#!/usr/bin/python

r'''more-reasonable core functionality for numpy

* SYNOPSIS
    >>> import numpy as np
    >>> import numpysane as nps

    >>> a   = np.arange(6).reshape(2,3)
    >>> b   = a + 100
    >>> row = a[0,:] + 1000

    >>> a
    array([[0, 1, 2],
           [3, 4, 5]])

    >>> b
    array([[100, 101, 102],
           [103, 104, 105]])

    >>> row
    array([1000, 1001, 1002])

    >>> nps.glue(a,b, axis=-1)
    array([[  0,   1,   2, 100, 101, 102],
           [  3,   4,   5, 103, 104, 105]])

    >>> nps.glue(a,b,row, axis=-2)
    array([[   0,    1,    2],
           [   3,    4,    5],
           [ 100,  101,  102],
           [ 103,  104,  105],
           [1000, 1001, 1002]])

    >>> nps.cat(a,b)
    array([[[  0,   1,   2],
            [  3,   4,   5]],

           [[100, 101, 102],
            [103, 104, 105]]])

    >>> @nps.broadcast_define( (('n',), ('n',)) )
    ... def inner_product(a, b):
    ...     return a.dot(b)

    >>> inner_product(a,b)
    array([ 305, 1250])

* DESCRIPTION

Numpy is a very widely used toolkit for numerical computation in Python. Despite
its popularity, some of its core functionality is mysterious and/or incomplete.
The numpysane library seeks to fill those gaps by providing its own replacement
routines. Many of the replacement functions are direct translations from PDL
(http://pdl.perl.org), a numerical computation library for perl. The functions
provided by this module fall into three broad categories:

- Broadcasting support
- Nicer array manipulation
- Basic linear algebra

** Broadcasting
Numpy has a limited support for broadcasting
(http://docs.scipy.org/doc/numpy/user/basics.broadcasting.html), a generic way
to vectorize functions. A broadcasting-aware function knows the dimensionality
of its inputs, and any extra dimensions in the input are automatically used for
vectorization.

*** Broadcasting rules
A basic example is an inner product: a function that takes in two
identically-sized 1-dimensional arrays (input prototype (('n',), ('n',)) ) and
returns a scalar (output prototype () ). If one calls a broadcasting-aware inner
product with two arrays of shape (2,3,4) as input, it would compute 6 inner
products of length-4 each, and report the output in an array of shape (2,3).

In short:

- The most significant dimension in a numpy array is the LAST one, so the
  prototype of an input argument must exactly match a given input's trailing
  shape. So a prototype shape of (a,b,c) accepts an argument shape of (......,
  a,b,c), with as many or as few leading dimensions as desired.
- The extra leading dimensions must be compatible across all the inputs. This
  means that each leading dimension must either
  - equal 1
  - be missing (thus assumed to equal 1)
  - equal to some positive integer >1, consistent across all arguments
- The output is collected into an array that's sized as a superset of the
  above-prototype shape of each argument

More involved example: A function with input prototype ( (3,), ('n',3), ('n',),
('m',) ) given inputs of shape

    (1,5,    3)
    (2,1,  8,3)
    (        8)
    (  5,    9)

will return an output array of shape (2,5, ...), where ... is the shape of each
output slice. Note again that the prototype dictates the TRAILING shape of the
inputs.

*** What about the stock broadcasting support?

The numpy documentation dedicates a whole page explaining the broadcasting
rules, but only a small number of numpy functions provide any broadcasting
support. It's fairly inconsistent, and most functions have no broadcasting
support and no mention of it in the documentation. And as a result, this is not
a prominent part of the numpy ecosystem and there's little user awareness that
it exists.

*** What this module provides
This module contains functionality to make any arbitrary function broadcastable,
in either C or Python. In both cases, the input and output prototypes are
declared, and these are used for shape-checking and vectorization each time the
function is called.

The functions can have either

- A single output, returned as a numpy array. The output specification in the
  prototype is a single shape tuple
- Multiple outputs, returned as a tuple of numpy arrays. The output
  specification in the prototype is a tuple of shape tuples

*** Broadcasting in python
This is invoked as a decorator, applied to any function. An example:

    >>> import numpysane as nps

    >>> @nps.broadcast_define( (('n',), ('n',)) )
    ... def inner_product(a, b):
    ...     return a.dot(b)

Here we have a simple inner product function to compute ONE inner product. The
'broadcast_define' decorator adds broadcasting-awareness: 'inner_product()'
expects two 1D vectors of length 'n' each (same 'n' for the two inputs),
vectorizing extra dimensions, as needed. The inputs are shape-checked, and
incompatible dimensions will trigger an exception. Example:

    >>> import numpy as np

    >>> a = np.arange(6).reshape(2,3)
    >>> b = a + 100

    >>> a
    array([[0, 1, 2],
           [3, 4, 5]])

    >>> b
    array([[100, 101, 102],
           [103, 104, 105]])

    >>> inner_product(a,b)
    array([ 305, 1250])

Another related function in this module broadcast_generate(). It's similar to
broadcast_define(), but instead of adding broadcasting-awareness to an existing
function, it returns a generator that produces tuples from a set of arguments
according to a given prototype.

Stock numpy has some rudimentary support for all this with its vectorize()
function, but it assumes only scalar inputs and outputs, which severely limits
its usefulness. See the docstrings for 'broadcast_define' and
'broadcast_generate' in the INTERFACE section below for usage details.

*** Broadcasting in C
The python broadcasting is useful, but it is a python loop, so the loop itself
is computationally expensive if we have many iterations. If the function being
wrapped is available in C, we can apply broadcasting awareness in C, which makes
a much faster loop.

The "numpysane_pywrap" module generates code to wrap arbitrary C code in a
broadcasting-aware wrapper callable from python. This is an analogue of PDL::PP
(http://pdl.perl.org/PDLdocs/PP.html). This generated code is compiled and
linked into a python extension module, as usual. This functionality documented
separately: https://github.com/dkogan/numpysane/blob/master/README-pywrap.org

After I wrote this, I realized there is some support for this in stock numpy:

https://docs.scipy.org/doc/numpy-1.13.0/reference/c-api.ufunc.html

Note: I have not tried using these APIs.

** Nicer array manipulation
Numpy functions that move dimensions around and concatenate matrices are
unintuitive. For instance, a simple concatenation of a row-vector or a
column-vector to a matrix requires arcane knowledge to accomplish reliably. This
module provides new functions that can be used for these basic operations. These
new functions do have well-defined and sensible behavior, and they largely come
from the interfaces in PDL (http://pdl.perl.org). These all respect the core
rules of numpy broadcasting:

- LEADING length-1 dimensions don't affect the meaning of an array, so the
  routines handle missing or extra length-1 dimensions at the front

- The inner-most dimensions of an array are the TRAILING ones, so whenever an
  axis specification is used, it is strongly recommended (sometimes required) to
  count the axes from the back by passing in axis<0

A high level description of the functionality is given here, and each function
is described in detail in the INTERFACE section below. In the following
examples, I use a function "arr" that returns a numpy array with given
dimensions:

    >>> def arr(*shape):
    ...     product = reduce( lambda x,y: x*y, shape)
    ...     return numpy.arange(product).reshape(*shape)

    >>> arr(1,2,3)
    array([[[0, 1, 2],
            [3, 4, 5]]])

    >>> arr(1,2,3).shape
    (1, 2, 3)

*** Concatenation
This module provides two functions to do this

**** glue
Concatenates some number of arrays along a given axis ('axis' must be given in a
kwarg). Implicit length-1 dimensions are added at the start as needed.
Dimensions other than the glueing axis must match exactly. Basic usage:

    >>> row_vector = arr(  3,)
    >>> col_vector = arr(5,1,)
    >>> matrix     = arr(5,3,)

    >>> numpysane.glue(matrix, row_vector, axis = -2).shape
    (6,3)

    >>> numpysane.glue(matrix, col_vector, axis = -1).shape
    (5,4)

**** cat
Concatenate some number of arrays along a new leading axis. Implicit length-1
dimensions are added, and the logical shapes of the inputs must match. This
function is a logical inverse of numpy array iteration: iteration splits an
array over its leading dimension, while cat joins a number of arrays via a new
leading dimension. Basic usage:

    >>> numpysane.cat(arr(5,), arr(5,)).shape
    (2,5)

    >>> numpysane.cat(arr(5,), arr(1,1,5,)).shape
    (2,1,1,5)

*** Manipulation of dimensions
Several functions are available, all being fairly direct ports of their PDL
(http://pdl.perl.org) equivalents
**** clump
Reshapes the array by grouping together 'n' dimensions, where 'n' is given in a
kwarg. If 'n' > 0, then n leading dimensions are clumped; if 'n' < 0, then -n
trailing dimensions are clumped. Basic usage:

    >>> numpysane.clump( arr(2,3,4), n = -2).shape
    (2, 12)

    >>> numpysane.clump( arr(2,3,4), n =  2).shape
    (6, 4)

**** atleast_dims
Adds length-1 dimensions at the front of an array so that all the given
dimensions are in-bounds. Any axis<0 may expand the shape. Adding new leading
dimensions (axis>=0) is never useful, since numpy broadcasts from the end, so
clump() treats axis>0 as a check only: the requested axis MUST already be
in-bounds, or an exception is thrown. This function always preserves the meaning
of all the axes in the array: axis=-1 is the same axis before and after the
call. Basic usage:

    >>> numpysane.atleast_dims(arr(2,3), -1).shape
    (2, 3)

    >>> numpysane.atleast_dims(arr(2,3), -2).shape
    (2, 3)

    >>> numpysane.atleast_dims(arr(2,3), -3).shape
    (1, 2, 3)

    >>> numpysane.atleast_dims(arr(2,3), 0).shape
    (2, 3)

    >>> numpysane.atleast_dims(arr(2,3), 1).shape
    (2, 3)

    >>> numpysane.atleast_dims(arr(2,3), 2).shape
    [exception]

**** mv
Moves a dimension from one position to another. Basic usage to move the last
dimension (-1) to the front (0)

    >>> numpysane.mv(arr(2,3,4), -1, 0).shape
    (4, 2, 3)

Or to move a dimension -5 (added implicitly) to the end

    >>> numpysane.mv(arr(2,3,4), -5, -1).shape
    (1, 2, 3, 4, 1)

**** xchg
Exchanges the positions of two dimensions. Basic usage to move the last
dimension (-1) to the front (0), and the front to the back.

