<commit_before>#!/usr/bin/env python
"""
fitpack (dierckx in netlib) --- A Python-C wrapper to FITPACK (by <NAME>).
        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 <NAME> all rights reserved,
<NAME> <<EMAIL>>          
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.

<NAME>

Running test programs:
    $ python fitpack.py 1 3    # run test programs 1, and 3
    $ python fitpack.py        # run all available test programs

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']
__version__ = "$Revision$"[10:-1]
import _fitpack
from numpy import *
myasarray = atleast_1d

_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 strorage space.
    Probably causes: nest to 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. (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 occured""",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 strorage space.
    Probably causes: nxest or nyest to 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 to 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 occured""",TypeError]}

_parcur_cache = {'t': array([],'d'), 'wrk': array([],'d'), 'iwrk':array([],'i'),
                 'u': array([],'d'),'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.

    Description:

      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

    Inputs:

      x -- A list of sample vector arrays representing the curve. 
      u -- 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 -- The end-points of the parameters interval.  Defaults to
                u[0] and u[-1].
      k -- Degree of the spline.  Cubic splines are recommended.  Even values of
           k should be avoided especially with a small s-value.
           1 <= k <= 5.
      task -- 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.
              If task=-1 find the weighted least square spline for a given set of
                knots, t.
      s -- A smoothing condition.  The amount of smoothness is determined by
           satisfying the conditions: sum((w * (y - g))**2) <= 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.
      t -- The knots needed for task=-1.
      full_output -- If non-zero, then return optional outputs.
      nest -- 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 -- 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 -- Non-zero to suppress messages.

    Outputs: (tck, u, {fp, ier, msg})

      tck -- (t,c,k) a tuple containing the vector of knots, the B-spline
             coefficients, and the degree of the spline.
      u -- An array of the values of the parameter.
             
      fp -- The weighted sum of squared residuals of the spline approximation.
      ier -- 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 -- A message corresponding to the integer flag, ier.          
 
    Remarks:

      SEE splev for evaluation of the spline and its derivatives.      
    """
    if task<=0:
        _parcur_cache = {'t': array([],'d'), 'wrk': array([],'d'),
                         'iwrk':array([],'i'),'u': array([],'d'),'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,'d')
    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,'d')
    if not (1<=k<=5): raise TypeError, '1<=k=%d<=5 must hold'%(k)
    if not (-1<=task<=1): raise TypeError, 'task must be either -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

    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([],'d'), 'wrk': array([],'d'), 'iwrk':array([],'i')}
def splrep(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):
    """Find the B-spline representation of 1-D curve.

    Description:

      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.

    Inputs:

      x, y -- The data points defining a curve y = f(x).
      w -- 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 -- The interval to fit.  If None, these default to x[0] and x[-1]
                respectively.
      k -- 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 -- 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.
              If task=-1 find the weighted least square spline for a given set of
                knots, t.
      s -- A smoothing condition.  The amount of smoothness is determined by
           satisfying the conditions: sum((w * (y - g))**2) <= 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.
      t -- The knots needed for task=-1.
      full_output -- If non-zero, then return optional outputs.
      nest -- An over-estimate of the total number of knots of the spline to
              help in determining the storage space.  By default nest=m/2.
      per -- 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 -- Non-zero to suppress messages.

    Outputs: (tck, {fp, ier, msg})

      tck -- (t,c,k) a tuple containing the vector of knots, the B-spline
             coefficients, and the degree of the spline.
             
      fp -- The weighted sum of squared residuals of the spline approximation.
      ier -- 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 -- A message corresponding to the integer flag, ier.

    Remarks:

      SEE splev for evaluation of the spline and its derivatives.      
    """
    if task<=0:
        _curfit_cache = {'t': array([],'d'), 'wrk': array([],'d'),
                         'iwrk':array([],'i')}
    x,y=map(myasarray,[x,y])
    m=len(x)
    if w is None: w=ones(m,'d')
    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[0]
    if xe is None: xe=x[-1]
    if not (-1<=task<=1): raise TypeError, 'task must be either -1,0, or 1'
    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: _curfit_cache['t']=myasarray(t)
    n=len(_curfit_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 != 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 nest is None: nest=m/2
    if nest<0:
        if per: nest=m+2*k
        else: nest=m+k+1
    nest=max(nest,2*k+3)
    if task>=0 and s==0:
        if per: nest=m+2*k
        else: nest=m+k+1
    if task==-1:
        _curfit_cache['t']=myasarray(t)
        if not (2*k+2<=len(t)<=min(nest,m+k+1)):
            raise TypeError, 'Number of knots n is not acceptable (2*k+2<=n<=min(nest,m+l+1))'
    t=_curfit_cache['t']
    wrk=_curfit_cache['wrk']
    iwrk=_curfit_cache['iwrk']
    t,c,o = _fitpack._curfit(x,y,w,xb,xe,k,task,s,t,nest,wrk,iwrk,per)
    _curfit_cache['t']=t
    _curfit_cache['wrk']=o['wrk']
    _curfit_cache['iwrk']=o['iwrk']
    ier,fp=o['ier'],o['fp']
    tck = [t,c,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):
    """Evaulate a B-spline and its derivatives.

