = empty((numknots + 2*k+2,),float)
_curfit_cache['t'][k+1:-k-1] = t
nest = len(_curfit_cache['t'])
elif task == 0:
if per:
nest = max(m+2*k,2*k+3)
else:
nest = max(m+k+1,2*k+3)
t = empty((nest,),float)
_curfit_cache['t'] = t
if task <= 0:
if per: _curfit_cache['wrk'] = empty((m*(k+1)+nest*(8+5*k),),float)
else: _curfit_cache['wrk'] = empty((m*(k+1)+nest*(7+3*k),),float)
_curfit_cache['iwrk'] = empty((nest,),int32)
try:
t=_curfit_cache['t']
wrk=_curfit_cache['wrk']
iwrk=_curfit_cache['iwrk']
except KeyError:
raise TypeError("must call with task=1 only after"
" call with task=0,-1")
if not per:
n,c,fp,ier = dfitpack.curfit(task, x, y, w, t, wrk, iwrk, xb, xe, k, s)
else:
n,c,fp,ier = dfitpack.percur(task, x, y, w, t, wrk, iwrk, k, s)
tck = (t[:n],c[:n],k)
if ier<=0 and not quiet:
print _iermess[ier][0]
print "\tk=%d n=%d m=%d fp=%f s=%f"%(k,len(t),m,fp,s)
if ier>0 and not full_output:
if ier in [1,2,3]:
print "Warning: "+_iermess[ier][0]
else:
try:
raise _iermess[ier][1](_iermess[ier][0])
except KeyError:
raise _iermess['unknown'][1](_iermess['unknown'][0])
if full_output:
try:
return tck,fp,ier,_iermess[ier][0]
except KeyError:
return tck,fp,ier,_iermess['unknown'][0]
else:
return tck
def _ntlist(l): # return non-trivial list
return l
#if len(l)>1: return l
#return l[0]
def splev(x, tck, der=0, ext=0):
"""
Evaluate a B-spline or its derivatives.
Given the knots and coefficients of a B-spline representation, evaluate
the value of the smoothing polynomial and its derivatives. This is a
wrapper around the FORTRAN routines splev and splder of FITPACK.
Parameters
----------
x : array_like
A 1-D array of points at which to return the value of the smoothed
spline or its derivatives. If `tck` was returned from `splprep`,
then the parameter values, u should be given.
tck : tuple
A sequence of length 3 returned by `splrep` or `splprep` containing
the knots, coefficients, and degree of the spline.
der : int
The order of derivative of the spline to compute (must be less than
or equal to k).
ext : int
Controls the value returned for elements of ``x`` not in the
interval defined by the knot sequence.
* if ext=0, return the extrapolated value.
* if ext=1, return 0
* if ext=2, raise a ValueError
The default value is 0.
Returns
-------
y : ndarray or list of ndarrays
An array of values representing the spline function evaluated at
the points in ``x``. If `tck` was returned from splrep, then this
is a list of arrays representing the curve in N-dimensional space.
See Also
--------
splprep, splrep, sproot, spalde, splint
bisplrep, bisplev
References
----------
.. [1] C. de Boor, "On calculating with b-splines", J. Approximation
Theory, 6, p.50-62, 1972.
.. [2] M.G. Cox, "The numerical evaluation of b-splines", J. Inst. Maths
Applics, 10, p.134-149, 1972.
.. [3] P. Dierckx, "Curve and surface fitting with splines", Monographs
on Numerical Analysis, Oxford University Press, 1993.
"""
t,c,k = tck
try:
c[0][0]
parametric = True
except:
parametric = False
if parametric:
return map(lambda c, x=x, t=t, k=k, der=der : splev(x, [t,c,k], der, ext), c)
else:
if not (0 <= der <= k):
raise ValueError("0<=der=%d<=k=%d must hold"%(der,k))
if not ext in (0,1,2):
raise ValueError("ext not in (0, 1, 2)")
x = asarray(x)
shape = x.shape
x = atleast_1d(x)
y, ier =_fitpack._spl_(x, der, t, c, k, ext)
if ier == 10:
raise ValueError("Invalid input data")
if ier == 1:
raise ValueError("Found x value not in the domain")
if ier:
raise TypeError("An error occurred")
return y.reshape(shape)
def splint(a,b,tck,full_output=0):
"""
Evaluate the definite integral of a B-spline.
