arraytools — pyFormex 1.0.5 documentation

# Source code for arraytools

    #
    ##
    ##  This file is part of pyFormex 1.0.5  (Sat Feb 16 10:40:32 CET 2019)
    ##  pyFormex is a tool for generating, manipulating and transforming 3D
    ##  geometrical models by sequences of mathematical operations.
    ##  Home page: http://pyformex.org
    ##  Project page:  http://savannah.nongnu.org/projects/pyformex/
    ##  Copyright 2004-2018 (C) Benedict Verhegghe (benedict.verhegghe@ugent.be)
    ##  Distributed under the GNU General Public License version 3 or later.
    ##
    ##  This program is free software: you can redistribute it and/or modify
    ##  it under the terms of the GNU General Public License as published by
    ##  the Free Software Foundation, either version 3 of the License, or
    ##  (at your option) any later version.
    ##
    ##  This program is distributed in the hope that it will be useful,
    ##  but WITHOUT ANY WARRANTY; without even the implied warranty of
    ##  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
    ##  GNU General Public License for more details.
    ##
    ##  You should have received a copy of the GNU General Public License
    ##  along with this program.  If not, see http://www.gnu.org/licenses/.
    ##

    """A collection of numerical utilities.

    This module contains a large collection of numerical utilities.
    Many of them are related to processing arrays. Some are similar
    to existing NumPy functions but offer some extended functionality.

    Note
    ----
        While these functions were historically developed for pyFormex,
        this module only depends on numpy and can be used outside
        of pyFormex without changes.

    Definitions imported from numpy
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    These all have the obvious meaning. Seed numpy documentation for details:
    **pi**, **sin**, **cos**, **tan**,
    **arcsin**, **arccos**, **arctan**, **arctan2**,
    **sqrt**, **abs**
    **linalg.norm**

    """
    from __future__ import absolute_import, division, print_function

    import warnings
    from itertools import combinations, permutations
    import numpy as np
    from numpy import random

    # TODO: REMOVE
    # For now, we need it for the doctests, and for other modules
    # that import * from arraytools
    from numpy import *

    # TODO: move this to script globals
    from numpy import pi, sin, cos, tan, arcsin, arccos, arctan, arctan2, sqrt, abs

    # TODO: this should be merged with the norm2, norm, min, max functions
    # from plugins.postproc
    from numpy.linalg import norm

    _attributes_ = [ 'Float', 'Int', 'DEG', 'RAD', 'golden_ratio' ]

    # default float and int types
    Float = np.float32
    Int = np.int32

    # Some constants
    DEG = np.pi/180.
    """float: multiplier to transform degrees to radians = pi/180."""
    RAD = 1.
    """float: multiplier to transform radians to radians"""

    golden_ratio = 0.5 * (1.0 + sqrt(5.))
    """golden ratio is defined as  0.5 * (1.0 + sqrt(5.))"""

    #
    # Functions for checking types, values and array dimensions
    ###########################################################

    def isInt(obj):
        """Test if an object is an integer number.

        Returns
        -------
        bool
            True if the object is a single integer number (a Python ``int``
            or a ``numpy.integer`` type), else False.

        Examples
        --------
        >>> isInt(1)
        True
        >>> isInt(np.arange(3)[1])
        True
        """
        return isinstance(obj, (int, np.integer))

    def isFloat(obj):
        """Test if an object is a floating point number.

        Returns
        -------
        bool
            True if the object is a single floating point number (a Python
            ``float`` or a ``numpy.floating`` type), else False.

        Examples
        --------
        >>> isFloat(1.)
        True
        >>> isFloat(array([1,2],dtype=np.float32)[1])
        True
        """
        return isinstance(obj, (float, np.floating))

    def isNum(obj):
        """Test if an object is an integer or a floating point number.

        Returns
        -------
        bool
            True if the object is a single integer or floating point
            number, else False. The type of the object can be either
            a Python ``int`` or ``float`` or a numpy ``integer`` or ``floating``.

        Examples
        --------
        >>> isNum(1)
        True
        >>> isNum(1.0)
        True
        >>> isNum(array([1,2],dtype=np.int32)[1])
        True
        >>> isNum(array([1,2],dtype=np.float32)[1])
        True
        """
        return isInt(obj) or isFloat(obj)

    def checkInt(value,min=None,max=None):
        """Check that a value is an int in the range min..max.

        Parameters
        ----------
        value: int-like
            The value to check.
        min: int, optional
            If provided, minimal value to be accepted.
        max: int, optional
            If provided, maximal value to be accepted.

        Returns
        -------
        checked_int: int
            An integer not exceeding the provided boundaries.

        Raises
        ------
        ValueError:
            If the value is not convertible to an integer type or exceeds one
            of the specified boundaries.

        Examples
        --------
        >>> checkInt(1)
        1
        >>> checkInt(1,min=0,max=1)
        1
        >>> checkInt('2',min=0)
        2
        """
        try:
            a = int(value)
            if min is not None and a < min:
                raise ValueError
            if max is not None and a > max:
                raise ValueError
            return a
        except ValueError:
            raise ValueError("Expected an int in the range(%s, %s), got: %s" % (min, max, value))

    def checkFloat(value,min=None,max=None):
        """Check that a value is a float in the range min..max.

        Parameters
        ----------
        value: float-like
            The value to check
        min: float-like, optional
            If provided, minimal value to be accepted.
        max: float-like, optional
            If provided, maximal value to be accepted.

        Returns
        -------
        checked_float: float
            A float not exceeding the provided boundaries.

        Raises
        ------
        ValueError:
            If the value is not convertible to a float type or exceeds one of the
            specified boundaries.

        Examples
        --------
        >>> checkFloat(1)
        1.0
        >>> checkFloat(1,min=0,max=1)
        1.0
        >>> checkFloat('2',min=0)
        2.0
        """
        try:
            a = float(value)
            if min is not None and a < min:
                raise ValueError
            if max is not None and a > max:
                raise ValueError
            return a
        except:
            raise ValueError("Expected a float in the range(%s, %s), got: %s" % (min, max, value))

    def checkBroadcast(shape1,shape2):
        """Check that two array shapes are broadcast compatible.

        In many numerical operations, NumPy will automatically broadcast
        arrays of different shapes to a single shape, if they have broadcast
        compatible shapes. Two array shapes are broadcast compatible if, in
        all the last dimensions that exist in both arrays, either the shape
        of both arrays has the same length, or one of the shapes has a length
        1.

        Parameters
        ----------
        shape1: tuple of ints
            Shape of first array
        shape2: tuple of ints
            Shape of second array

        Returns
        -------
        tuple of ints
            The broadcasted shape of the arrays.

        Raises
        ------
        ValueError: Shapes are not broadcast compatible
            If the two shapes can not be broadcast to a single one.

        Examples
        --------
        >>> checkBroadcast((8,1,6,1),(7,1,5))
        (8, 7, 6, 5)
        >>> checkBroadcast((5,4),(1,))
        (5, 4)
        >>> checkBroadcast((5,4),(4,))
        (5, 4)
        >>> checkBroadcast((15,3,5),(15,1,5))
        (15, 3, 5)
        >>> checkBroadcast((15,3,5),(3,5))
        (15, 3, 5)
        >>> checkBroadcast((15,3,5),(3,1))
        (15, 3, 5)
        >>> checkBroadcast((7,1,5),(8,1,6,1))
        (8, 7, 6, 5)

        """
        len1, len2 = len(shape1), len(shape2)
        if len1 < len2:
            shape1, shape2 = shape2, shape1
            len1, len2 = len2, len1
        shape = list(shape1[:len1-len2])
        for n1,n2 in zip(shape1[len1-len2:],shape2):
            if n1 == 1 or n2 == 1 or n1==n2:
                shape.append(max(n1,n2))
            else:
                raise ValueError("Shapes are not broadcast compatible")
        return tuple(shape)

    def checkArray(a,shape=None,kind=None,allow=None,size=None,ndim=None,bcast=None):
        """Check that an array a has the correct shape, type and/or size.

        Parameters
        ----------
        a: array-like
            Anything that can be converted into a numpy array.
        shape: tuple of ints, optional
            If provided, the shape of the array should match this value
            along each axis for which a nonzero value is specified. The
            length of the shape tuple should also match.
        kind: dtype.kind character code, optional
            If provided, the array's dtype.kind should match this value,
            or one of the values in ``allow``, if provided.
        allow: string of dtype.kind character codes, optional
            If provided, and ``kind`` is also specified, any of the specified
            array types will also be accepted if it is convertible to the
            specified ``kind``. See also Notes below.
        size: int, optional
            If provided, the total array size should match this value.
        ndim: int, optional
            If provided, the array should have precisely ``ndim`` dimensions.
        bcast: tuple of ints, optional
            If provided, the array's shape should be broadcast comaptible with
            the specified shape.

        Returns
        -------
        array
            The checked_array is equivalent to the input data. It has the same
            contents and shape. It also has the same type, unless ``kind`` is
            is provided, in which case the result is converted to this type.

        Raises
        ------
        ValueError: invalid array
            The input data failed for one of more of the tests requested by the
            provided parameters.

        Notes
        -----
        Currently, the only allowed conversion from an ``allow`` type to ``kind``
        type, is 'i' to 'f'. Thus specifiying ``kind='f', allow='i'`` will accept
        integer input but return float32 output.

        See Also
        --------
        :func:`checkArray1D`

        Examples
        --------
        >>> checkArray([1,2])
        array([1, 2])
        >>> checkArray([1,2],shape=(2,))
        array([1, 2])
        >>> checkArray([[1,2],[3,4]],shape=(2,-1))
        array([[1, 2], [3, 4]])
        >>> checkArray([1,2],kind='i')
        array([1, 2])
        >>> checkArray([1,2],kind='f',allow='i')
        array([ 1., 2.], dtype=float32)
        >>> checkArray([1,2],size=2)
        array([1, 2])
        >>> checkArray([1,2],ndim=1)
        array([1, 2])
        >>> checkArray([[1,2],[3,4]],bcast=(3,1,2))
        array([[1, 2], [3, 4]])

        """
        try:
            a = np.asarray(a)
            if shape is not None:
                shape = np.asarray(shape)
                w = np.where(shape >= 0)[0]
                if (np.asarray(a.shape)[w] != shape[w]).any():
                    raise ValueError
            if kind is not None:
                if allow is None and a.dtype.kind != kind:
                    raise ValueError
                if kind == 'f':
                    a = a.astype(Float)
            if size is not None:
                if a.size != size:
                    raise ValueError
            if ndim is not None:
                if a.ndim != ndim:
                    raise ValueError
            if bcast is not None:
                checkBroadcast(a.shape,bcast)
            return a
        except ValueError:
            raise ValueError("Invalid array\n  Expected shape %s, kind %s, size %s, ndim %s\n  Got: shape %s, kind %s, size %s, ndim %s" % (shape, kind, size, ndim, a.shape, a.dtype.kind, a.size, a.ndim))

    def checkArray1D(a,kind=None,allow=None,size=None):
        """Check and force an array to be 1D.

        This is equivalent to calling :func:`checkArray` without the ``shape`` and
        ``ndim`` parameters, and then turning the result into a 1D array.

        Parameters
        ----------
        See :func:`checkArray`.

        Returns
        -------
        1D array
            The checked_array holds the same data as the input, but the shape
            is rveled to 1D. It also has the same type, unless ``kind`` is
            is provided, in which case the result is converted to this type.

        Examples
        --------
        >>> checkArray1D([[1,2],[3,4]],size=4)
        array([1, 2, 3, 4])

        """
        return checkArray(a, kind=kind, allow=allow, size=size).ravel()

    def checkUniqueNumbers(nrs,nmin=0,nmax=None):
        """Check that an array contains a set of unique integers in a given range.

        This functions tests that all integer numbers in the array are within the
        range math:`nmin <= i < nmax`. Default range is [0,unlimited].

        Parameters
        ----------
        nrs: array-like, int
            Input array with integers to check against provided limits.
        nmin: int or None, optional
            If not None, no value in ``a`` should be lower than this.
        nmax: int, optionallMinimum allowed value.
        - `nmax`: maximum allowed value + 1! If set to None, the test is skipped.
            If provided, no value in ``a`` should be higher than this.

        Returns
        -------
        1D int array
            Containing the sorted unique numbers from the input.

        Raises
        ------
        ValueError
            If the numbers are not unique or some input value surpasses one of
            the specified limmits.

        Examples
        --------
        >>> checkUniqueNumbers([0,5,1,7,2])
        array([0, 1, 2, 5, 7])
        >>> checkUniqueNumbers([0,5,1,7,-2],nmin=None)
        array([-2, 0, 1, 5, 7])
        """
        nrs = np.asarray(nrs)
        uniq = np.unique(nrs)
        if uniq.size != nrs.size or \
               (nmin is not None and uniq.min() < nmin) or \
               (nmax is not None and uniq.max() > nmax):
            raise ValueError("Values not unique or not in range")
        return uniq

    ###########################################################################
    ##
    ##   Some generic array functions
    ##
    #########################

    def addAxis(a,axis,warn=True):
        """Add an additional axis with length 1 to an array.

        Parameters
        ----------
        a : array-like
            The array in wich to add an extra axis.
        axis: int
            Position in the expanded array where the new axis is created.
            Should be in the range [-a.ndim,a.ndim].

        Returns
        -------
        array-like
            Same type and data as `a`, but shape has one extra axis with length 1
            in the specified position.

        Note
        ----
        This function is now equivalent to numpy.expand_dims. If a negative value
        is specified, a warning will be issued because of broken compatibility
        with older pyFormex versions (<= 1.0.4).

        Examples
        --------
        >>> A = array([[1,2,3],[4,5,6]])
        >>> addAxis(A,0)
        array([[[1, 2, 3],
                [4, 5, 6]]])
        >>> addAxis(A,1)
        array([[[1, 2, 3]],
        <BLANKLINE>
               [[4, 5, 6]]])
        >>> addAxis(A,-1)
        array([[[1],
                [2],
                [3]],
        <BLANKLINE>
               [[4],
                [5],
                [6]]])
        """
        if warn and axis < 0:
            warnings.warn('warn_addAxis_changed')
        return np.expand_dims(a,axis)

    def growAxis(a,add,axis=-1,fill=0):
        """Increase the length of an array axis.

        Parameters
        ----------
        a : array-like
            The array in which to extend n axis.
        add: int
            The length over which the specified axis should grow.
            If add<=0, the array is returned unchanged.
        axis: int
            Position of the target axis in the array. Default is last (-1).
        fill: int or float
            Value to set the new elements along the grown axis to.

        Returns
        -------
        array-like
            Same type and data as `a`, but length of specified axis has been
            increased with a value `add` and the new elements are filled with
            the value `fill`.

        Raises
        ------
        ValueError:
            If the specified axis exceeds the array dimensions.

        Examples
        --------
        >>> growAxis([[1,2,3],[4,5,6]],2)
        array([[1, 2, 3, 0, 0],
               [4, 5, 6, 0, 0]])
        >>> growAxis([[1,2,3],[4,5,6]],1,axis=0,fill=-3)
        array([[ 1,  2,  3],
               [ 4,  5,  6],
               [-3, -3, -3]])
        >>> growAxis([[1,2,3],[4,5,6]],-1)
        array([[1, 2, 3],
               [4, 5, 6]])
        """
        a = np.asarray(a)
        if axis >= a.ndim or axis < -a.ndim:
            raise ValueError("Array with ndim %s has no axis %s" % (a.ndim,axis))
        if add <= 0:
            return a
        else:
            missing = list(a.shape)
            missing[axis] = add
            return np.concatenate([a, fill * np.ones(missing, dtype=a.dtype)], axis=axis)

    def reorderAxis(a,order,axis=-1):
        """Reorder the planes of an array along the specified axis.

        Parameters
        ----------
        a : array-like
            The array in which to reorder the elements.
        order: int array-like | str
            Specifies how to reorder the elements. It can be an integer index
            which should be a permutation of `arange(a.shape[axis])`. Each value
            in the index specified the old index of the elements that should be
            placed at its position. This is equivalent to `a.take(order,axis)`.

            `order` can also be one of the following predefined sting values,
            resulting in the corresponding renumbering scheme being generated:

            - 'reverse': the elements along axis are placed in reverse order
            - 'random': the elements along axis are placed in random order

        axis: int
            The axis of the array along which the elements are to be reordered.
            Default is last (-1).

        Returns
        -------
        array-like
            Same type and data as `a`, but the element planes are along `axis`
            have been reordered.

