<sect1 id="MultiArray"><title>MultiArray Concept</title>


<para>The MultiArray
concept defines an interface to hierarchically nested
containers.  It specifies operations for accessing elements,
traversing containers, and creating views
of array data.
MultiArray defines
a flexible memory model that accomodates
a variety of data layouts.
</para>

<para>
At each level (or dimension) of a MultiArray's
container hierarchy lie a set of ordered containers, each of which
contains the same number and type of values. The depth of this
container hierarchy is the MultiArray's <emphasis>dimensionality</emphasis>. 
MultiArray is recursively defined; the
containers at each level of the container hierarchy model
MultiArray as well. While each dimension of a MultiArray
has its own size, the list of sizes for all dimensions 
defines the <emphasis>shape</emphasis> of the entire MultiArray.
At the base of this hierarchy lie 1-dimensional
MultiArrays.  Their values are the contained
objects of interest and not part of the container hierarchy. These are
the MultiArray's elements.
</para>


<para>
Like other container concepts, MultiArray exports
iterators to traverse its values. In addition, values can be
addressed directly using the familiar bracket notation.
</para>

<para>
MultiArray also specifies
routines for creating
specialized views. A <emphasis>view</emphasis> lets you treat a 
subset of the underlying
elements in a MultiArray as though it were a separate
MultiArray. Since a view refers to the same underlying elements,
changes made to a view's elements will be reflected in the original
MultiArray. For
example, given a 3-dimensional "cube" of elements, a 2-dimensional
slice can be viewed as if it were an independent
MultiArray.

Views are created using <literal>index_gen</literal> and
<literal>index_range</literal> objects.
<literal>index_range</literal>s denote elements from a certain
dimension that are to be included in a
view. <literal>index_gen</literal> aggregates range data and performs
bookkeeping to determine the view type to be returned.

MultiArray's <literal>operator[]</literal>
 must be passed the result
of <literal>N</literal> chained calls to 
<literal>index_gen::operator[]</literal>, i.e.

<programlisting>indices[a0][a1]...[aN];
</programlisting>

where <literal>N</literal> is the 
MultiArray's dimensionality and
<literal>indices</literal> an object of type <literal>index_gen</literal>.

The view type is dependent upon the number of degenerate dimensions
specified to <literal>index_gen</literal>.  A degenerate dimension
occurs when a single-index is specified to
<literal>index_gen</literal> for a certain dimension.  For example, if
<literal>indices</literal> is an object of type
<literal>index_gen</literal>, then the following example:

<programlisting>indices[index_range(0,5)][2][index_range(0,4)];
</programlisting>

has a degenerate second dimension.  The view generated from the above
specification will have 2 dimensions with shape <literal>5 x 4</literal>.
If the "<literal>2</literal>" above were replaced with
another <literal>index_range</literal> object, for example:

<programlisting>indices[index_range(0,5)][index_range(0,2)][index_range(0,4)];
</programlisting>

then the view would have 3 dimensions.</para>

<para>
MultiArray exports
information regarding the memory
layout of its contained elements. Its memory model for elements is
completely defined by 4 properties: the origin, shape, index bases,
and strides.  The origin is the address in memory of the element
accessed as <literal>a[0][0]...[0]</literal>, where
<literal>a</literal> is a MultiArray. The shape is a list of numbers
specifying the size of containers at each dimension.  For example, the
first extent is the size of the outermost container, the second extent
is the size of its subcontainers, and so on. The index bases are a
list of signed values specifying the index of the first value in a
container. All containers at the same dimension share the same index
base.  Note that since positive index bases are
possible, the origin need not exist in order to determine the location
in memory of the MultiArray's elements.
  The strides determine how index values are mapped to memory offsets. 
They accomodate a
number of possible element layouts.  For example, the elements of a 2
dimensional array can be stored by row (i.e., the elements of each row
are stored contiguously) or by column (i.e., the elements of each
column are stored contiguously).
</para>