    >>> numpysane.xchg(arr(2,3,4), -1, 0).shape
    (4, 3, 2)

Or to swap a dimension -5 (added implicitly) with dimension -2

    >>> numpysane.xchg(arr(2,3,4), -5, -2).shape
    (3, 1, 2, 1, 4)

**** transpose
Reverses the order of the two trailing dimensions in an array. The whole array
is seen as being an array of 2D matrices, each matrix living in the 2 most
significant dimensions, which implies this definition. Basic usage:

    >>> numpysane.transpose( arr(2,3) ).shape
    (3,2)

    >>> numpysane.transpose( arr(5,2,3) ).shape
    (5,3,2)

    >>> numpysane.transpose( arr(3,) ).shape
    (3,1)

Note that in the second example we had 5 matrices, and we transposed each one.
And in the last example we turned a row vector into a column vector by adding an
implicit leading length-1 dimension before transposing.

**** dummy
Adds a single length-1 dimension at the given position. Basic usage:

    >>> numpysane.dummy(arr(2,3,4), -1).shape
    (2, 3, 4, 1)

**** reorder
Reorders the dimensions in an array using the given order. Basic usage:

    >>> numpysane.reorder( arr(2,3,4), -1, -2, -3 ).shape
    (4, 3, 2)

    >>> numpysane.reorder( arr(2,3,4), 0, -1, 1 ).shape
    (2, 4, 3)

    >>> numpysane.reorder( arr(2,3,4), -2 , -1, 0 ).shape
    (3, 4, 2)

    >>> numpysane.reorder( arr(2,3,4), -4 , -2, -5, -1, 0 ).shape
    (1, 3, 1, 4, 2)

** Basic linear algebra
*** inner
Broadcast-aware inner product. Identical to numpysane.dot(). Basic usage to
compute 4 inner products of length 3 each:

    >>> numpysane.inner(arr(  3,),
                        arr(4,3,)).shape
    (4,)

    >>> numpysane.inner(arr(  3,),
                        arr(4,3,))
    array([5, 14, 23, 32])

*** dot
Broadcast-aware non-conjugating dot product. Identical to numpysane.inner().

*** vdot
Broadcast-aware conjugating dot product. Same as numpysane.dot(), except this
one conjugates complex input, which numpysane.dot() does not

*** outer
Broadcast-aware outer product. Basic usage to compute 4 outer products of length
3 each:

    >>> numpysane.outer(arr(  3,),
                        arr(4,3,)).shape
    array(4, 3, 3)

*** norm2
Broadcast-aware 2-norm. numpysane.norm2(x) is identical to numpysane.inner(x,x):

    >>> numpysane.norm2(arr(4,3))
    array([5, 50, 149, 302])

*** mag
Broadcast-aware vector magnitude. mag(x) is functionally identical to
sqrt(numpysane.norm2(x)) and sqrt(numpysane.inner(x,x))

    >>> numpysane.mag(arr(4,3))
    array([ 2.23606798,  7.07106781, 12.20655562, 17.3781472 ])

*** trace
Broadcast-aware matrix trace.

    >>> numpysane.trace(arr(4,3,3))
    array([12., 39., 66., 93.])

*** matmult
Broadcast-aware matrix multiplication. This accepts an arbitrary number of
inputs, and adds leading length-1 dimensions as needed. Multiplying a row-vector
by a matrix

    >>> numpysane.matmult( arr(3,), arr(3,2) ).shape
    (2,)

Multiplying a row-vector by 5 different matrices:

    >>> numpysane.matmult( arr(3,), arr(5,3,2) ).shape
    (5, 2)

Multiplying a matrix by a col-vector:

    >>> numpysane.matmult( arr(3,2), arr(2,1) ).shape
    (3, 1)

Multiplying a row-vector by a matrix by a col-vector:

    >>> numpysane.matmult( arr(3,), arr(3,2), arr(2,1) ).shape
    (1,)

** What's wrong with existing numpy functions?
Why did I go through all the trouble to reimplement all this? Doesn't numpy
already do all these things? Yes, it does. But in a very nonintuitive way.

*** Concatenation
**** hstack()
hstack() performs a "horizontal" concatenation. When numpy prints an array, this
is the last dimension (the most significant dimensions in numpy are at the end).
So one would expect that this function concatenates arrays along this last
dimension. In the special case of 1D and 2D arrays, one would be right:

    >>> numpy.hstack( (arr(3), arr(3))).shape
    (6,)

    >>> numpy.hstack( (arr(2,3), arr(2,3))).shape
    (2, 6)

but in any other case, one would be wrong:

    >>> numpy.hstack( (arr(1,2,3), arr(1,2,3))).shape
    (1, 4, 3)     <------ I expect (1, 2, 6)

    >>> numpy.hstack( (arr(1,2,3), arr(1,2,4))).shape
    [exception]   <------ I expect (1, 2, 7)

    >>> numpy.hstack( (arr(3), arr(1,3))).shape
    [exception]   <------ I expect (1, 6)

    >>> numpy.hstack( (arr(1,3), arr(3))).shape
    [exception]   <------ I expect (1, 6)

The above should all succeed, and should produce the shapes as indicated. Cases
such as "numpy.hstack( (arr(3), arr(1,3)))" are maybe up for debate, but
broadcasting rules allow adding as many extra length-1 dimensions as we want
without changing the meaning of the object, so I claim this should work. Either
way, if you print out the operands for any of the above, you too would expect a
"horizontal" stack() to work as stated above.

It turns out that normally hstack() concatenates along axis=1, unless the first
argument only has one dimension, in which case axis=0 is used. In a system where
the most significant dimension is the last one, this is only correct if everyone
has only 2D arrays. The correct way to do this is to concatenate along axis=-1.
It works for n-dimensionsal objects, and doesn't require the special case logic
for 1-dimensional objects.

**** vstack()
Similarly, vstack() performs a "vertical" concatenation. When numpy prints an
array, this is the second-to-last dimension (remember, the most significant
dimensions in numpy are at the end). So one would expect that this function
concatenates arrays along this second-to-last dimension. Again, in the special
case of 1D and 2D arrays, one would be right:

    >>> numpy.vstack( (arr(2,3), arr(2,3))).shape
    (4, 3)

    >>> numpy.vstack( (arr(3), arr(3))).shape
    (2, 3)

    >>> numpy.vstack( (arr(1,3), arr(3))).shape
    (2, 3)

    >>> numpy.vstack( (arr(3), arr(1,3))).shape
    (2, 3)

    >>> numpy.vstack( (arr(2,3), arr(3))).shape
    (3, 3)

Note that this function appears to tolerate some amount of shape mismatches. It
does it in a form one would expect, but given the state of the rest of this
system, I found it surprising. For instance "numpy.hstack( (arr(1,3), arr(3)))"
fails, so one would think that "numpy.vstack( (arr(1,3), arr(3)))" would fail
too.

And once again, adding more dimensions make it confused, for the same reason:

    >>> numpy.vstack( (arr(1,2,3), arr(2,3))).shape
    [exception]   <------ I expect (1, 4, 3)

    >>> numpy.vstack( (arr(1,2,3), arr(1,2,3))).shape
    (2, 2, 3)     <------ I expect (1, 4, 3)

Similarly to hstack(), vstack() concatenates along axis=0, which is "vertical"
only for 2D arrays, but not for any others. And similarly to hstack(), the 1D
case has special-cased logic to make it work properly.

The correct way to do this is to concatenate along axis=-2. It works for
n-dimensionsal objects, and doesn't require the special case for 1-dimensional
objects.

**** dstack()
I'll skip the detailed description, since this is similar to hstack() and
vstack(). The intent was to concatenate across axis=-3, but the implementation
takes axis=2 instead. This is wrong, as before. And I find it strange that these
3 functions even exist, since they are all special-cases: the concatenation axis
should be an argument, and at most, the edge special case (hstack()) should
exist. This brings us to the next function

**** concatenate()
This is a more general function, and unlike hstack(), vstack() and dstack(), it
takes as input a list of arrays AND the concatenation dimension. It accepts
negative concatenation dimensions to allow us to count from the end, so things
should work better. And in many cases that failed previously, they do:

    >>> numpy.concatenate( (arr(1,2,3), arr(1,2,3)), axis=-1).shape
    (1, 2, 6)

    >>> numpy.concatenate( (arr(1,2,3), arr(1,2,4)), axis=-1).shape
    (1, 2, 7)

    >>> numpy.concatenate( (arr(1,2,3), arr(1,2,3)), axis=-2).shape
    (1, 4, 3)

But many things still don't work as I would expect:

    >>> numpy.concatenate( (arr(1,3), arr(3)), axis=-1).shape
    [exception]   <------ I expect (1, 6)

    >>> numpy.concatenate( (arr(3), arr(1,3)), axis=-1).shape
    [exception]   <------ I expect (1, 6)

    >>> numpy.concatenate( (arr(1,3), arr(3)), axis=-2).shape
    [exception]   <------ I expect (3, 3)

    >>> numpy.concatenate( (arr(3), arr(1,3)), axis=-2).shape
    [exception]   <------ I expect (2, 3)

    >>> numpy.concatenate( (arr(2,3), arr(2,3)), axis=-3).shape
    [exception]   <------ I expect (2, 2, 3)

This function works as expected only if

- All inputs have the same number of dimensions
- All inputs have a matching shape, except for the dimension along which we're
  concatenating
- All inputs HAVE the dimension along which we're concatenating

A common use case that violates these conditions: I have an object that contains
N 3D vectors, and I want to add another 3D vector to it. This is essentially the
first failing example above.

**** stack()
The name makes it sound exactly like concatenate(), and it takes the same
arguments, but it is very different. stack() requires that all inputs have
EXACTLY the same shape. It then concatenates all the inputs along a new
dimension, and places that dimension in the location given by the 'axis' input.
If this is the exact type of concatenation you want, this function works fine.
But it's one of many things a user may want to do.

**** Thoughts on concatenation
This module introduces numpysane.glue() and numpysane.cat() to replace all the
above functions. These do not refer to anything being "horizontal" or
"vertical", nor do they talk about "rows" or "columns": these concepts simply
don't apply in a generic N-dimensional system. These functions are very explicit
about the dimensionality of the inputs/outputs, and fit well into a
broadcasting-aware system.