        Examples
        --------
        >>> reorderAxis([[1,2,3],[4,5,6]], [2,0,1])
        array([[3, 1, 2],
               [6, 4, 5]])

        """
        a = np.asarray(a)
        n = a.shape[axis]
        if order == 'reverse':
            order = arange(n-1, -1, -1)
        elif order == 'random':
            order = random.permutation(n)
        else:
            order = np.asarray(order)
        return a.take(order, axis)

    def reverseAxis(a,axis=-1):
        """Reverse the order of the elements along an axis.

        Parameters
        ----------
        a : array-like
            The array in which to reorder the elements.
        axis: int
            The axis of the array along which the elements are to be reordered.
            Default is last (-1).

        Returns
        -------
        array-like
            Same type and data as `a`, but the elements along `axis` are now
            in reversed order.

        Note
        ----
        This function is especially useful if axis has a computed value.
        If the axis is known in advance, it is more efficient to use
        an indexing operation. Thus **reverseAxis(A,-1)** is equivalent to
        **A[...,::-1]**.

        Examples
        --------
        >>> A = array([[1,2,3],[4,5,6]])
        >>> reverseAxis(A)
        array([[3, 2, 1],
               [6, 5, 4]])
        >>> A[...,::-1]
        array([[3, 2, 1],
               [6, 5, 4]])

        """
        return reorderAxis(a, 'reverse', axis)

    def interleave(a,b):
        """Interleave two arrays along their first axis.

        Parameters
        ----------
        a: array-like
            First array
        b: array-like
            Second array, with same type and shape as `a`, except that the first
            dimension may be one less than that of `a`, and if data type of `b`
            can be one that is convertible to that of `a`.

        Returns
        -------
        array-like
            An array with interleaved rows from `a` and `b`. The array has the
            datatype of `a` and its first axis has the combined length of that
            of `a` and `b`.

        Examples
        --------
        >>> interleave(arange(4), 10*arange(3))
        array([ 0,  0,  1, 10,  2, 20,  3])
        >>> a = arange(8).reshape(2,4)
        >>> print(interleave(a,10*a))
        [[ 0  1  2  3]
         [ 0 10 20 30]
         [ 4  5  6  7]
         [40 50 60 70]]

        """
        a = np.asarray(a)
        b = np.asarray(b)
        if a.shape[0]-b.shape[0] not in (0,1) or a.shape[1:] != b.shape[1:]:
            raise ValueError("Array sizes not compatible for interleave")
        c = empty((a.shape[0]+b.shape[0],)+a.shape[1:],dtype=a.dtype)
        c[0::2] = a
        c[1::2] = b
        return c

    def multiplex(a,n,axis,warn=True):
        """Multiplex an array over a length n in direction of a new axis.

        Inserts a new axis in the array at the specified position and repeats
        the data of the array `n` times in the direction of the new axis.

        Parameters
        ----------
        a: array-like
            The input array.
        n: int
            Number of times to repeat the data in direction of `axis`.
        axis: int, optional
            Position of the new axis in the expanded array. Should be
            in the range -a.ndim..a.ndim.

        Returns
        -------
        array
            An array with n times the original data repeated in the direction
            of the specified axis.

        See Also
        --------
        repeatValues: Repeat values in a 1-dim array a number of times

        Examples
        --------
        >>> a = arange(6).reshape(2,3)
        >>> print(a)
        [[0 1 2]
         [3 4 5]]
        >>> print(multiplex(a,4,-1))
        [[[0 0 0 0]
          [1 1 1 1]
          [2 2 2 2]]
        <BLANKLINE>
         [[3 3 3 3]
          [4 4 4 4]
          [5 5 5 5]]]
        >>> print(multiplex(a,4,-2))
        [[[0 1 2]
          [0 1 2]
          [0 1 2]
          [0 1 2]]
        <BLANKLINE>
         [[3 4 5]
          [3 4 5]
          [3 4 5]
          [3 4 5]]]

        """
        if warn and axis < 0:
            warnings.warn('warn_multiplex_changed')
        return addAxis(a, axis, warn=False).repeat(n, axis=axis)

    def repeatValues(a,n):
        """Repeat values in a 1-dim array a number of times.

        Parameters
        ----------
        a: array-like, 1-dim
            The input array. Can be a list or a single element.
        n: int array-like, 1-dim
            Number of times to repeat the corresponding value of ``a``.
            If ``n`` has less elements than ``a``, it is reused until
            the end of ``a`` is reached.

        Returns
        -------
        array
            An 1-dim array of the same dtype as ``a`` with the value ``a[i]``
            repeated ``n[i]`` times.

        See Also
        --------
        multiplex: Multiplex an array over a length n in direction of a new axis

        Examples
        --------
        >>> repeatValues(2,3)
        array([2, 2, 2])
        >>> repeatValues([2,3],2)
        array([2, 2, 3, 3])
        >>> repeatValues([2,3,4],[2,3])
        array([2, 2, 3, 3, 3, 4, 4])
        >>> repeatValues(1.6,[3,5])
        array([ 1.6,  1.6,  1.6])

        """
        a = checkArray1D(a)
        n = checkArray1D(n,kind='i')
        n = np.resize(n, a.shape)
        return np.concatenate([np.resize(ai, ni) for ai, ni in zip(a, n)])

    def concat(al,axis=0):
        """Smart array concatenation ignoring empty arrays.

        Parameters
        ----------
        al: list of arrays
            All arrays should have same shape except for the length of
            the concatenation axis, or be empty arrays.
        axis: int
            The axis along which the arrays are concatenated.

        Returns
        -------
        :array
            The concatenation of all non-empty arrays in the list,
            or an empty array if all arrays in the list are empty.

        Note
        ----
        This is just like numpy.concatenate, but allows empty arrays
        in the list and silently ignores them.

        Examples
        --------
        >>> concat([array([0,1]),array([]),array([2,3])])
        array([0, 1, 2, 3])

        """
        al = [a for a in al if a.size > 0]
        if len(al) > 0:
            return concatenate(al, axis=axis)
        else:
            return array([])

    def splitrange(n, nblk):
        """Split a range of integers 0..n in almost equal sized slices.

        Parameters
        ----------
        n: int
            Highest integer value in the range.
        nblk: int
            Number of blocks to split into. Should be <= n to allow splitting.

        Returns
        -------
        : 1-dim int array
            If nblk <= n, returns the boundaries that divide the integers
            in the range 0..n in nblk almost equal slices.
            The outer boundaries 0 and n are included, so the length of the
            array is nblk+1.
            If nblk >= n, returns range(n+1), thus all slices have length 1.

        Examples
        --------
        >>> splitrange(7,3)
        array([0, 2, 5, 7])
        """
        if n > nblk:
            ndata = (arange(nblk+1) * n * 1.0 / nblk).round().astype(int)
        else:
            ndata = arange(n+1)
        return ndata

    def splitar(a,nblk,axis=0,close=False):
        """Split an array in nblk subarrays along a given axis.

        Parameters
        ----------
        a: array-like
            Array to be divided in subarrays.
        nblk: int
            Number of subarrays to obtain. The subarrays will be
            of almost the same size.
        axis: int:
            Axis along which to split the array (default 0)
        close: bool
            If True, the last item of each block will be repeated
            as the first item of the next block.

        Returns
        -------
        : list of arrays
            A list of subarrays obtained by splitting a along the specified axis.
            All arrays have almost the same shape. The number of arrays is equal
            to nblk, unless nblk is larger than a.shape[axis], in which case a
            a list with only the original array is returned.

        Examples
        --------
        >>> splitar(arange(7),3)
        [array([0, 1]), array([2, 3, 4]), array([5, 6])]
        >>> splitar(arange(7),3,close=True)
        [array([0, 1, 2]), array([2, 3, 4]), array([4, 5, 6])]
        >>> X = array([[0.,1.,2.],[3.,4.,5.]])
        >>> splitar(X,2)
        [array([[ 0.,  1.,  2.]]), array([[ 3.,  4.,  5.]])]
        >>> splitar(np.matrix(X),2,axis=-1)
        [matrix([[ 0.,  1.],
                 [ 3.,  4.]]), matrix([[ 2.],
                 [ 5.]])]
        >>> splitar(X,3)
        [array([[ 0.,  1.,  2.],
               [ 3.,  4.,  5.]])]
        """
        a = asanyarray(a).swapaxes(axis,0)
        na = a.shape[0]
        if close:
            na -= 1
        if nblk > na:
            return [a]

        ndata = splitrange(na, nblk)
        k = 1 if close else 0
        return [ a[i:j+k].swapaxes(0,axis) for i, j in zip(ndata[:-1], ndata[1:])]

    def minmax(a,axis=-1):
        """Compute the minimum and maximum along an axis.

        Parameters
        ----------
        a: array-like
            The data array for which to compute the minimum and maximum.
        axis: int
            The array axis along which to compute the minimum and maximum.

        Returns
        -------
        : array
            The array has the same dtype as `a`. It also has the same shape,
            except for the specified axis, which will have a length of 2.
            The first value along this axis holds the minimum value of the input,
            the second holds the maximum value.

        Examples
        --------
        >>> a = array([[ [1.,0.,0.], [0.,1.,0.] ],
        ...            [ [2.,0.,0.], [0.,2.,0.] ] ])
        >>> print(minmax(a,axis=1))
        [[[ 0. 0. 0.]
          [ 1. 1. 0.]]
        <BLANKLINE>
         [[ 0. 0. 0.]
          [ 2. 2. 0.]]]
        """
        return np.stack([a.min(axis=axis), a.max(axis=axis)], axis=axis)

    def stretch(a,min=None,max=None,axis=None):
        """Scale the values of an array to fill a given range.

        Parameters
        ----------
        a: array_like, int or float
            Input data.
        min: int or float, optional
            The targeted minimum value in the array. Same type as `a`.
            If not provided, the minimum of a is used.
        max: int or float, optional
            The targeted maximum value in the array. Same type as `a`.
            If not provided, the maximum of a is used.
        axis: int, optional
            If provided, each slice along the specified axis is
            independently scaled.

        Returns
        -------
        : array
            Array of the same type and size as the input array, but in which
            the values have been linearly scaled to fill the specified range.

        Examples
        --------
        >>> stretch([1.,2.,3.],min=0,max=1)
        array([ 0. , 0.5, 1. ])
        >>> A = arange(6).reshape(2,3)
        >>> stretch(A,min=20,max=30)
        array([[20, 22, 24],
               [26, 28, 30]])
        >>> stretch(A,min=20,max=30,axis=1)
        array([[20, 25, 30],
               [20, 25, 30]])
        >>> stretch(A,max=30)
        array([[ 0,  6, 12],
               [18, 24, 30]])
        >>> stretch(A,min=2,axis=1)
        array([[2, 4, 5],
               [2, 4, 5]])
        >>> stretch(A.astype(Float),min=2,axis=1)
        array([[ 2. ,  3.5,  5. ],
               [ 2. ,  3.5,  5. ]], dtype=float32)
        """
        a = np.asarray(a)
        atype = a.dtype
        if min is None:
            min = a.min()
        if max is None:
            max = a.max()
        if not min < max:
            raise ValueError('max must be larger than min in `rng` parameter.')

        amin = a.min(axis=axis)
        amax = a.max(axis=axis)
        if axis is not None:
            amin = addAxis(amin,axis)
            amax = addAxis(amax,axis)
        sc = amax-amin
        if atype.kind == 'i':
            sc = sc.astype(Float)
        b = (a-amin) / sc * (max-min) + min
        if atype.kind == 'i':
            b = b.round()
        return b.astype(atype)

    def stringar(s,a):
        """Nicely format a string followed by an array.

        Parameters
        ----------
        s: str
            String to appear before the formatted array
        a: array
            Array to be formatted after the string, with proper vertical
            alignment

        Returns
        -------
        : str
           A multiline string where the first line consists of the
           string s and the first line of the formatted array, and the next lines
           hold the remainder of the array lines, properly indented to align with
           the first line of the array.

        Examples
        --------
        >>> print(stringar("Indented array: ",np.arange(4).reshape(2,2)))
        Indented array: [[0 1]
                         [2 3]]

        """
        s = str(s)
        n = len(s)
        repl = ' '*n
        return s + str(a).replace('\n','\n'+repl)

    def printar(s,a):
        """Print a string followed by a vertically aligned array.

        Parameters
        ----------
        s: str
            String to appear before the formatted array
        a: array
            Array to be formatted after the string, with proper vertical
            alignment

        Note
        ----
        This is a shorthand for ``print(stringar(s,a))``.

        Examples
        --------
        >>> printar("Indented array: ",np.arange(4).reshape(2,2))
        Indented array: [[0 1]
                         [2 3]]

        """
        print(stringar(s,a))

    def writeArray(fil,array,sep=' '):
        """Write an array to an open file.

        This uses :func:`numpy.tofile` to write an array to an open file.

        Parameters
        ----------
        fil: file or str
            Open file object or filename.
        array: array-like
            The array to write to the file.
        sep: str
            If empty, the array is written in binary mode.
            If not empty, the array is written in text mode, with this
            string as separator between the elements.

        See also
        --------
        readArray

        """
        array.tofile(fil, sep=sep)

    def readArray(fil,dtype,shape,sep=' '):
        """Read data for an array with known size and type from an open file.

        This uses :func:`numpy.fromfile` to read an array with known shape and
        data type from an open file.

        Parameters
        ----------
        fil: file or str
            Open file object or filename.
        dtype: data-type
            Data type of the array to be read.
        shape: tuple of ints
            The shape of the array to be read.
        sep: str
            If not empty, the array is read in text mode, with this
            string as separator between the elements.
            If empty, the array is read in binary mode and an extra '\\n'
            after the data will be stripped off

        See Also
        --------
        writeArray
        """
        shape = np.asarray(shape)
        size = shape.prod()
        data = fromfile(fil, dtype=dtype, count=size, sep=sep).reshape(shape)
        if sep == '':
            pos = fil.tell()
            byte = fil.read(1)
            if not ord(byte) == 10:
                # not a newline: push back
                fil.seek(pos)
        return data

    def powers(x, n):
        """Compute all the powers of x from zero up to n.

        Parameters
        ----------
        x: int, float or array (int,float)
            The number or numbers to be raised to the specified powers.
        n: int
            Maximal power to raise the numbers to.

        Returns
        -------
        powers: list
            A list of numbers or arrays of the same shape and type as the input.
            The list contains ``N+1`` items, being the input raised to the powers
            in ``range(n+1)``.

        Examples
        --------
        >>> powers(2,5)
        [1, 2, 4, 8, 16, 32]
        >>> powers(array([1.0,2.0]),5)
        [array([ 1.,  1.]), array([ 1.,  2.]), array([ 1.,  4.]), \
        array([ 1.,  8.]), array([  1.,  16.]), array([  1.,  32.])]
        """
        return [ x ** i for i in range(n+1) ]

    ###########################################################################
    ##
    ##   some math functions
    ##
    #########################

    # Convenience functions: trigonometric functions with argument in degrees

    def sind(arg,angle_spec=DEG):
        """Return the sine of an angle in degrees.

        Parameters
        ----------
        arg: float number or array
            Angle(s) for which the sine is to be returned.
            By default, angles are specified in degrees (see ``angle_spec``).
        angle_spec: :py:attr:`DEG`, :py:attr:`RAD` or float
            Multiplier to apply to ``arg`` before taking the sine.
            The default multiplier DEG makes the argument
            being intrepreted as an angle in degrees. Use RAD when angles
            are specified in radians.

        Returns
        -------
        :float number or array
            The sine of the input angle(s)

        See also
        --------
        cosd
        tand
        arcsind
        arccosd
        arctand
        arctand2

        Examples
        --------
        >>> print(sind(30), sind(pi/6,RAD))
        0.5 0.5
        >>> sind(array([0.,30.,45.,60.,90.]))
        array([ 0.  ,  0.5 ,  0.71,  0.87,  1.  ])
        """
        return np.sin(arg*angle_spec)

    def cosd(arg,angle_spec=DEG):
        """Return the cosine of an angle in degrees.

        Parameters
        ----------
        arg: float number or array
            Angle(s) for which the cosine is to be returned.
            By default, angles are specified in degrees (see ``angle_spec``).
        angle_spec: :py:attr:`DEG`, :py:attr:`RAD` or float
            Multiplier to apply to ``arg`` before taking the sine.
            The default multiplier DEG makes the argument
            being intrepreted as an angle in degrees. Use RAD when angles
            are specified in radians.