<para>
Two concept checking classes for the MultiArray concepts
(<literal>ConstMultiArrayConcept</literal> and
<literal>MutableMultiArrayConcept</literal>) are in the namespace
<literal>boost::multi_array_concepts</literal> in
<literal>&lt;boost/multi_array/concept_checks.hpp&gt;</literal>.
</para>


<sect2><title>Notation</title>
<para>What follows are the descriptions of symbols that will be used
to describe the MultiArray interface.</para>
    <table>
      <title>Notation</title>
      <tgroup cols="2">
	<tbody>
	  <row>
	      <entry><literal>A</literal></entry>
	      <entry>A type that is a model of MultiArray
</entry>
	    </row>
	    <row>
	      <entry><literal>a,b</literal></entry>
	      <entry>Objects of type <literal>A</literal></entry>
	    </row>
	    <row>
	      <entry><literal>NumDims</literal></entry>
	      <entry>The numeric dimension parameter associated with
<literal>A</literal>.</entry>
	    </row>
	    <row>
	      <entry><literal>Dims</literal></entry>
	      <entry>Some numeric dimension parameter such that 
<literal>0&lt;Dims&lt;NumDims</literal>.
</entry>
	  </row>
	  <row>
	      <entry><literal>indices</literal></entry>
	      <entry>An object created by some number of chained calls
to <literal>index_gen::operator[](index_range)</literal>.</entry>
	    </row>
	    <row>
	      <entry><literal>index_list</literal></entry>
	      <entry>An object whose type models
<ulink url="../../utility/Collection.html">Collection</ulink>
</entry>
	    </row>
	    <row>
	      <entry><literal>idx</literal></entry>
	      <entry>A signed integral value.</entry>
	    </row>
	    <row>
	      <entry><literal>tmp</literal></entry> 
	      <entry>An object of type
	      <literal>boost::array&lt;index,NumDims&gt;</literal></entry>
	    </row>
	</tbody>
      </tgroup>
    </table>
</sect2>

<sect2><title>Associated Types</title>
<para>
</para>
<table><title>Associated Types</title>
<tgroup cols="2">

<thead>
<row>
<entry>Type</entry>
<entry>Description</entry>
</row>
</thead>

<tbody>

<row>
<entry><literal>value_type</literal></entry>

<entry>This is the value type of the container.
  If <literal>NumDims == 1</literal>, then this is
<literal>element</literal>. Otherwise, this is the value type of the
immediately nested containers.
</entry>
</row>

<row>
<entry>
<literal>reference</literal>
</entry>

<entry>
This is the reference type of the contained value. 
If <literal>NumDims == 1</literal>, then this is 
<literal>element&amp;</literal>. Otherwise, this is the same type as
<literal>template subarray&lt;NumDims-1&gt;::type</literal>.
</entry>
</row>

<row>
<entry>
<literal>const_reference</literal>
</entry>
<entry>
This is the const reference type of the contained value.
If <literal>NumDims == 1</literal>, then this is  
<literal>const element&amp;</literal>. Otherwise, this is the same
type as
<literal>template const_subarray&lt;NumDims-1&gt;::type</literal>.
</entry>
</row>

<row>
<entry>
<literal>size_type</literal>
</entry>
<entry>
This is an unsigned integral type.  It is primarily used to specify array shape.
</entry>
</row>


<row>
<entry>
<literal>difference_type</literal>
</entry>
<entry>
This is a signed integral type used to represent the distance between two
iterators. It is the same type as
<literal>std::iterator_traits&lt;iterator&gt;::difference_type</literal>.
</entry>
</row>

<row>
<entry><literal>iterator</literal></entry>
<entry>
This is an iterator over the values of <literal>A</literal>.
If <literal>NumDims == 1</literal>, then it models 
<ulink url="http://www.boost.org/doc/html/RandomAccessIterator.html">
<literal>Random Access Iterator</literal></ulink>. 
Otherwise it models 
<ulink url="./iterator_categories.html#concept_RandomAccessTraversalIterator">
Random Access Traversal Iterator</ulink>,
<ulink url="./iterator_categories.html#concept_ReadableIterator">
Readable Iterator</ulink>,
<ulink url="./iterator_categories.html#concept_WritableIterator">
Writable Iterator</ulink>, and 
<ulink url="http://www.boost.org/doc/html/OutputIterator.html">
<literal>Output Iterator</literal></ulink>.
</entry>
</row>

