def Walk(root='<STR_LIT:.>', recurse=True, pattern='<STR_LIT:*>'):

for path, subdirs, files in os.walk(root):<EOL><INDENT>for name in files:<EOL><INDENT>if fnmatch.fnmatch(name, pattern):<EOL><INDENT>yield os.path.join(path, name)<EOL><DEDENT><DEDENT>if not recurse:<EOL><INDENT>break<EOL><DEDENT><DEDENT>

Generator for walking a directory tree.
Starts at specified root folder, returning files that match our pattern.
Optionally will also recurse through sub-folders.

Parameters
----------
root : string (default is *'.'*)
    Path for the root folder to look in.
recurse : bool (default is *True*)
    If *True*, will also look in the subfolders.
pattern : string (default is :emphasis:`'*'`, which means all the files are concerned)
    The pattern to look for in the files' name.

Returns
-------
generator
    **Walk** yields a generator from the matching files paths.


def scan_path(root='<STR_LIT:.>', recurse=False, pattern='<STR_LIT:*>'):

path_list = []<EOL>for path in Walk(root=root, recurse=recurse, pattern=pattern):<EOL><INDENT>path_list.append(path)<EOL><DEDENT>return path_list<EOL>

Runs a loop over the :doc:`Walk<relpy.utils.Walk>` Generator
to find all file paths in the root directory with the given
pattern. If recurse is *True*: matching paths are identified
for all sub directories.

Parameters
----------
root : string (default is *'.'*)
    Path for the root folder to look in.
recurse : bool (default is *True*)
    If *True*, will also look in the subfolders.
pattern : string (default is :emphasis:`'*'`, which means all the files are concerned)
    The pattern to look for in the files' name.

Returns
-------
path_list : list
    list of all the matching files paths.


def displayAll(elapsed, display_amt, est_end, nLoops, count, numPrints):

if numPrints > nLoops:<EOL><INDENT>display_amt = <NUM_LIT:1><EOL><DEDENT>else:<EOL><INDENT>display_amt = round(nLoops / numPrints)<EOL><DEDENT>if count % display_amt == <NUM_LIT:0>:<EOL><INDENT>avg = elapsed / count<EOL>est_end = round(avg * nLoops)<EOL>(disp_elapsed,<EOL>disp_avg,<EOL>disp_est) = timeUnit(int(round(elapsed)),<EOL>int(round(avg)),<EOL>int(round(est_end)))<EOL>print("<STR_LIT>" % str(round(count / float(nLoops) * <NUM_LIT:100>)), "<STR_LIT:@>" + str(count), end='<STR_LIT:U+0020>')<EOL>totalTime = disp_est[<NUM_LIT:0>]<EOL>unit = disp_est[<NUM_LIT:1>]<EOL>if str(unit) == "<STR_LIT>":<EOL><INDENT>remain = totalTime - round(elapsed)<EOL>remainUnit = "<STR_LIT>"<EOL><DEDENT>elif str(unit) == "<STR_LIT>":<EOL><INDENT>remain = totalTime - round(elapsed) / <NUM_LIT><EOL>remainUnit = "<STR_LIT>"<EOL><DEDENT>elif str(unit) == "<STR_LIT>":<EOL><INDENT>remain = totalTime - round(elapsed) / <NUM_LIT><EOL>remainUnit = "<STR_LIT>"<EOL><DEDENT>print("<STR_LIT>" % (str(remain), remainUnit))<EOL>print()<EOL><DEDENT>return<EOL>

Displays time if verbose is true and count is within the display amount


minute = <NUM_LIT><EOL>hr = <NUM_LIT><EOL>day = <NUM_LIT><EOL>if elapsed <= <NUM_LIT:3> * minute:<EOL><INDENT>unit_elapsed = (elapsed, "<STR_LIT>")<EOL><DEDENT>if elapsed > <NUM_LIT:3> * minute:<EOL><INDENT>unit_elapsed = ((elapsed / <NUM_LIT>), "<STR_LIT>")<EOL><DEDENT>if elapsed > <NUM_LIT:3> * hr:<EOL><INDENT>unit_elapsed = ((elapsed / <NUM_LIT>), "<STR_LIT>")<EOL><DEDENT>if avg <= <NUM_LIT:3> * minute:<EOL><INDENT>unit_avg = (avg, "<STR_LIT>")<EOL><DEDENT>if avg > <NUM_LIT:3> * minute:<EOL><INDENT>unit_avg = ((avg / <NUM_LIT>), "<STR_LIT>")<EOL><DEDENT>if avg > <NUM_LIT:3> * hr:<EOL><INDENT>unit_avg = ((avg / <NUM_LIT>), "<STR_LIT>")<EOL><DEDENT>if est_end <= <NUM_LIT:3> * minute:<EOL><INDENT>unit_estEnd = (est_end, "<STR_LIT>")<EOL><DEDENT>if est_end > <NUM_LIT:3> * minute:<EOL><INDENT>unit_estEnd = ((est_end / <NUM_LIT>), "<STR_LIT>")<EOL><DEDENT>if est_end > <NUM_LIT:3> * hr:<EOL><INDENT>unit_estEnd = ((est_end / <NUM_LIT>), "<STR_LIT>")<EOL><DEDENT>return [unit_elapsed, unit_avg, unit_estEnd]<EOL>


self.count += <NUM_LIT:1><EOL>self.tf = time.time()<EOL>self.elapsed = self.tf - self.ti<EOL>if self.verbose:<EOL><INDENT>displayAll(self.elapsed, self.display_amt, self.est_end,<EOL>self.nLoops, self.count, self.numPrints)<EOL><DEDENT>

Tracks the time in a loop. The estimated time to completion
can be calculated and if verbose is set to *True*, the object will print
estimated time to completion, and percent complete.
Actived in every loop to keep track


def __init__(self, verbose=False, infunc=False):

if infunc:<EOL><INDENT>_level = <NUM_LIT:2><EOL><DEDENT>else:<EOL><INDENT>_level = <NUM_LIT:1><EOL><DEDENT>self.parent_frame = frame = sys._getframe(_level)<EOL>self.parent_name = eval('<STR_LIT>', frame.f_globals, frame.f_locals)<EOL>parent_path = eval('<STR_LIT>', frame.f_globals, frame.f_locals)<EOL>if isinstance(parent_path, str):<EOL><INDENT>parent_path = [parent_path]<EOL><DEDENT>self.parent_path = parent_path<EOL>if '<STR_LIT>' not in frame.f_locals:<EOL><INDENT>exec('<STR_LIT>', frame.f_globals, frame.f_locals)<EOL><DEDENT>self.parent_export_names = eval('<STR_LIT>', frame.f_globals, frame.f_locals)<EOL>self.info_modules = {}<EOL>self.imported_packages = []<EOL>self.verbose = None<EOL>


def _get_info_files(self, package_dir, parent_path, parent_package=None):

from glob import glob<EOL>files = glob(os.path.join(parent_path, package_dir, '<STR_LIT>'))<EOL>for info_file in glob(os.path.join(parent_path, package_dir, '<STR_LIT>')):<EOL><INDENT>if info_file[:-<NUM_LIT:1>] not in files:<EOL><INDENT>files.append(info_file)<EOL><DEDENT><DEDENT>info_files = []<EOL>for info_file in files:<EOL><INDENT>package_name = os.path.dirname(info_file[len(parent_path)+<NUM_LIT:1>:]).replace(os.sep, '<STR_LIT:.>')<EOL>if parent_package:<EOL><INDENT>package_name = parent_package + '<STR_LIT:.>' + package_name<EOL><DEDENT>info_files.append((package_name, info_file))<EOL>info_files.extend(self._get_info_files('<STR_LIT:*>',<EOL>os.path.dirname(info_file),<EOL>package_name))<EOL><DEDENT>return info_files<EOL>

Return list of (package name,info.py file) from parent_path subdirectories.


import imp<EOL>info_files = []<EOL>info_modules = self.info_modules<EOL>if packages is None:<EOL><INDENT>for path in self.parent_path:<EOL><INDENT>info_files.extend(self._get_info_files('<STR_LIT:*>', path))<EOL><DEDENT><DEDENT>else:<EOL><INDENT>for package_name in packages:<EOL><INDENT>package_dir = os.path.join(*package_name.split('<STR_LIT:.>'))<EOL>for path in self.parent_path:<EOL><INDENT>names_files = self._get_info_files(package_dir, path)<EOL>if names_files:<EOL><INDENT>info_files.extend(names_files)<EOL>break<EOL><DEDENT><DEDENT>else:<EOL><INDENT>try:<EOL><INDENT>exec('<STR_LIT>' % (package_name))<EOL>info_modules[package_name] = info<EOL><DEDENT>except ImportError as msg:<EOL><INDENT>self.warn('<STR_LIT>''<STR_LIT>'% (package_name, '<STR_LIT::>'.join(self.parent_path), msg))<EOL><DEDENT><DEDENT><DEDENT><DEDENT>for package_name, info_file in info_files:<EOL><INDENT>if package_name in info_modules:<EOL><INDENT>continue<EOL><DEDENT>fullname = self.parent_name +'<STR_LIT:.>'+ package_name<EOL>if info_file[-<NUM_LIT:1>]=='<STR_LIT:c>':<EOL><INDENT>filedescriptor = ('<STR_LIT>', '<STR_LIT:rb>', <NUM_LIT:2>)<EOL><DEDENT>else:<EOL><INDENT>filedescriptor = ('<STR_LIT>', '<STR_LIT>', <NUM_LIT:1>)<EOL><DEDENT>try:<EOL><INDENT>info_module = imp.load_module(fullname+'<STR_LIT>',<EOL>open(info_file, filedescriptor[<NUM_LIT:1>]),<EOL>info_file,<EOL>filedescriptor)<EOL><DEDENT>except Exception as msg:<EOL><INDENT>self.error(msg)<EOL>info_module = None<EOL><DEDENT>if info_module is None or getattr(info_module, '<STR_LIT:ignore>', False):<EOL><INDENT>info_modules.pop(package_name, None)<EOL><DEDENT>else:<EOL><INDENT>self._init_info_modules(getattr(info_module, '<STR_LIT>', []))<EOL>info_modules[package_name] = info_module<EOL><DEDENT><DEDENT>return<EOL>

Initialize info_modules = {<package_name>: <package info.py module>}.


depend_dict = {}<EOL>for name, info_module in self.info_modules.items():<EOL><INDENT>depend_dict[name] = getattr(info_module, '<STR_LIT>', [])<EOL><DEDENT>package_names = []<EOL>for name in list(depend_dict.keys()):<EOL><INDENT>if not depend_dict[name]:<EOL><INDENT>package_names.append(name)<EOL>del depend_dict[name]<EOL><DEDENT><DEDENT>while depend_dict:<EOL><INDENT>for name, lst in list(depend_dict.items()):<EOL><INDENT>new_lst = [n for n in lst if n in depend_dict]<EOL>if not new_lst:<EOL><INDENT>package_names.append(name)<EOL>del depend_dict[name]<EOL><DEDENT>else:<EOL><INDENT>depend_dict[name] = new_lst<EOL><DEDENT><DEDENT><DEDENT>return package_names<EOL>

Return package names sorted in the order as they should be
        imported due to dependence relations between packages.


frame = self.parent_frame<EOL>self.info_modules = {}<EOL>if options.get('<STR_LIT>', False):<EOL><INDENT>self.imported_packages = []<EOL><DEDENT>self.verbose = verbose = options.get('<STR_LIT>', -<NUM_LIT:1>)<EOL>postpone = options.get('<STR_LIT>', None)<EOL>self._init_info_modules(packages or None)<EOL>self.log('<STR_LIT>'% self.parent_name)<EOL>for package_name in self._get_sorted_names():<EOL><INDENT>if package_name in self.imported_packages:<EOL><INDENT>continue<EOL><DEDENT>info_module = self.info_modules[package_name]<EOL>global_symbols = getattr(info_module, '<STR_LIT>', [])<EOL>postpone_import = getattr(info_module, '<STR_LIT>', False)<EOL>if (postpone and not global_symbols)or (postpone_import and postpone is not None):<EOL><INDENT>continue<EOL><DEDENT>old_object = frame.f_locals.get(package_name, None)<EOL>cmdstr = '<STR_LIT>'+package_name<EOL>if self._execcmd(cmdstr):<EOL><INDENT>continue<EOL><DEDENT>self.imported_packages.append(package_name)<EOL>if verbose!=-<NUM_LIT:1>:<EOL><INDENT>new_object = frame.f_locals.get(package_name)<EOL>if old_object is not None and old_object is not new_object:<EOL><INDENT>self.warn('<STR_LIT>'% (package_name, self._obj2repr(new_object),<EOL>self._obj2repr(old_object)))<EOL><DEDENT><DEDENT>if '<STR_LIT:.>' not in package_name:<EOL><INDENT>self.parent_export_names.append(package_name)<EOL><DEDENT>for symbol in global_symbols:<EOL><INDENT>if symbol=='<STR_LIT:*>':<EOL><INDENT>symbols = eval('<STR_LIT>'% (package_name),<EOL>frame.f_globals, frame.f_locals)<EOL>if symbols is None:<EOL><INDENT>symbols = eval('<STR_LIT>' % (package_name),<EOL>frame.f_globals, frame.f_locals)<EOL>symbols = [s for s in symbols if not s.startswith('<STR_LIT:_>')]<EOL><DEDENT><DEDENT>else:<EOL><INDENT>symbols = [symbol]<EOL><DEDENT>if verbose!=-<NUM_LIT:1>:<EOL><INDENT>old_objects = {}<EOL>for s in symbols:<EOL><INDENT>if s in frame.f_locals:<EOL><INDENT>old_objects[s] = frame.f_locals[s]<EOL><DEDENT><DEDENT><DEDENT>cmdstr = '<STR_LIT>'+package_name+'<STR_LIT>'+symbol<EOL>if self._execcmd(cmdstr):<EOL><INDENT>continue<EOL><DEDENT>if verbose!=-<NUM_LIT:1>:<EOL><INDENT>for s, old_object in old_objects.items():<EOL><INDENT>new_object = frame.f_locals[s]<EOL>if new_object is not old_object:<EOL><INDENT>self.warn('<STR_LIT>'% (s, self._obj2repr(new_object),<EOL>self._obj2repr(old_object)))<EOL><DEDENT><DEDENT><DEDENT>if symbol=='<STR_LIT:*>':<EOL><INDENT>self.parent_export_names.extend(symbols)<EOL><DEDENT>else:<EOL><INDENT>self.parent_export_names.append(symbol)<EOL><DEDENT><DEDENT><DEDENT>return<EOL>

Load one or more packages into parent package top-level namespace.