        Returns
        -------
        :float number or array
            The cosine of the input angle(s)

        See also
        --------
        sind
        tand
        arcsind
        arccosd
        arctand
        arctand2

        Examples
        --------
        >>> print(cosd(60), cosd(pi/3,RAD))
        0.5 0.5
        """
        return np.cos(arg*angle_spec)

    def tand(arg,angle_spec=DEG):
        """Return the tangens of an angle in degrees.

        Parameters
        ----------
        arg: float number or array
            Angle(s) for which the tangens is to be returned.
            By default, angles are specified in degrees (see ``angle_spec``).
        angle_spec: :py:attr:`DEG`, :py:attr:`RAD` or float
            Multiplier to apply to ``arg`` before taking the sine.
            The default multiplier DEG makes the argument
            being intrepreted as an angle in degrees. Use RAD when angles
            are specified in radians.

        Returns
        -------
        :float number or array
            The tangens of the input angle(s)

        See also
        --------
        sind
        cosd
        arcsind
        arccosd
        arctand
        arctand2

        Examples
        --------
        >>> print(tand(45), tand(pi/4,RAD))
        1.0 1.0
        """
        return np.tan(arg*angle_spec)

    def arcsind(arg,angle_spec=DEG):
        """Return the angle whose sine is equal to the argument.

        Parameters
        ----------
        arg: float number or array, in the range -1.0 to 1.0.
            Value(s) for which to return the arcsine.
        angle_spec: :py:attr:`DEG`, :py:attr:`RAD` or float, nonzero.
            Divisor applied to the resulting angles before returning.
            The default divisor DEG makes the angles be returned in
            degrees. Use RAD to get angles in radians.

        Returns
        -------
        :float number or array
            The angle(s) for which the input value(s) is/are the cosine.
            The default ``angle_spec=DEG`` returns values in the range
            -90 to +90.

        See also
        --------
        sind
        cosd
        tand
        arccosd
        arctand
        arctand2

        Examples
        --------
        >>> print("{:.1f} {:.4f}".format(arcsind(0.5), arcsind(1.0,RAD)))
        30.0 1.5708
        >>> arcsind(-1)
        -90.0
        >>> arcsind(1)
        90.0
        """
        return np.arcsin(arg)/angle_spec

    def arccosd(arg,angle_spec=DEG):
        """Return the angle whose cosine is equal to the argument.

        Parameters
        ----------
        arg: float number or array, in the range -1.0 to 1.0.
            Value(s) for which to return the arccos.
        angle_spec: :py:attr:`DEG`, :py:attr:`RAD` or float, nonzero.
            Divisor applied to the resulting angles before returning.
            The default divisor DEG makes the angles be
            returned in degrees. Use RAD to get angles in radians.

        Returns
        -------
        :float number or array
            The angle(s) for which the input value(s) is/are the cosine.
            The default ``angle_spec=DEG`` returns values in the range
            0 to 180.

        See also
        --------
        sind
        cosd
        tand
        arcsind
        arctand
        arctand2

        Examples
        --------
        >>> print("{:.1f} {:.4f}".format(arccosd(0.5), arccosd(-1.0,RAD)))
        60.0 3.1416
        >>> arccosd(array([-1,0,1]))
        array([ 180.,   90.,    0.])
        """
        return np.arccos(arg)/angle_spec

    def arctand(arg,angle_spec=DEG):
        """Return the angle whose tangens is equal to the argument.

        Parameters
        ----------
        arg: float number or array.
            Value(s) for which to return the arctan.
        angle_spec: :py:attr:`DEG`, :py:attr:`RAD` or float, nonzero.
            Divisor applied to the resulting angles before returning.
            The default divisor DEG makes the angles be
            returned in degrees. Use RAD to get angles in radians.

        Returns
        -------
        :float number or array
            The angle(s) for which the input value(s) is/are the tangens.
            The default ``angle_spec=DEG`` returns values in the range
            -90 to +90.

        See also
        --------
        sind
        cosd
        tand
        arcsind
        arccosd
        arctand2

        Examples
        --------

        >>> print("{:.1f} {:.4f}".format(arctand(1.0), arctand(-1.0,RAD)))
        45.0 -0.7854
        >>> arctand(array([-inf,-1,0,1,inf]))
        array([-90., -45.,   0.,  45.,  90.])
        """
        return np.arctan(arg)/angle_spec

    def arctand2(sin,cos,angle_spec=DEG):
        """Return the angle whose sine and cosine values are given.

        Parameters
        ----------
        sin: float number or array with same shape as ``cos``.
            Sine value(s) for which to return the corresponding angle.
        cos: float number or array with same shape as ``sin``
            Cosine value(s) for which to return the corresponding angle.
        angle_spec: :py:attr:`DEG`, :py:attr:`RAD` or float, nonzero.
            Divisor applied to the resulting angles before returning.
            The default divisor DEG makes the angles be returned in degrees.
            Use RAD to get angles in radians.

        Returns
        -------
        :float number or array with same shape as ``sin`` and ``cos``.
            The angle(s) for which the input value(s) are the sine and
            cosine.
            The default ``angle_spec=DEG`` returns values in the range
            [-180, 180].

        Note
        ----
        The input values ``sin`` and ``cos`` are not restricted to the [-1.,1.]
        range. The returned angle is that for which the tangens is given by
        ``sin/cos``, but with a sine and cosine that have the same sign as the
        ``sin`` and ``cos`` values.

        See also
        --------
        sind
        cosd
        tand
        arcsind
        arccosd
        arctand

        Examples
        --------
        >>> print("{:.1f} {:.4f}".format(arctand2(0.0,-1.0), arctand2(-sqrt(0.5),-sqrt(0.5),RAD)))
        180.0 -2.3562
        >>> arctand2(array([0., 1., 0., -1.]), np.array([1., 0., -1., 0.]))
        array([   0.,   90.,  180.,  -90.])
        >>> arctand2(2.,2.)
        45.0
        """
        return np.arctan2(sin, cos)/angle_spec

    def niceLogSize(f):
        """Return an integer estimate of the magnitude of a float number.

        Parameters
        ----------
        f: float
            Value for which the integer magnitude has to be computed. The
            sign of the value is disregarded.

        Returns
        -------
        :int
            An integer magnitude estimator for the input.

        Note
        ----
        The returned value is the smallest integer ``e`` such that
        ``10**e > abs(f)``.
        If positive, it is equal to the number of digits
        before the decimal point; if negative, it is equal to the number of
        leading zeros after the decimal point.

        See also
        --------
        nicenumber

        Examples
        --------
        >>> print([ niceLogSize(a) for a in [1.3, 35679.23, 0.4, 0.0004567, -1.3] ])
        [1, 5, 0, -3, 1]
        """
        return int(np.ceil(np.log10(abs(f))))

    def niceNumber(f,round=np.ceil):
        """Return a nice number close to abs(f).

        A nice number is a number which only has only one significant digit
        (in the decimal system).

        Parameters
        ----------
        f: float
            A float number to approximate with a nice number. The sign of ``f``
            is disregarded.
        round: callable
            A function that rounds a float to the nearest integer. Useful
            functions are ``ceil``, ``floor`` and ``round`` from either
            NumPy or Python's math module. Default is ``numpy.ceil``.

        Returns
        -------
        :float
            A float value close to the input value, but having only
            a single decimal digit.

        Examples
        --------
        >>> numbers = [ 0.0837, 0.867, 8.5, 83.7, 93.7]
        >>> [ str(niceNumber(f)) for f in numbers ]
        ['0.09', '0.9', '9.0', '90.0', '100.0']
        >>> [ str(niceNumber(f,round=np.floor)) for f in numbers ]
        ['0.08', '0.8', '8.0', '80.0', '90.0']
        >>> [ str(niceNumber(f,round=np.round)) for f in numbers ]
        ['0.08', '0.9', '8.0', '80.0', '90.0']
        """
        fa = abs(f)
        s = "%.1e" % fa
        m, n = s.split('e')
        m = int(round(float(m)))
        n = int(n)
        return m*10.**n

    def isqrt(n):
        """Compute the square root of an integer number.

        Parameters
        ----------
        n: int
            An integer number that is a perfect square.

        Returns
        -------
        :int
            The square root from the input number

        Raises
        ------
        ValueError:
            If the input integer is not a perfect square.

        Examples
        --------
        >>> isqrt(36)
        6
        """
        import math
        i = int(math.sqrt(n) + 0.5)
        if i*i != n:
            raise ValueError('Input is not a perfect square: %s' % n)
        return i

    ###########################################################################
    ##
    ##   Vector operations
    ##
    #########################

    def dotpr (A,B,axis=-1):
        """Return the dot product of vectors of A and B in the direction of axis.

        Parameters
        ----------
        A: float array-like
            Array containing vectors in the direction of axis.
        B: float array-like
            Array containing vectors in the direction of axis. Same shape
            as A, or broadcast-compatible.
        axis: int
            Axis of A and B in which direction the vectors are layed out.
            Default is the last axis. A and B should have the same length
            along this axis.

        Returns
        -------
        : float array-like, shape as A and B with axis direction removed.
            The elements contain the dot product of the vectors of A and B at
            that position.

        Note
        ----
        This multiplies the elements of the A and B and then sums them
        in the direction of the specified axis.

        Examples
        --------
        >>> A = array( [[1.0, 1.0], [1.0,-1.0], [0.0, 5.0]] )
        >>> B = array( [[5.0, 3.0], [2.0, 3.0], [1.33,2.0]] )
        >>> print(dotpr(A,B))
        [ 8. -1. 10.]
        >>> print(dotpr(A,B,0))
        [ 7. 10.]

        """
        A = np.asarray(A)
        B = np.asarray(B)
        return (A*B).sum(axis)

    def length(A,axis=-1):
        """Returns the length of the vectors of A in the direction of axis.

        Parameters
        ----------
        A: float array-like
            Array containing vectors in the direction of axis.
        axis: int
            Axis of A in which direction the vectors are layed out.
            Default is the last axis. A and B shoud have the same length
            along this axis.

        Returns
        -------
        : float array-like, shape of A with axis direction removed.
            The elements contain the length of the vector in A at that position.

        Note
        ----
        This is equivalent with ``sqrt(dotpr(A,A))``.

        Examples
        --------
        >>> A = array( [[1.0, 1.0], [1.0,-1.0], [0.0, 5.0]] )
        >>> print(length(A))
        [ 1.41 1.41 5.  ]
        >>> print(length(A,0))
        [ 1.41 5.2 ]

        """
        A = np.asarray(A)
        return sqrt((A*A).sum(axis))

    def normalize(A,axis=-1,on_zeros='n',return_length=False,ignore_zeros=False):
        """Normalize the vectors of A in the direction of axis.

        Parameters
        ----------
        A: float array-like
            Array containing vectors in the direction of axis.
        axis: int
            Axis of A in which direction the vectors are layed out.
        on_zeros: 'n', 'e' or 'i'
            Specifies how to treat occurrences of zero length vectors (having
            all components equal to zero):

            - 'n': return a vector of nan values
            - 'e': raise a ValueError
            - 'i': ignore zero vectors and return them as such.

        return_length: bool
            If True, also returns also the length of the original vectors.

        ignore_zeros: bool
             (Deprecated) Equivalent to specifying ``on_zeros='i'``.

        Returns
        -------
        norm: float array
            Array with same shape as A but where each vector along axis
            has been rescaled so that its length is 1.
        len: float array, optional
            Array with shape like A but with axis removed. The length of
            the original vectors in the direction of axis. Only returned
            if ``return_length=True`` provided.

        Raises
        ------
        ValueError: Can not normalize zero-length vector
            If any of the  vectors of B is a zero vector.

        Examples
        --------
        >>> A = array( [[3.0, 3.0], [4.0,-3.0], [0.0, 0.0]] )
        >>> print(normalize(A))
        [[ 0.71  0.71]
         [ 0.8  -0.6 ]
         [  nan   nan]]
        >>> print(normalize(A,on_zeros='i'))
        [[ 0.71  0.71]
         [ 0.8  -0.6 ]
         [ 0.    0.  ]]
        >>> print(normalize(A,0))
        [[ 0.6   0.71]
         [ 0.8  -0.71]
         [ 0.    0.  ]]
        >>> n,l = normalize(A,return_length=True)
        >>> print(n)
        [[ 0.71  0.71]
         [ 0.8  -0.6 ]
         [  nan   nan]]
        >>> print(l)
        [ 4.24  5.    0.  ]

        """
        if ignore_zeros:
            on_zeros = 'i'
        A = np.asarray(A)
        Al = length(A,axis)
        if on_zeros != 'n' :
            if (Al == 0.).any():
                if on_zeros=='i':
                    Al[Al==0.] = 1.
                else:
                    raise ValueError("Can not normalize zero-length vector.")
        with np.errstate(divide='ignore',invalid='ignore'):
            res = A / np.expand_dims(Al,axis)
        if return_length:
            return res, Al
        else:
            return res

    def projection(A,B,axis=-1):
        """Return the (signed) length of the projection of vectors of A on B.

        Parameters
        ----------
        A: float array-like
            Array containing vectors in the direction of axis.
        B: float array-like
            Array containing vectors in the direction of axis. Same shape
            as A, or broadcast-compatible.
        axis: int
            Axis of A and B in which direction the vectors are layed out.
            Default is the last axis. A and B should have the same length
            along this axis.

        Returns
        -------
        : float array-like, shape as A and B with axis direction removed.
            The elements contain the length of the projections of vectors of A
            on the directions of the corresponding vectors of B.

        Raises
        ------
        ValueError: Can not normalize zero-length vector
            If any of the  vectors of B is a zero vector.

        Note
        ----
        This returns ``dotpr(A,normalize(B))``.

        Examples
        --------
        >>> A = [[2.,0.],[1.,1.],[0.,1.]]
        >>> projection(A,[1.,0.])
        array([ 2., 1., 0.])
        >>> projection(A,[1.,1.])
        array([ 1.41, 1.41, 0.71])
        >>> projection(A,[[1.],[1.],[0.]],axis=0)
        array([ 2.12,  0.71])

        """
        return dotpr(A,normalize(B,axis=axis,on_zeros='e'),axis=axis)

    def parallel(A,B,axis=-1):
        """Return the component of vector of A that is parallel to B.

        Parameters
        ----------
        A, B: float array-like
            Broadcast compatible arrays containing vectors in the direction of
            axis.
        axis: int
            Axis of A and B in which direction the vectors are layed out.
            Default is the last axis. A and B should have the same length
            along this axis.

        Returns
        -------
        : float array-like, same shape as A and B.
            The vectors in the axis direction are the vectors of A projected
            on the direction of the corresponding vectors of B.

        See also
        --------
        orthog

        Examples
        --------
        >>> A = [[2.,0.],[1.,1.],[0.,1.]]
        >>> parallel(A,[1.,0.])
        array([[ 2.,  0.],
               [ 1.,  0.],
               [ 0.,  0.]])
        >>> parallel(A,A)
        array([[ 2.,  0.],
               [ 1.,  1.],
               [ 0.,  1.]])
        >>> parallel(A,[[1.],[1.],[0.]],axis=0)
        array([[ 1.5,  0.5],
               [ 1.5,  0.5],
               [ 0. ,  0. ]])

        """
        Bn = normalize(B,axis=axis,on_zeros='e')
        p = dotpr(A, Bn, axis=axis)
        return np.expand_dims(p,axis) * Bn

    def orthog(A,B,axis=-1):
        """Return the component of vector of A that is orthogonal to B.

        Parameters
        ----------
        A: float array-like
            Array containing vectors in the direction of axis.
        B: float array-like
            Array containing vectors in the direction of axis. Same shape
            as A, or broadcast-compatible.
        axis: int
            Axis of A and B in which direction the vectors are layed out.
            Default is the last axis. A and B should have the same length
            along this axis.

        Returns
        -------
        : float array-like, same shape as A and B.
            The vectors in the axis direction are the components of the vectors
            of A orthogonal to the direction of the corresponding vectors of B.

        See also
        --------
        parallel

        Examples
        --------
        >>> A = [[2.,0.],[1.,1.],[0.,1.]]
        >>> orthog(A,[1.,0.])
        array([[ 0.,  0.],
               [ 0.,  1.],
               [ 0.,  1.]])
        >>> orthog(A,[[1.],[1.],[0.]],axis=0)
        array([[ 0.5, -0.5],
               [-0.5,  0.5],
               [ 0. ,  1. ]])

        """
        return A - parallel(A,B,axis=axis)

    def inside(p, mi, ma):
        """Return true if point p is inside bbox defined by points mi and ma.