    Description:

      Given the knots and coefficients of a B-spline representation, evaluate
      the value of the smoothing polynomial and it's derivatives.
      This is a wrapper around the FORTRAN routines splev and splder of FITPACK.

    Inputs:

      x (u) -- 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 -- A sequence of length 3 returned by splrep or splprep containg the
             knots, coefficients, and degree of the spline.
      der -- The order of derivative of the spline to compute (must be less than
             or equal to k).

    Outputs: (y, )

      y -- an array of values representing the spline function or curve.  If tck
           was returned from splrep, then this is a list of arrays representing
           the curve in N-dimensional space.
    """
    t,c,k=tck
    try:
        c[0][0]
        return map(lambda c,x=x,t=t,k=k,der=der:splev(x,[t,c,k],der),c)
    except: pass
    if not (0<=der<=k):
        raise ValueError,"0<=der=%d<=k=%d must hold"%(der,k)
    x=myasarray(x)
    y,ier=_fitpack._spl_(x,der,t,c,k)
    if ier==10: raise ValueError,"Invalid input data"
    if ier: raise TypeError,"An error occurred"
    if len(y)>1: return y
    return y[0]

def splint(a,b,tck,full_output=0):
    """Evaluate the definite integral of a B-spline.

    Description:

      Given the knots and coefficients of a B-spline, evaluate the definite
      integral of the smoothing polynomial between two given points.

    Inputs:

      a, b -- The end-points of the integration interval.
      tck -- A length 3 sequence describing the given spline (See splev).
      full_output -- Non-zero to return optional output.

    Outputs: (integral, {wrk})

      integral -- The resulting integral.
      wrk -- An array containing the integrals of the normalized B-splines defined
             on the set of knots.

    """      
    t,c,k=tck
    try: c[0][0];return _ntlist(map(lambda c,a=a,b=b,t=t,k=k:splint(a,b,[t,c,k]),c))
    except: pass
    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.

    Description:

      Given the knots (>=8) and coefficients of a cubic B-spline return the
      roots of the spline.

    Inputs:

      tck -- A length 3 sequence describing the given spline (See splev).
             The number of knots must be >= 8.  The knots must be a montonically
             increasing sequence.
      mest -- An estimate of the number of zeros (Default is 10).

    Outputs: (zeros, )

      zeros -- An array giving the roots of the spline.
    """
    t,c,k=tck
    if k==4: t=t[1:-1]
    if k==5: t=t[2:-2]
    try: c[0][0];return _ntlist(map(lambda c,t=t,k=k,mest=mest:sproot([t,c,k],mest),c))
    except: pass
    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.

    Description:

      Given the knots and coefficients of a cubic B-spline compute all
      derivatives up to order k at a point (or set of points).

    Inputs:

      tck -- A length 3 sequence describing the given spline (See splev).
      x -- 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.

    Outputs: (results, )

      results -- An array (or a list of arrays) containing all derivatives
                 up to order k inclusive for each point x.
    """    
    t,c,k=tck
    try:
        c[0][0]
        return _ntlist(map(lambda c,x=x,t=t,k=k:spalde(x,[t,c,k]),c))
    except: pass
    try: x=x.tolist()
    except:
        try: x=list(x)
        except: x=[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([],'d'),'ty': array([],'d'),
                 'wrk': array([],'d'), 'iwrk':array([],'i')}
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.

    Description:

      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.

    Inputs:

      x, y, z -- Rank-1 arrays of data points.
      w -- Rank-1 array of weights. By default w=ones(len(x)).
      xb, xe -- End points of approximation interval in x. 
      yb, ye -- End points of approximation interval in y.
                By default xb, xe, yb, ye = x[0], x[-1], y[0], y[-1]
      kx, ky -- The degrees of the spline (1 <= kx, ky <= 5).  Third order
                (kx=ky=3) is recommended.
      task -- 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 -- 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 -- A threshold for determining the effective rank of an over-determined
             linear system of equations (0 < eps < 1) --- not likely to need
             changing.
      tx, ty -- Rank-1 arrays of the knots of the spline for task=-1
      full_output -- Non-zero to return optional outputs.
      nxest, nyest -- 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 -- Non-zero to suppress printing of messages.