       This function is intended to shorten the need to import many
       subpackages, say of scipy, constantly with statements such as

         import scipy.linalg, scipy.fftpack, scipy.etc...

       Instead, you can say:

         import scipy
         scipy.pkgload('linalg','fftpack',...)

       or

         scipy.pkgload()

       to load all of them in one call.

       If a name which doesn't exist in scipy's namespace is
       given, a warning is shown.

       Parameters
       ----------
        *packages : arg-tuple
             the names (one or more strings) of all the modules one
             wishes to load into the top-level namespace.
        verbose= : integer
             verbosity level [default: -1].
             verbose=-1 will suspend also warnings.
        force= : bool
             when True, force reloading loaded packages [default: False].
        postpone= : bool
             when True, don't load packages [default: False]


frame = self.parent_frame<EOL>try:<EOL><INDENT>exec (cmdstr, frame.f_globals, frame.f_locals)<EOL><DEDENT>except Exception as msg:<EOL><INDENT>self.error('<STR_LIT>' % (cmdstr, msg))<EOL>return True<EOL><DEDENT>else:<EOL><INDENT>self.log('<STR_LIT>' % (cmdstr))<EOL><DEDENT>return<EOL>


module = getattr(obj, '<STR_LIT>', None)<EOL>file = getattr(obj, '<STR_LIT>', None)<EOL>if module is not None:<EOL><INDENT>return repr(obj) + '<STR_LIT>' + module<EOL><DEDENT>if file is not None:<EOL><INDENT>return repr(obj) + '<STR_LIT>' + file<EOL><DEDENT>return repr(obj)<EOL>


title = getattr(info_module, '<STR_LIT>', None)<EOL>if title is not None:<EOL><INDENT>return title<EOL><DEDENT>title = getattr(info_module, '<STR_LIT>', None)<EOL>if title is not None:<EOL><INDENT>title = title.lstrip().split('<STR_LIT:\n>', <NUM_LIT:1>)[<NUM_LIT:0>]<EOL>return title<EOL><DEDENT>return '<STR_LIT>'<EOL>

Get the title from a package info.py file.


import sys<EOL>self.info_modules = {}<EOL>self._init_info_modules(None)<EOL>titles = []<EOL>symbols = []<EOL>for package_name, info_module in self.info_modules.items():<EOL><INDENT>global_symbols = getattr(info_module, '<STR_LIT>', [])<EOL>fullname = self.parent_name +'<STR_LIT:.>'+ package_name<EOL>note = '<STR_LIT>'<EOL>if fullname not in sys.modules:<EOL><INDENT>note = '<STR_LIT>'<EOL><DEDENT>titles.append((fullname, self._get_doc_title(info_module) + note))<EOL>if global_symbols:<EOL><INDENT>symbols.append((package_name, '<STR_LIT:U+002CU+0020>'.join(global_symbols)))<EOL><DEDENT><DEDENT>retstr = self._format_titles(titles) +'<STR_LIT>'<EOL>if symbols:<EOL><INDENT>retstr += """<STR_LIT>""""""<STR_LIT>""" +self._format_titles(symbols, '<STR_LIT>')<EOL><DEDENT>return retstr<EOL>

Return documentation summary of subpackages.


frame = self.parent_frame<EOL>print('<STR_LIT>', repr(cmdstr), '<STR_LIT>', end='<STR_LIT:U+0020>')<EOL>sys.stdout.flush()<EOL>exec (cmdstr, frame.f_globals, frame.f_locals)<EOL>print('<STR_LIT>')<EOL>sys.stdout.flush()<EOL>return<EOL>


return RandomState(seed=<NUM_LIT:0>)<EOL>

Return a RandomState instance.

    This function exists solely to assist (un)pickling.

    Note that the state of the RandomState returned here is irrelevant, as this function's
    entire purpose is to return a newly allocated RandomState whose state pickle can set.
    Consequently the RandomState returned by this function is a freshly allocated copy
    with a seed=0.

    See https://github.com/numpy/numpy/issues/4763 for a detailed discussion


n = c.size<EOL>zs = np.zeros(<NUM_LIT:2>*n-<NUM_LIT:1>, dtype=c.dtype)<EOL>zs[n-<NUM_LIT:1>:] = c/<NUM_LIT:2><EOL>return zs + zs[::-<NUM_LIT:1>]<EOL>

Covert Chebyshev series to z-series.

    Covert a Chebyshev series to the equivalent z-series. The result is
    never an empty array. The dtype of the return is the same as that of
    the input. No checks are run on the arguments as this routine is for
    internal use.

    Parameters
    ----------
    c : 1-D ndarray
        Chebyshev coefficients, ordered from low to high

    Returns
    -------
    zs : 1-D ndarray
        Odd length symmetric z-series, ordered from  low to high.


n = (zs.size + <NUM_LIT:1>)//<NUM_LIT:2><EOL>c = zs[n-<NUM_LIT:1>:].copy()<EOL>c[<NUM_LIT:1>:n] *= <NUM_LIT:2><EOL>return c<EOL>

Covert z-series to a Chebyshev series.

    Covert a z series to the equivalent Chebyshev series. The result is
    never an empty array. The dtype of the return is the same as that of
    the input. No checks are run on the arguments as this routine is for
    internal use.

    Parameters
    ----------
    zs : 1-D ndarray
        Odd length symmetric z-series, ordered from  low to high.

    Returns
    -------
    c : 1-D ndarray
        Chebyshev coefficients, ordered from  low to high.


return np.convolve(z1, z2)<EOL>

Multiply two z-series.

    Multiply two z-series to produce a z-series.

    Parameters
    ----------
    z1, z2 : 1-D ndarray
        The arrays must be 1-D but this is not checked.

    Returns
    -------
    product : 1-D ndarray
        The product z-series.

    Notes
    -----
    This is simply convolution. If symmetric/anti-symmetric z-series are
    denoted by S/A then the following rules apply:

    S*S, A*A -> S
    S*A, A*S -> A


z1 = z1.copy()<EOL>z2 = z2.copy()<EOL>len1 = len(z1)<EOL>len2 = len(z2)<EOL>if len2 == <NUM_LIT:1>:<EOL><INDENT>z1 /= z2<EOL>return z1, z1[:<NUM_LIT:1>]*<NUM_LIT:0><EOL><DEDENT>elif len1 < len2:<EOL><INDENT>return z1[:<NUM_LIT:1>]*<NUM_LIT:0>, z1<EOL><DEDENT>else:<EOL><INDENT>dlen = len1 - len2<EOL>scl = z2[<NUM_LIT:0>]<EOL>z2 /= scl<EOL>quo = np.empty(dlen + <NUM_LIT:1>, dtype=z1.dtype)<EOL>i = <NUM_LIT:0><EOL>j = dlen<EOL>while i < j:<EOL><INDENT>r = z1[i]<EOL>quo[i] = z1[i]<EOL>quo[dlen - i] = r<EOL>tmp = r*z2<EOL>z1[i:i+len2] -= tmp<EOL>z1[j:j+len2] -= tmp<EOL>i += <NUM_LIT:1><EOL>j -= <NUM_LIT:1><EOL><DEDENT>r = z1[i]<EOL>quo[i] = r<EOL>tmp = r*z2<EOL>z1[i:i+len2] -= tmp<EOL>quo /= scl<EOL>rem = z1[i+<NUM_LIT:1>:i-<NUM_LIT:1>+len2].copy()<EOL>return quo, rem<EOL><DEDENT>

Divide the first z-series by the second.

    Divide `z1` by `z2` and return the quotient and remainder as z-series.
    Warning: this implementation only applies when both z1 and z2 have the
    same symmetry, which is sufficient for present purposes.

    Parameters
    ----------
    z1, z2 : 1-D ndarray
        The arrays must be 1-D and have the same symmetry, but this is not
        checked.

    Returns
    -------

    (quotient, remainder) : 1-D ndarrays
        Quotient and remainder as z-series.

    Notes
    -----
    This is not the same as polynomial division on account of the desired form
    of the remainder. If symmetric/anti-symmetric z-series are denoted by S/A
    then the following rules apply:

    S/S -> S,S
    A/A -> S,A

    The restriction to types of the same symmetry could be fixed but seems like
    unneeded generality. There is no natural form for the remainder in the case
    where there is no symmetry.


n = len(zs)//<NUM_LIT:2><EOL>ns = np.array([-<NUM_LIT:1>, <NUM_LIT:0>, <NUM_LIT:1>], dtype=zs.dtype)<EOL>zs *= np.arange(-n, n+<NUM_LIT:1>)*<NUM_LIT:2><EOL>d, r = _zseries_div(zs, ns)<EOL>return d<EOL>

Differentiate a z-series.

    The derivative is with respect to x, not z. This is achieved using the
    chain rule and the value of dx/dz given in the module notes.

    Parameters
    ----------
    zs : z-series
        The z-series to differentiate.

    Returns
    -------
    derivative : z-series
        The derivative

    Notes
    -----
    The zseries for x (ns) has been multiplied by two in order to avoid
    using floats that are incompatible with Decimal and likely other
    specialized scalar types. This scaling has been compensated by
    multiplying the value of zs by two also so that the two cancels in the
    division.


n = <NUM_LIT:1> + len(zs)//<NUM_LIT:2><EOL>ns = np.array([-<NUM_LIT:1>, <NUM_LIT:0>, <NUM_LIT:1>], dtype=zs.dtype)<EOL>zs = _zseries_mul(zs, ns)<EOL>div = np.arange(-n, n+<NUM_LIT:1>)*<NUM_LIT:2><EOL>zs[:n] /= div[:n]<EOL>zs[n+<NUM_LIT:1>:] /= div[n+<NUM_LIT:1>:]<EOL>zs[n] = <NUM_LIT:0><EOL>return zs<EOL>

Integrate a z-series.

    The integral is with respect to x, not z. This is achieved by a change
    of variable using dx/dz given in the module notes.

    Parameters
    ----------
    zs : z-series
        The z-series to integrate

    Returns
    -------
    integral : z-series
        The indefinite integral

    Notes
    -----
    The zseries for x (ns) has been multiplied by two in order to avoid
    using floats that are incompatible with Decimal and likely other
    specialized scalar types. This scaling has been compensated by
    dividing the resulting zs by two.


[pol] = pu.as_series([pol])<EOL>deg = len(pol) - <NUM_LIT:1><EOL>res = <NUM_LIT:0><EOL>for i in range(deg, -<NUM_LIT:1>, -<NUM_LIT:1>):<EOL><INDENT>res = chebadd(chebmulx(res), pol[i])<EOL><DEDENT>return res<EOL>

Convert a polynomial to a Chebyshev series.

Convert an array representing the coefficients of a polynomial (relative
to the "standard" basis) ordered from lowest degree to highest, to an
array of the coefficients of the equivalent Chebyshev series, ordered
from lowest to highest degree.

Parameters
----------
pol : array_like
    1-D array containing the polynomial coefficients

Returns
-------
c : ndarray
    1-D array containing the coefficients of the equivalent Chebyshev
    series.

See Also
--------
cheb2poly

Notes
-----
The easy way to do conversions between polynomial basis sets
is to use the convert method of a class instance.

Examples
--------
>>> from numpy import polynomial as P
>>> p = P.Polynomial(range(4))
>>> p
Polynomial([ 0.,  1.,  2.,  3.], [-1.,  1.])
>>> c = p.convert(kind=P.Chebyshev)
>>> c
Chebyshev([ 1.  ,  3.25,  1.  ,  0.75], [-1.,  1.])
>>> P.poly2cheb(range(4))
array([ 1.  ,  3.25,  1.  ,  0.75])


from .polynomial import polyadd, polysub, polymulx<EOL>[c] = pu.as_series([c])<EOL>n = len(c)<EOL>if n < <NUM_LIT:3>:<EOL><INDENT>return c<EOL><DEDENT>else:<EOL><INDENT>c0 = c[-<NUM_LIT:2>]<EOL>c1 = c[-<NUM_LIT:1>]<EOL>for i in range(n - <NUM_LIT:1>, <NUM_LIT:1>, -<NUM_LIT:1>):<EOL><INDENT>tmp = c0<EOL>c0 = polysub(c[i - <NUM_LIT:2>], c1)<EOL>c1 = polyadd(tmp, polymulx(c1)*<NUM_LIT:2>)<EOL><DEDENT>return polyadd(c0, polymulx(c1))<EOL><DEDENT>

Convert a Chebyshev series to a polynomial.

Convert an array representing the coefficients of a Chebyshev series,
ordered from lowest degree to highest, to an array of the coefficients
of the equivalent polynomial (relative to the "standard" basis) ordered
from lowest to highest degree.

Parameters
----------
c : array_like
    1-D array containing the Chebyshev series coefficients, ordered
    from lowest order term to highest.

Returns
-------
pol : ndarray
    1-D array containing the coefficients of the equivalent polynomial
    (relative to the "standard" basis) ordered from lowest order term
    to highest.

See Also
--------
poly2cheb

Notes
-----
The easy way to do conversions between polynomial basis sets
is to use the convert method of a class instance.

Examples
--------
>>> from numpy import polynomial as P
>>> c = P.Chebyshev(range(4))
>>> c
Chebyshev([ 0.,  1.,  2.,  3.], [-1.,  1.])
>>> p = c.convert(kind=P.Polynomial)
>>> p
Polynomial([ -2.,  -8.,   4.,  12.], [-1.,  1.])
>>> P.cheb2poly(range(4))
array([ -2.,  -8.,   4.,  12.])


if scl != <NUM_LIT:0>:<EOL><INDENT>return np.array([off, scl])<EOL><DEDENT>else:<EOL><INDENT>return np.array([off])<EOL><DEDENT>

Chebyshev series whose graph is a straight line.

Parameters
----------
off, scl : scalars
    The specified line is given by ``off + scl*x``.

Returns
-------
y : ndarray
    This module's representation of the Chebyshev series for
    ``off + scl*x``.

See Also
--------
polyline

Examples
--------
>>> import numpy.polynomial.chebyshev as C
>>> C.chebline(3,2)
array([3, 2])
>>> C.chebval(-3, C.chebline(3,2)) # should be -3
-3.0


if len(roots) == <NUM_LIT:0>:<EOL><INDENT>return np.ones(<NUM_LIT:1>)<EOL><DEDENT>else:<EOL><INDENT>[roots] = pu.as_series([roots], trim=False)<EOL>roots.sort()<EOL>p = [chebline(-r, <NUM_LIT:1>) for r in roots]<EOL>n = len(p)<EOL>while n > <NUM_LIT:1>:<EOL><INDENT>m, r = divmod(n, <NUM_LIT:2>)<EOL>tmp = [chebmul(p[i], p[i+m]) for i in range(m)]<EOL>if r:<EOL><INDENT>tmp[<NUM_LIT:0>] = chebmul(tmp[<NUM_LIT:0>], p[-<NUM_LIT:1>])<EOL><DEDENT>p = tmp<EOL>n = m<EOL><DEDENT>return p[<NUM_LIT:0>]<EOL><DEDENT>

Generate a Chebyshev series with given roots.