Given the knots and coefficients of a B-spline, evaluate the definite
integral of the smoothing polynomial between two given points.
Parameters
----------
a, b : float
The end-points of the integration interval.
tck : tuple
A tuple (t,c,k) containing the vector of knots, the B-spline
coefficients, and the degree of the spline (see `splev`).
full_output : int, optional
Non-zero to return optional output.
Returns
-------
integral : float
The resulting integral.
wrk : ndarray
An array containing the integrals of the normalized B-splines
defined on the set of knots.
See Also
--------
splprep, splrep, sproot, spalde, splev
bisplrep, bisplev
UnivariateSpline, BivariateSpline
References
----------
.. [1] P.W. Gaffney, The calculation of indefinite integrals of b-splines",
J. Inst. Maths Applics, 17, p.37-41, 1976.
.. [2] P. Dierckx, "Curve and surface fitting with splines", Monographs
on Numerical Analysis, Oxford University Press, 1993.
"""
t,c,k=tck
try:
c[0][0]
parametric = True
except:
parametric = False
if parametric:
return _ntlist(map(lambda c,a=a,b=b,t=t,k=k:splint(a,b,[t,c,k]),c))
else:
aint,wrk=_fitpack._splint(t,c,k,a,b)
if full_output: return aint,wrk
else: return aint
def sproot(tck,mest=10):
"""
Find the roots of a cubic B-spline.
Given the knots (>=8) and coefficients of a cubic B-spline return the
roots of the spline.
Parameters
----------
tck : tuple
A tuple (t,c,k) containing the vector of knots,
the B-spline coefficients, and the degree of the spline.
The number of knots must be >= 8.
The knots must be a montonically increasing sequence.
mest : int
An estimate of the number of zeros (Default is 10).
Returns
-------
zeros : ndarray
An array giving the roots of the spline.
See also
--------
splprep, splrep, splint, spalde, splev
bisplrep, bisplev
UnivariateSpline, BivariateSpline
References
----------
.. [1] C. de Boor, "On calculating with b-splines", J. Approximation
Theory, 6, p.50-62, 1972.
.. [2] M.G. Cox, "The numerical evaluation of b-splines", J. Inst. Maths
Applics, 10, p.134-149, 1972.
.. [3] P. Dierckx, "Curve and surface fitting with splines", Monographs
on Numerical Analysis, Oxford University Press, 1993.
"""
t,c,k=tck
if k==4: t=t[1:-1]
if k==5: t=t[2:-2]
try:
c[0][0]
parametric = True
except:
parametric = False
if parametric:
return _ntlist(map(lambda c,t=t,k=k,mest=mest:sproot([t,c,k],mest),c))
else:
if len(t)<8:
raise TypeError("The number of knots %d>=8" % len(t))
z,ier=_fitpack._sproot(t,c,k,mest)
if ier==10:
raise TypeError("Invalid input data. t1<=..<=t4<t5<..<tn-3<=..<=tn must hold.")
if ier==0: return z
if ier==1:
print "Warning: the number of zeros exceeds mest"
return z
raise TypeError("Unknown error")
def spalde(x,tck):
"""
Evaluate all derivatives of a B-spline.
Given the knots and coefficients of a cubic B-spline compute all
derivatives up to order k at a point (or set of points).
Parameters
----------
tck : tuple
A tuple (t,c,k) containing the vector of knots,
the B-spline coefficients, and the degree of the spline.
x : array_like
A point or a set of points at which to evaluate the derivatives.
Note that ``t(k) <= x <= t(n-k+1)`` must hold for each `x`.
Returns
-------
results : array_like
An array (or a list of arrays) containing all derivatives
up to order k inclusive for each point x.
See Also
--------
splprep, splrep, splint, sproot, splev, bisplrep, bisplev,
UnivariateSpline, BivariateSpline
References
----------
.. [1] de Boor C : On calculating with b-splines, J. Approximation Theory
6 (1972) 50-62.
.. [2] Cox M.G. : The numerical evaluation of b-splines, J. Inst. Maths
applics 10 (1972) 134-149.
.. [3] Dierckx P. : Curve and surface fitting with splines, Monographs on
Numerical Analysis, Oxford University Press, 1993.