    Outputs: (tck, {fp, ier, msg})

      tck -- 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 -- The weighted sum of squared residuals of the spline approximation.
      ier -- 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 -- A message corresponding to the integer flag, ier.

    Remarks:

      SEE bisplev to evaluate the value of the B-spline given its tck
      representation.           
    """
    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,'d')
    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[0]
    if xe is None: xe=x[-1]
    if yb is None: yb=y[0]
    if ye is None: ye=y[-1]
    if not (-1<=task<=1): raise TypeError, 'task must be either -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: _curfit_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: _curfit_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=kx+sqrt(m/2)
    if nyest is None: nyest=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
    lwrk1=u*v*(2+b1+b2)+2*(u+v+km*(m+ne)+ne-kx-ky)+b2+1
    lwrk2=u*v*(b2+1)+b2
    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]
    if ier<=0 and not quiet:
        print _iermess2[ier][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)
    ierm=min(11,max(-3,ier))
    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.

    Description:

      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.

    Inputs:

      x, y -- Rank-1 arrays specifying the domain over which to evaluate the
              spline or its derivative.
      tck -- 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 -- The orders of the partial derivatives in x and y respectively.

    Outputs: (vals, )

      vals -- The B-pline or its derivative evaluated over the set formed by
              the cross-product of x and y.
    """
    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]