The function returns the coefficients of the polynomial

.. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n),

in Chebyshev form, where the `r_n` are the roots specified in `roots`.
If a zero has multiplicity n, then it must appear in `roots` n times.
For instance, if 2 is a root of multiplicity three and 3 is a root of
multiplicity 2, then `roots` looks something like [2, 2, 2, 3, 3]. The
roots can appear in any order.

If the returned coefficients are `c`, then

.. math:: p(x) = c_0 + c_1 * T_1(x) + ... +  c_n * T_n(x)

The coefficient of the last term is not generally 1 for monic
polynomials in Chebyshev form.

Parameters
----------
roots : array_like
    Sequence containing the roots.

Returns
-------
out : ndarray
    1-D array of coefficients.  If all roots are real then `out` is a
    real array, if some of the roots are complex, then `out` is complex
    even if all the coefficients in the result are real (see Examples
    below).

See Also
--------
polyfromroots, legfromroots, lagfromroots, hermfromroots,
hermefromroots.

Examples
--------
>>> import numpy.polynomial.chebyshev as C
>>> C.chebfromroots((-1,0,1)) # x^3 - x relative to the standard basis
array([ 0.  , -0.25,  0.  ,  0.25])
>>> j = complex(0,1)
>>> C.chebfromroots((-j,j)) # x^2 + 1 relative to the standard basis
array([ 1.5+0.j,  0.0+0.j,  0.5+0.j])


<EOL>[c1, c2] = pu.as_series([c1, c2])<EOL>if len(c1) > len(c2):<EOL><INDENT>c1[:c2.size] += c2<EOL>ret = c1<EOL><DEDENT>else:<EOL><INDENT>c2[:c1.size] += c1<EOL>ret = c2<EOL><DEDENT>return pu.trimseq(ret)<EOL>

Add one Chebyshev series to another.

Returns the sum of two Chebyshev series `c1` + `c2`.  The arguments
are sequences of coefficients ordered from lowest order term to
highest, i.e., [1,2,3] represents the series ``T_0 + 2*T_1 + 3*T_2``.

Parameters
----------
c1, c2 : array_like
    1-D arrays of Chebyshev series coefficients ordered from low to
    high.

Returns
-------
out : ndarray
    Array representing the Chebyshev series of their sum.

See Also
--------
chebsub, chebmul, chebdiv, chebpow

Notes
-----
Unlike multiplication, division, etc., the sum of two Chebyshev series
is a Chebyshev series (without having to "reproject" the result onto
the basis set) so addition, just like that of "standard" polynomials,
is simply "component-wise."

Examples
--------
>>> from numpy.polynomial import chebyshev as C
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> C.chebadd(c1,c2)
array([ 4.,  4.,  4.])


<EOL>[c1, c2] = pu.as_series([c1, c2])<EOL>if len(c1) > len(c2):<EOL><INDENT>c1[:c2.size] -= c2<EOL>ret = c1<EOL><DEDENT>else:<EOL><INDENT>c2 = -c2<EOL>c2[:c1.size] += c1<EOL>ret = c2<EOL><DEDENT>return pu.trimseq(ret)<EOL>

Subtract one Chebyshev series from another.

Returns the difference of two Chebyshev series `c1` - `c2`.  The
sequences of coefficients are from lowest order term to highest, i.e.,
[1,2,3] represents the series ``T_0 + 2*T_1 + 3*T_2``.

Parameters
----------
c1, c2 : array_like
    1-D arrays of Chebyshev series coefficients ordered from low to
    high.

Returns
-------
out : ndarray
    Of Chebyshev series coefficients representing their difference.

See Also
--------
chebadd, chebmul, chebdiv, chebpow

Notes
-----
Unlike multiplication, division, etc., the difference of two Chebyshev
series is a Chebyshev series (without having to "reproject" the result
onto the basis set) so subtraction, just like that of "standard"
polynomials, is simply "component-wise."

Examples
--------
>>> from numpy.polynomial import chebyshev as C
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> C.chebsub(c1,c2)
array([-2.,  0.,  2.])
>>> C.chebsub(c2,c1) # -C.chebsub(c1,c2)
array([ 2.,  0., -2.])


<EOL>[c] = pu.as_series([c])<EOL>if len(c) == <NUM_LIT:1> and c[<NUM_LIT:0>] == <NUM_LIT:0>:<EOL><INDENT>return c<EOL><DEDENT>prd = np.empty(len(c) + <NUM_LIT:1>, dtype=c.dtype)<EOL>prd[<NUM_LIT:0>] = c[<NUM_LIT:0>]*<NUM_LIT:0><EOL>prd[<NUM_LIT:1>] = c[<NUM_LIT:0>]<EOL>if len(c) > <NUM_LIT:1>:<EOL><INDENT>tmp = c[<NUM_LIT:1>:]/<NUM_LIT:2><EOL>prd[<NUM_LIT:2>:] = tmp<EOL>prd[<NUM_LIT:0>:-<NUM_LIT:2>] += tmp<EOL><DEDENT>return prd<EOL>

Multiply a Chebyshev series by x.

    Multiply the polynomial `c` by x, where x is the independent
    variable.

    Parameters
    ----------
    c : array_like
        1-D array of Chebyshev series coefficients ordered from low to
        high.

    Returns
    -------
    out : ndarray
        Array representing the result of the multiplication.

    Notes
    -----

    .. versionadded:: 1.5.0


<EOL>[c1, c2] = pu.as_series([c1, c2])<EOL>z1 = _cseries_to_zseries(c1)<EOL>z2 = _cseries_to_zseries(c2)<EOL>prd = _zseries_mul(z1, z2)<EOL>ret = _zseries_to_cseries(prd)<EOL>return pu.trimseq(ret)<EOL>

Multiply one Chebyshev series by another.

Returns the product of two Chebyshev series `c1` * `c2`.  The arguments
are sequences of coefficients, from lowest order "term" to highest,
e.g., [1,2,3] represents the series ``T_0 + 2*T_1 + 3*T_2``.

Parameters
----------
c1, c2 : array_like
    1-D arrays of Chebyshev series coefficients ordered from low to
    high.

Returns
-------
out : ndarray
    Of Chebyshev series coefficients representing their product.

See Also
--------
chebadd, chebsub, chebdiv, chebpow

Notes
-----
In general, the (polynomial) product of two C-series results in terms
that are not in the Chebyshev polynomial basis set.  Thus, to express
the product as a C-series, it is typically necessary to "reproject"
the product onto said basis set, which typically produces
"unintuitive live" (but correct) results; see Examples section below.

Examples
--------
>>> from numpy.polynomial import chebyshev as C
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> C.chebmul(c1,c2) # multiplication requires "reprojection"
array([  6.5,  12. ,  12. ,   4. ,   1.5])


<EOL>[c1, c2] = pu.as_series([c1, c2])<EOL>if c2[-<NUM_LIT:1>] == <NUM_LIT:0>:<EOL><INDENT>raise ZeroDivisionError()<EOL><DEDENT>lc1 = len(c1)<EOL>lc2 = len(c2)<EOL>if lc1 < lc2:<EOL><INDENT>return c1[:<NUM_LIT:1>]*<NUM_LIT:0>, c1<EOL><DEDENT>elif lc2 == <NUM_LIT:1>:<EOL><INDENT>return c1/c2[-<NUM_LIT:1>], c1[:<NUM_LIT:1>]*<NUM_LIT:0><EOL><DEDENT>else:<EOL><INDENT>z1 = _cseries_to_zseries(c1)<EOL>z2 = _cseries_to_zseries(c2)<EOL>quo, rem = _zseries_div(z1, z2)<EOL>quo = pu.trimseq(_zseries_to_cseries(quo))<EOL>rem = pu.trimseq(_zseries_to_cseries(rem))<EOL>return quo, rem<EOL><DEDENT>

Divide one Chebyshev series by another.

Returns the quotient-with-remainder of two Chebyshev series
`c1` / `c2`.  The arguments are sequences of coefficients from lowest
order "term" to highest, e.g., [1,2,3] represents the series
``T_0 + 2*T_1 + 3*T_2``.

Parameters
----------
c1, c2 : array_like
    1-D arrays of Chebyshev series coefficients ordered from low to
    high.

Returns
-------
[quo, rem] : ndarrays
    Of Chebyshev series coefficients representing the quotient and
    remainder.

See Also
--------
chebadd, chebsub, chebmul, chebpow

Notes
-----
In general, the (polynomial) division of one C-series by another
results in quotient and remainder terms that are not in the Chebyshev
polynomial basis set.  Thus, to express these results as C-series, it
is typically necessary to "reproject" the results onto said basis
set, which typically produces "unintuitive" (but correct) results;
see Examples section below.

Examples
--------
>>> from numpy.polynomial import chebyshev as C
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> C.chebdiv(c1,c2) # quotient "intuitive," remainder not
(array([ 3.]), array([-8., -4.]))
>>> c2 = (0,1,2,3)
>>> C.chebdiv(c2,c1) # neither "intuitive"
(array([ 0.,  2.]), array([-2., -4.]))


def chebpow(c, pow, maxpower=<NUM_LIT:16>):

<EOL>[c] = pu.as_series([c])<EOL>power = int(pow)<EOL>if power != pow or power < <NUM_LIT:0>:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>elif maxpower is not None and power > maxpower:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>elif power == <NUM_LIT:0>:<EOL><INDENT>return np.array([<NUM_LIT:1>], dtype=c.dtype)<EOL><DEDENT>elif power == <NUM_LIT:1>:<EOL><INDENT>return c<EOL><DEDENT>else:<EOL><INDENT>zs = _cseries_to_zseries(c)<EOL>prd = zs<EOL>for i in range(<NUM_LIT:2>, power + <NUM_LIT:1>):<EOL><INDENT>prd = np.convolve(prd, zs)<EOL><DEDENT>return _zseries_to_cseries(prd)<EOL><DEDENT>

Raise a Chebyshev series to a power.

    Returns the Chebyshev series `c` raised to the power `pow`. The
    argument `c` is a sequence of coefficients ordered from low to high.
    i.e., [1,2,3] is the series  ``T_0 + 2*T_1 + 3*T_2.``

    Parameters
    ----------
    c : array_like
        1-D array of Chebyshev series coefficients ordered from low to
        high.
    pow : integer
        Power to which the series will be raised
    maxpower : integer, optional
        Maximum power allowed. This is mainly to limit growth of the series
        to unmanageable size. Default is 16

    Returns
    -------
    coef : ndarray
        Chebyshev series of power.

    See Also
    --------
    chebadd, chebsub, chebmul, chebdiv

    Examples
    --------


def chebder(c, m=<NUM_LIT:1>, scl=<NUM_LIT:1>, axis=<NUM_LIT:0>):

c = np.array(c, ndmin=<NUM_LIT:1>, copy=<NUM_LIT:1>)<EOL>if c.dtype.char in '<STR_LIT>':<EOL><INDENT>c = c.astype(np.double)<EOL><DEDENT>cnt, iaxis = [int(t) for t in [m, axis]]<EOL>if cnt != m:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>if cnt < <NUM_LIT:0>:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>if iaxis != axis:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>if not -c.ndim <= iaxis < c.ndim:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>if iaxis < <NUM_LIT:0>:<EOL><INDENT>iaxis += c.ndim<EOL><DEDENT>if cnt == <NUM_LIT:0>:<EOL><INDENT>return c<EOL><DEDENT>c = np.rollaxis(c, iaxis)<EOL>n = len(c)<EOL>if cnt >= n:<EOL><INDENT>c = c[:<NUM_LIT:1>]*<NUM_LIT:0><EOL><DEDENT>else:<EOL><INDENT>for i in range(cnt):<EOL><INDENT>n = n - <NUM_LIT:1><EOL>c *= scl<EOL>der = np.empty((n,) + c.shape[<NUM_LIT:1>:], dtype=c.dtype)<EOL>for j in range(n, <NUM_LIT:2>, -<NUM_LIT:1>):<EOL><INDENT>der[j - <NUM_LIT:1>] = (<NUM_LIT:2>*j)*c[j]<EOL>c[j - <NUM_LIT:2>] += (j*c[j])/(j - <NUM_LIT:2>)<EOL><DEDENT>if n > <NUM_LIT:1>:<EOL><INDENT>der[<NUM_LIT:1>] = <NUM_LIT:4>*c[<NUM_LIT:2>]<EOL><DEDENT>der[<NUM_LIT:0>] = c[<NUM_LIT:1>]<EOL>c = der<EOL><DEDENT><DEDENT>c = np.rollaxis(c, <NUM_LIT:0>, iaxis + <NUM_LIT:1>)<EOL>return c<EOL>

Differentiate a Chebyshev series.

Returns the Chebyshev series coefficients `c` differentiated `m` times
along `axis`.  At each iteration the result is multiplied by `scl` (the
scaling factor is for use in a linear change of variable). The argument
`c` is an array of coefficients from low to high degree along each
axis, e.g., [1,2,3] represents the series ``1*T_0 + 2*T_1 + 3*T_2``
while [[1,2],[1,2]] represents ``1*T_0(x)*T_0(y) + 1*T_1(x)*T_0(y) +
2*T_0(x)*T_1(y) + 2*T_1(x)*T_1(y)`` if axis=0 is ``x`` and axis=1 is
``y``.

Parameters
----------
c : array_like
    Array of Chebyshev series coefficients. If c is multidimensional
    the different axis correspond to different variables with the
    degree in each axis given by the corresponding index.
m : int, optional
    Number of derivatives taken, must be non-negative. (Default: 1)
scl : scalar, optional
    Each differentiation is multiplied by `scl`.  The end result is
    multiplication by ``scl**m``.  This is for use in a linear change of
    variable. (Default: 1)
axis : int, optional
    Axis over which the derivative is taken. (Default: 0).

    .. versionadded:: 1.7.0

Returns
-------
der : ndarray
    Chebyshev series of the derivative.

See Also
--------
chebint

Notes
-----
In general, the result of differentiating a C-series needs to be
"reprojected" onto the C-series basis set. Thus, typically, the
result of this function is "unintuitive," albeit correct; see Examples
section below.