<row>
<entry>
<literal>const_iterator</literal>
</entry>
<entry>
This is the const iterator over the values of <literal>A</literal>.
</entry>
</row>
<row>

<entry>
<literal>reverse_iterator</literal>
</entry>
<entry>
This is the reversed iterator, used to iterate backwards over the values of 
<literal>A</literal>.
</entry>
</row>

<row>
<entry>
<literal>const_reverse_iterator</literal>
</entry>
<entry>
This is the reversed const iterator.
<literal>A</literal>.
</entry>
</row>
<row>

<entry>
<literal>element</literal>
</entry>
<entry>
This is the type of objects stored at the base of the
hierarchy of MultiArrays. It is the same as
<literal>template subarray&lt;1&gt;::value_type</literal>
</entry>
</row>

<row>
<entry>
<literal>index</literal>
</entry>
<entry>
This is a signed integral type used for indexing into <literal>A</literal>. It 
is also used to represent strides and index bases.
</entry>
</row>

<row>
<entry>
<literal>index_gen</literal>
</entry>
<entry>
This type is used to create a tuple of <literal>index_range</literal>s 
passed to <literal>operator[]</literal> to create
an <literal>array_view&lt;Dims&gt;::type</literal> object.
</entry>
</row>

<row>
<entry>
<literal>index_range</literal>
</entry>
<entry>
This type specifies a range of indices over some dimension of a
MultiArray.  This range will be visible through an 
<literal>array_view&lt;Dims&gt;::type</literal> object.
</entry>
</row>

<row>
<entry>
<literal>template subarray&lt;Dims&gt;::type</literal>
</entry>
<entry>
This is subarray type with <literal>Dims</literal> dimensions.
It is the reference type of the <literal>(NumDims - Dims)</literal>
dimension of <literal>A</literal> and also models
MultiArray.
</entry>
</row>

<row>
<entry>
<literal>template const_subarray&lt;Dims&gt;::type</literal>
</entry>
<entry>
This is the const subarray type.
</entry>
</row>

<row>
<entry>
<literal>template array_view&lt;Dims&gt;::type</literal>
</entry>
<entry>
This is the view type with <literal>Dims</literal> dimensions.  It is
returned by calling <literal>operator[](<literal>indices</literal>)</literal>.
It models MultiArray.
</entry>
</row>

<row>
<entry>
<literal>template
const_array_view&lt;Dims&gt;::type</literal>
</entry>
<entry>
This is the const view type with <literal>Dims</literal> dimensions.
</entry>
</row>

</tbody>
</tgroup>
</table>

</sect2>


  <sect2><title>Valid expressions</title>
    
    <table><title>Valid Expressions</title>
      <tgroup cols="3">
	<thead>
	  <row>
	    <entry>Expression</entry>
	    <entry>Return type</entry>
	    <entry>Semantics</entry>
	  </row>
	</thead>
	<tbody>
	  <row>
	    <entry><literal>A::dimensionality</literal></entry>
	    <entry><literal>size_type</literal></entry>
	    <entry>This compile-time constant represents the number of
dimensions of the array (note that 
<literal>A::dimensionality == NumDims</literal>).</entry>
	  </row>
	  <row>
	    <entry><literal>a.shape()</literal></entry>
	    <entry><literal>const size_type*</literal></entry>
	    <entry>
This returns a list of <literal>NumDims</literal> elements specifying the
extent of each array dimension.
</entry>
	  </row>

	  <row>
	    <entry><literal>a.strides()</literal></entry>
	    <entry><literal>const index*</literal></entry>
	    <entry>
This returns a list of <literal>NumDims</literal> elements specifying the
stride associated with each array dimension. When accessing values,
strides is used to calculate an element's location in memory.
</entry>
	  </row>