if __name__ == "__main__":
    import sys,string
    runtest=range(10)
    if len(sys.argv[1:])>0:
        runtest=map(string.atoi,sys.argv[1:])
    put=sys.stdout.write
    def norm2(x):
        return matrixmultiply(transpose(x),x)
    def f1(x,d=0):
        if d is None: return "sin"
        if x is None: return "sin(x)"
        if d%4 == 0: return sin(x)
        if d%4 == 1: return cos(x)
        if d%4 == 2: return -sin(x)
        if d%4 == 3: return -cos(x)
    def f2(x,y=0,dx=0,dy=0):
        if x is None: return "sin(x+y)"
        d=dx+dy
        if d%4 == 0: return sin(x+y)
        if d%4 == 1: return cos(x+y)
        if d%4 == 2: return -sin(x+y)
        if d%4 == 3: return -cos(x+y)
    def test1(f=f1,per=0,s=0,a=0,b=2*pi,N=20,at=0,xb=None,xe=None):
        if xb is None: xb=a
        if xe is None: xe=b
        x=a+(b-a)*arange(N+1,typecode='d')/float(N)    # nodes
        x1=a+(b-a)*arange(1,N,typecode='d')/float(N-1) # middle points of the nodes
        v,v1=f(x),f(x1)
        nk=[]
        for k in range(1,6):
            tck=splrep(x,v,s=s,per=per,k=k,nest=-1,xe=xe)
            if at:t=tck[0][k:-k]
            else: t=x1
            nd=[]
            for d in range(k+1):
                nd.append(norm2(f(t,d)-splev(t,tck,d)))
            nk.append(nd)
        print "\nf = %s  s=S_k(x;t,c)  x in [%s, %s] > [%s, %s]"%(f(None),
                                                        `round(xb,3)`,`round(xe,3)`,
                                                          `round(a,3)`,`round(b,3)`)
        if at: str="at knots"
        else: str="at the middle of nodes"
        print " per=%d s=%s Evaluation %s"%(per,`s`,str)
        print " k :  |f-s|^2  |f'-s'| |f''-.. |f'''-. |f''''- |f'''''"
        k=1
        for l in nk:
            put(' %d : '%k)
            for r in l:
                put(' %.1e'%r)
            put('\n')
            k=k+1
    def test2(f=f1,per=0,s=0,a=0,b=2*pi,N=20,xb=None,xe=None,
              ia=0,ib=2*pi,dx=0.2*pi):
        if xb is None: xb=a
        if xe is None: xe=b
        x=a+(b-a)*arange(N+1,typecode='d')/float(N)    # nodes
        v=f(x)
        nk=[]
        for k in range(1,6):
            tck=splrep(x,v,s=s,per=per,k=k,nest=-1,xe=xe)
            nk.append([splint(ia,ib,tck),spalde(dx,tck)])
        print "\nf = %s  s=S_k(x;t,c)  x in [%s, %s] > [%s, %s]"%(f(None),
                                                   `round(xb,3)`,`round(xe,3)`,
                                                    `round(a,3)`,`round(b,3)`)
        print " per=%d s=%s N=%d [a, b] = [%s, %s]  dx=%s"%(per,`s`,N,`round(ia,3)`,`round(ib,3)`,`round(dx,3)`)
        print " k :  int(s,[a,b]) Int.Error   Rel. error of s^(d)(dx) d = 0, .., k"
        k=1
        for r in nk:
            if r[0]<0: sr='-'
            else: sr=' '
            put(" %d   %s%.8f   %.1e "%(k,sr,abs(r[0]),
                                         abs(r[0]-(f(ib,-1)-f(ia,-1)))))
            d=0
            for dr in r[1]:
                put(" %.1e "%(abs(1-dr/f(dx,d))))
                d=d+1
            put("\n")
            k=k+1
    def test3(f=f1,per=0,s=0,a=0,b=2*pi,N=20,xb=None,xe=None,
              ia=0,ib=2*pi,dx=0.2*pi):
        if xb is None: xb=a
        if xe is None: xe=b
        x=a+(b-a)*arange(N+1,typecode='d')/float(N)    # nodes
        v=f(x)
        nk=[]
        print "  k  :     Roots of s(x) approx %s  x in [%s,%s]:"%\
              (f(None),`round(a,3)`,`round(b,3)`)
        for k in range(1,6):
            tck=splrep(x,v,s=s,per=per,k=k,nest=-1,xe=xe)
            print '  %d  : %s'%(k,`sproot(tck).tolist()`)
    def test4(f=f1,per=0,s=0,a=0,b=2*pi,N=20,xb=None,xe=None,
              ia=0,ib=2*pi,dx=0.2*pi):
        if xb is None: xb=a
        if xe is None: xe=b
        x=a+(b-a)*arange(N+1,typecode='d')/float(N)    # nodes
        x1=a+(b-a)*arange(1,N,typecode='d')/float(N-1) # middle points of the nodes
        v,v1=f(x),f(x1)
        nk=[]
        print " u = %s   N = %d"%(`round(dx,3)`,N)
        print "  k  :  [x(u), %s(x(u))]  Error of splprep  Error of splrep "%(f(0,None))
        for k in range(1,6):
            tckp,u=splprep([x,v],s=s,per=per,k=k,nest=-1)
            tck=splrep(x,v,s=s,per=per,k=k,nest=-1)
            uv=splev(dx,tckp)
            print "  %d  :  %s    %.1e           %.1e"%\
                  (k,`map(lambda x:round(x,3),uv)`,
                   abs(uv[1]-f(uv[0])),
                   abs(splev(uv[0],tck)-f(uv[0])))
        print "Derivatives of parametric cubic spline at u (first function):"
        k=3
        tckp,u=splprep([x,v],s=s,per=per,k=k,nest=-1)
        for d in range(1,k+1):
            uv=splev(dx,tckp,d)
            put(" %s "%(`uv[0]`))
        print
    def makepairs(x,y):
        x,y=map(myasarray,[x,y])
        xy=array(map(lambda x,y:map(None,len(y)*[x],y),x,len(x)*[y]))
        sh=xy.shape
        xy.shape=sh[0]*sh[1],sh[2]
        return transpose(xy)
    def test5(f=f2,kx=3,ky=3,xb=0,xe=2*pi,yb=0,ye=2*pi,Nx=20,Ny=20,s=0):
        x=xb+(xe-xb)*arange(Nx+1,typecode='d')/float(Nx)
        y=yb+(ye-yb)*arange(Ny+1,typecode='d')/float(Ny)
        xy=makepairs(x,y)
        tck=bisplrep(xy[0],xy[1],f(xy[0],xy[1]),s=s,kx=kx,ky=ky)
        tt=[tck[0][kx:-kx],tck[1][ky:-ky]]
        t2=makepairs(tt[0],tt[1])
        v1=bisplev(tt[0],tt[1],tck)
        v2=f2(t2[0],t2[1])
        v2.shape=len(tt[0]),len(tt[1])
        print norm2(ravel(v1-v2))
    if 1 in runtest:
        print """\
******************************************
\tTests of splrep and splev
******************************************"""
        test1(s=1e-6)
        test1()
        test1(at=1)
        test1(per=1)
        test1(per=1,at=1)
        test1(b=1.5*pi)
        test1(b=1.5*pi,xe=2*pi,per=1,s=1e-1)
    if 2 in runtest:
        print """\
******************************************
\tTests of splint and spalde
******************************************"""
        test2()
        test2(per=1)
        test2(ia=0.2*pi,ib=pi)
        test2(ia=0.2*pi,ib=pi,N=50)
    if 3 in runtest:
        print """\
******************************************
\tTests of sproot
******************************************"""
        test3(a=0,b=15)
        print "Note that if k is not 3, some roots are missed or incorrect"
    if 4 in runtest:
        print """\
******************************************
\tTests of splprep, splrep, and splev
******************************************"""
        test4()
        test4(N=50)
    if 5 in runtest:
        print """\
******************************************
\tTests of bisplrep, bisplev
******************************************"""
        test5()







<commit_msg>change 'import *' to explicit imports<commit_after>#!/usr/bin/env python
"""
fitpack (dierckx in netlib) --- A Python-C wrapper to FITPACK (by <NAME>).
        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 <NAME> all rights reserved,
<NAME> <<EMAIL>>          
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.