Examples
--------
>>> from numpy.polynomial import chebyshev as C
>>> c = (1,2,3,4)
>>> C.chebder(c)
array([ 14.,  12.,  24.])
>>> C.chebder(c,3)
array([ 96.])
>>> C.chebder(c,scl=-1)
array([-14., -12., -24.])
>>> C.chebder(c,2,-1)
array([ 12.,  96.])


def chebint(c, m=<NUM_LIT:1>, k=[], lbnd=<NUM_LIT:0>, scl=<NUM_LIT:1>, axis=<NUM_LIT:0>):

c = np.array(c, ndmin=<NUM_LIT:1>, copy=<NUM_LIT:1>)<EOL>if c.dtype.char in '<STR_LIT>':<EOL><INDENT>c = c.astype(np.double)<EOL><DEDENT>if not np.iterable(k):<EOL><INDENT>k = [k]<EOL><DEDENT>cnt, iaxis = [int(t) for t in [m, axis]]<EOL>if cnt != m:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>if cnt < <NUM_LIT:0>:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>if len(k) > cnt:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>if iaxis != axis:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>if not -c.ndim <= iaxis < c.ndim:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>if iaxis < <NUM_LIT:0>:<EOL><INDENT>iaxis += c.ndim<EOL><DEDENT>if cnt == <NUM_LIT:0>:<EOL><INDENT>return c<EOL><DEDENT>c = np.rollaxis(c, iaxis)<EOL>k = list(k) + [<NUM_LIT:0>]*(cnt - len(k))<EOL>for i in range(cnt):<EOL><INDENT>n = len(c)<EOL>c *= scl<EOL>if n == <NUM_LIT:1> and np.all(c[<NUM_LIT:0>] == <NUM_LIT:0>):<EOL><INDENT>c[<NUM_LIT:0>] += k[i]<EOL><DEDENT>else:<EOL><INDENT>tmp = np.empty((n + <NUM_LIT:1>,) + c.shape[<NUM_LIT:1>:], dtype=c.dtype)<EOL>tmp[<NUM_LIT:0>] = c[<NUM_LIT:0>]*<NUM_LIT:0><EOL>tmp[<NUM_LIT:1>] = c[<NUM_LIT:0>]<EOL>if n > <NUM_LIT:1>:<EOL><INDENT>tmp[<NUM_LIT:2>] = c[<NUM_LIT:1>]/<NUM_LIT:4><EOL><DEDENT>for j in range(<NUM_LIT:2>, n):<EOL><INDENT>t = c[j]/(<NUM_LIT:2>*j + <NUM_LIT:1>)<EOL>tmp[j + <NUM_LIT:1>] = c[j]/(<NUM_LIT:2>*(j + <NUM_LIT:1>))<EOL>tmp[j - <NUM_LIT:1>] -= c[j]/(<NUM_LIT:2>*(j - <NUM_LIT:1>))<EOL><DEDENT>tmp[<NUM_LIT:0>] += k[i] - chebval(lbnd, tmp)<EOL>c = tmp<EOL><DEDENT><DEDENT>c = np.rollaxis(c, <NUM_LIT:0>, iaxis + <NUM_LIT:1>)<EOL>return c<EOL>

Integrate a Chebyshev series.

Returns the Chebyshev series coefficients `c` integrated `m` times from
`lbnd` along `axis`. At each iteration the resulting series is
**multiplied** by `scl` and an integration constant, `k`, is added.
The scaling factor is for use in a linear change of variable.  ("Buyer
beware": note that, depending on what one is doing, one may want `scl`
to be the reciprocal of what one might expect; for more information,
see the Notes section below.)  The argument `c` is an array of
coefficients from low to high degree along each axis, e.g., [1,2,3]
represents the series ``T_0 + 2*T_1 + 3*T_2`` while [[1,2],[1,2]]
represents ``1*T_0(x)*T_0(y) + 1*T_1(x)*T_0(y) + 2*T_0(x)*T_1(y) +
2*T_1(x)*T_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``.

Parameters
----------
c : array_like
    Array of Chebyshev series coefficients. If c is multidimensional
    the different axis correspond to different variables with the
    degree in each axis given by the corresponding index.
m : int, optional
    Order of integration, must be positive. (Default: 1)
k : {[], list, scalar}, optional
    Integration constant(s).  The value of the first integral at zero
    is the first value in the list, the value of the second integral
    at zero is the second value, etc.  If ``k == []`` (the default),
    all constants are set to zero.  If ``m == 1``, a single scalar can
    be given instead of a list.
lbnd : scalar, optional
    The lower bound of the integral. (Default: 0)
scl : scalar, optional
    Following each integration the result is *multiplied* by `scl`
    before the integration constant is added. (Default: 1)
axis : int, optional
    Axis over which the integral is taken. (Default: 0).

    .. versionadded:: 1.7.0

Returns
-------
S : ndarray
    C-series coefficients of the integral.

Raises
------
ValueError
    If ``m < 1``, ``len(k) > m``, ``np.isscalar(lbnd) == False``, or
    ``np.isscalar(scl) == False``.

See Also
--------
chebder

Notes
-----
Note that the result of each integration is *multiplied* by `scl`.
Why is this important to note?  Say one is making a linear change of
variable :math:`u = ax + b` in an integral relative to `x`.  Then
.. math::`dx = du/a`, so one will need to set `scl` equal to
:math:`1/a`- perhaps not what one would have first thought.

Also note that, in general, the result of integrating a C-series needs
to be "reprojected" onto the C-series basis set.  Thus, typically,
the result of this function is "unintuitive," albeit correct; see
Examples section below.

Examples
--------
>>> from numpy.polynomial import chebyshev as C
>>> c = (1,2,3)
>>> C.chebint(c)
array([ 0.5, -0.5,  0.5,  0.5])
>>> C.chebint(c,3)
array([ 0.03125   , -0.1875    ,  0.04166667, -0.05208333,  0.01041667,
        0.00625   ])
>>> C.chebint(c, k=3)
array([ 3.5, -0.5,  0.5,  0.5])
>>> C.chebint(c,lbnd=-2)
array([ 8.5, -0.5,  0.5,  0.5])
>>> C.chebint(c,scl=-2)
array([-1.,  1., -1., -1.])


c = np.array(c, ndmin=<NUM_LIT:1>, copy=<NUM_LIT:1>)<EOL>if c.dtype.char in '<STR_LIT>':<EOL><INDENT>c = c.astype(np.double)<EOL><DEDENT>if isinstance(x, (tuple, list)):<EOL><INDENT>x = np.asarray(x)<EOL><DEDENT>if isinstance(x, np.ndarray) and tensor:<EOL><INDENT>c = c.reshape(c.shape + (<NUM_LIT:1>,)*x.ndim)<EOL><DEDENT>if len(c) == <NUM_LIT:1>:<EOL><INDENT>c0 = c[<NUM_LIT:0>]<EOL>c1 = <NUM_LIT:0><EOL><DEDENT>elif len(c) == <NUM_LIT:2>:<EOL><INDENT>c0 = c[<NUM_LIT:0>]<EOL>c1 = c[<NUM_LIT:1>]<EOL><DEDENT>else:<EOL><INDENT>x2 = <NUM_LIT:2>*x<EOL>c0 = c[-<NUM_LIT:2>]<EOL>c1 = c[-<NUM_LIT:1>]<EOL>for i in range(<NUM_LIT:3>, len(c) + <NUM_LIT:1>):<EOL><INDENT>tmp = c0<EOL>c0 = c[-i] - c1<EOL>c1 = tmp + c1*x2<EOL><DEDENT><DEDENT>return c0 + c1*x<EOL>

Evaluate a Chebyshev series at points x.

If `c` is of length `n + 1`, this function returns the value:

.. math:: p(x) = c_0 * T_0(x) + c_1 * T_1(x) + ... + c_n * T_n(x)

The parameter `x` is converted to an array only if it is a tuple or a
list, otherwise it is treated as a scalar. In either case, either `x`
or its elements must support multiplication and addition both with
themselves and with the elements of `c`.

If `c` is a 1-D array, then `p(x)` will have the same shape as `x`.  If
`c` is multidimensional, then the shape of the result depends on the
value of `tensor`. If `tensor` is true the shape will be c.shape[1:] +
x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that
scalars have shape (,).

Trailing zeros in the coefficients will be used in the evaluation, so
they should be avoided if efficiency is a concern.

Parameters
----------
x : array_like, compatible object
    If `x` is a list or tuple, it is converted to an ndarray, otherwise
    it is left unchanged and treated as a scalar. In either case, `x`
    or its elements must support addition and multiplication with
    with themselves and with the elements of `c`.
c : array_like
    Array of coefficients ordered so that the coefficients for terms of
    degree n are contained in c[n]. If `c` is multidimensional the
    remaining indices enumerate multiple polynomials. In the two
    dimensional case the coefficients may be thought of as stored in
    the columns of `c`.
tensor : boolean, optional
    If True, the shape of the coefficient array is extended with ones
    on the right, one for each dimension of `x`. Scalars have dimension 0
    for this action. The result is that every column of coefficients in
    `c` is evaluated for every element of `x`. If False, `x` is broadcast
    over the columns of `c` for the evaluation.  This keyword is useful
    when `c` is multidimensional. The default value is True.

    .. versionadded:: 1.7.0

Returns
-------
values : ndarray, algebra_like
    The shape of the return value is described above.

See Also
--------
chebval2d, chebgrid2d, chebval3d, chebgrid3d

Notes
-----
The evaluation uses Clenshaw recursion, aka synthetic division.

Examples
--------


try:<EOL><INDENT>x, y = np.array((x, y), copy=<NUM_LIT:0>)<EOL><DEDENT>except:<EOL><INDENT>raise ValueError('<STR_LIT>')<EOL><DEDENT>c = chebval(x, c)<EOL>c = chebval(y, c, tensor=False)<EOL>return c<EOL>

Evaluate a 2-D Chebyshev series at points (x, y).

This function returns the values:

.. math:: p(x,y) = \\sum_{i,j} c_{i,j} * T_i(x) * T_j(y)

The parameters `x` and `y` are converted to arrays only if they are
tuples or a lists, otherwise they are treated as a scalars and they
must have the same shape after conversion. In either case, either `x`
and `y` or their elements must support multiplication and addition both
with themselves and with the elements of `c`.

If `c` is a 1-D array a one is implicitly appended to its shape to make
it 2-D. The shape of the result will be c.shape[2:] + x.shape.

Parameters
----------
x, y : array_like, compatible objects
    The two dimensional series is evaluated at the points `(x, y)`,
    where `x` and `y` must have the same shape. If `x` or `y` is a list
    or tuple, it is first converted to an ndarray, otherwise it is left
    unchanged and if it isn't an ndarray it is treated as a scalar.
c : array_like
    Array of coefficients ordered so that the coefficient of the term
    of multi-degree i,j is contained in ``c[i,j]``. If `c` has
    dimension greater than 2 the remaining indices enumerate multiple
    sets of coefficients.

Returns
-------
values : ndarray, compatible object
    The values of the two dimensional Chebyshev series at points formed
    from pairs of corresponding values from `x` and `y`.

See Also
--------
chebval, chebgrid2d, chebval3d, chebgrid3d

Notes
-----

.. versionadded::1.7.0


c = chebval(x, c)<EOL>c = chebval(y, c)<EOL>return c<EOL>

Evaluate a 2-D Chebyshev series on the Cartesian product of x and y.

This function returns the values:

.. math:: p(a,b) = \sum_{i,j} c_{i,j} * T_i(a) * T_j(b),

where the points `(a, b)` consist of all pairs formed by taking
`a` from `x` and `b` from `y`. The resulting points form a grid with
`x` in the first dimension and `y` in the second.

The parameters `x` and `y` are converted to arrays only if they are
tuples or a lists, otherwise they are treated as a scalars. In either
case, either `x` and `y` or their elements must support multiplication
and addition both with themselves and with the elements of `c`.

If `c` has fewer than two dimensions, ones are implicitly appended to
its shape to make it 2-D. The shape of the result will be c.shape[2:] +
x.shape + y.shape.

Parameters
----------
x, y : array_like, compatible objects
    The two dimensional series is evaluated at the points in the
    Cartesian product of `x` and `y`.  If `x` or `y` is a list or
    tuple, it is first converted to an ndarray, otherwise it is left
    unchanged and, if it isn't an ndarray, it is treated as a scalar.
c : array_like
    Array of coefficients ordered so that the coefficient of the term of
    multi-degree i,j is contained in `c[i,j]`. If `c` has dimension
    greater than two the remaining indices enumerate multiple sets of
    coefficients.

Returns
-------
values : ndarray, compatible object
    The values of the two dimensional Chebyshev series at points in the
    Cartesian product of `x` and `y`.

See Also
--------
chebval, chebval2d, chebval3d, chebgrid3d

Notes
-----

.. versionadded::1.7.0


try:<EOL><INDENT>x, y, z = np.array((x, y, z), copy=<NUM_LIT:0>)<EOL><DEDENT>except:<EOL><INDENT>raise ValueError('<STR_LIT>')<EOL><DEDENT>c = chebval(x, c)<EOL>c = chebval(y, c, tensor=False)<EOL>c = chebval(z, c, tensor=False)<EOL>return c<EOL>

Evaluate a 3-D Chebyshev series at points (x, y, z).

This function returns the values:

.. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * T_i(x) * T_j(y) * T_k(z)

The parameters `x`, `y`, and `z` are converted to arrays only if
they are tuples or a lists, otherwise they are treated as a scalars and
they must have the same shape after conversion. In either case, either
`x`, `y`, and `z` or their elements must support multiplication and
addition both with themselves and with the elements of `c`.

If `c` has fewer than 3 dimensions, ones are implicitly appended to its
shape to make it 3-D. The shape of the result will be c.shape[3:] +
x.shape.

Parameters
----------
x, y, z : array_like, compatible object
    The three dimensional series is evaluated at the points
    `(x, y, z)`, where `x`, `y`, and `z` must have the same shape.  If
    any of `x`, `y`, or `z` is a list or tuple, it is first converted
    to an ndarray, otherwise it is left unchanged and if it isn't an
    ndarray it is  treated as a scalar.
c : array_like
    Array of coefficients ordered so that the coefficient of the term of
    multi-degree i,j,k is contained in ``c[i,j,k]``. If `c` has dimension
    greater than 3 the remaining indices enumerate multiple sets of
    coefficients.

Returns
-------
values : ndarray, compatible object
    The values of the multidimensional polynomial on points formed with
    triples of corresponding values from `x`, `y`, and `z`.

See Also
--------
chebval, chebval2d, chebgrid2d, chebgrid3d

Notes
-----

.. versionadded::1.7.0


c = chebval(x, c)<EOL>c = chebval(y, c)<EOL>c = chebval(z, c)<EOL>return c<EOL>

Evaluate a 3-D Chebyshev series on the Cartesian product of x, y, and z.