"""
t,c,k=tck
try:
c[0][0]
parametric = True
except:
parametric = False
if parametric:
return _ntlist(map(lambda c,x=x,t=t,k=k:spalde(x,[t,c,k]),c))
else:
x = myasarray(x)
if len(x)>1:
return map(lambda x,tck=tck:spalde(x,tck),x)
d,ier=_fitpack._spalde(t,c,k,x[0])
if ier==0: return d
if ier==10:
raise TypeError("Invalid input data. t(k)<=x<=t(n-k+1) must hold.")
raise TypeError("Unknown error")
#def _curfit(x,y,w=None,xb=None,xe=None,k=3,task=0,s=None,t=None,
# full_output=0,nest=None,per=0,quiet=1):
_surfit_cache = {'tx': array([],float),'ty': array([],float),
'wrk': array([],float), 'iwrk':array([],int32)}
def bisplrep(x,y,z,w=None,xb=None,xe=None,yb=None,ye=None,kx=3,ky=3,task=0,
s=None,eps=1e-16,tx=None,ty=None,full_output=0,
nxest=None,nyest=None,quiet=1):
"""
Find a bivariate B-spline representation of a surface.
Given a set of data points (x[i], y[i], z[i]) representing a surface
z=f(x,y), compute a B-spline representation of the surface. Based on
the routine SURFIT from FITPACK.
Parameters
----------
x, y, z : ndarray
Rank-1 arrays of data points.
w : ndarray, optional
Rank-1 array of weights. By default ``w=np.ones(len(x))``.
xb, xe : float, optional
End points of approximation interval in `x`.
By default ``xb = x.min(), xe=x.max()``.
yb, ye : float, optional
End points of approximation interval in `y`.
By default ``yb=y.min(), ye = y.max()``.
kx, ky : int, optional
The degrees of the spline (1 <= kx, ky <= 5).
Third order (kx=ky=3) is recommended.
task : int, optional
If task=0, find knots in x and y and coefficients for a given
smoothing factor, s.
If task=1, find knots and coefficients for another value of the
smoothing factor, s. bisplrep must have been previously called
with task=0 or task=1.
If task=-1, find coefficients for a given set of knots tx, ty.
s : float, optional
A non-negative smoothing factor. If weights correspond
to the inverse of the standard-deviation of the errors in z,
then a good s-value should be found in the range
``(m-sqrt(2*m),m+sqrt(2*m))`` where m=len(x).
eps : float, optional
A threshold for determining the effective rank of an
over-determined linear system of equations (0 < eps < 1).
`eps` is not likely to need changing.
tx, ty : ndarray, optional
Rank-1 arrays of the knots of the spline for task=-1
full_output : int, optional
Non-zero to return optional outputs.
nxest, nyest : int, optional
Over-estimates of the total number of knots. If None then
``nxest = max(kx+sqrt(m/2),2*kx+3)``,
``nyest = max(ky+sqrt(m/2),2*ky+3)``.
quiet : int, optional
Non-zero to suppress printing of messages.
Returns
-------
tck : array_like
A list [tx, ty, c, kx, ky] containing the knots (tx, ty) and
coefficients (c) of the bivariate B-spline representation of the
surface along with the degree of the spline.
fp : ndarray
The weighted sum of squared residuals of the spline approximation.
ier : int
An integer flag about splrep success. Success is indicated if
ier<=0. If ier in [1,2,3] an error occurred but was not raised.
Otherwise an error is raised.
msg : str
A message corresponding to the integer flag, ier.
See Also
--------
splprep, splrep, splint, sproot, splev
UnivariateSpline, BivariateSpline
Notes
-----
See `bisplev` to evaluate the value of the B-spline given its tck
representation.
References
----------
.. [1] Dierckx P.:An algorithm for surface fitting with spline functions
Ima J. Numer. Anal. 1 (1981) 267-283.
.. [2] Dierckx P.:An algorithm for surface fitting with spline functions
report tw50, Dept. Computer Science,K.U.Leuven, 1980.
.. [3] Dierckx P.:Curve and surface fitting with splines, Monographs on
Numerical Analysis, Oxford University Press, 1993.