<NAME>

Running test programs:
    $ python fitpack.py 1 3    # run test programs 1, and 3
    $ python fitpack.py        # run all available test programs

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']
__version__ = "$Revision$"[10:-1]
import _fitpack
from numpy import atleast_1d, array, ones, zeros, sqrt, ravel, transpose, \
     matrixmultiply, sin, cos, pi, arange
myasarray = atleast_1d

_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 strorage space.
    Probably causes: nest to 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. (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 occured""",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 strorage space.
    Probably causes: nxest or nyest to 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 to 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 occured""",TypeError]}

_parcur_cache = {'t': array([],'d'), 'wrk': array([],'d'), 'iwrk':array([],'i'),
                 'u': array([],'d'),'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.

    Description:

      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

    Inputs:

      x -- A list of sample vector arrays representing the curve. 
      u -- 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 -- The end-points of the parameters interval.  Defaults to
                u[0] and u[-1].
      k -- Degree of the spline.  Cubic splines are recommended.  Even values of
           k should be avoided especially with a small s-value.
           1 <= k <= 5.
      task -- 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.
              If task=-1 find the weighted least square spline for a given set of
                knots, t.
      s -- A smoothing condition.  The amount of smoothness is determined by
           satisfying the conditions: sum((w * (y - g))**2) <= 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.
      t -- The knots needed for task=-1.
      full_output -- If non-zero, then return optional outputs.
      nest -- 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 -- 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 -- Non-zero to suppress messages.

    Outputs: (tck, u, {fp, ier, msg})

      tck -- (t,c,k) a tuple containing the vector of knots, the B-spline
             coefficients, and the degree of the spline.
      u -- An array of the values of the parameter.
             
      fp -- The weighted sum of squared residuals of the spline approximation.
      ier -- 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 -- A message corresponding to the integer flag, ier.          
 
    Remarks:

      SEE splev for evaluation of the spline and its derivatives.      
    """
    if task<=0:
        _parcur_cache = {'t': array([],'d'), 'wrk': array([],'d'),
                         'iwrk':array([],'i'),'u': array([],'d'),'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,'d')
    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,'d')
    if not (1<=k<=5): raise TypeError, '1<=k=%d<=5 must hold'%(k)
    if not (-1<=task<=1): raise TypeError, 'task must be either -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

    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([],'d'), 'wrk': array([],'d'), 'iwrk':array([],'i')}
def splrep(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):
    """Find the B-spline representation of 1-D curve.

    Description:

      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.

    Inputs:

      x, y -- The data points defining a curve y = f(x).
      w -- 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 -- The interval to fit.  If None, these default to x[0] and x[-1]
                respectively.
      k -- 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 -- 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.
              If task=-1 find the weighted least square spline for a given set of
                knots, t.
      s -- A smoothing condition.  The amount of smoothness is determined by
           satisfying the conditions: sum((w * (y - g))**2) <= 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.
      t -- The knots needed for task=-1.
      full_output -- If non-zero, then return optional outputs.
      nest -- An over-estimate of the total number of knots of the spline to
              help in determining the storage space.  By default nest=m/2.
      per -- 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 -- Non-zero to suppress messages.

    Outputs: (tck, {fp, ier, msg})

      tck -- (t,c,k) a tuple containing the vector of knots, the B-spline
             coefficients, and the degree of the spline.
             
      fp -- The weighted sum of squared residuals of the spline approximation.
      ier -- 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 -- A message corresponding to the integer flag, ier.

    Remarks:

      SEE splev for evaluation of the spline and its derivatives.      
    """
    if task<=0:
        _curfit_cache = {'t': array([],'d'), 'wrk': array([],'d'),
                         'iwrk':array([],'i')}
    x,y=map(myasarray,[x,y])
    m=len(x)
    if w is None: w=ones(m,'d')
    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[0]
    if xe is None: xe=x[-1]
    if not (-1<=task<=1): raise TypeError, 'task must be either -1,0, or 1'
    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: _curfit_cache['t']=myasarray(t)
    n=len(_curfit_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 != 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 nest is None: nest=m/2
    if nest<0:
        if per: nest=m+2*k
        else: nest=m+k+1
    nest=max(nest,2*k+3)
    if task>=0 and s==0:
        if per: nest=m+2*k
        else: nest=m+k+1
    if task==-1:
        _curfit_cache['t']=myasarray(t)
        if not (2*k+2<=len(t)<=min(nest,m+k+1)):
            raise TypeError, 'Number of knots n is not acceptable (2*k+2<=n<=min(nest,m+l+1))'
    t=_curfit_cache['t']
    wrk=_curfit_cache['wrk']
    iwrk=_curfit_cache['iwrk']
    t,c,o = _fitpack._curfit(x,y,w,xb,xe,k,task,s,t,nest,wrk,iwrk,per)
    _curfit_cache['t']=t
    _curfit_cache['wrk']=o['wrk']
    _curfit_cache['iwrk']=o['iwrk']
    ier,fp=o['ier'],o['fp']
    tck = [t,c,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):
    """Evaulate a B-spline and its derivatives.