This function returns the values:

.. math:: p(a,b,c) = \\sum_{i,j,k} c_{i,j,k} * T_i(a) * T_j(b) * T_k(c)

where the points `(a, b, c)` consist of all triples formed by taking
`a` from `x`, `b` from `y`, and `c` from `z`. The resulting points form
a grid with `x` in the first dimension, `y` in the second, and `z` in
the third.

The parameters `x`, `y`, and `z` are converted to arrays only if they
are tuples or a lists, otherwise they are treated as a scalars. In
either case, either `x`, `y`, and `z` or their elements must support
multiplication and addition both with themselves and with the elements
of `c`.

If `c` has fewer than three dimensions, ones are implicitly appended to
its shape to make it 3-D. The shape of the result will be c.shape[3:] +
x.shape + y.shape + z.shape.

Parameters
----------
x, y, z : array_like, compatible objects
    The three dimensional series is evaluated at the points in the
    Cartesian product of `x`, `y`, and `z`.  If `x`,`y`, or `z` is a
    list or tuple, it is first converted to an ndarray, otherwise it is
    left unchanged and, if it isn't an ndarray, it is treated as a
    scalar.
c : array_like
    Array of coefficients ordered so that the coefficients for terms of
    degree i,j are contained in ``c[i,j]``. If `c` has dimension
    greater than two the remaining indices enumerate multiple sets of
    coefficients.

Returns
-------
values : ndarray, compatible object
    The values of the two dimensional polynomial at points in the Cartesian
    product of `x` and `y`.

See Also
--------
chebval, chebval2d, chebgrid2d, chebval3d

Notes
-----

.. versionadded::1.7.0


ideg = int(deg)<EOL>if ideg != deg:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>if ideg < <NUM_LIT:0>:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>x = np.array(x, copy=<NUM_LIT:0>, ndmin=<NUM_LIT:1>) + <NUM_LIT:0.0><EOL>dims = (ideg + <NUM_LIT:1>,) + x.shape<EOL>dtyp = x.dtype<EOL>v = np.empty(dims, dtype=dtyp)<EOL>v[<NUM_LIT:0>] = x*<NUM_LIT:0> + <NUM_LIT:1><EOL>if ideg > <NUM_LIT:0>:<EOL><INDENT>x2 = <NUM_LIT:2>*x<EOL>v[<NUM_LIT:1>] = x<EOL>for i in range(<NUM_LIT:2>, ideg + <NUM_LIT:1>):<EOL><INDENT>v[i] = v[i-<NUM_LIT:1>]*x2 - v[i-<NUM_LIT:2>]<EOL><DEDENT><DEDENT>return np.rollaxis(v, <NUM_LIT:0>, v.ndim)<EOL>

Pseudo-Vandermonde matrix of given degree.

    Returns the pseudo-Vandermonde matrix of degree `deg` and sample points
    `x`. The pseudo-Vandermonde matrix is defined by

    .. math:: V[..., i] = T_i(x),

    where `0 <= i <= deg`. The leading indices of `V` index the elements of
    `x` and the last index is the degree of the Chebyshev polynomial.

    If `c` is a 1-D array of coefficients of length `n + 1` and `V` is the
    matrix ``V = chebvander(x, n)``, then ``np.dot(V, c)`` and
    ``chebval(x, c)`` are the same up to roundoff.  This equivalence is
    useful both for least squares fitting and for the evaluation of a large
    number of Chebyshev series of the same degree and sample points.

    Parameters
    ----------
    x : array_like
        Array of points. The dtype is converted to float64 or complex128
        depending on whether any of the elements are complex. If `x` is
        scalar it is converted to a 1-D array.
    deg : int
        Degree of the resulting matrix.

    Returns
    -------
    vander : ndarray
        The pseudo Vandermonde matrix. The shape of the returned matrix is
        ``x.shape + (deg + 1,)``, where The last index is the degree of the
        corresponding Chebyshev polynomial.  The dtype will be the same as
        the converted `x`.


ideg = [int(d) for d in deg]<EOL>is_valid = [id == d and id >= <NUM_LIT:0> for id, d in zip(ideg, deg)]<EOL>if is_valid != [<NUM_LIT:1>, <NUM_LIT:1>]:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>degx, degy = ideg<EOL>x, y = np.array((x, y), copy=<NUM_LIT:0>) + <NUM_LIT:0.0><EOL>vx = chebvander(x, degx)<EOL>vy = chebvander(y, degy)<EOL>v = vx[..., None]*vy[..., None,:]<EOL>return v.reshape(v.shape[:-<NUM_LIT:2>] + (-<NUM_LIT:1>,))<EOL>

Pseudo-Vandermonde matrix of given degrees.

    Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
    points `(x, y)`. The pseudo-Vandermonde matrix is defined by

    .. math:: V[..., deg[1]*i + j] = T_i(x) * T_j(y),

    where `0 <= i <= deg[0]` and `0 <= j <= deg[1]`. The leading indices of
    `V` index the points `(x, y)` and the last index encodes the degrees of
    the Chebyshev polynomials.

    If ``V = chebvander2d(x, y, [xdeg, ydeg])``, then the columns of `V`
    correspond to the elements of a 2-D coefficient array `c` of shape
    (xdeg + 1, ydeg + 1) in the order

    .. math:: c_{00}, c_{01}, c_{02} ... , c_{10}, c_{11}, c_{12} ...

    and ``np.dot(V, c.flat)`` and ``chebval2d(x, y, c)`` will be the same
    up to roundoff. This equivalence is useful both for least squares
    fitting and for the evaluation of a large number of 2-D Chebyshev
    series of the same degrees and sample points.

    Parameters
    ----------
    x, y : array_like
        Arrays of point coordinates, all of the same shape. The dtypes
        will be converted to either float64 or complex128 depending on
        whether any of the elements are complex. Scalars are converted to
        1-D arrays.
    deg : list of ints
        List of maximum degrees of the form [x_deg, y_deg].

    Returns
    -------
    vander2d : ndarray
        The shape of the returned matrix is ``x.shape + (order,)``, where
        :math:`order = (deg[0]+1)*(deg([1]+1)`.  The dtype will be the same
        as the converted `x` and `y`.

    See Also
    --------
    chebvander, chebvander3d. chebval2d, chebval3d

    Notes
    -----

    .. versionadded::1.7.0


ideg = [int(d) for d in deg]<EOL>is_valid = [id == d and id >= <NUM_LIT:0> for id, d in zip(ideg, deg)]<EOL>if is_valid != [<NUM_LIT:1>, <NUM_LIT:1>, <NUM_LIT:1>]:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>degx, degy, degz = ideg<EOL>x, y, z = np.array((x, y, z), copy=<NUM_LIT:0>) + <NUM_LIT:0.0><EOL>vx = chebvander(x, degx)<EOL>vy = chebvander(y, degy)<EOL>vz = chebvander(z, degz)<EOL>v = vx[..., None, None]*vy[..., None,:, None]*vz[..., None, None,:]<EOL>return v.reshape(v.shape[:-<NUM_LIT:3>] + (-<NUM_LIT:1>,))<EOL>

Pseudo-Vandermonde matrix of given degrees.

    Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
    points `(x, y, z)`. If `l, m, n` are the given degrees in `x, y, z`,
    then The pseudo-Vandermonde matrix is defined by

    .. math:: V[..., (m+1)(n+1)i + (n+1)j + k] = T_i(x)*T_j(y)*T_k(z),

    where `0 <= i <= l`, `0 <= j <= m`, and `0 <= j <= n`.  The leading
    indices of `V` index the points `(x, y, z)` and the last index encodes
    the degrees of the Chebyshev polynomials.

    If ``V = chebvander3d(x, y, z, [xdeg, ydeg, zdeg])``, then the columns
    of `V` correspond to the elements of a 3-D coefficient array `c` of
    shape (xdeg + 1, ydeg + 1, zdeg + 1) in the order

    .. math:: c_{000}, c_{001}, c_{002},... , c_{010}, c_{011}, c_{012},...

    and ``np.dot(V, c.flat)`` and ``chebval3d(x, y, z, c)`` will be the
    same up to roundoff. This equivalence is useful both for least squares
    fitting and for the evaluation of a large number of 3-D Chebyshev
    series of the same degrees and sample points.

    Parameters
    ----------
    x, y, z : array_like
        Arrays of point coordinates, all of the same shape. The dtypes will
        be converted to either float64 or complex128 depending on whether
        any of the elements are complex. Scalars are converted to 1-D
        arrays.
    deg : list of ints
        List of maximum degrees of the form [x_deg, y_deg, z_deg].

    Returns
    -------
    vander3d : ndarray
        The shape of the returned matrix is ``x.shape + (order,)``, where
        :math:`order = (deg[0]+1)*(deg([1]+1)*(deg[2]+1)`.  The dtype will
        be the same as the converted `x`, `y`, and `z`.

    See Also
    --------
    chebvander, chebvander3d. chebval2d, chebval3d

    Notes
    -----

    .. versionadded::1.7.0


def chebfit(x, y, deg, rcond=None, full=False, w=None):

order = int(deg) + <NUM_LIT:1><EOL>x = np.asarray(x) + <NUM_LIT:0.0><EOL>y = np.asarray(y) + <NUM_LIT:0.0><EOL>if deg < <NUM_LIT:0>:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>if x.ndim != <NUM_LIT:1>:<EOL><INDENT>raise TypeError("<STR_LIT>")<EOL><DEDENT>if x.size == <NUM_LIT:0>:<EOL><INDENT>raise TypeError("<STR_LIT>")<EOL><DEDENT>if y.ndim < <NUM_LIT:1> or y.ndim > <NUM_LIT:2>:<EOL><INDENT>raise TypeError("<STR_LIT>")<EOL><DEDENT>if len(x) != len(y):<EOL><INDENT>raise TypeError("<STR_LIT>")<EOL><DEDENT>lhs = chebvander(x, deg).T<EOL>rhs = y.T<EOL>if w is not None:<EOL><INDENT>w = np.asarray(w) + <NUM_LIT:0.0><EOL>if w.ndim != <NUM_LIT:1>:<EOL><INDENT>raise TypeError("<STR_LIT>")<EOL><DEDENT>if len(x) != len(w):<EOL><INDENT>raise TypeError("<STR_LIT>")<EOL><DEDENT>lhs = lhs * w<EOL>rhs = rhs * w<EOL><DEDENT>if rcond is None:<EOL><INDENT>rcond = len(x)*np.finfo(x.dtype).eps<EOL><DEDENT>if issubclass(lhs.dtype.type, np.complexfloating):<EOL><INDENT>scl = np.sqrt((np.square(lhs.real) + np.square(lhs.imag)).sum(<NUM_LIT:1>))<EOL><DEDENT>else:<EOL><INDENT>scl = np.sqrt(np.square(lhs).sum(<NUM_LIT:1>))<EOL><DEDENT>scl[scl == <NUM_LIT:0>] = <NUM_LIT:1><EOL>c, resids, rank, s = la.lstsq(lhs.T/scl, rhs.T, rcond)<EOL>c = (c.T/scl).T<EOL>if rank != order and not full:<EOL><INDENT>msg = "<STR_LIT>"<EOL>warnings.warn(msg, pu.RankWarning)<EOL><DEDENT>if full:<EOL><INDENT>return c, [resids, rank, s, rcond]<EOL><DEDENT>else:<EOL><INDENT>return c<EOL><DEDENT>

Least squares fit of Chebyshev series to data.

Return the coefficients of a Legendre series of degree `deg` that is the
least squares fit to the data values `y` given at points `x`. If `y` is
1-D the returned coefficients will also be 1-D. If `y` is 2-D multiple
fits are done, one for each column of `y`, and the resulting
coefficients are stored in the corresponding columns of a 2-D return.
The fitted polynomial(s) are in the form

.. math::  p(x) = c_0 + c_1 * T_1(x) + ... + c_n * T_n(x),

where `n` is `deg`.

Parameters
----------
x : array_like, shape (M,)
    x-coordinates of the M sample points ``(x[i], y[i])``.
y : array_like, shape (M,) or (M, K)
    y-coordinates of the sample points. Several data sets of sample
    points sharing the same x-coordinates can be fitted at once by
    passing in a 2D-array that contains one dataset per column.
deg : int
    Degree of the fitting series
rcond : float, optional
    Relative condition number of the fit. Singular values smaller than
    this relative to the largest singular value will be ignored. The
    default value is len(x)*eps, where eps is the relative precision of
    the float type, about 2e-16 in most cases.
full : bool, optional
    Switch determining nature of return value. When it is False (the
    default) just the coefficients are returned, when True diagnostic
    information from the singular value decomposition is also returned.
w : array_like, shape (`M`,), optional
    Weights. If not None, the contribution of each point
    ``(x[i],y[i])`` to the fit is weighted by `w[i]`. Ideally the
    weights are chosen so that the errors of the products ``w[i]*y[i]``
    all have the same variance.  The default value is None.

    .. versionadded:: 1.5.0

Returns
-------
coef : ndarray, shape (M,) or (M, K)
    Chebyshev coefficients ordered from low to high. If `y` was 2-D,
    the coefficients for the data in column k  of `y` are in column
    `k`.

[residuals, rank, singular_values, rcond] : list
    These values are only returned if `full` = True

    resid -- sum of squared residuals of the least squares fit
    rank -- the numerical rank of the scaled Vandermonde matrix
    sv -- singular values of the scaled Vandermonde matrix
    rcond -- value of `rcond`.

    For more details, see `linalg.lstsq`.

Warns
-----
RankWarning
    The rank of the coefficient matrix in the least-squares fit is
    deficient. The warning is only raised if `full` = False.  The
    warnings can be turned off by

    >>> import warnings
    >>> warnings.simplefilter('ignore', RankWarning)

See Also
--------
polyfit, legfit, lagfit, hermfit, hermefit
chebval : Evaluates a Chebyshev series.
chebvander : Vandermonde matrix of Chebyshev series.
chebweight : Chebyshev weight function.
linalg.lstsq : Computes a least-squares fit from the matrix.
scipy.interpolate.UnivariateSpline : Computes spline fits.

Notes
-----
The solution is the coefficients of the Chebyshev series `p` that
minimizes the sum of the weighted squared errors

.. math:: E = \\sum_j w_j^2 * |y_j - p(x_j)|^2,

where :math:`w_j` are the weights. This problem is solved by setting up
as the (typically) overdetermined matrix equation

.. math:: V(x) * c = w * y,

where `V` is the weighted pseudo Vandermonde matrix of `x`, `c` are the
coefficients to be solved for, `w` are the weights, and `y` are the
observed values.  This equation is then solved using the singular value
decomposition of `V`.