Since these functions assume that broadcasting is an important concept in the
system, the given axis indices should be counted from the most significant
dimension: the last dimension in numpy. This means that where an axis index is
specified, negative indices are encouraged. glue() forbids axis>=0 outright.

***** Example for further justification
An array containing N 3D vectors would have shape (N,3). Another array
containing a single 3D vector would have shape (3,). Counting the dimensions
from the end, each vector is indexed in dimension -1. However, counting from the
front, the vector is indexed in dimension 0 or 1, depending on which of the two
arrays we're looking at. If we want to add the single vector to the array
containing the N vectors, and we mistakenly try to concatenate along the first
dimension, it would fail if N != 3. But if we're unlucky, and N=3, then we'd get
a nonsensical output array of shape (3,4). Why would an array of N 3D vectors
have shape (N,3) and not (3,N)? Because if we apply python iteration to it, we'd
expect to get N iterates of arrays with shape (3,) each, and numpy iterates from
the first dimension:

    >>> a = numpy.arange(2*3).reshape(2,3)

    >>> a
    array([[0, 1, 2],
           [3, 4, 5]])

    >>> [x for x in a]
    [array([0, 1, 2]), array([3, 4, 5])]

*** Manipulation of dimensions
**** atleast_xd()
Numpy has 3 special-case functions atleast_1d(), atleast_2d() and atleast_3d().
For 4d and higher, you need to do something else. These do surprising things:

    >>> numpy.atleast_3d(arr(3)).shape
    (1, 3, 1)

**** transpose()
Given a matrix (a 2D array), numpy.transpose() swaps the two dimensions, as
expected. Given anything else, it does not do what is expected:

    >>> numpy.transpose(arr(3,      )).shape
    (3,)

    >>> numpy.transpose(arr(3,4,    )).shape
    (4, 3)

    >>> numpy.transpose(arr(3,4,5,6,)).shape
    (6, 5, 4, 3)

I.e. numpy.transpose() reverses the order of ALL dimensions in the array. So if
we have N 2D matrices in a single array, numpy.transpose() doesn't transpose
each matrix.

*** Basic linear algebra
**** inner() and dot()
numpy.inner() and numpy.dot() are strange. In a real-valued n-dimensional
Euclidean space, a "dot product" is just another name for an "inner product".
Numpy disagrees.

It looks like numpy.dot() is matrix multiplication, with some wonky behaviors
when given higher-dimension objects, and with some special-case behaviors for
1-dimensional and 0-dimensional objects:

    >>> numpy.dot( arr(4,5,2,3), arr(3,5)).shape
    (4, 5, 2, 5) <--- expected result for a broadcasted matrix multiplication

    >>> numpy.dot( arr(3,5), arr(4,5,2,3)).shape
    [exception] <--- numpy.dot() is not commutative.
                     Expected for matrix multiplication, but not for a dot
                     product

    >>> numpy.dot( arr(4,5,2,3), arr(1,3,5)).shape
    (4, 5, 2, 1, 5) <--- don't know where this came from at all

    >>> numpy.dot( arr(4,5,2,3), arr(3)).shape
    (4, 5, 2) <--- 1D special case. This is a dot product.

    >>> numpy.dot( arr(4,5,2,3), 3).shape
    (4, 5, 2, 3) <--- 0D special case. This is a scaling.

It looks like numpy.inner() is some sort of quasi-broadcastable inner product, also
with some funny dimensioning rules. In many cases it looks like numpy.dot(a,b) is
the same as numpy.inner(a, transpose(b)) where transpose() swaps the last two
dimensions:


    >>> numpy.inner( arr(4,5,2,3), arr(5,3)).shape
    (4, 5, 2, 5) <--- All the length-3 inner products collected into a shape
                      with not-quite-broadcasting rules

    >>> numpy.inner( arr(5,3), arr(4,5,2,3)).shape
    (5, 4, 5, 2) <--- numpy.inner() is not commutative. Unexpected
                      for an inner product

    >>> numpy.inner( arr(4,5,2,3), arr(1,5,3)).shape
    (4, 5, 2, 1, 5) <--- No idea

    >>> numpy.inner( arr(4,5,2,3), arr(3)).shape
    (4, 5, 2) <--- 1D special case. This is a dot product.

    >>> numpy.inner( arr(4,5,2,3), 3).shape
    (4, 5, 2, 3) <--- 0D special case. This is a scaling.

'''

import numpy as np
from functools import reduce
import itertools
import types
import inspect
from distutils.version import StrictVersion

# setup.py assumes the version is a simple string in '' quotes
__version__ = '0.34'

def _product(l):
    r'''Returns product of all values in the given list'''
    return reduce( lambda a,b: a*b, l )


def _clone_function(f, name):
    r'''Returns a clone of a given function.

    This is useful to copy a function, updating its metadata, such as the
    documentation, name, etc. There are also differences here between python 2
    and python 3 that this function handles.

    '''
    def get(f, what):
        what2 = 'func_{}'.format(what)
        what3 = '__{}__' .format(what)
        try:
            return getattr(f, what2)
        except:
            try:
                return getattr(f, what3)
            except:
                pass
        return None

    return types.FunctionType(get(f, 'code'),
                              get(f, 'globals'),
                              name,
                              get(f, 'defaults'),
                              get(f, 'closure'))



class NumpysaneError(Exception):
    def __init__(self, err): self.err = err
    def __str__(self):       return self.err


def _eval_broadcast_dims( args, prototype ):
    r'''Helper function to evaluate a given list of arguments in respect to a given
    broadcasting prototype. This function will flag any errors in the
    dimensionality of the inputs. If no errors are detected, it returns

      dims_extra,dims_named

    where

      dims_extra is the outer shape of the broadcast
        This is a list: the union of all the leading shapes of all the
        arguments, after the trailing shapes of the prototype have been stripped

      dims_named is the sizes of the named dimensions
        This is a dict mapping dimension names to their sizes

    '''

    # First I initialize dims_extra: the array containing the broadcasted
    # slices. Each argument calls for some number of extra dimensions, and the
    # overall array is as large as the biggest one of those
    Ndims_extra = 0
    for i_arg in range(len(args)):
        Ndims_extra_here = len(args[i_arg].shape) - len(prototype[i_arg])
        if Ndims_extra_here > Ndims_extra:
            Ndims_extra = Ndims_extra_here
    dims_extra = [1] * Ndims_extra


    def parse_dim( name_arg,
                   shape_prototype, shape_arg, dims_named ):

        def range_rev(n):
            r'''Returns a range from -1 to -n.

            Useful to index variable-sized lists while aligning their ends.'''
            return range(-1, -n-1, -1)



        # first, I make sure the input is at least as dimension-ful as the
        # prototype. I do this by prepending dummy dimensions of length-1 as
        # necessary
        if len(shape_prototype) > len(shape_arg):
            ndims_missing_here = len(shape_prototype) - len(shape_arg)
            shape_arg = (1,) * ndims_missing_here + shape_arg

        # MAKE SURE THE PROTOTYPE DIMENSIONS MATCH (the trailing dimensions)
        #
        # Loop through the dimensions. Set the dimensionality of any new named
        # argument to whatever the current argument has. Any already-known
        # argument must match
        for i_dim in range_rev(len(shape_prototype)):

            dim_prototype = shape_prototype[i_dim]

            if not isinstance(dim_prototype, int):
                # This is a named dimension. These can have any value, but ALL
                # dimensions of the same name must thave the SAME value
                # EVERYWHERE
                if dim_prototype not in dims_named:
                    dims_named[dim_prototype] = shape_arg[i_dim]
                dim_prototype = dims_named[dim_prototype]

            # The prototype dimension (named or otherwise) now has a numeric
            # value. Make sure it matches what I have
            if dim_prototype != shape_arg[i_dim]:
                raise NumpysaneError("Argument {} dimension '{}': expected {} but got {}".
                    format(name_arg,
                           shape_prototype[i_dim],
                           dim_prototype,
                           shape_arg[i_dim]))

        # I now know that this argument matches the prototype. I look at the
        # extra dimensions to broadcast, and make sure they match with the
        # dimensions I saw previously
        Ndims_extra_here = len(shape_arg) - len(shape_prototype)

        # MAKE SURE THE BROADCASTED DIMENSIONS MATCH (the leading dimensions)
        #
        # This argument has Ndims_extra_here dimensions to broadcast. The
        # current shape to broadcast must be at least as large, and must match
        for i_dim in range_rev(Ndims_extra_here):
            dim_arg = shape_arg[i_dim - len(shape_prototype)]
            if dim_arg != 1:
                if dims_extra[i_dim] == 1:
                    dims_extra[i_dim] = dim_arg
                elif dims_extra[i_dim] != dim_arg:
                    raise NumpysaneError("Argument {} prototype {} extra broadcast dim {} mismatch: previous arg set this to {}, but this arg wants {}".
                        format(name_arg,
                               shape_prototype,
                               i_dim,
                               dims_extra[i_dim],
                               dim_arg))


    dims_named = {} # parse_dim() adds to this
    for i_arg in range(len(args)):
        parse_dim( i_arg,
                   prototype[i_arg], args[i_arg].shape,
                   dims_named )

    return dims_extra,dims_named



def _broadcast_iter_dim( args, prototype, dims_extra ):
    r'''Generator to iterate through all the broadcasting slices.
    '''

    # pad the dimension of each arg with ones. This lets me use the full
    # dims_extra index on each argument, without worrying about overflow
    args = [ atleast_dims(args[i], -(len(prototype[i])+len(dims_extra)) ) for i in range(len(args)) ]

    # per-arg dims_extra indexing varies: len-1 dimensions always index at 0. I
    # make a mask that I apply each time
    idx_slice_mask = np.ones( (len(args), len(dims_extra)), dtype=int)
    for i in range(len(args)):
        idx_slice_mask[i, np.array(args[i].shape,dtype=int)[:len(dims_extra)]==1] = 0

    for idx_slice in itertools.product( *(range(x) for x in dims_extra) ):
        # tuple(idx) because of wonky behavior differences:
        #     >>> a
        #     array([[0, 1, 2],
        #            [3, 4, 5]])
        #
        #     >>> a[tuple((1,1))]
        #     4
        #
        #     >>> a[list((1,1))]
        #     array([[3, 4, 5],
        #            [3, 4, 5]])
        yield tuple( args[i][tuple(idx_slice *
                                   idx_slice_mask[i])] for i in range(len(args)) )


def broadcast_define(prototype, prototype_output=None, out_kwarg=None):
    r'''Vectorizes an arbitrary function, expecting input as in the given prototype.