    Description:

      Given the knots and coefficients of a B-spline representation, evaluate
      the value of the smoothing polynomial and it's derivatives.
      This is a wrapper around the FORTRAN routines splev and splder of FITPACK.

    Inputs:

      x (u) -- 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 -- A sequence of length 3 returned by splrep or splprep containg the
             knots, coefficients, and degree of the spline.
      der -- The order of derivative of the spline to compute (must be less than
             or equal to k).

    Outputs: (y, )

      y -- an array of values representing the spline function or curve.  If tck
           was returned from splrep, then this is a list of arrays representing
           the curve in N-dimensional space.
    """
    t,c,k=tck
    try:
        c[0][0]
        return map(lambda c,x=x,t=t,k=k,der=der:splev(x,[t,c,k],der),c)
    except: pass
    if not (0<=der<=k):
        raise ValueError,"0<=der=%d<=k=%d must hold"%(der,k)
    x=myasarray(x)
    y,ier=_fitpack._spl_(x,der,t,c,k)
    if ier==10: raise ValueError,"Invalid input data"
    if ier: raise TypeError,"An error occurred"
    if len(y)>1: return y
    return y[0]

def splint(a,b,tck,full_output=0):
    """Evaluate the definite integral of a B-spline.

    Description:

      Given the knots and coefficients of a B-spline, evaluate the definite
      integral of the smoothing polynomial between two given points.

    Inputs:

      a, b -- The end-points of the integration interval.
      tck -- A length 3 sequence describing the given spline (See splev).
      full_output -- Non-zero to return optional output.

    Outputs: (integral, {wrk})

      integral -- The resulting integral.
      wrk -- An array containing the integrals of the normalized B-splines defined
             on the set of knots.

    """      
    t,c,k=tck
    try: c[0][0];return _ntlist(map(lambda c,a=a,b=b,t=t,k=k:splint(a,b,[t,c,k]),c))
    except: pass
    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.

    Description:

      Given the knots (>=8) and coefficients of a cubic B-spline return the
      roots of the spline.

    Inputs:

      tck -- A length 3 sequence describing the given spline (See splev).
             The number of knots must be >= 8.  The knots must be a montonically
             increasing sequence.
      mest -- An estimate of the number of zeros (Default is 10).

    Outputs: (zeros, )

      zeros -- An array giving the roots of the spline.
    """
    t,c,k=tck
    if k==4: t=t[1:-1]
    if k==5: t=t[2:-2]
    try: c[0][0];return _ntlist(map(lambda c,t=t,k=k,mest=mest:sproot([t,c,k],mest),c))
    except: pass
    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.

    Description:

      Given the knots and coefficients of a cubic B-spline compute all
      derivatives up to order k at a point (or set of points).

    Inputs:

      tck -- A length 3 sequence describing the given spline (See splev).
      x -- 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.

    Outputs: (results, )

      results -- An array (or a list of arrays) containing all derivatives
                 up to order k inclusive for each point x.
    """    
    t,c,k=tck
    try:
        c[0][0]
        return _ntlist(map(lambda c,x=x,t=t,k=k:spalde(x,[t,c,k]),c))
    except: pass
    try: x=x.tolist()
    except:
        try: x=list(x)
        except: x=[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([],'d'),'ty': array([],'d'),
                 'wrk': array([],'d'), 'iwrk':array([],'i')}
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.

    Description:

      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.

    Inputs:

      x, y, z -- Rank-1 arrays of data points.
      w -- Rank-1 array of weights. By default w=ones(len(x)).
      xb, xe -- End points of approximation interval in x. 
      yb, ye -- End points of approximation interval in y.
                By default xb, xe, yb, ye = x[0], x[-1], y[0], y[-1]
      kx, ky -- The degrees of the spline (1 <= kx, ky <= 5).  Third order
                (kx=ky=3) is recommended.
      task -- 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 -- 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 -- A threshold for determining the effective rank of an over-determined
             linear system of equations (0 < eps < 1) --- not likely to need
             changing.
      tx, ty -- Rank-1 arrays of the knots of the spline for task=-1
      full_output -- Non-zero to return optional outputs.
      nxest, nyest -- 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 -- Non-zero to suppress printing of messages.