If some of the singular values of `V` are so small that they are
neglected, then a `RankWarning` will be issued. This means that the
coefficient values may be poorly determined. Using a lower order fit
will usually get rid of the warning.  The `rcond` parameter can also be
set to a value smaller than its default, but the resulting fit may be
spurious and have large contributions from roundoff error.

Fits using Chebyshev series are usually better conditioned than fits
using power series, but much can depend on the distribution of the
sample points and the smoothness of the data. If the quality of the fit
is inadequate splines may be a good alternative.

References
----------
.. [1] Wikipedia, "Curve fitting",
       http://en.wikipedia.org/wiki/Curve_fitting

Examples
--------


<EOL>[c] = pu.as_series([c])<EOL>if len(c) < <NUM_LIT:2>:<EOL><INDENT>raise ValueError('<STR_LIT>')<EOL><DEDENT>if len(c) == <NUM_LIT:2>:<EOL><INDENT>return np.array([[-c[<NUM_LIT:0>]/c[<NUM_LIT:1>]]])<EOL><DEDENT>n = len(c) - <NUM_LIT:1><EOL>mat = np.zeros((n, n), dtype=c.dtype)<EOL>scl = np.array([<NUM_LIT:1.>] + [np.sqrt(<NUM_LIT>)]*(n-<NUM_LIT:1>))<EOL>top = mat.reshape(-<NUM_LIT:1>)[<NUM_LIT:1>::n+<NUM_LIT:1>]<EOL>bot = mat.reshape(-<NUM_LIT:1>)[n::n+<NUM_LIT:1>]<EOL>top[<NUM_LIT:0>] = np.sqrt(<NUM_LIT>)<EOL>top[<NUM_LIT:1>:] = <NUM_LIT:1>/<NUM_LIT:2><EOL>bot[...] = top<EOL>mat[:, -<NUM_LIT:1>] -= (c[:-<NUM_LIT:1>]/c[-<NUM_LIT:1>])*(scl/scl[-<NUM_LIT:1>])*<NUM_LIT><EOL>return mat<EOL>

Return the scaled companion matrix of c.

    The basis polynomials are scaled so that the companion matrix is
    symmetric when `c` is aa Chebyshev basis polynomial. This provides
    better eigenvalue estimates than the unscaled case and for basis
    polynomials the eigenvalues are guaranteed to be real if
    `numpy.linalg.eigvalsh` is used to obtain them.

    Parameters
    ----------
    c : array_like
        1-D array of Chebyshev series coefficients ordered from low to high
        degree.

    Returns
    -------
    mat : ndarray
        Scaled companion matrix of dimensions (deg, deg).

    Notes
    -----

    .. versionadded::1.7.0


<EOL>[c] = pu.as_series([c])<EOL>if len(c) < <NUM_LIT:2>:<EOL><INDENT>return np.array([], dtype=c.dtype)<EOL><DEDENT>if len(c) == <NUM_LIT:2>:<EOL><INDENT>return np.array([-c[<NUM_LIT:0>]/c[<NUM_LIT:1>]])<EOL><DEDENT>m = chebcompanion(c)<EOL>r = la.eigvals(m)<EOL>r.sort()<EOL>return r<EOL>

Compute the roots of a Chebyshev series.

Return the roots (a.k.a. "zeros") of the polynomial

.. math:: p(x) = \\sum_i c[i] * T_i(x).

Parameters
----------
c : 1-D array_like
    1-D array of coefficients.

Returns
-------
out : ndarray
    Array of the roots of the series. If all the roots are real,
    then `out` is also real, otherwise it is complex.

See Also
--------
polyroots, legroots, lagroots, hermroots, hermeroots

Notes
-----
The root estimates are obtained as the eigenvalues of the companion
matrix, Roots far from the origin of the complex plane may have large
errors due to the numerical instability of the series for such
values. Roots with multiplicity greater than 1 will also show larger
errors as the value of the series near such points is relatively
insensitive to errors in the roots. Isolated roots near the origin can
be improved by a few iterations of Newton's method.

The Chebyshev series basis polynomials aren't powers of `x` so the
results of this function may seem unintuitive.

Examples
--------
>>> import numpy.polynomial.chebyshev as cheb
>>> cheb.chebroots((-1, 1,-1, 1)) # T3 - T2 + T1 - T0 has real roots
array([ -5.00000000e-01,   2.60860684e-17,   1.00000000e+00])


ideg = int(deg)<EOL>if ideg != deg or ideg < <NUM_LIT:1>:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>x = np.cos(np.pi * np.arange(<NUM_LIT:1>, <NUM_LIT:2>*ideg, <NUM_LIT:2>) / (<NUM_LIT>*ideg))<EOL>w = np.ones(ideg)*(np.pi/ideg)<EOL>return x, w<EOL>

Gauss-Chebyshev quadrature.

Computes the sample points and weights for Gauss-Chebyshev quadrature.
These sample points and weights will correctly integrate polynomials of
degree :math:`2*deg - 1` or less over the interval :math:`[-1, 1]` with
the weight function :math:`f(x) = 1/\sqrt{1 - x^2}`.

Parameters
----------
deg : int
    Number of sample points and weights. It must be >= 1.

Returns
-------
x : ndarray
    1-D ndarray containing the sample points.
y : ndarray
    1-D ndarray containing the weights.

Notes
-----

.. versionadded:: 1.7.0

The results have only been tested up to degree 100, higher degrees may
be problematic. For Gauss-Chebyshev there are closed form solutions for
the sample points and weights. If n = `deg`, then

.. math:: x_i = \cos(\pi (2 i - 1) / (2 n))

.. math:: w_i = \pi / n


w = <NUM_LIT:1.>/(np.sqrt(<NUM_LIT:1.> + x) * np.sqrt(<NUM_LIT:1.> - x))<EOL>return w<EOL>

The weight function of the Chebyshev polynomials.

The weight function is :math:`1/\sqrt{1 - x^2}` and the interval of
integration is :math:`[-1, 1]`. The Chebyshev polynomials are
orthogonal, but not normalized, with respect to this weight function.

Parameters
----------
x : array_like
   Values at which the weight function will be computed.

Returns
-------
w : ndarray
   The weight function at `x`.

Notes
-----

.. versionadded:: 1.7.0


_npts = int(npts)<EOL>if _npts != npts:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>if _npts < <NUM_LIT:1>:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>x = np.linspace(-np.pi, <NUM_LIT:0>, _npts, endpoint=False) + np.pi/(<NUM_LIT:2>*_npts)<EOL>return np.cos(x)<EOL>

Chebyshev points of the first kind.

The Chebyshev points of the first kind are the points ``cos(x)``,
where ``x = [pi*(k + .5)/npts for k in range(npts)]``.

Parameters
----------
npts : int
    Number of sample points desired.

Returns
-------
pts : ndarray
    The Chebyshev points of the first kind.

See Also
--------
chebpts2

Notes
-----

.. versionadded:: 1.5.0


_npts = int(npts)<EOL>if _npts != npts:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>if _npts < <NUM_LIT:2>:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>x = np.linspace(-np.pi, <NUM_LIT:0>, _npts)<EOL>return np.cos(x)<EOL>

Chebyshev points of the second kind.

The Chebyshev points of the second kind are the points ``cos(x)``,
where ``x = [pi*k/(npts - 1) for k in range(npts)]``.

Parameters
----------
npts : int
    Number of sample points desired.

Returns
-------
pts : ndarray
    The Chebyshev points of the second kind.

Notes
-----

.. versionadded:: 1.5.0


[pol] = pu.as_series([pol])<EOL>deg = len(pol) - <NUM_LIT:1><EOL>res = <NUM_LIT:0><EOL>for i in range(deg, -<NUM_LIT:1>, -<NUM_LIT:1>):<EOL><INDENT>res = legadd(legmulx(res), pol[i])<EOL><DEDENT>return res<EOL>

Convert a polynomial to a Legendre series.

Convert an array representing the coefficients of a polynomial (relative
to the "standard" basis) ordered from lowest degree to highest, to an
array of the coefficients of the equivalent Legendre series, ordered
from lowest to highest degree.

Parameters
----------
pol : array_like
    1-D array containing the polynomial coefficients

Returns
-------
c : ndarray
    1-D array containing the coefficients of the equivalent Legendre
    series.

See Also
--------
leg2poly

Notes
-----
The easy way to do conversions between polynomial basis sets
is to use the convert method of a class instance.

Examples
--------
>>> from numpy import polynomial as P
>>> p = P.Polynomial(np.arange(4))
>>> p
Polynomial([ 0.,  1.,  2.,  3.], [-1.,  1.])
>>> c = P.Legendre(P.poly2leg(p.coef))
>>> c
Legendre([ 1.  ,  3.25,  1.  ,  0.75], [-1.,  1.])


from .polynomial import polyadd, polysub, polymulx<EOL>[c] = pu.as_series([c])<EOL>n = len(c)<EOL>if n < <NUM_LIT:3>:<EOL><INDENT>return c<EOL><DEDENT>else:<EOL><INDENT>c0 = c[-<NUM_LIT:2>]<EOL>c1 = c[-<NUM_LIT:1>]<EOL>for i in range(n - <NUM_LIT:1>, <NUM_LIT:1>, -<NUM_LIT:1>):<EOL><INDENT>tmp = c0<EOL>c0 = polysub(c[i - <NUM_LIT:2>], (c1*(i - <NUM_LIT:1>))/i)<EOL>c1 = polyadd(tmp, (polymulx(c1)*(<NUM_LIT:2>*i - <NUM_LIT:1>))/i)<EOL><DEDENT>return polyadd(c0, polymulx(c1))<EOL><DEDENT>

Convert a Legendre series to a polynomial.

Convert an array representing the coefficients of a Legendre series,
ordered from lowest degree to highest, to an array of the coefficients
of the equivalent polynomial (relative to the "standard" basis) ordered
from lowest to highest degree.

Parameters
----------
c : array_like
    1-D array containing the Legendre series coefficients, ordered
    from lowest order term to highest.

Returns
-------
pol : ndarray
    1-D array containing the coefficients of the equivalent polynomial
    (relative to the "standard" basis) ordered from lowest order term
    to highest.

See Also
--------
poly2leg

Notes
-----
The easy way to do conversions between polynomial basis sets
is to use the convert method of a class instance.

Examples
--------
>>> c = P.Legendre(range(4))
>>> c
Legendre([ 0.,  1.,  2.,  3.], [-1.,  1.])
>>> p = c.convert(kind=P.Polynomial)
>>> p
Polynomial([-1. , -3.5,  3. ,  7.5], [-1.,  1.])
>>> P.leg2poly(range(4))
array([-1. , -3.5,  3. ,  7.5])


if scl != <NUM_LIT:0>:<EOL><INDENT>return np.array([off, scl])<EOL><DEDENT>else:<EOL><INDENT>return np.array([off])<EOL><DEDENT>

Legendre series whose graph is a straight line.

Parameters
----------
off, scl : scalars
    The specified line is given by ``off + scl*x``.

Returns
-------
y : ndarray
    This module's representation of the Legendre series for
    ``off + scl*x``.

See Also
--------
polyline, chebline

Examples
--------
>>> import numpy.polynomial.legendre as L
>>> L.legline(3,2)
array([3, 2])
>>> L.legval(-3, L.legline(3,2)) # should be -3
-3.0


if len(roots) == <NUM_LIT:0>:<EOL><INDENT>return np.ones(<NUM_LIT:1>)<EOL><DEDENT>else:<EOL><INDENT>[roots] = pu.as_series([roots], trim=False)<EOL>roots.sort()<EOL>p = [legline(-r, <NUM_LIT:1>) for r in roots]<EOL>n = len(p)<EOL>while n > <NUM_LIT:1>:<EOL><INDENT>m, r = divmod(n, <NUM_LIT:2>)<EOL>tmp = [legmul(p[i], p[i+m]) for i in range(m)]<EOL>if r:<EOL><INDENT>tmp[<NUM_LIT:0>] = legmul(tmp[<NUM_LIT:0>], p[-<NUM_LIT:1>])<EOL><DEDENT>p = tmp<EOL>n = m<EOL><DEDENT>return p[<NUM_LIT:0>]<EOL><DEDENT>

Generate a Legendre series with given roots.

The function returns the coefficients of the polynomial

.. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n),

in Legendre form, where the `r_n` are the roots specified in `roots`.
If a zero has multiplicity n, then it must appear in `roots` n times.
For instance, if 2 is a root of multiplicity three and 3 is a root of
multiplicity 2, then `roots` looks something like [2, 2, 2, 3, 3]. The
roots can appear in any order.

If the returned coefficients are `c`, then

.. math:: p(x) = c_0 + c_1 * L_1(x) + ... +  c_n * L_n(x)

The coefficient of the last term is not generally 1 for monic
polynomials in Legendre form.

Parameters
----------
roots : array_like
    Sequence containing the roots.

Returns
-------
out : ndarray
    1-D array of coefficients.  If all roots are real then `out` is a
    real array, if some of the roots are complex, then `out` is complex
    even if all the coefficients in the result are real (see Examples
    below).

See Also
--------
polyfromroots, chebfromroots, lagfromroots, hermfromroots,
hermefromroots.

Examples
--------
>>> import numpy.polynomial.legendre as L
>>> L.legfromroots((-1,0,1)) # x^3 - x relative to the standard basis
array([ 0. , -0.4,  0. ,  0.4])
>>> j = complex(0,1)
>>> L.legfromroots((-j,j)) # x^2 + 1 relative to the standard basis
array([ 1.33333333+0.j,  0.00000000+0.j,  0.66666667+0.j])


<EOL>[c1, c2] = pu.as_series([c1, c2])<EOL>if len(c1) > len(c2):<EOL><INDENT>c1[:c2.size] += c2<EOL>ret = c1<EOL><DEDENT>else:<EOL><INDENT>c2[:c1.size] += c1<EOL>ret = c2<EOL><DEDENT>return pu.trimseq(ret)<EOL>

Add one Legendre series to another.

Returns the sum of two Legendre series `c1` + `c2`.  The arguments
are sequences of coefficients ordered from lowest order term to
highest, i.e., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.

Parameters
----------
c1, c2 : array_like
    1-D arrays of Legendre series coefficients ordered from low to
    high.

Returns
-------
out : ndarray
    Array representing the Legendre series of their sum.

See Also
--------
legsub, legmul, legdiv, legpow

Notes
-----
Unlike multiplication, division, etc., the sum of two Legendre series
is a Legendre series (without having to "reproject" the result onto
the basis set) so addition, just like that of "standard" polynomials,
is simply "component-wise."