    SYNOPSIS

        >>> import numpy as np
        >>> import numpysane as nps

        >>> @nps.broadcast_define( (('n',), ('n',)) )
        ... def inner_product(a, b):
        ...     return a.dot(b)

        >>> a = np.arange(6).reshape(2,3)
        >>> b = a + 100

        >>> a
        array([[0, 1, 2],
               [3, 4, 5]])

        >>> b
        array([[100, 101, 102],
               [103, 104, 105]])

        >>> inner_product(a,b)
        array([ 305, 1250])

    The prototype defines the dimensionality of the inputs. In the inner product
    example above, the input is two 1D n-dimensional vectors. In particular, the
    'n' is the same for the two inputs. This function is intended to be used as
    a decorator, applied to a function defining the operation to be vectorized.
    Each element in the prototype list refers to each input, in order. In turn,
    each such element is a list that describes the shape of that input. Each of
    these shape descriptors can be any of

    - a positive integer, indicating an input dimension of exactly that length
    - a string, indicating an arbitrary, but internally consistent dimension

    The normal numpy broadcasting rules (as described elsewhere) apply. In
    summary:

    - Dimensions are aligned at the end of the shape list, and must match the
      prototype

    - Extra dimensions left over at the front must be consistent for all the
      input arguments, meaning:

      - All dimensions of length != 1 must match
      - Dimensions of length 1 match corresponding dimensions of any length in
        other arrays
      - Missing leading dimensions are implicitly set to length 1

    - The output(s) have a shape where
      - The trailing dimensions are whatever the function being broadcasted
        returns
      - The leading dimensions come from the extra dimensions in the inputs

    Calling a function wrapped with broadcast_define() with extra arguments
    (either positional or keyword), passes these verbatim to the inner function.
    Only the arguments declared in the prototype are broadcast.

    Scalars are represented as 0-dimensional numpy arrays: arrays with shape (),
    and these broadcast as one would expect:

        >>> @nps.broadcast_define( (('n',), ('n',), ()))
        ... def scaled_inner_product(a, b, scale):
        ...     return a.dot(b)*scale

        >>> a = np.arange(6).reshape(2,3)
        >>> b = a + 100
        >>> scale = np.array((10,100))

        >>> a
        array([[0, 1, 2],
               [3, 4, 5]])

        >>> b
        array([[100, 101, 102],
               [103, 104, 105]])

        >>> scale
        array([ 10, 100])

        >>> scaled_inner_product(a,b,scale)
        array([[  3050],
               [125000]])

    Let's look at a more involved example. Let's say we have a function that
    takes a set of points in R^2 and a single center point in R^2, and finds a
    best-fit least-squares line that passes through the given center point. Let
    it return a 3D vector containing the slope, y-intercept and the RMS residual
    of the fit. This broadcasting-enabled function can be defined like this:

        import numpy as np
        import numpysane as nps

        @nps.broadcast_define( (('n',2), (2,)) )
        def fit(xy, c):
            # line-through-origin-model: y = m*x
            # E = sum( (m*x - y)**2 )
            # dE/dm = 2*sum( (m*x-y)*x ) = 0
            # ----> m = sum(x*y)/sum(x*x)
            x,y = (xy - c).transpose()
            m = np.sum(x*y) / np.sum(x*x)
            err = m*x - y
            err **= 2
            rms = np.sqrt(err.mean())
            # I return m,b because I need to translate the line back
            b = c[1] - m*c[0]

            return np.array((m,b,rms))

    And I can use broadcasting to compute a number of these fits at once. Let's
    say I want to compute 4 different fits of 5 points each. I can do this:

        n = 5
        m = 4
        c = np.array((20,300))
        xy = np.arange(m*n*2, dtype=np.float64).reshape(m,n,2) + c
        xy += np.random.rand(*xy.shape)*5

        res = fit( xy, c )
        mb  = res[..., 0:2]
        rms = res[..., 2]
        print "RMS residuals: {}".format(rms)

    Here I had 4 different sets of points, but a single center point c. If I
    wanted 4 different center points, I could pass c as an array of shape (4,2).
    I can use broadcasting to plot all the results (the points and the fitted
    lines):

        import gnuplotlib as gp

        gp.plot( *nps.mv(xy,-1,0), _with='linespoints',
                 equation=['{}*x + {}'.format(mb_single[0],
                                              mb_single[1]) for mb_single in mb],
                 unset='grid', square=1)

    The examples above all create a separate output array for each broadcasted
    slice, and copy the contents from each such slice into the larger output
    array that contains all the results. This is inefficient, and it is possible
    to pre-allocate an array to forgo these extra allocation and copy
    operations. There are several settings to control this. If the function
    being broadcasted can write its output to a given array instead of creating
    a new one, most of the inefficiency goes away. broadcast_define() supports
    the case where this function takes this array in a kwarg: the name of this
    kwarg can be given to broadcast_define() like so:

        @nps.broadcast_define( ....., out_kwarg = "out" )
        def func( ....., out):
            .....
            out[:] = result

    When used this way, the return value of the broadcasted function is ignored.
    In order for broadcast_define() to pass such an output array to the inner
    function, this output array must be available, which means that it must be
    given to us somehow, or we must create it.

    The most efficient way to make a broadcasted call is to create the full
    output array beforehand, and to pass that to the broadcasted function. In
    this case, nothing extra will be allocated, and no unnecessary copies will
    be made. This can be done like this:

        @nps.broadcast_define( (('n',), ('n',)), ....., out_kwarg = "out" )
        def inner_product(a, b, out):
            .....
            out.setfield(a.dot(b), out.dtype)

        out = np.empty((2,4), np.float)
        inner_product( np.arange(3), np.arange(2*4*3).reshape(2,4,3), out=out)

    In this example, the caller knows that it's calling an inner_product
    function, and that the shape of each output slice would be (). The caller
    also knows the input dimensions and that we have an extra broadcasting
    dimension (2,4), so the output array will have shape (2,4) + () = (2,4).
    With this knowledge, the caller preallocates the array, and passes it to the
    broadcasted function call. Furthermore, in this case the inner function will
    be called with an output array EVERY time, and this is the only mode the
    inner function needs to support.

    If the caller doesn't know (or doesn't want to pre-compute) the shape of the
    output, it can let the broadcasting machinery create this array for them. In
    order for this to be possible, the shape of the output should be
    pre-declared, and the dtype of the output should be known:

        @nps.broadcast_define( (('n',), ('n',)),
                               (),
                               out_kwarg = "out" )
        def inner_product(a, b, out):
            .....
            out.setfield(a.dot(b), out.dtype)

        out = inner_product( np.arange(3), np.arange(2*4*3).reshape(2,4,3), dtype=int)

    Note that the caller didn't need to specify the prototype of the output or
    the extra broadcasting dimensions (output prototype is in the
    broadcast_define() call, but not the inner_product() call). Specifying the
    dtype here is optional: it defaults to float if omitted. If we want the
    output array to be pre-allocated, the output prototype (it is () in this
    example) is required: we must know the shape of the output array in order to
    create it.

    Without a declared output prototype, we can still make mostly- efficient
    calls: the broadcasting mechanism can call the inner function for the first
    slice as we showed earlier, by creating a new array for the slice. This new
    array required an extra allocation and copy, but it contains the required
    shape information. This infomation will be used to allocate the output, and
    the subsequent calls to the inner function will be efficient:

        @nps.broadcast_define( (('n',), ('n',)),
                               out_kwarg = "out" )
        def inner_product(a, b, out=None):
            .....
            if out is None:
                return a.dot(b)
            out.setfield(a.dot(b), out.dtype)
            return out

        out = inner_product( np.arange(3), np.arange(2*4*3).reshape(2,4,3))

    Here we were slighly inefficient, but the ONLY required extra specification
    was out_kwarg: that's all you need. Also it is important to note that in
    this case the inner function is called both with passing it an output array
    to fill in, and with asking it to create a new one (by passing out=None to
    the inner function). This inner function then must support both modes of
    operation. If the inner function does not support filling in an output
    array, none of these efficiency improvements are possible.

    It is possible for a function to return more than one output, and this is
    supported by broadcast_define(). This case works exactly like the one-output
    case, except the output prototype is REQUIRED, and this output prototype
    contains multiple tuples, one for each output. The inner function must
    return the outputs in a tuple, and each individual output will be
    broadcasted as expected.

    broadcast_define() is analogous to thread_define() in PDL.

    '''
    def inner_decorator_for_some_reason(func):
        # args broadcast, kwargs do not. All auxillary data should go into the
        # kwargs
        def broadcast_loop(*args, **kwargs):

            if len(args) < len(prototype):
                raise NumpysaneError("Mismatched number of input arguments. Wanted at least {} but got {}". \
                                      format(len(prototype), len(args)))

            args_passthru = args[  len(prototype):]
            args          = args[0:len(prototype) ]

            # make sure all the arguments are numpy arrays
            args = tuple(np.asarray(arg) for arg in args)

            # dims_extra: extra dimensions to broadcast through
            # dims_named: values of the named dimensions
            dims_extra,dims_named = \
                _eval_broadcast_dims( args, prototype)

            # If None, the single output is either returned, or stored into
            # out_kwarg. If an integer, then a tuple is returned (or stored into
            # out_kwarg). If Noutputs==1 then we return a TUPLE of length 1
            Noutputs = None

            # substitute named variable values into the output prototype
            prototype_output_expanded = None
            if prototype_output is not None:
                # If a single prototype_output is given, wrap it in a tuple to indicate
                # that we only have one output
                if all( type(o) is int or type(o) is str for o in prototype_output ):
                    prototype_output_expanded = \
                        [d if type(d) is int else dims_named[d] \
                         for d in prototype_output]
                else:
                    Noutputs = len(prototype_output)
                    prototype_output_expanded = \
                        [ [d if type(d) is int else dims_named[d] \
                           for d in _prototype_output] for \
                          _prototype_output in prototype_output ]