	  <row>
	    <entry><literal>a.index_bases()</literal></entry>
	    <entry><literal>const index*</literal></entry>
	    <entry>
This returns a list of <literal>NumDims</literal> elements specifying the
numeric index of the first element for each array dimension.
</entry>
	  </row>
	  <row>
	    <entry><literal>a.origin()</literal></entry>
	    <entry>
<literal>element*</literal> if <literal>a</literal> is mutable,
<literal>const element*</literal> otherwise.
</entry>
<entry>
This returns the address of the element accessed by the expression
<literal>a[0][0]...[0].</literal>. If the index bases are positive,
this element won't exist, but the address can still be used to locate
a valid element given its indices.
</entry>
	  </row>
	  <row>
	    <entry><literal>a.num_dimensions()</literal></entry>
	    <entry><literal>size_type</literal></entry>
	    <entry>This returns the number of dimensions of the array
(note that <literal>a.num_dimensions() == NumDims</literal>).</entry>
	  </row>

	    <row>
	      <entry><literal>a.num_elements()</literal></entry>
	      <entry><literal>size_type</literal></entry>
	      <entry>This returns the number of elements contained
in the array. It is equivalent to the following code:
<programlisting>
std::accumulate(a.shape(),a.shape+a.num_dimensions(),
    size_type(1),std::multiplies&lt;size_type&gt;());
</programlisting>
</entry>
	  </row>

	  <row>
	    <entry><literal>a.size()</literal></entry>
	    <entry><literal>size_type</literal></entry>
	    <entry>
This returns the number of values contained in
<literal>a</literal>. It is equivalent to <literal>a.shape()[0];</literal>
</entry>
	  </row>
	  <row>
	    <entry><literal>a(index_list)</literal></entry>
            <entry>
<literal>element&amp;</literal>;  if <literal>a</literal> is mutable,
<literal>const element&amp;</literal> otherwise.
            </entry>
	    <entry>
This expression accesses a specific element of
<literal>a</literal>.<literal>index_list</literal> is the unique set
of indices that address the element returned.  It is 
equivalent to the following code (disregarding intermediate temporaries):
<programlisting>
    // multiply indices by strides
    std::transform(index_list.begin(), index_list.end(),
      a.strides(), tmp.begin(), std::multiplies&lt;index&gt;()),

    // add the sum of the products to the origin
    *std::accumulate(tmp.begin(), tmp.end(), a.origin());
</programlisting>
</entry>
	  </row>

	  <row>
	    <entry><literal>a.begin()</literal></entry>
	    <entry>
<literal>iterator</literal> if <literal>a</literal> is mutable,
<literal>const_iterator</literal> otherwise.
            </entry>
	    <entry>This returns an iterator pointing to the beginning of
<literal>a</literal>.</entry>
	  </row>

	  <row>
	    <entry><literal>a.end()</literal></entry>
	    <entry>
<literal>iterator</literal> if <literal>a</literal> is mutable,
<literal>const_iterator</literal> otherwise.
            </entry>
	      <entry>This returns an iterator pointing to the end of
<literal>a</literal>.</entry>
	  </row>

	  <row>
	    <entry><literal>a.rbegin()</literal></entry>
	    <entry>
<literal>reverse_iterator</literal> if <literal>a</literal> is mutable, 
<literal>const_reverse_iterator</literal> otherwise.
            </entry>
	    <entry>This returns a reverse iterator pointing to the
beginning of <literal>a</literal> reversed.
</entry>
	  </row>

	  <row>
	    <entry><literal>a.rend()</literal></entry>
	    <entry>
<literal>reverse_iterator</literal> if <literal>a</literal> is mutable, 
<literal>const_reverse_iterator</literal> otherwise.
</entry>
	    <entry>
This returns a reverse iterator pointing to the end of <literal>a</literal>
reversed.
</entry>
</row>
	  <row>
	    <entry><literal>a[idx]</literal></entry>
	    <entry>
<literal>reference</literal> if <literal>a</literal> is mutable,
<literal>const_reference</literal> otherwise.
            </entry>
	    <entry>
This returns a reference type that is bound to the index
<literal>idx</literal> value of <literal>a</literal>.  Note that if
<literal>i</literal> is the index base for this dimension, the above
expression returns the <literal>(idx-i)</literal>th element (counting
from zero).  The expression is equivalent to
<literal>*(a.begin()+idx-a.index_bases()[0]);</literal>.
</entry>
	  </row>