        Parameters
        ----------
        p: float array-like with shape (ndim,)
            Point to test against the boundaries.
        mi: float array-like with shape (ndim,)
            Minimum values for the components of p
        ma: float array-like with shape (ndim,)
            Maximum values for the components of p

        Returns
        -------
        :bool
            True is all components are inside the specified limits, limits
            included. This means that the n-dimensional point p lies within
            the n-dimensional rectangular bounding box defined by the two
            n-dimensional points (mi,ma).

        Examples
        --------
        >>> inside([0.5,0.5],[0.,0.],[1.,1.])
        True
        >>> inside([0.,1.],[0.,0.],[1.,1.])
        True
        >>> inside([0.,1.1],[0.,0.],[1.,1.])
        False
        """
        p = np.asarray(p)
        return (p >= mi).all() and (p <= ma).all()

    def unitVector(v):
        """Return a unit vector in the direction of v.

        Parameters
        ----------
        v: a single integer or a (3,) shaped float array-like.
            If an int, it specifies one of the global axes (0,1,2).
            Else, it is a vector in 3D space.

        Returns
        -------
        : (3,) shaped float array
            A unit vector along the specified direction.

        Examples
        --------
        >>> unitVector(1)
        array([ 0.,  1.,  0.], dtype=float32)
        >>> unitVector([0.,3.,4.])
        array([ 0. ,  0.6,  0.8], dtype=float32)

        """
        if isInt(v):
            if v not in range(3):
                raise ValueError("v should be one of 0, 1 or 2")
            u = zeros((3), dtype=Float)
            u[v] = 1.0
        else:
            v = checkArray(v,shape=(3,),kind='f',allow='i')
            u = normalize(v,on_zeros='e')
        return u

    def rotationMatrix(angle,axis=None,angle_spec=DEG):
        """Create a 2D or 3D rotation matrix over angle, optionally around axis.

        Parameters
        ----------
        angle: float
            Rotation angle, by default in degrees.
        axis: int or (3,) float array-like, optional
            If not provided, a 2D rotation matrix is returned.
            If provided, it specifies the rotation axis in a 3D world. It is either
            one of 0,1,2, specifying a global axis, or a vector with 3 components
            specifying an axis through the origin. The returned matrix is 3D.
        angle_spec: float, DEG or RAD, optional
            The default (DEG) interpretes the angle in degrees. Use RAD to
            specify the angle in radians.

        Returns
        -------
        float array
            Rotation matrix which will rotate a vector over the specified angle.
            Shape is (3,3) if axis is specified, or (2,2) if not.

        See also
        --------
        rotationMatrix3: subsequent rotation around 3 axes
        rotmat: rotation matrix specified by three points in space
        trfmat: transformation matrix to transform 3 points
        rotMatrix: rotation matrix transforming global axis 0 into a given vector
        rotMatrix2: rotation matrix that transforms one vector into another

        Examples
        --------
        >>> rotationMatrix(30.,1)
        array([[ 0.87,  0.  , -0.5 ],
               [ 0.  ,  1.  ,  0.  ],
               [ 0.5 ,  0.  ,  0.87]])
        >>> rotationMatrix(45.,[1.,1.,0.])
        array([[ 0.85,  0.15, -0.5 ],
               [ 0.15,  0.85,  0.5 ],
               [ 0.5 , -0.5 ,  0.71]])
        """
        a = angle*angle_spec
        c = cos(a)
        s = sin(a)
        if axis is None:
            f = [[c, s], [-s, c]]
        elif array(axis).size == 1:
            f = zeros((3,3))
            axes = list(range(3))
            i, j, k = axes[axis:]+axes[:axis]
            f[i][i] = 1.0
            f[j][j] = c
            f[j][k] = s
            f[k][j] = -s
            f[k][k] = c
        else:
            X, Y, Z = unitVector(axis)
            t = 1.-c
            f = [ [ t*X*X + c, t*X*Y + s*Z, t*X*Z - s*Y ],
                  [ t*Y*X - s*Z, t*Y*Y + c, t*Y*Z + s*X ],
                  [ t*Z*X + s*Y, t*Z*Y - s*X, t*Z*Z + c   ] ]

        return array(f)

    def rotationMatrix3(rx,ry,rz, angle_spec=DEG):
        """Create a rotation matrix defined by three angles.

        This applies successive rotations about the 0, 1 and 2 axes, over
        the angles rx, ry and rx, respectively. These angles are also known
        as the cardan angles.

        Parameters
        ----------
        rx: float
            Rotation angle around the 0 axis.
        ry: float
            Rotation angle around the 1 axis.
        rz: float
            Rotation angle around the 2 axis.
        angle_spec: float, DEG or RAD, optional
            The default (DEG) interpretes the angles in degrees. Use RAD to
            specify the angle in radians.

        Returns
        -------
        : float array (3,3)
            Rotation matrix that performs the combined rotation equivalent
            to subsequent rotations around the three global axes.

        See Also
        --------
        rotationMatrix: rotation matrix specified by an axis and angle
        cardanAngles: find cardan angles that produce a given rotation matrix

        Examples
        --------
        >>> rotationMatrix3(60,45,30)
        array([[ 0.61,  0.35, -0.71],
               [ 0.28,  0.74,  0.61],
               [ 0.74, -0.57,  0.35]])

        """
        Rx = rotationMatrix(rx,0,angle_spec=angle_spec)
        Ry = rotationMatrix(ry,1,angle_spec=angle_spec)
        Rz = rotationMatrix(rz,2,angle_spec=angle_spec)
        return dot(Rx, dot(Ry, Rz))

    def cardanAngles(R,angle_spec=DEG):
        """Compute cardan angles from rotation matrix

        Computes the angles over which to rotate subsequently around
        the 0-axis, the 1-axis and the 2-axis to obtain the rotation
        corresponding to the given rotation matrix.

        Parameters
        ----------
        R: (3,3) float array_like
            Rotation matrix for post multiplication (see Notes)
        angle_spec: :py:attr:`DEG`, :py:attr:`RAD` or float, nonzero.
            Divisor applied to the resulting angles before returning.
            The default divisor DEG makes the angles be returned in
            degrees. Use RAD to get angles in radians.

        Returns
        -------
        (rx,ry,rz): tuple of floats
            The three rotation angles around that when applied subsequently
            around the global 0, 1 and 2 axes, yield the same rotation
            as the input. The default angle_spec=DEG returns the angles
            in degrees.

        Notes
        -----
        The returned angles are but one of many ways to obtain a given
        rotation by three subsequent rotations around frame axes. Look
        on the web for 'Euler angles' to get comprehensive information.
        Different sets of angles can be obtained depending on the sequence
        of rotation axes used, and whether fixed axes (extrinsic) or
        rotated axes (intrinsic) are used in subsequent rotations.
        The here obtained 'cardan' angles are commonly denoted as a zy'x''
        system with intrinsic angles or xyz with extrinsic angles. It is the
        latter angles that are returned.

        Because pyFormex stores rotation matrices as post-multiplication
        matrices (to be applied on row-vectors), the combined rotation
        around first the 0-axis, then the 1-axis and finally the 2-axis,
        is found as the matrix product Rx.Ry.Rz. (Traditionally,
        vectors were often written as column matrices, and rotation
        matrices were pre-multiplication matrices, so the subsequent
        rotation matrices would have to be multiplied in reverse order.)

        Even if one chooses a single frame system for the subsequent
        rotations, the resulting angles are not unique. There are infinitely
        many sets of angles that will result in the same rotation matrix.
        The implementation here results in angles rx and rz in the range
        [-pi,pi], while the angle ry will be in [-pi/2,pi/2]. Even then,
        there remain infinite solutions in the case where the elements
        R[0,2] == R[2,0] equal +1 or -1 (ry = +pi/2 or -pi/2). The result
        will then be the solution with rx==0.

        Examples
        --------
        >>> print("%8.2f  "*3 % cardanAngles(rotationMatrix3(60,45,30)))
           60.00     45.00     30.00
        >>> print("%8.2f  "*3 % cardanAngles(rotationMatrix3(0,90,77)))
            0.00     90.00     77.00
        >>> print("%8.2f  "*3 % cardanAngles(rotationMatrix3(0,-90,30)))
            0.00    -90.00     30.00

        But:

        >>> print("%8.2f  "*3 % cardanAngles(rotationMatrix3(20,-90,30)))
            0.00    -90.00     50.00

        """

        if abs(R[0,2]) < 1.:

            theta = -arcsin(R[0,2])
            if theta < -pi/2:
                theta = pi - theta
            c = cos(theta)
            psi = arctan2(R[1,2]/c, R[2,2]/c)
            phi = arctan2(R[0,1]/c, R[0,0]/c)

        else:
            psi = 0.
            if R[0,2] < 0.:
                theta = pi/2
                phi = psi - arctan2(R[1,0],R[2,0])
            else:
                theta = -pi/2
                phi = psi + arctan2(-R[1,0],-R[2,0])

        return psi/DEG, theta/DEG, phi/DEG

    def rotmat(x):
        """Create a rotation matrix defined by 3 points in space.

        Parameters
        ----------
        x: array-like (3,3)
            The rows contain the coordinates in 3D space of three non-colinear
            points x0, x1, x2.

        Returns
        -------
        rotmat: matrix(3,3)
            Rotation matrix which transforms the global axes
            into a new (orthonormal) coordinate system with the
            following properties:

            - the origin is at point x0,
            - the 0 axis is along the direction x1-x0
            - the 1 axis is in the plane (x0,x1,x2) with x2 lying
              at the positive side.

        Notes
        -----
        The rows of the rotation matrix represent the unit vectors
        of the resulting coordinate system.
        The coodinates in the rotated axes of any point are
        obtained by the reverse transformation, i.e. multiplying the
        point with the transpose of the rotation matrix.

        See also
        --------
        rotationMatrix: rotation matrix specified by angle and axis
        trfmat: transformation matrices defined by 2 sets of 3 points
        rotMatrix: rotation matrix transforming global axis 0 into a given vector
        rotMatrix2: rotation matrix that transforms one vector into another

        Examples
        --------
        >>> rotmat([[0,0,0],[1,0,0],[0,1,0]])
        array([[ 1.,  0.,  0.],
               [ 0.,  1.,  0.],
               [ 0.,  0.,  1.]], dtype=float32)
        >>> rotmat(eye(3,3))
        array([[-0.71,  0.71,  0.  ],
               [-0.41, -0.41,  0.82],
               [ 0.58,  0.58,  0.58]], dtype=float32)
        >>> s,c = sind(30),cosd(30)
        >>> R = rotmat([[0,0,0],[c,s,0],[0,1,0]])
        >>> print(R)
        [[ 0.87  0.5   0.  ]
         [-0.5   0.87  0.  ]
         [ 0.   -0.    1.  ]]
        >>> B = array([[2.,0.,0.],[3*s,3*c,3]])
        >>> D = dot(B,R)    # Rotate some vectors with the matrix R
        >>> print(D)
        [[ 1.73  1.    0.  ]
         [-0.    3.    3.  ]]
        """
        x = checkArray(x,shape=(3,3),kind='f',allow='i')
        u = normalize(x[1]-x[0])
        v = normalize(x[2]-x[0])
        v = normalize(orthog(v, u))
        w = cross(u, v) # is orthog and normalized
        m = row_stack([u, v, w])
        return m

    def trfmat(x, y):
        """Find the transformation matrices from 3 points x into y.

        Constructs the rotation matrix and translation vector that will
        transform the points x thus that:

        - point x0 coincides with point y0,
        - line x0,x1 coincides with line y0,y1
        - plane x0,x1,x2 coincides with plane y0,y1,y2

        Parameters
        ----------
        x: float array-like (3,3)
            Original coordinates of three  non-colinear points.
        y: float array-like (3,3)
            Final coordinates of the three points.

        Returns
        -------
        rot: float array (3,3)
            The rotation matrix for the transformation x to y.
        trf:
             float array(3,)
            The translation vector for the transformation x to y, Obviously,
            this is equal to y0-x0.

        The rotation is to be applied first and should be around the first
        point x0. The full transformation of a Coords object is thus obtained
        by ``(coords-x0)*rot+trl+x0 = coords*rot+(trl+x0-x0*rot)``.

        Examples
        --------
        >>> R,T = trfmat(eye(3,3), [[0,0,0],[1,0,0],[0,1,0]])
        >>> print(R)
        [[-0.71 -0.41  0.58]
         [ 0.71 -0.41  0.58]
         [ 0.    0.82  0.58]]
        >>> print(T)
        [ 0.71  0.41 -0.58]

        """
        # rotation matrices for both systems
        r1 = rotmat(x)
        r2 = rotmat(y)
        # combined rotation matrix
        r = dot(r1.transpose(), r2)
        # translation vector (in a rotate first operation
        t = y[0] - dot(x[0], r)
        return r, t

    # TODO: we could allow here a None value for w, like in rotMatrix2
    def rotMatrix(u,w=[0., 0., 1.]):
        """Create a rotation matrix that rotates global axis 0 to a given vector.

        Parameters
        ----------
        u: (3,) array-like
            Vector specifying the direction to which the global axis 0 should
            be rotated by the returned rotation matrix.
        w: (3,) array-like
            Vector that is not parallel to u. This vector is used to uniquely
            define the resulting rotation. It will be equivalen to rotating
            first around ``w``, until the target ``u`` lies in the plane
            of the rotated axis 0 and ``w``, then rotated in that plane until
            the rotated axis 0 coincides with ``u``. See also Note.
            If a parallel w is provided, it will be replaced with a non-parallel
            one.

        Returns
        -------
        : float array (3,3)
            Rotation matrix that transforms a vector [1.,0.,0.] into ``u``.
            The returned matrix should be used in postmultiplication to the
            coordinates.

        See Also
        --------
        rotMatrix2: rotation matrix that transforms one vector into another
        rotationMatrix: rotation matrix specified by an axis and angle
        rotmat: rotation matrix specified by three points in space
        trfmat: rotation and translation matrix that transform three points

        Examples
        --------
        >>> rotMatrix([1,0,0])
        array([[ 1.,  0.,  0.],
               [ 0.,  1.,  0.],
               [ 0.,  0.,  1.]], dtype=float32)
        >>> rotMatrix([0,1,0])
        array([[ 0.,  1.,  0.],
               [-1.,  0.,  0.],
               [ 0., -0.,  1.]], dtype=float32)
        >>> rotMatrix([0,0,1])
        array([[ 0.,  0.,  1.],
               [ 1.,  0.,  0.],
               [ 0.,  1.,  0.]], dtype=float32)
        >>> rotMatrix([0,1,1])
        array([[ 0.  ,  0.71,  0.71],
               [-1.  ,  0.  ,  0.  ],
               [ 0.  , -0.71,  0.71]], dtype=float32)
        >>> rotMatrix([1,0,1])
        array([[ 0.71,  0.  ,  0.71],
               [ 0.  ,  1.  ,  0.  ],
               [-0.71,  0.  ,  0.71]], dtype=float32)
        >>> rotMatrix([1,1,0])
        array([[ 0.71,  0.71,  0.  ],
               [-0.71,  0.71,  0.  ],
               [ 0.  , -0.  ,  1.  ]], dtype=float32)
        >>> rotMatrix([1,1,1])
        array([[ 0.58,  0.58,  0.58],
               [-0.71,  0.71,  0.  ],
               [-0.41, -0.41,  0.82]], dtype=float32)

        >>> dot([1,0,0], rotMatrix([1,1,1]))
        array([ 0.58,  0.58,  0.58])
        """
        u = unitVector(u)
        w = unitVector(w)
        v = cross(w,u)
        if length(v) == 0:
            # u and w are parallel
            x, y, z = u
            w = np.array([0., 1., 0.]) if x==0. and y==0. else array([-y, x, 0.])
            v = cross(w,u)
        v = unitVector(v)
        w = unitVector(cross(u, v))
        m = row_stack([u, v, w])
        return m

    def rotMatrix2(vec1,vec2,upvec=None):
        """Create a rotation matrix that rotates a vector vec1 to vec2.

        Parameters
        ----------
        vec1: (3,) array-like
            Original vector.
        vec2: (3,) array-like
            Direction of ``vec1`` after rotation.
        upvec: (3,) array-like, optional
            If provided, the rotation matrix will be such that the plane of
            vec2 and the rotated upvec will be parallel to the original upvec.
            If not provided, the rotation matrix will perform a rotation
            around the normal to the plane ov the two vectors.

        Returns
        -------
        : float array (3,3)
            Rotation matrix that transforms a vector [1.,0.,0.] into ``u``.
            The returned matrix should be used in postmultiplication to the
            coordinates.