            # I checked all the dimensions and aligned everything. I have my
            # to-broadcast dimension counts. Iterate through all the broadcasting
            # output, and gather the results
            output = None
            i_slice = 0

            if Noutputs is None:
                # We expect a SINGLE output

                # if the output was supposed to go to a particular place, set that
                if out_kwarg is not None and out_kwarg in kwargs:
                    output = kwargs[out_kwarg]
                    if prototype_output_expanded is not None:
                        expected_shape = dims_extra + prototype_output_expanded
                        if output.shape != tuple(expected_shape):
                            raise NumpysaneError("Inconsistent output shape: expected {}, but got {}".format(expected_shape, output.shape))
                # if we know enough to allocate the output, do that
                elif prototype_output_expanded is not None:
                    kwargs_dtype = {}
                    if 'dtype' in kwargs:
                        kwargs_dtype['dtype'] = kwargs['dtype']
                    output = np.empty(dims_extra + prototype_output_expanded,
                                      **kwargs_dtype)
                # else:
                #   We don't have an output and we don't know its dimensions, so
                #   we can't allocate an array for it. Leave output as None. I
                #   will allocate it as soon I get the first slice; this will let
                #   me know how large the whole thing should be

                # if no broadcasting involved, just call the function
                if not dims_extra:
                    # if the function knows how to write directly to an array,
                    # request that
                    if output is not None and out_kwarg is not None:
                        kwargs[out_kwarg] = output

                    sliced_args = args + args_passthru
                    result = func( *sliced_args, **kwargs )

                    if out_kwarg             is not None and \
                       kwargs.get(out_kwarg) is not None:
                        # We wrote the output in-place. Return the output array
                        return kwargs.get(out_kwarg)

                    # Using the returned output. Run some checks, and return the
                    # returned value
                    if isinstance(result, tuple):
                        raise NumpysaneError("Only a single output expected, but a tuple was returned!")
                    if prototype_output_expanded is not None and \
                       np.array(result).shape != tuple(prototype_output_expanded):
                        raise NumpysaneError("Inconsistent slice output shape: expected {}, but got {}".format(prototype_output_expanded,
                                                                                                               np.array(result).shape))
                    return result

                # reshaped output. I write to this array
                if output is not None:
                    output_flattened = clump(output, n=len(dims_extra))

                for x in _broadcast_iter_dim( args, prototype, dims_extra ):

                    # if the function knows how to write directly to an array,
                    # request that
                    if output is not None and out_kwarg is not None:
                        kwargs[out_kwarg] = output_flattened[i_slice, ...]

                    sliced_args = x + args_passthru
                    result = func( *sliced_args, **kwargs )

                    if output is None or out_kwarg is None:
                        # We weren't writing directly into the output, so check
                        # the output for validity
                        if isinstance(result, tuple):
                            raise NumpysaneError("Only a single output expected, but a tuple was returned!")
                        if not isinstance(result, np.ndarray):
                            result = np.array(result)

                        if prototype_output_expanded is None:
                            prototype_output_expanded = result.shape
                        else:
                            if result.shape != tuple(prototype_output_expanded):
                                raise NumpysaneError("Inconsistent slice output shape: expected {}, but got {}".format(prototype_output_expanded, result.shape))

                    if output is None:
                        # I didn't already have an output array because I didn't
                        # know how large it should be. But I now have the first
                        # slice, and I know how big the whole output should be.
                        # I create it
                        output = np.empty( dims_extra + list(result.shape),
                                           dtype = result.dtype)
                        output_flattened = output.reshape( (_product(dims_extra),) + result.shape)
                        output_flattened[i_slice, ...] = result

                    elif out_kwarg is None:
                        output_flattened[i_slice, ...] = result
                    # else:
                    #   I was writing directly to the output, so I don't need to
                    #   manually populate the slice

                    i_slice = i_slice+1

            else:
                # We expect MULTIPLE outputs: a tuple of length Noutputs

                # if the output was supposed to go to a particular place, set that
                if out_kwarg is not None and out_kwarg in kwargs:
                    output = kwargs[out_kwarg]
                    if prototype_output_expanded is not None:
                        for i in range(Noutputs):
                            expected_shape = dims_extra + prototype_output_expanded[i]
                            if output[i].shape != tuple(expected_shape):
                                raise NumpysaneError("Inconsistent output shape for output {}: expected {}, but got {}". \
                                                     format(i, expected_shape, output[i].shape))
                # if we know enough to allocate the output, do that
                elif prototype_output_expanded is not None:
                    kwargs_dtype = {}
                    if 'dtype' in kwargs:
                        kwargs_dtype['dtype'] = kwargs['dtype']
                    output = [np.empty(dims_extra + prototype_output_expanded[i],
                                       **kwargs_dtype) for i in range(Noutputs)]
                # else:
                #   We don't have an output and we don't know its dimensions, so
                #   we can't allocate an array for it. Leave output as None. I
                #   will allocate it as soon I get the first slice; this will let
                #   me know how large the whole thing should be

                # if no broadcasting involved, just call the function
                if not dims_extra:
                    # if the function knows how to write directly to an array,
                    # request that
                    if output is not None and out_kwarg is not None:
                        kwargs[out_kwarg] = tuple(output)

                    sliced_args = args + args_passthru
                    result = func( *sliced_args, **kwargs )
                    if out_kwarg             is not None and \
                       kwargs.get(out_kwarg) is not None:
                        # We wrote the output in-place. Return the output array
                        return kwargs.get(out_kwarg)

                    if not isinstance(result, tuple):
                        raise NumpysaneError("A tuple of {} outputs is expected, but an object of type {} was returned". \
                                             format(Noutputs, type(result)))
                    if len(result) != Noutputs:
                        raise NumpysaneError("A tuple of {} outputs is expected, but a length-{} tuple was returned". \
                                             format(Noutputs, len(result)))
                    if prototype_output_expanded is not None:
                        for i in range(Noutputs):
                            if np.array(result[i]).shape != tuple(prototype_output_expanded[i]):
                                raise NumpysaneError("Inconsistent output {} shape: expected {}, but got {}". \
                                                     format(i,
                                                            prototype_output_expanded[i],
                                                            np.array(result[i]).shape))
                    return result


                # reshaped output. I write to this array
                if output is not None:
                    output_flattened = [clump(output[i], n=len(dims_extra)) for i in range(Noutputs)]

                for x in _broadcast_iter_dim( args, prototype, dims_extra ):

                    # if the function knows how to write directly to an array,
                    # request that
                    if output is not None and out_kwarg is not None:
                        kwargs[out_kwarg] = tuple(o[i_slice, ...] for o in output_flattened)

                    sliced_args = x + args_passthru
                    result = func( *sliced_args, **kwargs )

                    if output is None or out_kwarg is None:
                        # We weren't writing directly into the output, so check
                        # the output for validity
                        if not isinstance(result, tuple):
                            raise NumpysaneError("A tuple of {} outputs is expected, but an object of type {} was returned". \
                                                 format(Noutputs, type(result)))
                        if len(result) != Noutputs:
                            raise NumpysaneError("A tuple of {} outputs is expected, but a length-{} tuple was returned". \
                                                 format(Noutputs, len(result)))

                        result = [x if isinstance(x, np.ndarray) else np.array(x) for x in result]

                        if prototype_output_expanded is None:
                            prototype_output_expanded = [result[i].shape for i in range(Noutputs)]
                        else:
                            for i in range(Noutputs):
                                if result[i].shape != tuple(prototype_output_expanded[i]):
                                    raise NumpysaneError("Inconsistent slice output {} shape: expected {}, but got {}". \
                                                         format(i, prototype_output_expanded[i], result[i].shape))

                    if output is None:
                        # I didn't already have an output array because I didn't
                        # know how large it should be. But I now have the first
                        # slice, and I know how big the whole output should be.
                        # I create it
                        output = [np.empty( dims_extra + list(result[i].shape),
                                            dtype = result[i].dtype) for i in range(Noutputs)]
                        output_flattened = [output[i].reshape( (_product(dims_extra),) + result[i].shape) for i in range(Noutputs)]
                        for i in range(Noutputs):
                            output_flattened[i][i_slice, ...] = result[i]

                    elif out_kwarg is None:
                        for i in range(Noutputs):
                            output_flattened[i][i_slice, ...] = result[i]
                    # else:
                    #   I was writing directly to the output, so I don't need to
                    #   manually populate the slice

                    i_slice = i_slice+1

            return output



        if out_kwarg is not None and not isinstance(out_kwarg, str):
            raise NumpysaneError("out_kwarg must be a string")

        # Make sure all dimensions are >=0 and that named output dimensions are
        # known from the input
        known_named_dims = set()
        if not isinstance(prototype, tuple):
            raise NumpysaneError("Input prototype must be given as a tuple")
        for dims_input in prototype:
            if not isinstance(dims_input, tuple):
                raise NumpysaneError("Input prototype dims must be given as a tuple")
            for dim in dims_input:
                if type(dim) is not int:
                    if type(dim) is not str:
                        raise NumpysaneError("Prototype dimensions must be integers > 0 or strings. Got '{}' of type '{}'". \
                                             format(dim, type(dim)))
                    known_named_dims.add(dim)
                else:
                    if dim < 0:
                        raise NumpysaneError("Prototype dimensions must be > 0. Got '{}'". \
                                             format(dim))

        if prototype_output is not None:

            if not isinstance(prototype_output, tuple):
                raise NumpysaneError("Output prototype dims must be given as a tuple")

            # If a single prototype_output is given, wrap it in a tuple to indicate
            # that we only have one output
            if all( type(o) is int or type(o) is str for o in prototype_output ):
                prototype_outputs = (prototype_output, )
            else:
                prototype_outputs = prototype_output
                if not all( isinstance(p,tuple) for p in prototype_outputs ):
                    raise NumpysaneError("Output dimensions must be integers > 0 or strings. Each output must be a tuple. Some given output aren't tuples: {}". \
                                         format(prototype_outputs))

            for dims_output in prototype_outputs:
                for dim in dims_output:
                    if type(dim) is not int:
                        if type(dim) is not str:
                            raise NumpysaneError("Output dimensions must be integers > 0 or strings. Got '{}' of type '{}'". \
                                                 format(dim, type(dim)))
                        if dim not in known_named_dims:
                            raise NumpysaneError("Output prototype has named dimension '{}' not seen in the input prototypes". \
                                                 format(dim))
                    else:
                        if dim < 0:
                            raise NumpysaneError("Prototype dimensions must be > 0. Got '{}'". \
                                                 format(dim))

        func_out = _clone_function( broadcast_loop, func.__name__ )
        func_out.__doc__  = inspect.getdoc(func)
        if func_out.__doc__ is None:
            func_out.__doc__ = ''
        func_out.__doc__+= \
'''\n\nThis function is broadcast-aware through numpysane.broadcast_define().
The expected inputs have input prototype:

    {prototype}

{output_prototype_text}

The first {nargs} positional arguments will broadcast. The trailing shape of
those arguments must match the input prototype; the leading shape must follow
the standard broadcasting rules. Positional arguments past the first {nargs} and
all the keyword arguments are passed through untouched.'''. \
        format(prototype = prototype,
               output_prototype_text = 'No output prototype is defined.' if prototype_output is None else
               'and output prototype\n\n    {}'.format(prototype_output),
               nargs = len(prototype))
        return func_out
    return inner_decorator_for_some_reason


def broadcast_generate(prototype, args):
    r'''A generator that produces broadcasted slices

    SYNOPSIS

        >>> import numpy as np
        >>> import numpysane as nps

        >>> a = np.arange(6).reshape(2,3)
        >>> b = a + 100

        >>> a
        array([[0, 1, 2],
               [3, 4, 5]])

        >>> b
        array([[100, 101, 102],
               [103, 104, 105]])

        >>> for s in nps.broadcast_generate( (('n',), ('n',)), (a,b)):
        ...     print "slice: {}".format(s)
        slice: (array([0, 1, 2]), array([100, 101, 102]))
        slice: (array([3, 4, 5]), array([103, 104, 105]))
    '''

    if len(args) != len(prototype):
        raise NumpysaneError("Mismatched number of input arguments. Wanted {} but got {}". \
                              format(len(prototype), len(args)))

    # make sure all the arguments are numpy arrays
    args = tuple(np.asarray(arg) for arg in args)

    # dims_extra: extra dimensions to broadcast through
    # dims_named: values of the named dimensions
    dims_extra,dims_named = \
        _eval_broadcast_dims( args, prototype )

    # I checked all the dimensions and aligned everything. I have my
    # to-broadcast dimension counts. Iterate through all the broadcasting
    # output, and gather the results
    for x in _broadcast_iter_dim( args, prototype, dims_extra ):
        yield x


def glue(*args, **kwargs):
    r'''Concatenates a given list of arrays along the given 'axis' keyword argument.

    SYNOPSIS

        >>> import numpy as np
        >>> import numpysane as nps

        >>> a = np.arange(6).reshape(2,3)
        >>> b = a + 100
        >>> row = a[0,:] + 1000

        >>> a
        array([[0, 1, 2],
               [3, 4, 5]])

        >>> b
        array([[100, 101, 102],
               [103, 104, 105]])

        >>> row
        array([1000, 1001, 1002])

        >>> nps.glue(a,b, axis=-1)
        array([[  0,   1,   2, 100, 101, 102],
               [  3,   4,   5, 103, 104, 105]])

        # empty arrays ignored when glueing. Useful for initializing an accumulation
        >>> nps.glue(a,b, np.array(()), axis=-1)
        array([[  0,   1,   2, 100, 101, 102],
               [  3,   4,   5, 103, 104, 105]])

        >>> nps.glue(a,b,row, axis=-2)
        array([[   0,    1,    2],
               [   3,    4,    5],
               [ 100,  101,  102],
               [ 103,  104,  105],
               [1000, 1001, 1002]])

        >>> nps.glue(a,b, axis=-3)
        array([[[  0,   1,   2],
                [  3,   4,   5]],

               [[100, 101, 102],
                [103, 104, 105]]])

    The 'axis' must be given in a keyword argument.

    In order to count dimensions from the inner-most outwards, this function accepts
    only negative axis arguments. This is because numpy broadcasts from the last
    dimension, and the last dimension is the inner-most in the (usual) internal
    storage scheme. Allowing glue() to look at dimensions at the start would allow
    it to unalign the broadcasting dimensions, which is never what you want.

    To glue along the last dimension, pass axis=-1; to glue along the second-to-last
    dimension, pass axis=-2, and so on.

    Unlike in PDL, this function refuses to create duplicated data to make the
    shapes fit. In my experience, this isn't what you want, and can create bugs.
    For instance, PDL does this:

        pdl> p sequence(3,2)
        [
         [0 1 2]
         [3 4 5]
        ]

        pdl> p sequence(3)
        [0 1 2]

        pdl> p PDL::glue( 0, sequence(3,2), sequence(3) )
        [
         [0 1 2 0 1 2]   <--- Note the duplicated "0,1,2"
         [3 4 5 0 1 2]
        ]

    while numpysane.glue() does this:

        >>> import numpy as np
        >>> import numpysane as nps

        >>> a = np.arange(6).reshape(2,3)
        >>> b = a[0:1,:]


        >>> a
        array([[0, 1, 2],
               [3, 4, 5]])

        >>> b
        array([[0, 1, 2]])

        >>> nps.glue(a,b,axis=-1)
        [exception]

    Finally, this function adds as many length-1 dimensions at the front as
    required. Note that this does not create new data, just new degenerate
    dimensions. Example:

        >>> import numpy as np
        >>> import numpysane as nps

        >>> a = np.arange(6).reshape(2,3)
        >>> b = a + 100

        >>> a
        array([[0, 1, 2],
               [3, 4, 5]])

        >>> b
        array([[100, 101, 102],
               [103, 104, 105]])

        >>> res = nps.glue(a,b, axis=-5)
        >>> res
        array([[[[[  0,   1,   2],
                  [  3,   4,   5]]]],



               [[[[100, 101, 102],
                  [103, 104, 105]]]]])

        >>> res.shape
        (2, 1, 1, 2, 3)

    In numpysane older than 0.10 the semantics were slightly different: the axis
    kwarg was optional, and glue(*args) would glue along a new leading
    dimension, and thus would be equivalent to cat(*args). This resulted in very
    confusing error messages if the user accidentally omitted the kwarg. To
    request the legacy behavior, do

        nps.glue.legacy_version = '0.9'

    '''
    legacy = \
        hasattr(glue, 'legacy_version') and \
        StrictVersion(glue.legacy_version) <= StrictVersion('0.9')

    axis = kwargs.get('axis')

    if legacy:
        if axis is not None and axis >= 0:
            raise NumpysaneError("axis >= 0 can make broadcasting dimensions inconsistent, and is thus not allowed")
    else:
        if axis is None:
            raise NumpysaneError("glue() requires the axis to be given in the 'axis' kwarg")
        if axis >= 0:
            raise NumpysaneError("axis >= 0 can make broadcasting dimensions inconsistent, and is thus not allowed")


    # deal with scalar (non-ndarray) args
    args = [ np.asarray(x) for x in args ]

    # Special case to support this common idiom:
    #
    #   accum = np.array(())
    #   while ...:
    #      x     = ...
    #      accum = nps.glue(accum, x,  axis = -2)
    #
    # Without special logic, this would throw an error since accum.shape starts
    # at (0,), which is almost certainly not compatible with x.shape. I support
    # both glue(empty,x) and glue(x,empty)
    if len(args) == 2:
        if args[0].shape == (0,) and args[1].size != 0:
            return atleast_dims(args[1], axis)
        if args[1].shape == (0,) and args[0].size != 0:
            return atleast_dims(args[0], axis)

    # Legacy behavior: if no axis is given, add a new axis at the front, and
    # glue along it
    max_ndim = max( x.ndim for x in args )
    if axis is None:
        axis = -1 - max_ndim

    # if we're glueing along a dimension beyond what we already have, expand the
    # target dimension count
    if max_ndim < -axis:
        max_ndim = -axis

    # Now I add dummy dimensions at the front of each array, to bring the source
    # arrays to the same dimensionality. After this is done, ndims for all the
    # matrices will be the same, and np.concatenate() should know what to do.
    args = [ x[(np.newaxis,)*(max_ndim - x.ndim) + (Ellipsis,)] for x in args ]
    return atleast_dims(np.concatenate( args, axis=axis ),
                        axis)


def cat(*args):
    r'''Concatenates a given list of arrays along a new first (outer) dimension.

    SYNOPSIS

        >>> import numpy as np
        >>> import numpysane as nps

        >>> a = np.arange(6).reshape(2,3)
        >>> b = a + 100
        >>> c = a - 100

        >>> a
        array([[0, 1, 2],
               [3, 4, 5]])

        >>> b
        array([[100, 101, 102],
               [103, 104, 105]])

        >>> c
        array([[-100,  -99,  -98],
               [ -97,  -96,  -95]])

        >>> res = nps.cat(a,b,c)
        >>> res
        array([[[   0,    1,    2],
                [   3,    4,    5]],

               [[ 100,  101,  102],
                [ 103,  104,  105]],

               [[-100,  -99,  -98],
                [ -97,  -96,  -95]]])

        >>> res.shape
        (3, 2, 3)

        >>> [x for x in res]
        [array([[0, 1, 2],
                [3, 4, 5]]),
         array([[100, 101, 102],
                [103, 104, 105]]),
         array([[-100,  -99,  -98],
                [ -97,  -96,  -95]])]
        ### Note that this is the same as [a,b,c]: cat is the reverse of
        ### iterating on an array

    This function concatenates the input arrays into an array shaped like the
    highest-dimensioned input, but with a new outer (at the start) dimension.
    The concatenation axis is this new dimension.

    As usual, the dimensions are aligned along the last one, so broadcasting
    will continue to work as expected. Note that this is the opposite operation
    from iterating a numpy array: iteration splits an array over its leading
    dimension, while cat joins a number of arrays via a new leading dimension.

    '''
    if len(args) == 0:
        return np.array(())
    max_ndim = max( x.ndim for x in args )
    return glue(*args, axis = -1 - max_ndim)


def clump(x, **kwargs):
    r'''Groups the given n dimensions together.