    Outputs: (tck, {fp, ier, msg})

      tck -- 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 -- The weighted sum of squared residuals of the spline approximation.
      ier -- 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 -- A message corresponding to the integer flag, ier.

    Remarks:

      SEE bisplev to evaluate the value of the B-spline given its tck
      representation.           
    """
    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,'d')
    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[0]
    if xe is None: xe=x[-1]
    if yb is None: yb=y[0]
    if ye is None: ye=y[-1]
    if not (-1<=task<=1): raise TypeError, 'task must be either -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: _curfit_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: _curfit_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=kx+sqrt(m/2)
    if nyest is None: nyest=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
    lwrk1=u*v*(2+b1+b2)+2*(u+v+km*(m+ne)+ne-kx-ky)+b2+1
    lwrk2=u*v*(b2+1)+b2
    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]
    if ier<=0 and not quiet:
        print _iermess2[ier][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)
    ierm=min(11,max(-3,ier))
    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.

    Description:

      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.

    Inputs:

      x, y -- Rank-1 arrays specifying the domain over which to evaluate the
              spline or its derivative.
      tck -- 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 -- The orders of the partial derivatives in x and y respectively.

    Outputs: (vals, )

      vals -- The B-pline or its derivative evaluated over the set formed by
              the cross-product of x and y.
    """
    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]