        See Also
        --------
        rotMatrix: rotation matrix transforming global axis 0 into a given vector
        rotationMatrix: rotation matrix specified by an axis and angle
        rotmat: rotation matrix specified by three points in space
        trfmat: rotation and translation matrix that transform three points

        Examples
        --------
        >>> rotMatrix2([1,0,0],[1,0,0])
        array([[ 1.,  0.,  0.],
               [ 0.,  1.,  0.],
               [ 0.,  0.,  1.]], dtype=float32)
        >>> rotMatrix2([1,0,0],[0,1,0])
        array([[ 0.,  1.,  0.],
               [-1.,  0.,  0.],
               [ 0.,  0.,  1.]], dtype=float32)
        >>> rotMatrix2([1,0,0],[0,0,1])
        array([[ 0.,  0.,  1.],
               [ 0.,  1.,  0.],
               [-1.,  0.,  0.]], dtype=float32)
        >>> rotMatrix2([1,0,0],[0,1,1])
        array([[ 0.  ,  0.71,  0.71],
               [-0.71,  0.5 , -0.5 ],
               [-0.71, -0.5 ,  0.5 ]], dtype=float32)
        >>> rotMatrix2([1,0,0],[1,0,1])
        array([[ 0.71,  0.  ,  0.71],
               [ 0.  ,  1.  ,  0.  ],
               [-0.71,  0.  ,  0.71]], dtype=float32)
        >>> rotMatrix2([1,0,0],[1,1,0])
        array([[ 0.71,  0.71,  0.  ],
               [-0.71,  0.71,  0.  ],
               [ 0.  ,  0.  ,  1.  ]], dtype=float32)
        >>> rotMatrix2([1,0,0],[1,1,1])
        array([[ 0.58,  0.58,  0.58],
               [-0.58,  0.79, -0.21],
               [-0.58, -0.21,  0.79]], dtype=float32)

        >>> rotMatrix2([1,0,0],[1,0,0],[0,0,1])
        array([[ 1.,  0.,  0.],
               [ 0.,  1.,  0.],
               [ 0.,  0.,  1.]], dtype=float32)
        >>> rotMatrix2([1,0,0],[0,1,0],[0,0,1])
        array([[ 0.,  1.,  0.],
               [-1.,  0.,  0.],
               [ 0.,  0.,  1.]], dtype=float32)
        >>> rotMatrix2([1,0,0],[0,0,1],[0,0,1])
        array([[ 0.,  0.,  1.],
               [ 1.,  0.,  0.],
               [ 0.,  1.,  0.]], dtype=float32)
        >>> rotMatrix2([1,0,0],[0,1,1],[0,0,1])
        array([[ 0.  ,  0.71,  0.71],
               [-1.  ,  0.  ,  0.  ],
               [ 0.  , -0.71,  0.71]], dtype=float32)
        >>> rotMatrix2([1,0,0],[1,0,1],[0,0,1])
        array([[ 0.71,  0.  ,  0.71],
               [ 0.  ,  1.  ,  0.  ],
               [-0.71,  0.  ,  0.71]], dtype=float32)
        >>> rotMatrix2([1,0,0],[1,1,0],[0,0,1])
        array([[ 0.71,  0.71,  0.  ],
               [-0.71,  0.71,  0.  ],
               [ 0.  ,  0.  ,  1.  ]], dtype=float32)
        >>> rotMatrix2([1,0,0],[1,1,1],[0,0,1])
        array([[ 0.58,  0.58,  0.58],
               [-0.71,  0.71,  0.  ],
               [-0.41, -0.41,  0.82]], dtype=float32)
        """
        vec1 = checkArray(vec1,shape=(3,),kind='f',allow='i')
        vec2 = checkArray(vec2,shape=(3,),kind='f',allow='i')
        if upvec is None:
            upvec = np.cross(vec1,vec2)
            if length(upvec) == 0.: # vec1 and vec2 are parallel
                return np.eye(3,3,dtype=Float)

        mat1 = rotMatrix(vec1,upvec)
        mat2 = rotMatrix(vec2,upvec)
        mat = np.dot(mat1.T, mat2)
        return mat

    def abat(a,b):
        """Compute the matrix product a * b * at.

        Parameters
        ----------
        a: array-like, 2-dim
            Array with shape (m,n).
        b: array-like, 2-dim
            Array with square shape (n,n).

        Returns
        -------
        :array
            Array with shape (m,m) holding the matrix product a * b * at.

        See Also
        --------
        atba

        Examples
        --------
        >>> abat([[1],[2]],[[3]])
        array([[ 3,  6],
               [ 6, 12]])
        >>> abat([[1,2]],[[0,1],[2,3]])
        array([[18]])
        """
        a = checkArray(a,ndim=2)
        b = checkArray(b,shape=(a.shape[1],a.shape[1]))
        return np.dot( np.dot(a,b) , a.T )

    def atba(a,b):
        """Compute the matrix product at * b * a

        Parameters
        ----------
        a: array-like, 2-dim
            Array with shape (n,m).
        b: array-like, 2-dim
            Array with square shape (n,n).

        Returns
        -------
        :array
            Array with shape (m,m) holding the matrix product at * b * a.

        Note
        ----
        This multiplication typically occurs when rotating a symmetric tensor b
        to axes defined by the rotation matrix a.

        See Also
        --------
        abat

        Examples
        --------
        >>> atba([[1,2]],[[3]])
        array([[ 3,  6],
               [ 6, 12]])
        >>> atba([[1],[2]],[[0,1],[2,3]])
        array([[18]])
        """
        a = checkArray(a,ndim=2)
        b = checkArray(b,shape=(a.shape[0],a.shape[0]))
        return np.dot( np.dot(a.T,b) , a )

    def horner(a, u):
        """Compute the value of a polynom using Horner's rule.

        Parameters
        ----------
        a: float array-like (n+1,nd)
            `nd`-dimensional coefficients of a polynom of degree `n` ``n`` *n* **n** in a scalar variable `u`. The coefficients are in order of increasing degree.
        u: float array-like (nu)
            Parametric values where the polynom is to be evaluated.

        Returns
        -------
        :float array(nu,nd)
            The nd-dimensional values of the polynom at the specified `nu`
            parameter values.

        Examples
        --------
        >>> print(horner([[1.,1.,1.],[1.,2.,3.]],[0.5,1.0]))
        [[ 1.5 2.  2.5]
         [ 2.  3.  4. ]]

        """
        a = checkArray(a,ndim=2,kind='f',allow='i')
        u = checkArray(u,ndim=1,kind='f',allow='i').reshape(-1,1)
        c = a[-1]
        for i in range(-2, -1-len(a), -1):
            c = c * u + a[i]
        return c

    def solveMany(A,b,direct=True):
        """Solve many systems of linear equations.

        Parameters
        ----------
        A: (ndof,ndof,nsys) shaped float array
            Coefficient matrices for nsys systems of ndof linear equations
            in ndof unknowns.
        b: (ndof,nrhs,nsys) shaped float array
            Right hand sides for each of the nsys systems of linear equations.
            Each of the nsys systems is solved simultaneously for nrhs right
            hand sides.
        direct: bool
            If True (default), systems with ndof=1, 2 or 3 are
            solved with a (faster) direct method instead of using the general
            linear equation solver.

        Returns
        -------
        x: float array with same shape as `b`
            The set of values ``x[:,i,j]`` that solve the systems of linear
            equations A[:,:,j].x[:,i,j] = b[:,i,j].

        """
        ndof, nrhs, nsys = b.shape
        if A.shape[:2] != (ndof, ndof) or A.shape[2] != nsys:
            raise ValueError("A(%s) and b(%s) have incompatible shape" % (A.shape, b.shape))

        if ndof < 4 and direct:
            A = addAxis(A, 2)
            b = addAxis(b, 1)

            if ndof == 1:
                x = b[0]/A[0, 0]

            elif ndof == 2:
                denom = cross(A[:, 0], A[:, 1], axis=0)
                As = roll(A, -1, axis=1)
                As[:, 1] *= -1.
                x = cross(b, As, axis=0) / denom

            elif ndof == 3:
                C = cross(roll(A, -1, axis=1), roll(A, -2, axis=1), axis=0)
                denom = dotpr(A[:, 0], C[:, 0], axis=0)
                x = dotpr(b, C, axis=0) / denom

        else:
            x = dstack([linalg.solve(A[:,:, i], b[:,:, i]) for i in range(nsys)])

        return x

    def cubicEquation(a, b, c, d):
        """Solve a cubiq equation using a direct method.

        Given a polynomial equation of the third degree with real coefficients::

          a*x**3 + b*x**2 + c*x + d = 0

        Such an equation (with a non-zero) always has exactly three roots, with
        some possibly being complex, or identical.
        This function computes all three roots of the equation and returns full
        information about their nature, multiplicity and sorting order.
        It uses scaling of the variables to enhance the accuracy.

        Parameters
        ----------
        a: float
            Coefficient of the third degree term.
        b: float
            Coefficient of the second degree term.
        c: float
            Coefficient of the first degree term.
        d: float
            Constant in the third degree polynom.

        Returns
        -------
        r1: float
            First real root of the cubiq equation
        r2: float
            Second real root of the cubiq equation or real part of the
            complex conjugate second and third root.
        r3: float
            Third real root of the cubiq equation or imaginary part of the
            complex conjugate second and third root.
        kind: int
            A value specifying the nature and ordering of the roots:

            ======   ============================================================
             kind      roots
            ======   ============================================================
              0       three real roots r1 < r2 < r3
              1       three real roots r1 < r2 = r3
              2       three real roots r1 = r2 < r3
              3       three real roots r1 = r2 = r3
              4       one real root r1 and two complex conjugate roots with real
                      part r2 and imaginary part r3; the complex roots are thus:
                      r2+i*r3 en r2-i*r3, where i=sqrt(-1).
            ======   ============================================================

        Raises
        ------
        ValueError:
            If the coefficient a==0 and the equation reduces to a second degree.

        Examples
        --------
        >>> cubicEquation(1.,-6.,11.,-6.)
        (array([ 1.,  2.,  3.]), 0)
        >>> cubicEquation(1.,-2.,1.,0.)
        (array([-0.,  1.,  1.]), 1)
        >>> cubicEquation(1.,-5.,8.,-4.)
        (array([ 1.,  2.,  2.]), 1)
        >>> cubicEquation(1.,-4.,5.,-2.)
        (array([ 1.,  1.,  2.]), 2)
        >>> cubicEquation(1.,-3.,3.,-1.)
        (array([ 1.,  1.,  1.]), 3)
        >>> cubicEquation(1.,-1.,1.,-1.)
        (array([ 1.,  0.,  1.]), 4)
        >>> cubicEquation(1.,-3.,4.,-2.)
        (array([ 1.,  1.,  1.]), 4)

        """
        #
        # BV: We should return the solution of a second degree equation if a==0
        #
        if a == 0.0:
            raise ValueError("Coefficient a of cubiq equation should not be 0")

        e3 = 1./3.
        pie = np.pi*2.*e3
        r = b/a
        s = c/a
        t = d/a

        # scale variable
        sc = max(abs(r), sqrt(abs(s)), abs(t)**e3)
        sc = 10**(int(log10(sc)))
        r = r/sc
        s = s/sc/sc
        t = t/sc/sc/sc
        gc = max(abs(r),abs(s),abs(t)) * 1.e-8

        rx = r*e3
        p3 = (s-r*rx)*e3
        q2 = rx**3-rx*s/2.+t/2.

        q2s = q2*q2
        p3c = p3**3
        som = q2s+p3c

        if abs(som) < gc:
            # two equal real roots
            ic = 1
            u = -q2
            r1 = sign(u) * abs(u)**e3
            r2 = -r1-rx
            r3 = r2
            r1 = r1+r1-rx
            if abs(r1-r2) < gc:
                ic = 3
            if r1 > r2:
                ic = 2
                r3,r1 = r1,r2
            roots = np.array([r1,r2,r3])

        elif som < 0.0:
            # 3 different roots
            ic = 0
            rt = sqrt(-p3c)
            roots = np.array([-rx] * 3)
            if abs(rt) > gc:
                phi = arccos(-q2/rt)*e3
                rt = 2.*sqrt(-p3)
                roots += rt*cos(phi + np.array([0., +pie, -pie]))

            # sort the 3 roots

            roots.sort()
            if roots[1] == roots[2]:
                ic += 1
            if roots[1] == roots[0]:
                ic += 2

        else: # som > 0.0
            #  1 real and 2 complex conjugate roots
            ic = 4
            som = sqrt(som)
            u = -q2+som
            u = sign(u) * abs(u)**e3
            v = -q2-som
            v = sign(v) * abs(v)**e3
            r1 = u+v
            r2 = -r1/2-rx
            r3 = (u-v)*sqrt(3.)/2.
            r1 = r1-rx
            roots = np.array([r1, r2, r3])

        # scale and return values
        roots *= sc
        return roots, ic

    ###########################################################################
    ##
    ##   Operations on integer arrays
    ##
    #########################

    def renumberIndex(index):
        """Renumber an index sequentially.

        Given a one-dimensional integer array with only non-negative values,
        and `nval` being the number of different values in it, and you want to
        replace its elements with values in the range `0..nval`, such that
        identical numbers are always replaced with the same number and the
        new values at their first occurrence form an increasing sequence `0..nval`.
        This function will give you the old numbers corresponding with each
        position `0..nval`.

        Parameters
        ----------
        index: 1-dim int array_like
            Array with non-negative integer values.

        Returns
        -------
        : int array
            A 1-dim int array with length equal to `nval`, where `nval`
            is the number of different values in `index`. The elements are
            the original values corresponding to the new values `0..nval`.

        See Also
        --------
        inverseUniqueIndex: get the inverse mapping.

        Examples
        --------
        >>> renumberIndex([0,5,2,2,6,0])
        array([0, 5, 2, 6])
        >>> inverseUniqueIndex(renumberIndex([0,5,2,2,6,0]))[[0,5,2,2,6,0]]
        array([0, 1, 2, 2, 3, 0])

        """
        un, pos = np.unique(index, True)
        srt = pos.argsort()
        old = un[srt]
        return old

    def inverseUniqueIndex(index):
        """Inverse an index.

        Given a 1-D integer array with *unique* non-negative values, and
        `max` being the highest value in it, this function returns the position
        in the array of the values `0..max`. Values not occurring in input index
        get a value -1 in the inverse index.

        Parameters
        ----------
        index: 1-dim int array_like
            Array with non-negative values, which all have to be unique.
            It's highest value is `max = index.max()`.

        Returns
        -------
        1-dim int array
            Array with length `max+1`, with the position in `index` of each
            of the values `0..max`, or -1 if the value does not occur in
            `index`.

        Note
        ----
        This is a low level function that does not check whether the input
        has indeed all unique values.

        The inverse index translates the unique index numbers in a
        sequential index, so that
        ``inverseUniqueIndex(index)[index] == arange(1+index.max())``.

        Examples
        --------
        >>> inverseUniqueIndex([0,5,2,6])
        array([ 0, -1,  2, -1, -1,  1,  3])
        >>> inverseUniqueIndex([0,5,2,6])[[0,5,2,6]]
        array([0, 1, 2, 3])

        """
        index = checkArray(index,ndim=1,kind='i')
        if np.unique(index).size != index.size:
            raise ValueError("The array does not contain unique values")
        inv = -ones(index.max()+1, dtype=index.dtype)
        inv[index] = arange(index.size, dtype=index.dtype)
        return inv

    def cumsum0(a):
        """Cumulative sum of a list of numbers preprended with a 0.

        Parameters
        ----------
        a: array-like, int
            List of integers to compute the cumulative sum for.

        Returns
        -------
        : array, int
             Array with ``len(a)+1`` integers holding the cumulative sum of the
             integers from ``a`` with a 0 prepended.

        Examples
        --------
        >>> cumsum0([2,4,3])
        array([0, 2, 6, 9])

        A common use case is when concatenating some blocks of different length.
        If the list `a` holds the length of each block, the cumsum0(a) holds
        the start and end of each block in the concatenation.

        >>> L = [ [0,1], [2,3,4,5], [6], [7,8,9] ]
        >>> n = cumsum0([len(i) for i in L])
        >>> print(n)
        [ 0  2  6  7 10]
        >>> C = concatenate(L)
        >>> print(C)
        [0 1 2 3 4 5 6 7 8 9]
        >>> for i,j in zip(n[:-1],n[1:]):
        ...     print("%s:%s = %s" % (i,j,C[i:j]))
        ...
        0:2 = [0 1]
        2:6 = [2 3 4 5]
        6:7 = [6]
        7:10 = [7 8 9]
        """
        return np.cumsum([0]+a)

    def multiplicity(a):
        """Return the multiplicity of the numbers in an array.