    SYNOPSIS

        >>> import numpysane as nps
        >>> nps.clump( arr(2,3,4), n = -2).shape
        (2, 12)

    Reshapes the array by grouping together 'n' dimensions, where 'n' is given
    in a kwarg. If 'n' > 0, then n leading dimensions are clumped; if 'n' < 0,
    then -n trailing dimensions are clumped

    So for instance, if x.shape is (2,3,4) then nps.clump(x, n = -2).shape is
    (2,12) and nps.clump(x, n = 2).shape is (6, 4)

    In numpysane older than 0.10 the semantics were different: n > 0 was
    required, and we ALWAYS clumped the trailing dimensions. Thus the new
    clump(-n) is equivalent to the old clump(n). To request the legacy behavior,
    do

        nps.clump.legacy_version = '0.9'

    '''
    legacy = \
        hasattr(clump, 'legacy_version') and \
        StrictVersion(clump.legacy_version) <= StrictVersion('0.9')


    n = kwargs.get('n')
    if n is None:
        raise NumpysaneError("clump() requires a dimension count in the 'n' kwarg")


    if legacy:
        # old PDL-like clump(). Takes positive dimension counts, and acts from
        # the most-significant dimension (from the back)
        if n < 0:
            raise NumpysaneError("clump() requires n > 0")
        if n <= 1:
            return x

        if x.ndim < n:
            n = x.ndim

        s = list(x.shape[:-n]) + [ _product(x.shape[-n:]) ]
        return x.reshape(s)


    if -1 <= n and n <= 1:
        return x

    if  x.ndim <  n:
        n = x.ndim
    if -x.ndim > n:
        n = -x.ndim

    if n < 0:
        s = list(x.shape[:n]) + [ _product(x.shape[n:]) ]
    else:
        s = [ _product(x.shape[:n]) ] + list(x.shape[n:])
    return x.reshape(s)

def atleast_dims(x, *dims):
    r'''Returns an array with extra length-1 dimensions to contain all given axes.

    SYNOPSIS

        >>> import numpy as np
        >>> import numpysane as nps

        >>> a = np.arange(6).reshape(2,3)
        >>> a
        array([[0, 1, 2],
               [3, 4, 5]])

        >>> nps.atleast_dims(a, -1).shape
        (2, 3)

        >>> nps.atleast_dims(a, -2).shape
        (2, 3)

        >>> nps.atleast_dims(a, -3).shape
        (1, 2, 3)

        >>> nps.atleast_dims(a, 0).shape
        (2, 3)

        >>> nps.atleast_dims(a, 1).shape
        (2, 3)

        >>> nps.atleast_dims(a, 2).shape
        [exception]

        >>> l = [-3,-2,-1,0,1]
        >>> nps.atleast_dims(a, l).shape
        (1, 2, 3)

        >>> l
        [-3, -2, -1, 1, 2]

    If the given axes already exist in the given array, the given array itself
    is returned. Otherwise length-1 dimensions are added to the front until all
    the requested dimensions exist. The given axis>=0 dimensions MUST all be
    in-bounds from the start, otherwise the most-significant axis becomes
    unaligned; an exception is thrown if this is violated. The given axis<0
    dimensions that are out-of-bounds result in new dimensions added at the
    front.

    If new dimensions need to be added at the front, then any axis>=0 indices
    become offset. For instance:

        >>> x.shape
        (2, 3, 4)

        >>> [x.shape[i] for i in (0,-1)]
        [2, 4]

        >>> x = nps.atleast_dims(x, 0, -1, -5)
        >>> x.shape
        (1, 1, 2, 3, 4)

        >>> [x.shape[i] for i in (0,-1)]
        [1, 4]

    Before the call, axis=0 refers to the length-2 dimension and axis=-1 refers
    to the length=4 dimension. After the call, axis=-1 refers to the same
    dimension as before, but axis=0 now refers to a new length=1 dimension. If
    it is desired to compensate for this offset, then instead of passing the
    axes as separate arguments, pass in a single list of the axes indices. This
    list will be modified to offset the axis>=0 appropriately. Ideally, you only
    pass in axes<0, and this does not apply. Doing this in the above example:

        >>> l
        [0, -1, -5]

        >>> x.shape
        (2, 3, 4)

        >>> [x.shape[i] for i in (l[0],l[1])]
        [2, 4]

        >>> x=nps.atleast_dims(x, l)
        >>> x.shape
        (1, 1, 2, 3, 4)

        >>> l
        [2, -1, -5]

        >>> [x.shape[i] for i in (l[0],l[1])]
        [2, 4]

    We passed the axis indices in a list, and this list was modified to reflect
    the new indices: The original axis=0 becomes known as axis=2. Again, if you
    pass in only axis<0, then you don't need to care about this.

    '''

    if any( not isinstance(d, int) for d in dims ):
        if len(dims) == 1 and isinstance(dims[0], list):
            dims = dims[0]
        else:
            raise NumpysaneError("atleast_dims() takes in axes as integers in separate arguments or\n"
                                 "as a single modifiable list")

    if max(dims) >= x.ndim:
        raise NumpysaneError("Axis {} out of bounds because x.ndim={}.\n"
                             "To keep the last dimension anchored, "
                             "only <0 out-of-bounds axes are allowed".format(max(dims), x.ndim))

    need_ndim = -min(d if d<0 else -1 for d in dims)
    if x.ndim >= need_ndim:
        return x
    num_new_axes = need_ndim-x.ndim

    # apply an offset to any axes that need it
    if isinstance(dims, list):
        dims[:] = [ d+num_new_axes if d >= 0 else d for d in dims ]

    return x[ (np.newaxis,)*(num_new_axes) ]

def mv(x, axis_from, axis_to):
    r'''Moves a given axis to a new position. Similar to numpy.moveaxis().

    SYNOPSIS

        >>> import numpy as np
        >>> import numpysane as nps

        >>> a = np.arange(24).reshape(2,3,4)
        >>> a.shape
        (2, 3, 4)

        >>> nps.mv( a, -1, 0).shape
        (4, 2, 3)

        >>> nps.mv( a, -1, -5).shape
        (4, 1, 1, 2, 3)

        >>> nps.mv( a, 0, -5).shape
        (2, 1, 1, 3, 4)

    New length-1 dimensions are added at the front, as required, and any axis>=0
    that are passed in refer to the array BEFORE these new dimensions are added.

    '''
    axes = [axis_from, axis_to]
    x = atleast_dims(x, axes)

    # The below is equivalent to
    #   return np.moveaxis( x, *axes )
    # but some older installs have numpy 1.8, where this isn't available

    axis_from = axes[0] if axes[0] >= 0 else x.ndim + axes[0]
    axis_to   = axes[1] if axes[1] >= 0 else x.ndim + axes[1]

    # python3 needs the list() cast
    order = list(range(0, axis_from)) + list(range((axis_from+1), x.ndim))
    order.insert(axis_to, axis_from)
    return np.transpose(x, order)

def xchg(x, axis_a, axis_b):
    r'''Exchanges the positions of the two given axes. Similar to numpy.swapaxes()

    SYNOPSIS

        >>> import numpy as np
        >>> import numpysane as nps

        >>> a = np.arange(24).reshape(2,3,4)
        >>> a.shape
        (2, 3, 4)

        >>> nps.xchg( a, -1, 0).shape
        (4, 3, 2)

        >>> nps.xchg( a, -1, -5).shape
        (4, 1, 2, 3, 1)

        >>> nps.xchg( a, 0, -5).shape
        (2, 1, 1, 3, 4)

    New length-1 dimensions are added at the front, as required, and any axis>=0
    that are passed in refer to the array BEFORE these new dimensions are added.

    '''
    axes = [axis_a, axis_b]
    x = atleast_dims(x, axes)
    return np.swapaxes( x, *axes )

def transpose(x):
    r'''Reverses the order of the last two dimensions.

    SYNOPSIS

        >>> import numpy as np
        >>> import numpysane as nps

        >>> a = np.arange(24).reshape(2,3,4)
        >>> a.shape
        (2, 3, 4)

        >>> nps.transpose(a).shape
        (2, 4, 3)

        >>> nps.transpose( np.arange(3) ).shape
        (3, 1)

    A "matrix" is generally seen as a 2D array that we can transpose by looking
    at the 2 dimensions in the opposite order. Here we treat an n-dimensional
    array as an n-2 dimensional object containing 2D matrices. As usual, the
    last two dimensions contain the matrix.

    New length-1 dimensions are added at the front, as required, meaning that 1D
    input of shape (n,) is interpreted as a 2D input of shape (1,n), and the
    transpose is 2 of shape (n,1).

    '''
    return xchg( atleast_dims(x, -2), -1, -2)


def dummy(x, axis, *axes_rest):
    r'''Adds length-1 dimensions at the given positions.

    SYNOPSIS

        >>> import numpy as np
        >>> import numpysane as nps

        >>> a = np.arange(24).reshape(2,3,4)
        >>> a.shape
        (2, 3, 4)

        >>> nps.dummy(a, 0).shape
        (1, 2, 3, 4)

        >>> nps.dummy(a, 1).shape
        (2, 1, 3, 4)

        >>> nps.dummy(a, -1).shape
        (2, 3, 4, 1)

        >>> nps.dummy(a, -2).shape
        (2, 3, 1, 4)

        >>> nps.dummy(a, -2, -2).shape
        (2, 3, 1, 1, 4)

        >>> nps.dummy(a, -5).shape
        (1, 1, 2, 3, 4)

    This is similar to numpy.expand_dims(), but handles out-of-bounds dimensions
    better. New length-1 dimensions are added at the front, as required, and any
    axis>=0 that are passed in refer to the array BEFORE these new dimensions
    are added.

    '''

    def dummy_inner(x, axis):
        need_ndim = axis+1 if axis >= 0 else -axis
        if x.ndim >= need_ndim:
            # referring to an axis that already exists. expand_dims() thus works
            return np.expand_dims(x, axis)

        # referring to a non-existing axis. I simply add sufficient new axes, and
        # I'm done
        return atleast_dims(x, axis)

    axes = (axis,) + axes_rest
    for axis in axes: x = dummy_inner(x, axis)
    return x


def reorder(x, *dims):
    r'''Reorders the dimensions of an array.

    SYNOPSIS

        >>> import numpy as np
        >>> import numpysane as nps

        >>> a = np.arange(24).reshape(2,3,4)
        >>> a.shape
        (2, 3, 4)

        >>> nps.reorder( a, 0, -1, 1 ).shape
        (2, 4, 3)

        >>> nps.reorder( a, -2 , -1, 0 ).shape
        (3, 4, 2)

        >>> nps.reorder( a, -4 , -2, -5, -1, 0 ).shape
        (1, 3, 1, 4, 2)

    This is very similar to numpy.transpose(), but handles out-of-bounds
    dimensions much better.

    New length-1 dimensions are added at the front, as required, and any axis>=0
    that are passed in refer to the array BEFORE these new dimensions are added.