if __name__ == "__main__":
    import sys,string
    runtest=range(10)
    if len(sys.argv[1:])>0:
        runtest=map(string.atoi,sys.argv[1:])
    put=sys.stdout.write
    def norm2(x):
        return matrixmultiply(transpose(x),x)
    def f1(x,d=0):
        if d is None: return "sin"
        if x is None: return "sin(x)"
        if d%4 == 0: return sin(x)
        if d%4 == 1: return cos(x)
        if d%4 == 2: return -sin(x)
        if d%4 == 3: return -cos(x)
    def f2(x,y=0,dx=0,dy=0):
        if x is None: return "sin(x+y)"
        d=dx+dy
        if d%4 == 0: return sin(x+y)
        if d%4 == 1: return cos(x+y)
        if d%4 == 2: return -sin(x+y)
        if d%4 == 3: return -cos(x+y)
    def test1(f=f1,per=0,s=0,a=0,b=2*pi,N=20,at=0,xb=None,xe=None):
        if xb is None: xb=a
        if xe is None: xe=b
        x=a+(b-a)*arange(N+1,typecode='d')/float(N)    # nodes
        x1=a+(b-a)*arange(1,N,typecode='d')/float(N-1) # middle points of the nodes
        v,v1=f(x),f(x1)
        nk=[]
        for k in range(1,6):
            tck=splrep(x,v,s=s,per=per,k=k,nest=-1,xe=xe)
            if at:t=tck[0][k:-k]
            else: t=x1
            nd=[]
            for d in range(k+1):
                nd.append(norm2(f(t,d)-splev(t,tck,d)))
            nk.append(nd)
        print "\nf = %s  s=S_k(x;t,c)  x in [%s, %s] > [%s, %s]"%(f(None),
                                                        `round(xb,3)`,`round(xe,3)`,
                                                          `round(a,3)`,`round(b,3)`)
        if at: str="at knots"
        else: str="at the middle of nodes"
        print " per=%d s=%s Evaluation %s"%(per,`s`,str)
        print " k :  |f-s|^2  |f'-s'| |f''-.. |f'''-. |f''''- |f'''''"
        k=1
        for l in nk:
            put(' %d : '%k)
            for r in l:
                put(' %.1e'%r)
            put('\n')
            k=k+1
    def test2(f=f1,per=0,s=0,a=0,b=2*pi,N=20,xb=None,xe=None,
              ia=0,ib=2*pi,dx=0.2*pi):
        if xb is None: xb=a
        if xe is None: xe=b
        x=a+(b-a)*arange(N+1,typecode='d')/float(N)    # nodes
        v=f(x)
        nk=[]
        for k in range(1,6):
            tck=splrep(x,v,s=s,per=per,k=k,nest=-1,xe=xe)
            nk.append([splint(ia,ib,tck),spalde(dx,tck)])
        print "\nf = %s  s=S_k(x;t,c)  x in [%s, %s] > [%s, %s]"%(f(None),
                                                   `round(xb,3)`,`round(xe,3)`,
                                                    `round(a,3)`,`round(b,3)`)
        print " per=%d s=%s N=%d [a, b] = [%s, %s]  dx=%s"%(per,`s`,N,`round(ia,3)`,`round(ib,3)`,`round(dx,3)`)
        print " k :  int(s,[a,b]) Int.Error   Rel. error of s^(d)(dx) d = 0, .., k"
        k=1
        for r in nk:
            if r[0]<0: sr='-'
            else: sr=' '
            put(" %d   %s%.8f   %.1e "%(k,sr,abs(r[0]),
                                         abs(r[0]-(f(ib,-1)-f(ia,-1)))))
            d=0
            for dr in r[1]:
                put(" %.1e "%(abs(1-dr/f(dx,d))))
                d=d+1
            put("\n")
            k=k+1
    def test3(f=f1,per=0,s=0,a=0,b=2*pi,N=20,xb=None,xe=None,
              ia=0,ib=2*pi,dx=0.2*pi):
        if xb is None: xb=a
        if xe is None: xe=b
        x=a+(b-a)*arange(N+1,typecode='d')/float(N)    # nodes
        v=f(x)
        nk=[]
        print "  k  :     Roots of s(x) approx %s  x in [%s,%s]:"%\
              (f(None),`round(a,3)`,`round(b,3)`)
        for k in range(1,6):
            tck=splrep(x,v,s=s,per=per,k=k,nest=-1,xe=xe)
            print '  %d  : %s'%(k,`sproot(tck).tolist()`)
    def test4(f=f1,per=0,s=0,a=0,b=2*pi,N=20,xb=None,xe=None,
              ia=0,ib=2*pi,dx=0.2*pi):
        if xb is None: xb=a
        if xe is None: xe=b
        x=a+(b-a)*arange(N+1,typecode='d')/float(N)    # nodes
        x1=a+(b-a)*arange(1,N,typecode='d')/float(N-1) # middle points of the nodes
        v,v1=f(x),f(x1)
        nk=[]
        print " u = %s   N = %d"%(`round(dx,3)`,N)
        print "  k  :  [x(u), %s(x(u))]  Error of splprep  Error of splrep "%(f(0,None))
        for k in range(1,6):
            tckp,u=splprep([x,v],s=s,per=per,k=k,nest=-1)
            tck=splrep(x,v,s=s,per=per,k=k,nest=-1)
            uv=splev(dx,tckp)
            print "  %d  :  %s    %.1e           %.1e"%\
                  (k,`map(lambda x:round(x,3),uv)`,
                   abs(uv[1]-f(uv[0])),
                   abs(splev(uv[0],tck)-f(uv[0])))
        print "Derivatives of parametric cubic spline at u (first function):"
        k=3
        tckp,u=splprep([x,v],s=s,per=per,k=k,nest=-1)
        for d in range(1,k+1):
            uv=splev(dx,tckp,d)
            put(" %s "%(`uv[0]`))
        print
    def makepairs(x,y):
        x,y=map(myasarray,[x,y])
        xy=array(map(lambda x,y:map(None,len(y)*[x],y),x,len(x)*[y]))
        sh=xy.shape
        xy.shape=sh[0]*sh[1],sh[2]
        return transpose(xy)
    def test5(f=f2,kx=3,ky=3,xb=0,xe=2*pi,yb=0,ye=2*pi,Nx=20,Ny=20,s=0):
        x=xb+(xe-xb)*arange(Nx+1,typecode='d')/float(Nx)
        y=yb+(ye-yb)*arange(Ny+1,typecode='d')/float(Ny)
        xy=makepairs(x,y)
        tck=bisplrep(xy[0],xy[1],f(xy[0],xy[1]),s=s,kx=kx,ky=ky)
        tt=[tck[0][kx:-kx],tck[1][ky:-ky]]
        t2=makepairs(tt[0],tt[1])
        v1=bisplev(tt[0],tt[1],tck)
        v2=f2(t2[0],t2[1])
        v2.shape=len(tt[0]),len(tt[1])
        print norm2(ravel(v1-v2))
    if 1 in runtest:
        print """\
******************************************
\tTests of splrep and splev
******************************************"""
        test1(s=1e-6)
        test1()
        test1(at=1)
        test1(per=1)
        test1(per=1,at=1)
        test1(b=1.5*pi)
        test1(b=1.5*pi,xe=2*pi,per=1,s=1e-1)
    if 2 in runtest:
        print """\
******************************************
\tTests of splint and spalde
******************************************"""
        test2()
        test2(per=1)
        test2(ia=0.2*pi,ib=pi)
        test2(ia=0.2*pi,ib=pi,N=50)
    if 3 in runtest:
        print """\
******************************************
\tTests of sproot
******************************************"""
        test3(a=0,b=15)
        print "Note that if k is not 3, some roots are missed or incorrect"
    if 4 in runtest:
        print """\
******************************************
\tTests of splprep, splrep, and splev
******************************************"""
        test4()
        test4(N=50)
    if 5 in runtest:
        print """\
******************************************
\tTests of bisplrep, bisplev
******************************************"""
        test5()








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