Examples
--------
>>> from numpy.polynomial import legendre as L
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> L.legadd(c1,c2)
array([ 4.,  4.,  4.])


<EOL>[c1, c2] = pu.as_series([c1, c2])<EOL>if len(c1) > len(c2):<EOL><INDENT>c1[:c2.size] -= c2<EOL>ret = c1<EOL><DEDENT>else:<EOL><INDENT>c2 = -c2<EOL>c2[:c1.size] += c1<EOL>ret = c2<EOL><DEDENT>return pu.trimseq(ret)<EOL>

Subtract one Legendre series from another.

Returns the difference of two Legendre series `c1` - `c2`.  The
sequences of coefficients are from lowest order term to highest, i.e.,
[1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.

Parameters
----------
c1, c2 : array_like
    1-D arrays of Legendre series coefficients ordered from low to
    high.

Returns
-------
out : ndarray
    Of Legendre series coefficients representing their difference.

See Also
--------
legadd, legmul, legdiv, legpow

Notes
-----
Unlike multiplication, division, etc., the difference of two Legendre
series is a Legendre series (without having to "reproject" the result
onto the basis set) so subtraction, just like that of "standard"
polynomials, is simply "component-wise."

Examples
--------
>>> from numpy.polynomial import legendre as L
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> L.legsub(c1,c2)
array([-2.,  0.,  2.])
>>> L.legsub(c2,c1) # -C.legsub(c1,c2)
array([ 2.,  0., -2.])


<EOL>[c] = pu.as_series([c])<EOL>if len(c) == <NUM_LIT:1> and c[<NUM_LIT:0>] == <NUM_LIT:0>:<EOL><INDENT>return c<EOL><DEDENT>prd = np.empty(len(c) + <NUM_LIT:1>, dtype=c.dtype)<EOL>prd[<NUM_LIT:0>] = c[<NUM_LIT:0>]*<NUM_LIT:0><EOL>prd[<NUM_LIT:1>] = c[<NUM_LIT:0>]<EOL>for i in range(<NUM_LIT:1>, len(c)):<EOL><INDENT>j = i + <NUM_LIT:1><EOL>k = i - <NUM_LIT:1><EOL>s = i + j<EOL>prd[j] = (c[i]*j)/s<EOL>prd[k] += (c[i]*i)/s<EOL><DEDENT>return prd<EOL>

Multiply a Legendre series by x.

    Multiply the Legendre series `c` by x, where x is the independent
    variable.

    Parameters
    ----------
    c : array_like
        1-D array of Legendre series coefficients ordered from low to
        high.

    Returns
    -------
    out : ndarray
        Array representing the result of the multiplication.

    Notes
    -----
    The multiplication uses the recursion relationship for Legendre
    polynomials in the form

    .. math::

      xP_i(x) = ((i + 1)*P_{i + 1}(x) + i*P_{i - 1}(x))/(2i + 1)


<EOL>[c1, c2] = pu.as_series([c1, c2])<EOL>if len(c1) > len(c2):<EOL><INDENT>c = c2<EOL>xs = c1<EOL><DEDENT>else:<EOL><INDENT>c = c1<EOL>xs = c2<EOL><DEDENT>if len(c) == <NUM_LIT:1>:<EOL><INDENT>c0 = c[<NUM_LIT:0>]*xs<EOL>c1 = <NUM_LIT:0><EOL><DEDENT>elif len(c) == <NUM_LIT:2>:<EOL><INDENT>c0 = c[<NUM_LIT:0>]*xs<EOL>c1 = c[<NUM_LIT:1>]*xs<EOL><DEDENT>else:<EOL><INDENT>nd = len(c)<EOL>c0 = c[-<NUM_LIT:2>]*xs<EOL>c1 = c[-<NUM_LIT:1>]*xs<EOL>for i in range(<NUM_LIT:3>, len(c) + <NUM_LIT:1>):<EOL><INDENT>tmp = c0<EOL>nd = nd - <NUM_LIT:1><EOL>c0 = legsub(c[-i]*xs, (c1*(nd - <NUM_LIT:1>))/nd)<EOL>c1 = legadd(tmp, (legmulx(c1)*(<NUM_LIT:2>*nd - <NUM_LIT:1>))/nd)<EOL><DEDENT><DEDENT>return legadd(c0, legmulx(c1))<EOL>

Multiply one Legendre series by another.

Returns the product of two Legendre series `c1` * `c2`.  The arguments
are sequences of coefficients, from lowest order "term" to highest,
e.g., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.

Parameters
----------
c1, c2 : array_like
    1-D arrays of Legendre series coefficients ordered from low to
    high.

Returns
-------
out : ndarray
    Of Legendre series coefficients representing their product.

See Also
--------
legadd, legsub, legdiv, legpow

Notes
-----
In general, the (polynomial) product of two C-series results in terms
that are not in the Legendre polynomial basis set.  Thus, to express
the product as a Legendre series, it is necessary to "reproject" the
product onto said basis set, which may produce "unintuitive" (but
correct) results; see Examples section below.

Examples
--------
>>> from numpy.polynomial import legendre as L
>>> c1 = (1,2,3)
>>> c2 = (3,2)
>>> P.legmul(c1,c2) # multiplication requires "reprojection"
array([  4.33333333,  10.4       ,  11.66666667,   3.6       ])


<EOL>[c1, c2] = pu.as_series([c1, c2])<EOL>if c2[-<NUM_LIT:1>] == <NUM_LIT:0>:<EOL><INDENT>raise ZeroDivisionError()<EOL><DEDENT>lc1 = len(c1)<EOL>lc2 = len(c2)<EOL>if lc1 < lc2:<EOL><INDENT>return c1[:<NUM_LIT:1>]*<NUM_LIT:0>, c1<EOL><DEDENT>elif lc2 == <NUM_LIT:1>:<EOL><INDENT>return c1/c2[-<NUM_LIT:1>], c1[:<NUM_LIT:1>]*<NUM_LIT:0><EOL><DEDENT>else:<EOL><INDENT>quo = np.empty(lc1 - lc2 + <NUM_LIT:1>, dtype=c1.dtype)<EOL>rem = c1<EOL>for i in range(lc1 - lc2, - <NUM_LIT:1>, -<NUM_LIT:1>):<EOL><INDENT>p = legmul([<NUM_LIT:0>]*i + [<NUM_LIT:1>], c2)<EOL>q = rem[-<NUM_LIT:1>]/p[-<NUM_LIT:1>]<EOL>rem = rem[:-<NUM_LIT:1>] - q*p[:-<NUM_LIT:1>]<EOL>quo[i] = q<EOL><DEDENT>return quo, pu.trimseq(rem)<EOL><DEDENT>

Divide one Legendre series by another.

Returns the quotient-with-remainder of two Legendre series
`c1` / `c2`.  The arguments are sequences of coefficients from lowest
order "term" to highest, e.g., [1,2,3] represents the series
``P_0 + 2*P_1 + 3*P_2``.

Parameters
----------
c1, c2 : array_like
    1-D arrays of Legendre series coefficients ordered from low to
    high.

Returns
-------
quo, rem : ndarrays
    Of Legendre series coefficients representing the quotient and
    remainder.

See Also
--------
legadd, legsub, legmul, legpow

Notes
-----
In general, the (polynomial) division of one Legendre series by another
results in quotient and remainder terms that are not in the Legendre
polynomial basis set.  Thus, to express these results as a Legendre
series, it is necessary to "reproject" the results onto the Legendre
basis set, which may produce "unintuitive" (but correct) results; see
Examples section below.

Examples
--------
>>> from numpy.polynomial import legendre as L
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> L.legdiv(c1,c2) # quotient "intuitive," remainder not
(array([ 3.]), array([-8., -4.]))
>>> c2 = (0,1,2,3)
>>> L.legdiv(c2,c1) # neither "intuitive"
(array([-0.07407407,  1.66666667]), array([-1.03703704, -2.51851852]))


def legpow(c, pow, maxpower=<NUM_LIT:16>):

<EOL>[c] = pu.as_series([c])<EOL>power = int(pow)<EOL>if power != pow or power < <NUM_LIT:0>:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>elif maxpower is not None and power > maxpower:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>elif power == <NUM_LIT:0>:<EOL><INDENT>return np.array([<NUM_LIT:1>], dtype=c.dtype)<EOL><DEDENT>elif power == <NUM_LIT:1>:<EOL><INDENT>return c<EOL><DEDENT>else:<EOL><INDENT>prd = c<EOL>for i in range(<NUM_LIT:2>, power + <NUM_LIT:1>):<EOL><INDENT>prd = legmul(prd, c)<EOL><DEDENT>return prd<EOL><DEDENT>

Raise a Legendre series to a power.

    Returns the Legendre series `c` raised to the power `pow`. The
    arguement `c` is a sequence of coefficients ordered from low to high.
    i.e., [1,2,3] is the series  ``P_0 + 2*P_1 + 3*P_2.``

    Parameters
    ----------
    c : array_like
        1-D array of Legendre series coefficients ordered from low to
        high.
    pow : integer
        Power to which the series will be raised
    maxpower : integer, optional
        Maximum power allowed. This is mainly to limit growth of the series
        to unmanageable size. Default is 16

    Returns
    -------
    coef : ndarray
        Legendre series of power.

    See Also
    --------
    legadd, legsub, legmul, legdiv

    Examples
    --------


def legder(c, m=<NUM_LIT:1>, scl=<NUM_LIT:1>, axis=<NUM_LIT:0>):

c = np.array(c, ndmin=<NUM_LIT:1>, copy=<NUM_LIT:1>)<EOL>if c.dtype.char in '<STR_LIT>':<EOL><INDENT>c = c.astype(np.double)<EOL><DEDENT>cnt, iaxis = [int(t) for t in [m, axis]]<EOL>if cnt != m:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>if cnt < <NUM_LIT:0>:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>if iaxis != axis:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>if not -c.ndim <= iaxis < c.ndim:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>if iaxis < <NUM_LIT:0>:<EOL><INDENT>iaxis += c.ndim<EOL><DEDENT>if cnt == <NUM_LIT:0>:<EOL><INDENT>return c<EOL><DEDENT>c = np.rollaxis(c, iaxis)<EOL>n = len(c)<EOL>if cnt >= n:<EOL><INDENT>c = c[:<NUM_LIT:1>]*<NUM_LIT:0><EOL><DEDENT>else:<EOL><INDENT>for i in range(cnt):<EOL><INDENT>n = n - <NUM_LIT:1><EOL>c *= scl<EOL>der = np.empty((n,) + c.shape[<NUM_LIT:1>:], dtype=c.dtype)<EOL>for j in range(n, <NUM_LIT:2>, -<NUM_LIT:1>):<EOL><INDENT>der[j - <NUM_LIT:1>] = (<NUM_LIT:2>*j - <NUM_LIT:1>)*c[j]<EOL>c[j - <NUM_LIT:2>] += c[j]<EOL><DEDENT>if n > <NUM_LIT:1>:<EOL><INDENT>der[<NUM_LIT:1>] = <NUM_LIT:3>*c[<NUM_LIT:2>]<EOL><DEDENT>der[<NUM_LIT:0>] = c[<NUM_LIT:1>]<EOL>c = der<EOL><DEDENT><DEDENT>c = np.rollaxis(c, <NUM_LIT:0>, iaxis + <NUM_LIT:1>)<EOL>return c<EOL>

Differentiate a Legendre series.

Returns the Legendre series coefficients `c` differentiated `m` times
along `axis`.  At each iteration the result is multiplied by `scl` (the
scaling factor is for use in a linear change of variable). The argument
`c` is an array of coefficients from low to high degree along each
axis, e.g., [1,2,3] represents the series ``1*L_0 + 2*L_1 + 3*L_2``
while [[1,2],[1,2]] represents ``1*L_0(x)*L_0(y) + 1*L_1(x)*L_0(y) +
2*L_0(x)*L_1(y) + 2*L_1(x)*L_1(y)`` if axis=0 is ``x`` and axis=1 is
``y``.

Parameters
----------
c : array_like
    Array of Legendre series coefficients. If c is multidimensional the
    different axis correspond to different variables with the degree in
    each axis given by the corresponding index.
m : int, optional
    Number of derivatives taken, must be non-negative. (Default: 1)
scl : scalar, optional
    Each differentiation is multiplied by `scl`.  The end result is
    multiplication by ``scl**m``.  This is for use in a linear change of
    variable. (Default: 1)
axis : int, optional
    Axis over which the derivative is taken. (Default: 0).

    .. versionadded:: 1.7.0

Returns
-------
der : ndarray
    Legendre series of the derivative.

See Also
--------
legint

Notes
-----
In general, the result of differentiating a Legendre series does not
resemble the same operation on a power series. Thus the result of this
function may be "unintuitive," albeit correct; see Examples section
below.

Examples
--------
>>> from numpy.polynomial import legendre as L
>>> c = (1,2,3,4)
>>> L.legder(c)
array([  6.,   9.,  20.])
>>> L.legder(c, 3)
array([ 60.])
>>> L.legder(c, scl=-1)
array([ -6.,  -9., -20.])
>>> L.legder(c, 2,-1)
array([  9.,  60.])


def legint(c, m=<NUM_LIT:1>, k=[], lbnd=<NUM_LIT:0>, scl=<NUM_LIT:1>, axis=<NUM_LIT:0>):

c = np.array(c, ndmin=<NUM_LIT:1>, copy=<NUM_LIT:1>)<EOL>if c.dtype.char in '<STR_LIT>':<EOL><INDENT>c = c.astype(np.double)<EOL><DEDENT>if not np.iterable(k):<EOL><INDENT>k = [k]<EOL><DEDENT>cnt, iaxis = [int(t) for t in [m, axis]]<EOL>if cnt != m:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>if cnt < <NUM_LIT:0>:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>if len(k) > cnt:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>if iaxis != axis:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>if not -c.ndim <= iaxis < c.ndim:<EOL><INDENT>raise ValueError("<STR_LIT>")<EOL><DEDENT>if iaxis < <NUM_LIT:0>:<EOL><INDENT>iaxis += c.ndim<EOL><DEDENT>if cnt == <NUM_LIT:0>:<EOL><INDENT>return c<EOL><DEDENT>c = np.rollaxis(c, iaxis)<EOL>k = list(k) + [<NUM_LIT:0>]*(cnt - len(k))<EOL>for i in range(cnt):<EOL><INDENT>n = len(c)<EOL>c *= scl<EOL>if n == <NUM_LIT:1> and np.all(c[<NUM_LIT:0>] == <NUM_LIT:0>):<EOL><INDENT>c[<NUM_LIT:0>] += k[i]<EOL><DEDENT>else:<EOL><INDENT>tmp = np.empty((n + <NUM_LIT:1>,) + c.shape[<NUM_LIT:1>:], dtype=c.dtype)<EOL>tmp[<NUM_LIT:0>] = c[<NUM_LIT:0>]*<NUM_LIT:0><EOL>tmp[<NUM_LIT:1>] = c[<NUM_LIT:0>]<EOL>if n > <NUM_LIT:1>:<EOL><INDENT>tmp[<NUM_LIT:2>] = c[<NUM_LIT:1>]/<NUM_LIT:3><EOL><DEDENT>for j in range(<NUM_LIT:2>, n):<EOL><INDENT>t = c[j]/(<NUM_LIT:2>*j + <NUM_LIT:1>)<EOL>tmp[j + <NUM_LIT:1>] = t<EOL>tmp[j - <NUM_LIT:1>] -= t<EOL><DEDENT>tmp[<NUM_LIT:0>] += k[i] - legval(lbnd, tmp)<EOL>c = tmp<EOL><DEDENT><DEDENT>c = np.rollaxis(c, <NUM_LIT:0>, iaxis + <NUM_LIT:1>)<EOL>return c<EOL>

Integrate a Legendre series.