"""
x,y,z=map(myasarray,[x,y,z])
x,y,z=map(ravel,[x,y,z]) # ensure 1-d arrays.
m=len(x)
if not (m==len(y)==len(z)):
raise TypeError('len(x)==len(y)==len(z) must hold.')
if w is None: w=ones(m,float)
else: w=myasarray(w)
if not len(w) == m:
raise TypeError('len(w)=%d is not equal to m=%d' % (len(w), m))
if xb is None: xb=x.min()
if xe is None: xe=x.max()
if yb is None: yb=y.min()
if ye is None: ye=y.max()
if not (-1<=task<=1):
raise TypeError('task must be -1, 0 or 1')
if s is None: s=m-sqrt(2*m)
if tx is None and task==-1:
raise TypeError('Knots_x must be given for task=-1')
if tx is not None: _surfit_cache['tx']=myasarray(tx)
nx=len(_surfit_cache['tx'])
if ty is None and task==-1:
raise TypeError('Knots_y must be given for task=-1')
if ty is not None: _surfit_cache['ty']=myasarray(ty)
ny=len(_surfit_cache['ty'])
if task==-1 and nx<2*kx+2:
raise TypeError('There must be at least 2*kx+2 knots_x for task=-1')
if task==-1 and ny<2*ky+2:
raise TypeError('There must be at least 2*ky+2 knots_x for task=-1')
if not ((1<=kx<=5) and (1<=ky<=5)):
raise TypeError('Given degree of the spline (kx,ky=%d,%d) is not supported. (1<=k<=5)' % (kx,ky))
if m<(kx+1)*(ky+1):
raise TypeError('m >= (kx+1)(ky+1) must hold')
if nxest is None: nxest=int(kx+sqrt(m/2))
if nyest is None: nyest=int(ky+sqrt(m/2))
nxest,nyest=max(nxest,2*kx+3),max(nyest,2*ky+3)
if task>=0 and s==0:
nxest=int(kx+sqrt(3*m))
nyest=int(ky+sqrt(3*m))
if task==-1:
_surfit_cache['tx']=myasarray(tx)
_surfit_cache['ty']=myasarray(ty)
tx,ty=_surfit_cache['tx'],_surfit_cache['ty']
wrk=_surfit_cache['wrk']
iwrk=_surfit_cache['iwrk']
u,v,km,ne=nxest-kx-1,nyest-ky-1,max(kx,ky)+1,max(nxest,nyest)
bx,by=kx*v+ky+1,ky*u+kx+1
b1,b2=bx,bx+v-ky
if bx>by: b1,b2=by,by+u-kx
try:
lwrk1=int32(u*v*(2+b1+b2)+2*(u+v+km*(m+ne)+ne-kx-ky)+b2+1)
lwrk2=int32(u*v*(b2+1)+b2)
except OverflowError:
raise OverflowError("Too many data points to interpolate")
tx,ty,c,o = _fitpack._surfit(x,y,z,w,xb,xe,yb,ye,kx,ky,task,s,eps,
tx,ty,nxest,nyest,wrk,lwrk1,lwrk2)
_curfit_cache['tx']=tx
_curfit_cache['ty']=ty
_curfit_cache['wrk']=o['wrk']
ier,fp=o['ier'],o['fp']
tck=[tx,ty,c,kx,ky]
ierm=min(11,max(-3,ier))
if ierm<=0 and not quiet:
print _iermess2[ierm][0]
print "\tkx,ky=%d,%d nx,ny=%d,%d m=%d fp=%f s=%f"%(kx,ky,len(tx),
len(ty),m,fp,s)
if ierm>0 and not full_output:
if ier in [1,2,3,4,5]:
print "Warning: "+_iermess2[ierm][0]
print "\tkx,ky=%d,%d nx,ny=%d,%d m=%d fp=%f s=%f"%(kx,ky,len(tx),
len(ty),m,fp,s)
else:
try:
raise _iermess2[ierm][1](_iermess2[ierm][0])
except KeyError:
raise _iermess2['unknown'][1](_iermess2['unknown'][0])
if full_output:
try:
return tck,fp,ier,_iermess2[ierm][0]
except KeyError:
return tck,fp,ier,_iermess2['unknown'][0]
else:
return tck
def bisplev(x,y,tck,dx=0,dy=0):
"""
Evaluate a bivariate B-spline and its derivatives.
Return a rank-2 array of spline function values (or spline derivative
values) at points given by the cross-product of the rank-1 arrays x and
y. In special cases, return an array or just a float if either x or y or
both are floats. Based on BISPEV from FITPACK.
Parameters
----------
x, y : ndarray
Rank-1 arrays specifying the domain over which to evaluate the
spline or its derivative.
tck : tuple
A sequence of length 5 returned by `bisplrep` containing the knot
locations, the coefficients, and the degree of the spline:
[tx, ty, c, kx, ky].
dx, dy : int, optional
The orders of the partial derivatives in `x` and `y` respectively.