    '''
    dims = list(dims)
    x = atleast_dims(x, dims)
    return np.transpose(x, dims)


# Note that this explicitly isn't done with @broadcast_define. Instead I
# implement the internals with core numpy routines. The advantage is that these
# are some of very few numpy functions that support broadcasting, and they do so
# in C, so their broadcasting loop is FAST. Much more so than my current
# @broadcast_define loop
def dot(a, b, out=None, dtype=None):
    r'''Non-conjugating dot product of two 1-dimensional n-long vectors.

    SYNOPSIS

        >>> import numpy as np
        >>> import numpysane as nps

        >>> a = np.arange(3)
        >>> b = a+5
        >>> a
        array([0, 1, 2])

        >>> b
        array([5, 6, 7])

        >>> nps.dot(a,b)
        20

    This is identical to numpysane.inner(). for a conjugating version of this
    function, use nps.vdot(). Note that the stock numpy dot() has some special
    handling when its dot() is given more than 1-dimensional input. THIS
    function has no special handling: normal broadcasting rules are applied, as
    expected.

    In-place operation is available with the "out" kwarg. The output dtype can
    be selected with the "dtype" kwarg. If omitted, the dtype of the input is
    used

    '''
    if out is not None and dtype is not None and out.dtype != dtype:
        raise NumpysaneError("'out' and 'dtype' given explicitly, but the dtypes are mismatched!")

    if dtype is not None:
        # Handle overflows. Cases that require this are checked in the tests
        v = np.sum(a.astype(dtype)*b.astype(dtype), axis=-1, out=out, dtype=dtype )
    else:
        v = np.sum(a*b, axis=-1, out=out, dtype=dtype )
    if out is None:
        return v
    return out

# nps.inner and nps.dot are equivalent. Set the functionality and update the
# docstring
inner = _clone_function( dot, "inner" )
doc = dot.__doc__
doc = doc.replace("vdot",  "aaa")
doc = doc.replace("dot",   "bbb")
doc = doc.replace("inner", "ccc")
doc = doc.replace("ccc",   "dot")
doc = doc.replace("bbb",   "inner")
doc = doc.replace("aaa",   "vdot")
inner.__doc__ = doc

# Note that this explicitly isn't done with @broadcast_define. Instead I
# implement the internals with core numpy routines. The advantage is that these
# are some of very few numpy functions that support broadcasting, and they do so
# on the C level, so their broadcasting loop is FAST. Much more so than my
# current @broadcast_define loop
def vdot(a, b, out=None, dtype=None):
    r'''Conjugating dot product of two 1-dimensional n-long vectors.

    vdot(a,b) is equivalent to dot(np.conj(a), b)

    SYNOPSIS

        >>> import numpy as np
        >>> import numpysane as nps

        >>> a = np.array(( 1 + 2j, 3 + 4j, 5 + 6j))
        >>> b = a+5
        >>> a
        array([ 1.+2.j,  3.+4.j,  5.+6.j])

        >>> b
        array([  6.+2.j,   8.+4.j,  10.+6.j])

        >>> nps.vdot(a,b)
        array((136-60j))

        >>> nps.dot(a,b)
        array((24+148j))

    For a non-conjugating version of this function, use nps.dot(). Note that the
    numpy vdot() has some special handling when its vdot() is given more than
    1-dimensional input. THIS function has no special handling: normal
    broadcasting rules are applied.

    '''
    return dot(np.conj(a), b, out=out, dtype=dtype)

@broadcast_define( (('n',), ('m',)), prototype_output=('n','m'), out_kwarg='out' )
def outer(a, b, out=None):
    r'''Outer product of two 1-dimensional n-long vectors.

    SYNOPSIS

        >>> import numpy as np
        >>> import numpysane as nps

        >>> a = np.arange(3)
        >>> b = a+5
        >>> a
        array([0, 1, 2])

        >>> b
        array([5, 6, 7])

        >>> nps.outer(a,b)
        array([[ 0,  0,  0],
               [ 5,  6,  7],
               [10, 12, 14]])
    '''
    if out is None:
        return np.outer(a,b)

    out.setfield(np.outer(a,b), out.dtype)
    return out

# Note that this explicitly isn't done with @broadcast_define. Instead I
# implement the internals with core numpy routines. The advantage is that these
# are some of very few numpy functions that support broadcasting, and they do so
# in C, so their broadcasting loop is FAST. Much more so than my current
# @broadcast_define loop
def norm2(a, **kwargs):
    r'''Broadcast-aware 2-norm. norm2(x) is identical to inner(x,x)

    SYNOPSIS

        >>> import numpy as np
        >>> import numpysane as nps

        >>> a = np.arange(3)
        >>> a
        array([0, 1, 2])

        >>> nps.norm2(a)
        5

    This is a convenience function to compute a 2-norm

    '''
    return inner(a,a, **kwargs)

def mag(a, out=None, dtype=None):
    r'''Magnitude of a vector. mag(x) is functionally identical to sqrt(inner(x,x))

    SYNOPSIS

        >>> import numpy as np
        >>> import numpysane as nps

        >>> a = np.arange(3)
        >>> a
        array([0, 1, 2])

        >>> nps.mag(a)
        2.23606797749979

    This is a convenience function to compute a magnitude of a vector, with full
    broadcasting support.

    In-place operation is available with the "out" kwarg. The output dtype can
    be selected with the "dtype" kwarg. If omitted, dtype=float is selected.

    '''

    if out is None:
        if dtype is None:
            dtype = float
        out = inner(a,a, dtype=dtype)

        if not isinstance(out, np.ndarray):
            # given two vectors, and without and 'out' array, inner() produces a
            # scalar, not an array. So I can't updated it inplace, and just
            # return a copy
            return np.sqrt(out)
    else:
        if dtype is None:
            dtype = out.dtype
        inner(a,a, out=out, dtype=dtype)

    # in-place sqrt
    np.sqrt.at(out,())
    return out

@broadcast_define( (('n','n',),), prototype_output=() )
def trace(a):
    r'''Broadcast-aware trace

    SYNOPSIS

        >>> import numpy as np
        >>> import numpysane as nps

        >>> a = np.arange(3*4*4).reshape(3,4,4)
        >>> a
        array([[[ 0,  1,  2,  3],
                [ 4,  5,  6,  7],
                [ 8,  9, 10, 11],
                [12, 13, 14, 15]],

               [[16, 17, 18, 19],
                [20, 21, 22, 23],
                [24, 25, 26, 27],
                [28, 29, 30, 31]],

               [[32, 33, 34, 35],
                [36, 37, 38, 39],
                [40, 41, 42, 43],
                [44, 45, 46, 47]]])

        >>> nps.trace(a)
        array([ 30,  94, 158])
    '''
    return np.trace(a)

# Could be implemented with a simple loop around np.dot():
#
#     @broadcast_define( (('n', 'm'), ('m', 'l')), prototype_output=('n','l'), out_kwarg='out' )
#     def matmult2(a, b, out=None):
#         return np.dot(a,b)
#
# but this would produce a python broadcasting loop, which is potentially slow.
# Instead I'm using the np.matmul() primitive to get C broadcasting loops. This
# function has stupid special-case rules for low-dimensional arrays, so I make
# sure to do the normal broadcasting thing in those cases
def matmult2(a, b, out=None):
    r'''Multiplication of two matrices

    SYNOPSIS

        >>> import numpy as np
        >>> import numpysane as nps

        >>> a = np.arange(6) .reshape(2,3)
        >>> b = np.arange(12).reshape(3,4)

        >>> a
        array([[0, 1, 2],
               [3, 4, 5]])

        >>> b
        array([[ 0,  1,  2,  3],
               [ 4,  5,  6,  7],
               [ 8,  9, 10, 11]])

        >>> nps.matmult2(a,b)
        array([[20, 23, 26, 29],
               [56, 68, 80, 92]])

    This function is exposed publically mostly for legacy compatibility. Use
    numpysane.matmult() instead

    '''

    if not isinstance(a, np.ndarray) and not isinstance(b, np.ndarray):
        # two non-arrays (assuming two scalars)
        if out is not None:
            o = a*b
            out.setfield(o, out.dtype)
            out.resize([])
            return out
        return a*b

    if not isinstance(a, np.ndarray) or len(a.shape) == 0:
        # one non-array (assuming one scalar)
        if out is not None:
            out.setfield(a*b, out.dtype)
            out.resize(b.shape)
            return out
        return a*b

    if not isinstance(b, np.ndarray) or len(b.shape) == 0:
        # one non-array (assuming one scalar)
        if out is not None:
            out.setfield(a*b, out.dtype)
            out.resize(a.shape)
            return out
        return a*b

    if len(b.shape) == 1:
        b = b[np.newaxis, :]

    o = np.matmul(a,b, out)
    return o

def matmult( a, *rest, **kwargs ):
    r'''Multiplication of N matrices

    SYNOPSIS

        >>> import numpy as np
        >>> import numpysane as nps

        >>> a = np.arange(6) .reshape(2,3)
        >>> b = np.arange(12).reshape(3,4)
        >>> c = np.arange(4) .reshape(4,1)

        >>> a
        array([[0, 1, 2],
               [3, 4, 5]])

        >>> b
        array([[ 0,  1,  2,  3],
               [ 4,  5,  6,  7],
               [ 8,  9, 10, 11]])

        >>> c
        array([[0],
               [1],
               [2],
               [3]])

        >>> nps.matmult(a,b,c)
        array([[162],
               [504]])

        >>> abc = np.zeros((2,1), dtype=float)
        >>> nps.matmult(a,b,c, out=abc)
        >>> abc
        array([[162],
               [504]])

    This multiplies N matrices together by repeatedly calling matmult2() for
    each adjacent pair. In-place output is supported with the 'out' keyword
    argument

    This function supports broadcasting fully, in C internally

    '''

    if len(rest) == 0:
        raise Exception("Need at least two terms to multiply")

    out = None
    if len(kwargs.keys()) > 1:
        raise Exception("Only ONE kwarg is supported: 'out'")
    if kwargs:
        # have exactly one kwarg
        if 'out' not in kwargs:
            raise Exception("The only supported kwarg is 'out'")
        out = kwargs['out']

    return matmult2(a,reduce(matmult2, rest), out=out)



# I use np.matmul at one point. This was added in numpy 1.10.0, but
# apparently I want to support even older releases. I thus provide a
# compatibility function in that case. This is slower (python loop instead of C
# loop), but at least it works
if not hasattr(np, 'matmul'):
    @broadcast_define( (('n','m'),('m','o')), ('n','o'))
    def matmul(a,b, out=None):
        return np.dot(a,b,out)
    np.matmul = matmul