        Parameters
        ----------
        a: array-like, 1-dim
            The data array, will be flattened if it is not 1-dim.

        Returns
        -------
        mult: 1-dim int array
            The multiplicity of the unique values in a
        uniq: 1-dim array
            Array of same type as a, with the sorted list of unique values in a.

        Examples
        --------
        >>> multiplicity([0,1,4,3,1,4,3,4,3,3])
        (array([1, 2, 4, 3]), array([0, 1, 3, 4]))
        >>> multiplicity([[1.0, 0.0, 0.5],[0.2,0.5,1.0]])
        (array([1, 1, 2, 2]), array([ 0. ,  0.2,  0.5,  1. ]))
        """
        a = checkArray1D(a)
        bins = unique(a)
        if bins.size > 0:
            mult, b = histogram(a, bins=concatenate([bins, [max(a)+1]]))
        else:
            mult = bins
        return mult, bins

    def subsets(a):
        """Split an array of integers into subsets.

        The subsets of an integer array are sets of elements
        with the same value.

        Parameters
        ----------
        a: int array-like, 1-dim
            Array with integer values to be split in subsets

        Returns
        -------
        val: array of ints
            The unique values in ``a``, sorted in increasing order.
        ind: :class:`varray.Varray`
            The Varray has the same number of rows as the number of values in
            ``ind``. Each row contains the indices in a of the elements with
            the corresponding value in ``val``.

        Examples
        --------
        >>> A = [0,1,4,3,1,4,3,4,3,3]
        >>> val,ind = subsets(A)
        >>> print(val)
        [0 1 3 4]
        >>> print(ind)
        Varray (4,4)
          [0]
          [1 4]
          [3 6 8 9]
          [2 5 7]
        <BLANKLINE>

        The inverse of ``ind`` can be used to restore A from val.

        >>> inv = inverseIndex(ind).reshape(-1)
        >>> print(inv)
        [0 1 3 2 1 3 2 3 2 2]
        >>> (val[inv] == A).all()
        True
        """
        from pyformex.varray import Varray
        a = checkArray(a,ndim=1,kind='i')
        val = unique(a)
        ind = Varray([where(a==v)[0] for v in val])
        return val,ind

    def sortSubsets(a,w=None):
        """Sort subsets of an integer array a.

        Subsets of an array are the sets of elements with equal values.
        By default the subsets are sorted according to decreasing number
        of elements in the set, or if a weight for each element is provided,
        according to decreasing sum of weights in the set.

        Parameters
        ----------
        a: 1-dim int array-like
            Input array containing non-negative integer sets to be sorted.
        w: 1-dim int or float array-like, optional
            If provided, it should have the same length as a. Each element of a
            will be attributed the corresponding weight.
            Specifying no weigth is equivalent to giving all elements the same
            weight.

        Returns
        -------
        : int array
            Array with same size as a, specifying for each element of a
            the index of its subset in the sorted list of subsets.

        Examples
        --------
        >>> sortSubsets([0,1,3,2,1,3,2,3,2,2])
        array([3, 2, 1, 0, 2, 1, 0, 1, 0, 0])
        >>> sortSubsets([0,1,4,3,1,4,3,4,3,3])
        array([3, 2, 1, 0, 2, 1, 0, 1, 0, 0])
        >>> sortSubsets([0,1,4,3,1,4,3,4,3,3],w=[9,8,7,6,5,4,3,2,1,0])
        array([3, 1, 0, 2, 1, 0, 2, 0, 2, 2])

        """
        a = checkArray(a,ndim=1,kind='i')
        # Make sure we have unique numbers
        a = inverseUniqueIndex(renumberIndex(a))[a]
        if w is None:
            h,u = multiplicity(a)
        else:
            w = checkArray(w,shape=a.shape,kind='f',allow='i')
            u = unique(a)
            h = [w[a==j].sum() for j in u]
        srt = argsort(h)[::-1]
        inv = inverseUniqueIndex(srt)
        return inv[a]

    def collectOnLength(items,return_index=False):
        """Separate items in a list according to their length.

        The length of all items in the list are determined and
        the items are put in separate lists according to their length.

        Parameters
        ----------
        items: list
             A list of any items that can be accepted
             as parameter of the len() function.
        return_index: bool
             If True, also return an index with the positions of the equal
             length items in the original iterable.

        Returns
        -------
        col: dict
            A dict whose keys are the item lengths and values are lists of
            items with this length.
        ind: dict, optional
            A dict with the same keys as ``col``, and the values being a
            list of indices in the list where the corresponding item
            of ``col`` appeared.

        Examples
        --------
        >>> collectOnLength(['a','bc','defg','hi','j','kl'])
        {1: ['a', 'j'], 2: ['bc', 'hi', 'kl'], 4: ['defg']}
        >>> collectOnLength(['a','bc','defg','hi','j','kl'],return_index=True)[1]
        {1: [0, 4], 2: [1, 3, 5], 4: [2]}
        """
        val,ind = subsets([len(e) for e in items])
        col = {}
        index = {}
        for v,i in zip(val,ind):
            col[v] = [items[j] for j in i]
            if return_index:
                index[v] = i.tolist()
        if return_index:
            return col,index
        else:
            return col

    def complement(index,n=-1):
        """Return the complement of an index in a range(0,n).

        The complement is the list of numbers from the range(0,n) that are
        not included in the index.

        Parameters
        ----------
        index: 1-dim array_like, of type int or bool.
            If integer, the array contains non-negative numbers in the range(0,n)
            and the return value will be the numbers in range(0,n) not included
            in index.
            If boolean, False value flag elements to be included (having a value
            True) in the output.
        n: int
            Upper limit for the range of numbers. If `index` is of
            type integer and `n` is not specified or is negative, it will be set
            equal to the largest number in `index` plus 1. If `index` is of type
            boolean and `n` is larger than the length of `index`, `index` will be
            padded with `False` values until length `n`.

        Returns
        -------
        : 1-dim array, type int or bool.
            The output array has the same dtype as the input.
            If `index` is integer: it is an array with the numbers from
            range(0,n) that are not included in `index`. If `index` is boolean,
            it is the negated input, padded to or cut at length `n`.

        Examples
        --------
        >>> print(complement([0,5,2,6]))
        [1 3 4]
        >>> print(complement([0,5,2,6],10))
        [1 3 4 7 8 9]
        >>> print(complement([False,True,True,True],6))
        [ True False False False  True  True]
        """
        index = np.asarray(index)
        if index.dtype == bool:
            m = index.shape[0]
            if n > m:
                comp = ones(n, dtype=bool)
                comp[:m] = ~index
            else:
                comp = ~index[:n]
        else:
            if n < 0:
                n = max(n, 1+index.max())
            comp = delete(arange(n), index)
        return comp

    def sortByColumns(a):
        """Sort an array on all its columns, from left to right.

        The rows of a 2-dimensional array are sorted, first on the first
        column, then on the second to resolve ties, etc..

        Parameters
        ----------
        a: array_like, 2-dim
            The array to be sorted

        Returns
        -------
        : int array, 1-dim
            Index specifying the order in which the rows have to
            be taken to obtain an array sorted by columns.

        Examples
        --------
        >>> sortByColumns([[1,2],[2,3],[3,2],[1,3],[2,3]])
        array([0, 3, 1, 4, 2])

        """
        a = checkArray(a, ndim=2)
        keys = [a[:, i] for i in range(a.shape[1]-1, -1, -1)]
        return lexsort(keys)

    def minroll(a):
        """Roll a 1-D array to get the lowest values in front

        If the lowest value occurs more than once, the one with the
        lowest next value is choosen, etcetera.

        Parameters
        ----------
        a: array, 1-dim
            The array to roll

        Returns
        -------
        m: int
            The index of the element that should be put in front. This
            means that the ``roll(a,-m)`` gives the rolled array with
            the lowest elements in front.

        Examples
        --------
        >>> minroll([1,3,5,1,2,6])
        3
        >>> minroll([0,0,2,0,0,1])
        3
        >>> minroll([0,0,0,0,0,0])
        0
        """
        a = checkArray(a,ndim=1)
        m = argmin(a)
        w = where(a==a[m])[0]
        m = w[0]
        for k in w[1:]:
            d = roll(a,-m) - roll(a,-k)
            u = where(d!=0)[0]
            if len(u) > 0 and d[u[0]] > 0:
                m = k
        return m

    def isroll(a,b):
        """Check that two 1-dim arrays can be rolled into eachother

        Parameters
        ----------
        a: array, 1-dim
            The first array
        b: array, 1-dim
            The second array, same length and dtype as a to be non-trivial.

        Returns
        -------
        m: int
            The number of positions (non-negative) that b has to be rolled
            to be equal to a, or -2 if the two arrays have a different length,
            or -1 if their elements are not the same or not in the same order.

        Examples
        --------
        >>> isroll(array([1,2,3,4]),array([2,3,4,1]))
        1
        >>> isroll(array([1,2,3,4]),array([2,3,1,4]))
        -1
        >>> isroll(array([1,2,3,4]),array([3,2,1,4]))
        -1
        >>> isroll(array([1,2,3,4]),array([1,2,3]))
        -2

        """
        a = asarray(a)
        b = asarray(b)
        if a.size != b.size:
            return -2

        for i in range(len(b)):
            if (a==roll(b,i)).all():
                return i

        return -1

    def findEqualRows(a,permutations='none'):
        """Find equal rows in a 2-dim array.

        Parameters
        ----------
        a: array-like, 2-dim
            The array in which to find the equal rows.
        permutations: str
            Defines which permutations of the row data are allowed while still
            considering the rows equal. Possible values are:

            - 'none': no permutations are allowed: rows must match the same date
              at the same positions. This is the default;
            - 'roll': rolling is allowed. Rows that can be transformed into
              each other by rolling are considered equal;
            - 'all': any permutation of the same data will be considered an
              equal row.

        Returns
        -------
        ind: 1-dim int array
            A row index sorting the rows in such order that equal rows are grouped
            together.
        ok: 1-dim bool array
            An array flagging the rows in the order of ``index`` with True if
            it is the first row of a group of equal rows, or with False if the
            row is equal to the previous.

        Notes
        -----
        This function provides the functionality for detecting equal rows,
        but is seldomly used directly. There are wrapper functions providing more
        paractical return values. See below.

        See Also
        --------
        equalRows: return the indices of the grouped equal rows
        uniqueRows: return the indices of the unique rows
        uniqueRowsIndex: like uniqueRows, but also returns index for all rows

        Examples
        --------
        >>> print(*findEqualRows([[1,2],[2,3],[3,2],[1,3],[2,3]]))
        [0 3 1 4 2] [ True  True  True False  True]
        >>> print(*findEqualRows([[1,2],[2,3],[3,2],[1,3],[2,3]],permutations='all'))
        [0 3 1 2 4] [ True  True  True False False]
        >>> print(*findEqualRows([[1,2,3],[3,2,1],[2,3,1],[1,2,3]]))
        [0 3 2 1] [ True False  True  True]
        >>> print(*findEqualRows([[1,2,3],[3,2,1],[2,3,1],[1,2,3]],permutations='all'))
        [0 1 2 3] [ True False False False]
        >>> print(*findEqualRows([[1,2,3],[3,2,1],[2,3,1],[1,2,3]],permutations='roll'))
        [0 2 3 1] [ True False False  True]
        """
        a = checkArray(a,ndim=2)
        if permutations == 'all':
            # Sort the rows
            a = a.copy()
            a.sort(axis=1)
        elif permutations == 'roll':
            # Roll the rows until smallest is in front
            a = np.stack([np.roll(ai,-minroll(ai)) for ai in a])
        ind = sortByColumns(a)  # groups the equal rows together
        a = a.take(ind, axis=0)
        ok = (a != np.roll(a, 1, axis=0)).any(axis=1)
        ok[0] = True
        return ind,ok

    def equalRows(a,permutations='none'):
        """Return equal rows in a 2-dim array.

        Parameters: see :meth:`findEqualRows`

        Returns
        -------
        V: :class:`varray.Varray`
            A Varray where each row contains a list of the row numbers
            from a that are considered equal. The entries in each row are
            sorted and the rows are sorted according to their first element.

        Notes
        -----
        The return Varray holds a lot of information:

        - ``V.col(0)`` gives the indices of the unique rows.
        - ``complement(V.col(0),len(a))`` gives the indices of duplicate rows.
        - ``V.col(0)[V.lengths==1]`` gives the indices of rows without duplicate.
        - ``Va.inverse(expand=True).reshape(-1)`` returns an index into the unique
          rows for each of the rows of ``a``.

        See Also
        --------
        findEqualRows: sorts and detects equal rows
        uniqueRows: return the indices of the unique rows
        uniqueRowsIndex: like uniqueRows, but also returns index for all rows

        Examples
        --------
        >>> equalRows([[1,2],[2,3],[3,2],[1,3],[2,3]])
        Varray([[0], [1, 4], [2], [3]])
        >>> equalRows([[1,2],[2,3],[3,2],[1,3],[2,3]],permutations='all')
        Varray([[0], [1, 2, 4], [3]])
        >>> equalRows([[1,2,3],[3,2,1],[2,3,1],[1,2,3]])
        Varray([[0, 3], [1], [2]])
        >>> equalRows([[1,2,3],[3,2,1],[2,3,1],[1,2,3]],permutations='all')
        Varray([[0, 1, 2, 3]])
        >>> equalRows([[1,2,3],[3,2,1],[2,3,1],[1,2,3]],permutations='roll')
        Varray([[0, 2, 3], [1]])
        """
        from pyformex import varray
        ind, ok = findEqualRows(a,permutations=permutations)
        w = np.where(ok)[0]
        w = np.concatenate([w,[len(ind)]])
        v = [ind[np.arange(i,j)] for i,j in zip (w[:-1],w[1:])]
        v = sorted(v,key=lambda a: a.min())
        return varray.Varray(v)

    def uniqueRows(a,permutations='none'):
        """Find the unique rows of a 2-D array.

        Parameters: see :meth:`findEqualRows`

        Returns
        -------
        uniq: 1-dim int array
            Contains the indices of the unique rows in `a`.

        See Also
        --------
        equalRows: return the indices of the grouped equal rows
        uniqueRowsIndex: like uniqueRows, but also returns index for all rows

        Examples
        --------
        >>> uniqueRows([[1,2],[2,3],[3,2],[1,3],[2,3]])
        array([0, 1, 2, 3])
        >>> uniqueRows([[1,2],[2,3],[3,2],[1,3],[2,3]],permutations='all')
        array([0, 1, 3])
        >>> uniqueRows([[1,2,3],[3,2,1],[2,3,1],[1,2,3]])
        array([0, 1, 2])
        >>> uniqueRows([[1,2,3],[3,2,1],[2,3,1],[1,2,3]],permutations='all')
        array([0])
        >>> uniqueRows([[1,2,3],[3,2,1],[2,3,1],[1,2,3]],permutations='roll')
        array([0, 1])
        >>> uniqueRows([[1,2,3],[3,2,1],[2,3,1],[1,2,3]])
        array([0, 1, 2])

        """
        ind, ok = findEqualRows(a,permutations=permutations)
        return np.sort(ind[ok])

    def uniqueRowsIndex(a,permutations='none'):
        """Return the unique rows of a 2-D array and an index for all rows.

        Parameters
        ----------
        a: array-like, 2-dim
            Array from which to find the unique rows.
        permutations: bool
             If True, rows which are permutations of the same data are considered
            equal. The default is to consider permutations as different.
        roll: bool
            If True, rows which can be rolled into the same contents are
            considered equal.

        Returns
        -------
        uniq: 1-dim int array
            Contains the indices of the unique rows in `a`.
            The order of the elements in `uniq` is determined by the sorting
            procedure: in the current implementation this is :func:`sortByColumns`.
            If `permutations==True`, `a` is sorted along its last axis -1 before
            calling this sorting function. If `roll=True`, the rows of ``a``
            are rolled to put the lowest values at the front.
        ind: 1-dim int array
            For each row of `a`, holds the index in `uniq` where the row with
            the same data is found.