Returns
-------
vals : ndarray
The B-spline or its derivative evaluated over the set formed by
the cross-product of `x` and `y`.
See Also
--------
splprep, splrep, splint, sproot, splev
UnivariateSpline, BivariateSpline
Notes
-----
See `bisplrep` to generate the `tck` representation.
References
----------
.. [1] Dierckx P. : An algorithm for surface fitting
with spline functions
Ima J. Numer. Anal. 1 (1981) 267-283.
.. [2] Dierckx P. : An algorithm for surface fitting
with spline functions
report tw50, Dept. Computer Science,K.U.Leuven, 1980.
.. [3] Dierckx P. : Curve and surface fitting with splines,
Monographs on Numerical Analysis, Oxford University Press, 1993.
"""
tx,ty,c,kx,ky=tck
if not (0<=dx<kx):
raise ValueError("0 <= dx = %d < kx = %d must hold" % (dx,kx))
if not (0<=dy<ky):
raise ValueError("0 <= dy = %d < ky = %d must hold" % (dy,ky))
x,y=map(myasarray,[x,y])
if (len(x.shape) != 1) or (len(y.shape) != 1):
raise ValueError("First two entries should be rank-1 arrays.")
z,ier=_fitpack._bispev(tx,ty,c,kx,ky,x,y,dx,dy)
if ier==10:
raise ValueError("Invalid input data")
if ier:
raise TypeError("An error occurred")
z.shape=len(x),len(y)
if len(z)>1: return z
if len(z[0])>1: return z[0]
return z[0][0]
def dblint(xa,xb,ya,yb,tck):
"""Evaluate the integral of a spline over area [xa,xb] x [ya,yb].
Parameters
----------
xa, xb : float
The end-points of the x integration interval.
ya, yb : float
The end-points of the y integration interval.
tck : list [tx, ty, c, kx, ky]
A sequence of length 5 returned by bisplrep containing the knot
locations tx, ty, the coefficients c, and the degrees kx, ky
of the spline.
Returns
-------
integ : float
The value of the resulting integral.
"""
tx,ty,c,kx,ky=tck
return dfitpack.dblint(tx,ty,c,kx,ky,xb,xe,yb,ye)
def insert(x,tck,m=1,per=0):
"""
Insert knots into a B-spline.
Given the knots and coefficients of a B-spline representation, create a
new B-spline with a knot inserted m times at point x.
This is a wrapper around the FORTRAN routine insert of FITPACK.
Parameters
----------
x (u) : array_like
A 1-D point at which to insert a new knot(s). If `tck` was returned
from `splprep`, then the parameter values, u should be given.
tck : tuple
A tuple (t,c,k) returned by `splrep` or `splprep` containing
the vector of knots, the B-spline coefficients,
and the degree of the spline.
m : int, optional
The number of times to insert the given knot (its multiplicity).
Default is 1.
per : int, optional
If non-zero, input spline is considered periodic.
Returns
-------
tck : tuple
A tuple (t,c,k) containing the vector of knots, the B-spline
coefficients, and the degree of the new spline.
``t(k+1) <= x <= t(n-k)``, where k is the degree of the spline.
In case of a periodic spline (`per` != 0) there must be
either at least k interior knots t(j) satisfying ``t(k+1)<t(j)<=x``
or at least k interior knots t(j) satisfying ``x<=t(j)<t(n-k)``.
Notes
-----
Based on algorithms from [1]_ and [2]_.
References
----------
.. [1] W. Boehm, "Inserting new knots into b-spline curves.",
Computer Aided Design, 12, p.199-201, 1980.
.. [2] P. Dierckx, "Curve and surface fitting with splines, Monographs on
Numerical Analysis", Oxford University Press, 1993.
"""
t,c,k=tck
try:
c[0][0]
parametric = True
except:
parametric = False
if parametric:
cc = []
for c_vals in c:
tt, cc_val, kk = insert(x, [t, c_vals, k], m)
cc.append(cc_val)
return (tt, cc, kk)
else:
tt, cc, ier = _fitpack._insert(per, t, c, k, x, m)
if ier==10:
raise ValueError("Invalid input data")
if ier:
raise TypeError("An error occurred")
return (tt, cc, k)
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