Returns the Legendre series coefficients `c` integrated `m` times from
`lbnd` along `axis`. At each iteration the resulting series is
**multiplied** by `scl` and an integration constant, `k`, is added.
The scaling factor is for use in a linear change of variable.  ("Buyer
beware": note that, depending on what one is doing, one may want `scl`
to be the reciprocal of what one might expect; for more information,
see the Notes section below.)  The argument `c` is an array of
coefficients from low to high degree along each axis, e.g., [1,2,3]
represents the series ``L_0 + 2*L_1 + 3*L_2`` while [[1,2],[1,2]]
represents ``1*L_0(x)*L_0(y) + 1*L_1(x)*L_0(y) + 2*L_0(x)*L_1(y) +
2*L_1(x)*L_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``.

Parameters
----------
c : array_like
    Array of Legendre series coefficients. If c is multidimensional the
    different axis correspond to different variables with the degree in
    each axis given by the corresponding index.
m : int, optional
    Order of integration, must be positive. (Default: 1)
k : {[], list, scalar}, optional
    Integration constant(s).  The value of the first integral at
    ``lbnd`` is the first value in the list, the value of the second
    integral at ``lbnd`` is the second value, etc.  If ``k == []`` (the
    default), all constants are set to zero.  If ``m == 1``, a single
    scalar can be given instead of a list.
lbnd : scalar, optional
    The lower bound of the integral. (Default: 0)
scl : scalar, optional
    Following each integration the result is *multiplied* by `scl`
    before the integration constant is added. (Default: 1)
axis : int, optional
    Axis over which the integral is taken. (Default: 0).

    .. versionadded:: 1.7.0

Returns
-------
S : ndarray
    Legendre series coefficient array of the integral.

Raises
------
ValueError
    If ``m < 0``, ``len(k) > m``, ``np.isscalar(lbnd) == False``, or
    ``np.isscalar(scl) == False``.

See Also
--------
legder

Notes
-----
Note that the result of each integration is *multiplied* by `scl`.
Why is this important to note?  Say one is making a linear change of
variable :math:`u = ax + b` in an integral relative to `x`.  Then
.. math::`dx = du/a`, so one will need to set `scl` equal to
:math:`1/a` - perhaps not what one would have first thought.

Also note that, in general, the result of integrating a C-series needs
to be "reprojected" onto the C-series basis set.  Thus, typically,
the result of this function is "unintuitive," albeit correct; see
Examples section below.

Examples
--------
>>> from numpy.polynomial import legendre as L
>>> c = (1,2,3)
>>> L.legint(c)
array([ 0.33333333,  0.4       ,  0.66666667,  0.6       ])
>>> L.legint(c, 3)
array([  1.66666667e-02,  -1.78571429e-02,   4.76190476e-02,
        -1.73472348e-18,   1.90476190e-02,   9.52380952e-03])
>>> L.legint(c, k=3)
array([ 3.33333333,  0.4       ,  0.66666667,  0.6       ])
>>> L.legint(c, lbnd=-2)
array([ 7.33333333,  0.4       ,  0.66666667,  0.6       ])
>>> L.legint(c, scl=2)
array([ 0.66666667,  0.8       ,  1.33333333,  1.2       ])


c = np.array(c, ndmin=<NUM_LIT:1>, copy=<NUM_LIT:0>)<EOL>if c.dtype.char in '<STR_LIT>':<EOL><INDENT>c = c.astype(np.double)<EOL><DEDENT>if isinstance(x, (tuple, list)):<EOL><INDENT>x = np.asarray(x)<EOL><DEDENT>if isinstance(x, np.ndarray) and tensor:<EOL><INDENT>c = c.reshape(c.shape + (<NUM_LIT:1>,)*x.ndim)<EOL><DEDENT>if len(c) == <NUM_LIT:1>:<EOL><INDENT>c0 = c[<NUM_LIT:0>]<EOL>c1 = <NUM_LIT:0><EOL><DEDENT>elif len(c) == <NUM_LIT:2>:<EOL><INDENT>c0 = c[<NUM_LIT:0>]<EOL>c1 = c[<NUM_LIT:1>]<EOL><DEDENT>else:<EOL><INDENT>nd = len(c)<EOL>c0 = c[-<NUM_LIT:2>]<EOL>c1 = c[-<NUM_LIT:1>]<EOL>for i in range(<NUM_LIT:3>, len(c) + <NUM_LIT:1>):<EOL><INDENT>tmp = c0<EOL>nd = nd - <NUM_LIT:1><EOL>c0 = c[-i] - (c1*(nd - <NUM_LIT:1>))/nd<EOL>c1 = tmp + (c1*x*(<NUM_LIT:2>*nd - <NUM_LIT:1>))/nd<EOL><DEDENT><DEDENT>return c0 + c1*x<EOL>

Evaluate a Legendre series at points x.

If `c` is of length `n + 1`, this function returns the value:

.. math:: p(x) = c_0 * L_0(x) + c_1 * L_1(x) + ... + c_n * L_n(x)

The parameter `x` is converted to an array only if it is a tuple or a
list, otherwise it is treated as a scalar. In either case, either `x`
or its elements must support multiplication and addition both with
themselves and with the elements of `c`.

If `c` is a 1-D array, then `p(x)` will have the same shape as `x`.  If
`c` is multidimensional, then the shape of the result depends on the
value of `tensor`. If `tensor` is true the shape will be c.shape[1:] +
x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that
scalars have shape (,).

Trailing zeros in the coefficients will be used in the evaluation, so
they should be avoided if efficiency is a concern.

Parameters
----------
x : array_like, compatible object
    If `x` is a list or tuple, it is converted to an ndarray, otherwise
    it is left unchanged and treated as a scalar. In either case, `x`
    or its elements must support addition and multiplication with
    with themselves and with the elements of `c`.
c : array_like
    Array of coefficients ordered so that the coefficients for terms of
    degree n are contained in c[n]. If `c` is multidimensional the
    remaining indices enumerate multiple polynomials. In the two
    dimensional case the coefficients may be thought of as stored in
    the columns of `c`.
tensor : boolean, optional
    If True, the shape of the coefficient array is extended with ones
    on the right, one for each dimension of `x`. Scalars have dimension 0
    for this action. The result is that every column of coefficients in
    `c` is evaluated for every element of `x`. If False, `x` is broadcast
    over the columns of `c` for the evaluation.  This keyword is useful
    when `c` is multidimensional. The default value is True.

    .. versionadded:: 1.7.0

Returns
-------
values : ndarray, algebra_like
    The shape of the return value is described above.

See Also
--------
legval2d, leggrid2d, legval3d, leggrid3d

Notes
-----
The evaluation uses Clenshaw recursion, aka synthetic division.

Examples
--------


try:<EOL><INDENT>x, y = np.array((x, y), copy=<NUM_LIT:0>)<EOL><DEDENT>except:<EOL><INDENT>raise ValueError('<STR_LIT>')<EOL><DEDENT>c = legval(x, c)<EOL>c = legval(y, c, tensor=False)<EOL>return c<EOL>

Evaluate a 2-D Legendre series at points (x, y).

This function returns the values:

.. math:: p(x,y) = \\sum_{i,j} c_{i,j} * L_i(x) * L_j(y)

The parameters `x` and `y` are converted to arrays only if they are
tuples or a lists, otherwise they are treated as a scalars and they
must have the same shape after conversion. In either case, either `x`
and `y` or their elements must support multiplication and addition both
with themselves and with the elements of `c`.

If `c` is a 1-D array a one is implicitly appended to its shape to make
it 2-D. The shape of the result will be c.shape[2:] + x.shape.

Parameters
----------
x, y : array_like, compatible objects
    The two dimensional series is evaluated at the points `(x, y)`,
    where `x` and `y` must have the same shape. If `x` or `y` is a list
    or tuple, it is first converted to an ndarray, otherwise it is left
    unchanged and if it isn't an ndarray it is treated as a scalar.
c : array_like
    Array of coefficients ordered so that the coefficient of the term
    of multi-degree i,j is contained in ``c[i,j]``. If `c` has
    dimension greater than two the remaining indices enumerate multiple
    sets of coefficients.

Returns
-------
values : ndarray, compatible object
    The values of the two dimensional Legendre series at points formed
    from pairs of corresponding values from `x` and `y`.

See Also
--------
legval, leggrid2d, legval3d, leggrid3d

Notes
-----

.. versionadded::1.7.0


c = legval(x, c)<EOL>c = legval(y, c)<EOL>return c<EOL>

Evaluate a 2-D Legendre series on the Cartesian product of x and y.

This function returns the values:

.. math:: p(a,b) = \sum_{i,j} c_{i,j} * L_i(a) * L_j(b)

where the points `(a, b)` consist of all pairs formed by taking
`a` from `x` and `b` from `y`. The resulting points form a grid with
`x` in the first dimension and `y` in the second.

The parameters `x` and `y` are converted to arrays only if they are
tuples or a lists, otherwise they are treated as a scalars. In either
case, either `x` and `y` or their elements must support multiplication
and addition both with themselves and with the elements of `c`.

If `c` has fewer than two dimensions, ones are implicitly appended to
its shape to make it 2-D. The shape of the result will be c.shape[2:] +
x.shape + y.shape.

Parameters
----------
x, y : array_like, compatible objects
    The two dimensional series is evaluated at the points in the
    Cartesian product of `x` and `y`.  If `x` or `y` is a list or
    tuple, it is first converted to an ndarray, otherwise it is left
    unchanged and, if it isn't an ndarray, it is treated as a scalar.
c : array_like
    Array of coefficients ordered so that the coefficient of the term of
    multi-degree i,j is contained in `c[i,j]`. If `c` has dimension
    greater than two the remaining indices enumerate multiple sets of
    coefficients.

Returns
-------
values : ndarray, compatible object
    The values of the two dimensional Chebyshev series at points in the
    Cartesian product of `x` and `y`.

See Also
--------
legval, legval2d, legval3d, leggrid3d

Notes
-----

.. versionadded::1.7.0


try:<EOL><INDENT>x, y, z = np.array((x, y, z), copy=<NUM_LIT:0>)<EOL><DEDENT>except:<EOL><INDENT>raise ValueError('<STR_LIT>')<EOL><DEDENT>c = legval(x, c)<EOL>c = legval(y, c, tensor=False)<EOL>c = legval(z, c, tensor=False)<EOL>return c<EOL>

Evaluate a 3-D Legendre series at points (x, y, z).

This function returns the values:

.. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * L_i(x) * L_j(y) * L_k(z)

The parameters `x`, `y`, and `z` are converted to arrays only if
they are tuples or a lists, otherwise they are treated as a scalars and
they must have the same shape after conversion. In either case, either
`x`, `y`, and `z` or their elements must support multiplication and
addition both with themselves and with the elements of `c`.

If `c` has fewer than 3 dimensions, ones are implicitly appended to its
shape to make it 3-D. The shape of the result will be c.shape[3:] +
x.shape.

Parameters
----------
x, y, z : array_like, compatible object
    The three dimensional series is evaluated at the points
    `(x, y, z)`, where `x`, `y`, and `z` must have the same shape.  If
    any of `x`, `y`, or `z` is a list or tuple, it is first converted
    to an ndarray, otherwise it is left unchanged and if it isn't an
    ndarray it is  treated as a scalar.
c : array_like
    Array of coefficients ordered so that the coefficient of the term of
    multi-degree i,j,k is contained in ``c[i,j,k]``. If `c` has dimension
    greater than 3 the remaining indices enumerate multiple sets of
    coefficients.

Returns
-------
values : ndarray, compatible object
    The values of the multidimensional polynomial on points formed with
    triples of corresponding values from `x`, `y`, and `z`.

See Also
--------
legval, legval2d, leggrid2d, leggrid3d

Notes
-----

.. versionadded::1.7.0


c = legval(x, c)<EOL>c = legval(y, c)<EOL>c = legval(z, c)<EOL>return c<EOL>

Evaluate a 3-D Legendre series on the Cartesian product of x, y, and z.

This function returns the values:

.. math:: p(a,b,c) = \\sum_{i,j,k} c_{i,j,k} * L_i(a) * L_j(b) * L_k(c)

where the points `(a, b, c)` consist of all triples formed by taking
`a` from `x`, `b` from `y`, and `c` from `z`. The resulting points form
a grid with `x` in the first dimension, `y` in the second, and `z` in
the third.

The parameters `x`, `y`, and `z` are converted to arrays only if they
are tuples or a lists, otherwise they are treated as a scalars. In
either case, either `x`, `y`, and `z` or their elements must support
multiplication and addition both with themselves and with the elements
of `c`.

If `c` has fewer than three dimensions, ones are implicitly appended to
its shape to make it 3-D. The shape of the result will be c.shape[3:] +
x.shape + y.shape + z.shape.

Parameters
----------
x, y, z : array_like, compatible objects
    The three dimensional series is evaluated at the points in the
    Cartesian product of `x`, `y`, and `z`.  If `x`,`y`, or `z` is a
    list or tuple, it is first converted to an ndarray, otherwise it is
    left unchanged and, if it isn't an ndarray, it is treated as a
    scalar.
c : array_like
    Array of coefficients ordered so that the coefficients for terms of
    degree i,j are contained in ``c[i,j]``. If `c` has dimension
    greater than two the remaining indices enumerate multiple sets of
    coefficients.

Returns
-------
values : ndarray, compatible object
    The values of the two dimensional polynomial at points in the Cartesian
    product of `x` and `y`.

See Also
--------
legval, legval2d, leggrid2d, legval3d

Notes
-----

.. versionadded::1.7.0