        See Also
        --------
        equalRows: return the indices of the grouped equal rows
        uniqueRows: return the indices of the unique rows

        Examples
        --------
        >>> print(*uniqueRowsIndex([[1,2],[2,3],[3,2],[1,3],[2,3]]))
        [0 1 2 3] [0 1 2 3 1]
        >>> print(*uniqueRowsIndex([[1,2],[2,3],[3,2],[1,3],[2,3]],permutations='all'))
        [0 1 3] [0 1 1 2 1]
        >>> print(*uniqueRowsIndex([[1,2,3],[3,2,1],[2,3,1],[1,2,3]]))
        [0 1 2] [0 1 2 0]
        >>> print(*uniqueRowsIndex([[1,2,3],[3,2,1],[2,3,1],[1,2,3]],permutations='all'))
        [0] [0 0 0 0]
        >>> print(*uniqueRowsIndex([[1,2,3],[3,2,1],[2,3,1],[1,2,3]],permutations='roll'))
        [0 1] [0 1 0 0]
        >>> print(*uniqueRowsIndex([[1,2,3],[3,2,1],[2,3,1],[1,2,3]]))
        [0 1 2] [0 1 2 0]

        """
        Va = equalRows(a,permutations=permutations)
        return Va.col(0), Va.inverse(expand=True).reshape(-1)

    def argNearestValue(values, target):
        """Return the index of the item nearest to target.

        Find in a list of floats the position of the value nearest to
        the target value.

        Parameters
        ----------
        values: list
            List of float values.
        target: float
            Target value to look up in list.

        Returns
        -------
        :int
            The index in `values` of the float value that is closest
            to `target`.

        See Also
        --------
        nearestValue

        Examples
        --------
        >>> argNearestValue([0.1,0.5,0.9],0.7)
        1
        """
        v = np.array(values).ravel()
        c = v - target
        return argmin(c*c)

    def nearestValue(values, target):
        """Return the float nearest to target.

        Find in a list of floats the value that is closest to
        the target value.

        Parameters
        ----------
        values: list
            List of float values.
        target: float
            Target value to look up in list.

        Returns
        -------
        :float
            The value from the list that is closest to `target`.

        See Also
        --------
        argNearestValue

        Examples
        --------
        >>> nearestValue([0.1,0.5,0.9],0.7)
        0.5
        """
        return values[argNearestValue(values, target)]

    def inverseIndex(a,sort=True,expand=True):
        """Create the inverse of a 2D index array.

        A 2D index array is a 2D integer array where only the nonnegative values.
        are relevant. Negative values are flagging a non-existent element. This
        allows for rows with different number of entries.
        While in most practical cases all (non-negative) values in a row are
        unique, this is not a requirement.

        Note
        ----
        This function is a wrapper around :func:`varray.inverseIndex` with
        other default values for the optional arguments. See there for parameters
        and return values.

        Examples
        --------
        >>> A = array([[0,1],[0,2],[1,2],[0,3]])
        >>> print(A)
        [[0 1]
         [0 2]
         [1 2]
         [0 3]]
        >>> inv = inverseIndex(A)
        >>> print(inv)
        [[ 0  1  3]
         [-1  0  2]
         [-1  1  2]
         [-1 -1  3]]

        We can use :class:`varray.Varray` to get rid of the -1 entries:

        >>> inv1 = inverseIndex([[0,1],[0,2],[1,2],[0,3]],expand=False)
        >>> print(inv1)
        Varray (4,3)
          [0 1 3]
          [0 2]
          [1 2]
          [3]
        <BLANKLINE>

        In both cases, the inverse of the inverse returns the original:

        >>> (inverseIndex(inv) == A).all()
        True
        >>> (inverseIndex(inv1) == A).all()
        True
        """
        from pyformex import varray
        return varray.inverseIndex(a, sort=sort, expand=expand)

    def findFirst(target, values):
        """Find first position of values in target.

        Find the first position in the array `target` of all the elements
        in the array `values`.

        Parameters
        ----------
        target: 1-dim int array
            Integer array with all non-negative values. If not 1-dim, it
            will be flattened.
        values: 1-dim int array
            Array with values to look up in target. If not 1-dim, it
            will be flattened.

        Returns
        -------
        : int array
            Array with same size as `values`. For each element in `values`,
            the return array contains the position of that value
            in the flattened `target`, or -1 if that number does not occur
            in `target`.
            If an element from `values` occurs more than once in `target`, it is
            currently undefined which of those positions is returned.

        Note
        ----
        After ``m = findIndex(target,values)`` the equality
        ``target[m] == values`` holds for all the non-negative positions of `m`.

        Examples
        --------
        >>> A = array([1,3,4,5,7,3,8,9])
        >>> B = array([0,7,1,3])
        >>> ind = findFirst(A,B)
        >>> print(ind)
        [-1  4  0 1]
        >>> (A[ind[ind>=0]] == B[ind>=0]).all()
        True
        """
        from pyformex import varray
        target = checkArray1D(target,kind='i').reshape(-1, 1)
        values = checkArray1D(values,kind='i')
        inv = varray.inverseIndex(target,sort=True)
        return array([inv[i][0] if i < inv.nrows and len(inv[i]) > 0 else -1 for i in values])

    def findAll(target, values):
        """Find all locations of values in target.

        Find the position in the array `target` of all occurrences of
        the elements in the array `values`.

        Parameters
        ----------
        target: 1-dim int array
            Integer array with all non-negative values. If not 1-dim, it
            will be flattened.
        values: 1-dim int array
            Array with values to look up in target. If not 1-dim, it
            will be flattened.

        Returns
        -------
        : list of int arrays.
            For each element in values, an array is returned with the indices
            in target of the elements with the same value.

        See Also
        --------
        findFirst

        Examples
        --------
        >>> gid = array([ 2, 1, 1, 6, 6, 1 ])
        >>> values = array([ 1, 2, 6 ])
        >>> print(findAll(gid,values))
        [array([1, 2, 5]), array([0]), array([3, 4])]
        """
        return [ where(target==i)[0] for i in values ]

    def groupArgmin(val, gid):
        """Compute the group minimum.

        Computes the minimum value per group of a set of values tagged with
        a group number.

        Parameters
        ----------
        val: 1-dim array
            Data values
        gid: 1-dim int array-like
            Array with same length as val, containing the group identifiers.

        Returns
        -------
        ugid: 1-dim int array
            (ngrp,) shaped array with unique group identifiers.
        minpos: 1-dim int array
            (ngrp,) shaped array giving the position in `val` of the minimum
            of all values with the corresponding group identifier in `ugid`.
            The minimum values corresponding to the groups in `ugid` can be
            obtained with ``val[minpos]``.

        Examples
        --------
        >>> val = array([ 0.0, 1.0, 2.0, 3.0, 4.0, -5.0 ])
        >>> gid = array([ 2, 1, 1, 6, 6, 1 ])
        >>> print(groupArgmin(val,gid))
        (array([1, 2, 6]), array([5, 0, 3]))
        """
        ugid = unique(gid)
        pos = findAll(gid, ugid)
        minid = hstack([ val[ind].argmin() for ind in pos ])
        minpos = hstack([ ind[k] for ind, k in zip(pos, minid) ])
        return ugid, minpos

    ###########################################################
    # Working with sets of vectors
    ###########################

    def vectorPairAreaNormals(vec1, vec2):
        """Compute area of and normals on parallellograms formed by vec1 and vec2.

        Parameters
        ----------
        vec1: (3,) or (n,3) shaped float array-like
            Array with 1 or n vectors in 3D space.
        vec2: (3,) or (n,3) shaped float array-like.
            Array with 1 or n vectors in 3D space.

        Returns
        -------
        area: (n,) shaped float array
            The area of the parallellograms formed by the vectors vec1 and vec2.
        normal: (n,3) shaped float array
            The unit length vectors normal to each vector pair (vec1,vec2).

        Note
        ----
        This first computes the cross product of vec1 and vec2, which is a normal
        vector with length equal to the area. Then :func:`normalize` produces
        the required results.

        Note that where two vectors are parallel, an area zero results and
        an axis with components NaN.

        See Also
        --------
        vectorPairNormals: only returns the normal vectors
        vectorPairArea: only returns the area

        Examples
        --------
        >>> a = array([[3.,4,0],[1,0,0],[1,-2,1]])
        >>> b = array([[1.,3.,0],[1,0,1],[-2,4,-2]])
        >>> l,v = vectorPairAreaNormals(a,b)
        >>> print(l)
        [ 5.  1.  0.]
        >>> print(v)
        [[  0.   0.   1.]
         [  0.  -1.   0.]
         [ nan  nan  nan]]
        """
        vec1 = checkArray(vec1,kind='f',allow='i')
        vec2 = checkArray(vec2,kind='f',allow='i')
        normal = cross(vec1.reshape(-1, 3), vec2.reshape(-1, 3))
        return normalize(normal,return_length=True)[::-1]

    def vectorPairNormals(vec1, vec2):
        """Create unit vectors normal to vec1 and vec2.

        Parameters
        ----------
        vec1: (3,) or (n,3) shaped float array-like
            Array with 1 or n vectors in 3D space.
        vec2: (3,) or (n,3) shaped float array-like.
            Array with 1 or n vectors in 3D space.

        Returns
        -------
        normal: (n,3) shaped float array
            The unit length vectors normal to each vector pair (vec1,vec2).

        See Also
        --------
        vectorPairAreaNormals: returns the normals and the area between vectors
        vectorPairArea: only returns the area between vectors

        Examples
        --------
        >>> a = array([[3.,4,0],[1,0,0],[1,-2,1]])
        >>> b = array([[1.,3.,0],[1,0,1],[-2,4,-2]])
        >>> v = vectorPairNormals(a,b)
        >>> print(v)
        [[  0.   0.   1.]
         [  0.  -1.   0.]
         [ nan  nan  nan]]
        """
        return vectorPairAreaNormals(vec1, vec2)[1]

    def vectorPairArea(vec1, vec2):
        """Compute area of the parallellogram formed by a vector pair vec1,vec2.

        Parameters
        ----------
        vec1: (3,) or (n,3) shaped float array-like
            Array with 1 or n vectors in 3D space.
        vec2: (3,) or (n,3) shaped float array-like.
            Array with 1 or n vectors in 3D space.

        Returns
        -------
        area: (n,) shaped float array
            The area of the parallellograms formed by the vectors vec1 and vec2.

        See Also
        --------
        vectorPairAreaNormals: returns the normals and the area between vectors
        vectorPairNormals: only returns the normal vectors

        Examples
        --------
        >>> a = array([[3.,4,0],[1,0,0],[1,-2,1]])
        >>> b = array([[1.,3.,0],[1,0,1],[-2,4,-2]])
        >>> l = vectorPairArea(a,b)
        >>> print(l)
        [ 5.  1.  0.]
        """
        vec1 = checkArray(vec1,kind='f',allow='i')
        vec2 = checkArray(vec2,kind='f',allow='i')
        normal = cross(vec1.reshape(-1, 3), vec2.reshape(-1, 3))
        return length(normal)

    def vectorTripleProduct(vec1, vec2, vec3):
        """Compute triple product vec1 . (vec2 x vec3).

        Parameters
        ----------
        (vec1, vec2, vec3): three (3,) or (n,3) shaped float array-like
            Three arrays with same shape holding 1 or n vectors in 3D space.

        Returns
        -------
        : (n,) shaped float array
            The triple product of each set of corresponding vectors from
            vec1, vec2, vec3.

        Note
        ----
        The triple product is the dot product of the first vector(s) and
        the normal(s) on the second and third vector(s).
        This is also twice the volume of the parallellepid formed by
        the 3 vectors.

        If vec1 has a unit length, the result is also the area of the parallellogram
        (vec2,vec3) projected in the direction vec1.

        This is functionally equivalent with `dotpr(vec1, cross(vec2, vec3))`
        but is implemented in a more efficient way, using the determinant formula.

        Examples
        --------
        >>> vectorTripleProduct([[1.,0.,0.],[2.,0.,0.]],
        ...                     [[1.,1.,0.],[2.,2.,0.]],
        ...                     [[1.,1.,1.],[2.,2.,2.]])
        array([ 1., 8.])
        """
        vec1 = np.asarray(vec1)
        vec2 = np.asarray(vec2)
        vec3 = np.asarray(vec3)
        a11 = vec1[...,0]
        a12 = vec1[...,1]
        a13 = vec1[...,2]
        a21 = vec2[...,0]
        a22 = vec2[...,1]
        a23 = vec2[...,2]
        a31 = vec3[...,0]
        a32 = vec3[...,1]
        a33 = vec3[...,2]
        return a11*(a22*a33-a32*a23) + a12*(a23*a31-a33*a21) + a13*(a21*a32-a22*a31)

    def vectorPairCosAngle(v1, v2):
        """Return the cosinus of the angle between the vectors v1 and v2.

        vec1 and vec2 are (n,3) shaped arrays holding collections of vectors.
        The result is an (n) shaped array with the cosinus of the angle between
        each pair of vectors (vec1,vec2).
        """
        v1 = np.asarray(v1)
        v2 = np.asarray(v2)
        cos = dotpr(v1, v2) / sqrt(dotpr(v1, v1)*dotpr(v2, v2))
        # clip to [-1.,1.] in case of rounding errors
        return cos.clip(min=-1., max=1.)

    def vectorPairAngle(v1, v2):
        """Return the angle (in radians) between the vectors v1 and v2.

        vec1 and vec2 are (n,3) shaped arrays holding collections of vectors.
        The result is an (n) shaped array with the angle between
        each pair of vectors (vec1,vec2).

        >>> vectorPairAngle([1,0,0],[0,1,0]) / DEG
        90.0
        >>> vectorPairAngle([[1,0,0],[0,1,0]],[[1,1,0],[1,1,1]]) / DEG
        array([ 45.  ,  54.74])
        """
        return arccos(vectorPairCosAngle(v1, v2))

    def det2(a):
        """Compute the determinant of 2x2 matrices.

        Parameters
        ----------
        a: int or float array-like(...,2,2)
            Array containing one or more (2,2) square matrices.

        Returns
        -------
        : int or float number or array(...)
             The determinant(s) of the matrices. The result has the same type
             as the input array.

        Note
        ----
        This method is faster than the generic numpy.linalg.det.

        See Also
        --------
        det3: determinant of (3,3) matrices
        det4: determinant of (4,4) matrices
        numpy.linalg.det: determinant of any size matrix

        Examples
        --------
        >>> det2([[1,2],[2,1]])
        -3
        >>> det2([[[1,2],[2,1]],[[4,2],[1,3]]])
        array([-3, 10])

        """
        a = np.asarray(a)
        a11 = a[...,0,0]
        a12 = a[...,0,1]
        a21 = a[...,1,0]
        a22 = a[...,1,1]
        return a11*a22 - a12*a21

    def det3(a):
        """Compute the determinant of 3x3 matrices.

        Parameters
        ----------
        a: int or float array-like(...,3,3)
            Array containing one or more (3,3) square matrices.

        Returns
        -------
        : int or float number or array(...)
             The determinant(s) of the matrices. The result has the same type
             as the input array.

        Note
        ----
        This method is faster than the generic numpy.linalg.det.

        See Also
        --------
        det2: determinant of (2,2) matrices
        det4: determinant of (4,4) matrices
        numpy.linalg.det: determinant of any size matrix

        Examples
        --------
        >>> det3([[1,2,3],[2,2,2],[3,2,1]])
        0
        >>> det3([[[1.,0.,0.],[1.,1.,0.],[1.,1.,1.]],
        ...       [[2.,0.,0.],[2.,2.,0.],[2.,2.,2.]]])
        array([ 1., 8.])
        """
        a = np.asarray(a)
        return vectorTripleProduct(a[...,0,:],a[...,1,:],a[...,2,:])

    def det4(a):
        """Compute the determinant of 4x4 matrices.

        Parameters
        ----------
        a: int or float array-like(...,4,4)
            Array containing one or more (4,4) square matrices.

        Returns
        -------
        : int or float number or array(...)
             The determinant(s) of the matrices. The result has the same type
             as the input array.

        Note
        ----
        This method is faster than the generic numpy.linalg.det.

        See Also
        --------
        det2: determinant of (2,2) matrices
        det3: determinant of (3,3) matrices
        numpy.linalg.det: determinant of any size matrix

        Examples
        --------
        >>> det4([[[1.,0.,0.,0.],[1.,1.,0.,0.],[1.,1.,1.,0.],[1.,1.,1.,1.]],
        ...       [[2.,0.,0.,0.],[2.,2.,0.,0.],[2.,2.,2.,0.],[2.,2.,2.,2.]]])
        array([ 1., 16.])
        """
        a = np.asarray(a)
        a00 = a[...,0,0]
        a01 = a[...,0,1]
        a02 = a[...,0,2]
        a03 = a[...,0,3]
        m00 = det3(a[...,1:,[1,2,3]])
        m01 = det3(a[...,1:,[0,2,3]])
        m02 = det3(a[...,1:,[0,1,3]])
        m03 = det3(a[...,1:,[0,1,2]])
        return a00*m00 - a01*m01 + a02*m02 - a03*m03

    def percentile(values,perc=[25., 50., 75.],wts=None):
        """Return percentiles of a set of values.

        A percentiles is the value such that at least a given percent of the
        values is lower or equal than the value.

        Parameters
        ----------
        values: 1-dim int or float array-like
            The set of values for which to compute the percentiles.
        perc: 1-dim int or float array-like
            One or multiple percentile values to compute. All values should be
            in the range [0,100]. By default, the quartiles are computed.
        wts: 1-dim array
            Array with same shape as values and all positive values. These are
            weights to be assigned to the values.

        Returns
        -------
        : 1-dim float array
            Array with the percentile value(s) that is/are greater or equal than
            `perc` percent of `values`. If the result lies between two items of
            `values`, it is obtained by interpolation.

        Examples
        --------
        >>> percentile(arange(100),[10,50,90])
        array([  9.,  49.,  89.])
        >>> percentile([1,1,1,1,1,2,2,2,3,5])
        array([ 1.,  1.,  2.])

        """
        values = checkArray1D(values)
        perc = checkArray1D(perc)
        if perc.min() < 0. or perc.max() > 100.:
            raise ValueError("Percentiles should be between 0 and 100, got %s." % perc)
        if wts is not None and wts.min() <= 0.:
            raise ValueError("Weights should be positive, got %s." % wts.min())
        ind = values.argsort()
        values = values[ind]
        if wts is None:
            wts = resize(1., values.shape)
        else:
            wts = wts[ind]
        wts = wts.cumsum()
        w = perc /100. * (wts[-1])
        ind = wts.searchsorted(w)
        val = where(ind>0, values[ind-1]+(w-wts[ind-1])/(wts[ind]-wts[ind-1])*(values[ind]-values[ind-1]), values[0])
        return val

    def histogram2(a,bins,range=None):
        """Compute the histogram of a set of data.

        This is similar to the numpy histogram function, but also returns
        the bin index for each individual entry in the data set.

        Parameters
        ----------
        a: array_like.
            Input data. The histogram is computed over the flattened array.
        bins: int or sequence of scalars.
            If `bins` is an int, it defines the number of equal-width bins
            in the given range (nbins).
            If `bins` is a sequence, it defines the bin edges, including the
            rightmost edge, allowing for non-uniform bin widths. The number of
            bins (nbins) is then equal to `len(bins) - 1`.
            A value `v` will be sorted in bin `i` if
            `bins[i] <= v < bins[i+1]`, except for the last bin, which will also
            contain the values equal to the right bin edge.
        range`: (float, float), optional.
            The lower and upper range of the bins.
            If not provided, range is simply (a.min(), a.max()). Values outside the
            range are ignored. This parameter is ignored if bins is a sequence.

        Returns
        -------
        hist: int array
            The number of elements from `a` sorted in each of the bins.
        ind: list of ``nbins`` int arrays
            Each array holds the indices the elements sorted in the
            corresponding bin.
        bin_edges: float array
            The array contains the ``len(hist)+1`` bin edges.

        Example
        -------
        >>> hist,ind,bins = histogram2([1,2,3,4,2,3,1],[1,2,3,4,5])
        >>> print(hist)
        [2 2 2 1]
        >>> for i in ind: print(i)
        [0 6]
        [1 4]
        [2 5]
        [3]
        >>> print(bins)
        [1 2 3 4 5]
        >>> hist,bins = histogram([1,2,3,4,2,3,1],5)
        >>> print(hist)
        [2 2 0 2 1]
        >>> hist,ind,bins = histogram2([1,2,3,4,2,3,1],5)
        >>> print(hist)
        [2 2 0 2 1]
        >>> for i in ind: print(i)
        [0 6]
        [1 4]
        []
        [2 5]
        [3]
        """
        a = np.asarray(a)
        if iterable(bins):
            bins = np.asarray(bins)
            if (np.diff(bins) < 0).any():
                raise ValueError("`bins` must increase monotonically.")
        else:
            if not isInt(bins) or bins < 1:
                raise ValueError("`bins` should be a positive integer.")
            if range is None:
                if a.size == 0:
                    # handle empty arrays. Can't determine range, so use 0-1.
                    range = (0, 1)
                else:
                    range = (a.min(), a.max())

                mn, mx = [mi+0.0 for mi in range]
                if mn == mx:
                    mn -= 0.5
                    mx += 0.5
                    bins = 1

            else:
                mn, mx = range
                if mn >= mx:
                    raise ValueError('max must be larger than min in `range` parameter.')
            bins = np.linspace(mn, mx, bins+1, endpoint=True)

            # Make sure to include the right edge of the last bin
            mx = bins[-1]
            bins[-1] = np.nextafter(mx,mx+0.5)

        d = digitize(a, bins)
        ind = [ where(d==i)[0] for i in arange(1, len(bins)) ]
        hist = np.asarray([ i.size for i in ind ])
        return hist, ind, bins

    def movingView(a, size):
        """Create a moving view along the first axis of an array.

        A moving view of an array is a view stacking a sequence of subarrays
        with fixed size along the 0 axis of the array, where each next subarray
        shifts one position down along the 0 axis.

        Parameters
        ----------
        a : array_like
            Array for which to create a moving view.
        size : int
            Size of the moving view: this is the number of rows to include
            in the subarray.

        Returns
        -------
        : view of the array ``a``
            The view of the original array has an extra first axis with length
            ``1 + a.shape[0] - size``, a second axis with length ``size``, and
            the remaining axes have the same length as those in ``a``.

        Note
        ----
        While this function limits the moving view to the direction of the 0 axis,
        using swapaxes(0,axis) allows to create moving views over any axis.

        See Also
        --------
        movingAverage: compute moving average values along axis 0

        Examples
        --------
        >>> x=arange(10).reshape((5,2))
        >>> print(x)
        [[0 1]
         [2 3]
         [4 5]
         [6 7]
         [8 9]]
        >>> print(movingView(x, 3))
        [[[0 1]
          [2 3]
          [4 5]]
        <BLANKLINE>
         [[2 3]
          [4 5]
          [6 7]]
        <BLANKLINE>
         [[4 5]
          [6 7]
          [8 9]]]

        Calculate rolling sum of first axis:

        >>> print(movingView(x, 3).sum(axis=0))
        [[ 6  9]
         [12 15]
         [18 21]]

        """
        from numpy.lib import stride_tricks
        if size < 1:
            raise ValueError("`size` must be at least 1.")
        if size > a.shape[0]:
            raise ValueError("`size` is too long.")
        shape = (size, a.shape[0] - size + 1) + a.shape[1:]
        strides = (a.strides[0],) + a.strides
        return stride_tricks.as_strided(a, shape=shape, strides=strides)

    def movingAverage(a,n,m0=None,m1=None):
        """Compute the moving average along the first axis of an array.

        Parameters
        ----------
        a: array_like
            The array to be averaged.
        n: int
            Sample length along axis 0 over which to compute the average.
        m0: int, optional
            If provided, the first data row of ``a`` will be prepended to ``a``
            this number of times.
        m1 : int, optional
            If provided, the last data row of ``a`` will  be appended to ``a``
            this number of times.

        Returns
        -------
        : float array
            Array containing the moving average over data sets of length ``n``
            along the first axis of ``a``.
            The array has a shape like ``a`` except for its first axis,
            which may have a different length.
            If neither m0 nor m1 are set, the first axis will have a length of
            1 + a.shape[0] - n.
            If both m0 and m1 are given, the first axis will have a length of
            1 + a.shape[0] - n + m0 + m1.
            If either m0 or m1 are set and the other not, the missing value m0
            or m1 will be computed thus that the return array has a first axis
            with length a.shape[0].

        Examples
        --------
        >>> x=arange(10).reshape((5,2))
        >>> print(x)
        [[0 1]
         [2 3]
         [4 5]
         [6 7]
         [8 9]]
        >>> print(movingAverage(x,3))
        [[ 2. 3.]
         [ 4. 5.]
         [ 6. 7.]]
        >>> print(movingAverage(x,3,2))
        [[ 0.   1.  ]
         [ 0.67 1.67]
         [ 2.   3.  ]
         [ 4.   5.  ]
         [ 6.   7.  ]]
        """
        if m0 is None and m1 is None:
            ae = a
        else:
            if m0 is None:
                m0 = n-1 - m1
            elif m1 is None:
                m1 = n-1 - m0
            if m0 < 0 or m1 < 0:
                raise ValueError("Invalid value a m0 or m1")
            ae = [a[:1]] * m0 + [ a ] + [a[-1:]] * m1
            ae = concatenate(ae, axis=0)
        return movingView(ae, n).mean(axis=0)

    def randomNoise(shape,min=0.0,max=1.0):
        """Create an array with random float values between min and max

        Parameters
        ----------
        shape: tuple of ints
            Shape of the array to create.
        min: float
            Minimum value of the random numbers.
        max: float
            Maximum value of the random numbers.

        Returns
        -------
        : float array
            An array of the requested shape filled with random numbers
            in the specified range.

        Examples
        --------
        >>> x = randomNoise((3,4))
        >>> x.shape == (3,4)
        True
        >>> (x >= 0.0).all()
        True
        >>> (x <= 1.0).all()
        True
        """
        return random.random(shape) * (max-min) + min

    def unitDivisor(div,start=0):
        """Divide a unit interval in equal parts.

        This function is intended to be used by interpolation functions
        that accept an input as either an int or a list of floats.

        Parameters
        ----------
        div: int, or list of floats in the range [0.0, 1.0].
            If it is an integer, it specifies the number of equal sized parts
            in which the interval [0.0, 1.0] is to be divided.
            If a list of floats, its values should be monotonuously increasing
            from 0.0 to 1.0. The values are returned unchanged.
        start: 0 or 1
            Set to 1 to omit the first value (0.0) of the interval in the output.

        Returns
        -------
        : 1-dim float array
            The float values that border the parts of the interval.
            If `div` is a an integer, returns the floating point values
        dividing the unit interval in div equal parts. If `div` is a list,
        just returns `div` as a 1D array.

        Examples
        --------
        >>> unitDivisor(4)
        array([ 0.  , 0.25, 0.5 , 0.75, 1.  ])
        >>> unitDivisor(4,1)
        array([ 0.25, 0.5 , 0.75, 1.  ])
        >>> unitDivisor([0., 0.3, 0.7, 1.0])
        array([ 0. ,  0.3,  0.7,  1. ])
        """
        div = np.asarray(div).ravel()
        if div.size == 1 and div.dtype.kind=='i':
            n = div[0]
            div = arange(start, n+1) / float(n)
        return div

    def uniformParamValues(n,umin=0.0,umax=1.0):
        """Create a set of uniformly distributed parameter values in a range.

        Parameters
        ----------
        n: int
            Number of intervals in which the range should be divided.
            The number of values returned is ``n+1``.
        umin: float
            Starting value of the interval.
        umax: float
            Ending value of the interval.

        Returns
        -------
        : 1-dim float array
            The array contains n+1 equidistant values in the range [umin, umax].
            For n > 0, both of the endpoints are included. For n=0, a single
            value at the center of the interval will be returned. For n<0, an
            empty array is returned.

        Examples
        --------
        >>> uniformParamValues(4).tolist()
        [0.0, 0.25, 0.5, 0.75, 1.0]
        >>> uniformParamValues(0).tolist()
        [0.5]
        >>> uniformParamValues(-1).tolist()
        []
        >>> uniformParamValues(2,1.5,2.5).tolist()
        [1.5, 2.0, 2.5]
        """
        if n == 0:
            return array([0.5*(umax+umin)])
        else:
            return umin + arange(n+1) * (umax-umin) / float(n)

    def nodalSum(val, elems, nnod=-1):
        """Compute the nodal sum of values defined on element nodes.

        Parameters
        ----------
        val: float array (nelems,nplex,nval)
            Array with ``nval`` values at ``nplex`` nodes of ``nelems`` elements.
        elems: int array (nelems,nplex)
            The node indices of the elements.
        nnod: int, optional
            If provided, the length of the output arrays will be set to
            this value. It should be higher than the highest node number
            appering in elems. The default will set it automatically to
            ``elems.max() + 1``.

        Returns
        sum: float array (nnod, nval)
            The sum of all the values at the same node.
        cnt: int array (nnod)
            The number of values summed at each node.

        See Also
        --------
        nodalAvg: compute the nodal average of values defined on element nodes
        """
        from pyformex.lib import misc
        if val.ndim != 3:
            val = val.reshape(val.shape+(1,))
        if elems.shape != val.shape[:2]:
            raise RuntimeError("shapes of val %s and elems %s do not match"%(elems.shape, val.shape[:2]))
        val = val.astype(Float)
        elems = elems.astype(Int)
        return misc.nodalSum(val, elems, nnod)

    def nodalAvg(val, elems, nnod=-1):
        """Compute the nodal average of values defined on element nodes.

        Parameters
        ----------
        val: float array (nelems,nplex,nval)
            Array with ``nval`` values at ``nplex`` nodes of ``nelems`` elements.
        elems: int array (nelems,nplex)
            The node indices of the elements.
        nnod: int, optional
            If provided, the length of the output arrays will be set to
            this value. It should be higher than the highest node number
            appering in elems. The default will set it automatically to
            ``elems.max() + 1``.

        Returns
        avg: float array (nnod, nval)
            The average of all the values at the same node.

        See Also
        --------
        nodalSum: compute the nodal sum of values defined on element nodes
        """
        sum, cnt = nodalSum(val, elems, nnod)
        return sum/cnt[:,newaxis]

    def fmtData1d(data,npl=8,sep=', ',linesep='\n',fmt=str):
        """Format data in lines with maximum npl items.

        Formats a list or array of data items in groups containing
        a maximum number of items. The data items are converted to
        strings using the `fmt` function, concatenated
        in groups of `npl` items using `sep` as a separator between them.
        Finally, the groups are concatenated with a `linesep` separator.

        Parameters
        ----------
        data: list or array.
            List or array with data. If an array, if will be flattened.
        npl: int
            Maximum number of items per group. Items will be concatenated
            groups of this number of items. The last group
            may contain less items.
        sep: str
            Separator to add between individual items in a group.
        linesep: str
            Separator to add between groups. The default (newline) will
            put each group of `npl` items on a separate line.
        fmt: callable
            Used to convert a single item to a string. Default
            is the Python built-in string converter.

        Returns
        -------
        : str
            Multiline string with the formatted data.

        Examples
        --------
        >>> print(fmtData1d(arange(10)))
        0, 1, 2, 3, 4, 5, 6, 7
        8, 9
        >>> print(fmtData1d([1.25, 3, 'no', 2.50, 4, 'yes'],npl=3))
        1.25, 3, no
        2.5, 4, yes
        >>> myformat = lambda x: "%10s" % str(x)
        >>> print(fmtData1d([1.25, 3, 'no', 2.50, 4, 'yes'],npl=3,fmt=myformat))
              1.25,          3,         no
               2.5,          4,        yes

        """
        if isinstance(data,ndarray):
            data = data.flat
        return linesep.join([
            sep.join(map(fmt, data[i:i+npl])) for i in range(0, len(data), npl)
            ])

    # These are retained for compatibility, but deprecated
    def trfMatrix(*args,**kargs):
        warnings.warn("warn_trfMatrix_deprecated")
        return trfmat(*args,**kargs)
    def vectorRotation(*args,**kargs):
        warnings.warn("warn_vectorRotation_deprecated")
        return rotMatrix2(*args,**kargs)
    def rotationAnglesFromMatrix(*args,**kargs):
        warnings.warn("warn_rotationAnglesFromMatrix_deprecated")
        return cardanAngles(*args,**kargs)
    def findIndex(*args,**kargs):
       warnings.warn("warn_findIndex_deprecated")
       return findFirst(*args,**kargs)
    def groupPositions(*args,**kargs):
        warnings.warn("warn_groupPositions_deprecated")
        return findAll(*args,**kargs)
    def vectorLength(*args,**kargs):
        warnings.warn("warn_vectorLength_deprecated")
        return length(*args,**kargs)
    def vectorNormalize(A):
        warnings.warn("warn_vectorNormalize_deprecated")
        v,l = normalize(A,return_length=True)
        return l,v

    # Removed
    #def matchIndex(*args,**kargs):
    #    warnings.warn("warn_matchIndex_deprecated")
    #    return findFirst(*args,**kargs)

    # End