	  <row>
	    <entry><literal>a[indices]</literal></entry>
	    <entry>
<literal>array_view&lt;Dims&gt;::type</literal> if
<literal>a</literal> is mutable,
<literal>const_array_view&lt;Dims&gt;::type</literal> otherwise.
            </entry>
	<entry>
This expression generates a view of the array determined by the
<literal>index_range</literal> and <literal>index</literal> values
 used to construct <literal>indices</literal>.
</entry>
      </row>
	    <row>
	      <entry><literal>a == b</literal></entry>
	      <entry>bool</entry>
	      <entry>This performs a lexicographical comparison of the
values of <literal>a</literal> and <literal>b</literal>.  The element
type must model <ulink url="https://www.boost.org/sgi/stl/EqualityComparable.html">EqualityComparable</ulink> for this
expression to be valid.</entry>
	    </row>
	    <row>
	      <entry><literal>a &lt; b</literal></entry>
	      <entry>bool</entry>
	      <entry>This performs a lexicographical comparison of the
values of <literal>a</literal> and <literal>b</literal>.  The element
type must model <ulink url="https://www.boost.org/sgi/stl/LessThanComparable.html">LessThanComparable</ulink> for this
expression to be valid.</entry>
	    </row>
	    <row>
	      <entry><literal>a &lt;= b</literal></entry>
	      <entry>bool</entry>
	      <entry>This performs a lexicographical comparison of the
values of <literal>a</literal> and <literal>b</literal>.  The element
type must model <ulink url="https://www.boost.org/sgi/stl/EqualityComparable.html">EqualityComparable</ulink> and
<ulink url="https://www.boost.org/sgi/stl/LessThanComparable.html">LessThanComparable</ulink> for this
expression to be valid.</entry>
	    </row>
	    <row>
	      <entry><literal>a &gt; b</literal></entry>
	      <entry>bool</entry>
	      <entry>This performs a lexicographical comparison of the
values of <literal>a</literal> and <literal>b</literal>.  The element
type must model <ulink url="https://www.boost.org/sgi/stl/EqualityComparable.html">EqualityComparable</ulink> and 
<ulink url="https://www.boost.org/sgi/stl/LessThanComparable.html">LessThanComparable</ulink> for this
expression to be valid.</entry>
	    </row>
	    <row>
	      <entry><literal>a &gt;= b</literal></entry>
	      <entry>bool</entry>
	      <entry>This performs a lexicographical comparison of the
values of <literal>a</literal> and <literal>b</literal>.  The element
type must model <ulink url="https://www.boost.org/sgi/stl/LessThanComparable.html">LessThanComparable</ulink> for this
expression to be valid.</entry>
	    </row>
    </tbody>
    </tgroup>
    </table>
    </sect2>


<sect2><title>Complexity guarantees</title>

<literal>begin()</literal> and <literal>end()</literal> execute in amortized
constant time.
<literal>size()</literal> executes in at most linear time in the 
MultiArray's size. 
</sect2>

<sect2>
<title>Invariants</title>
<table><title>Invariants</title>
      <tgroup cols="2">
	<tbody>
	  <row>
	    <entry>Valid range</entry>
	    <entry><literal>[a.begin(),a.end())</literal> is a valid range.
            </entry>
	  </row>

	  <row>
	    <entry>Range size</entry>
	    <entry>
<literal>a.size() == std::distance(a.begin(),a.end());</literal>.
</entry>
	  </row>

	  <row>
	    <entry>Completeness</entry>
	    <entry>
Iteration through the range 
<literal>[a.begin(),a.end())</literal> will traverse across every
<literal>value_type</literal> of <literal>a</literal>.
</entry>
	    </row>
	    <row>
	      <entry>Accessor Equivalence</entry>
	      <entry>
Calling <literal>a[a1][a2]...[aN]</literal> where <literal>N==NumDims</literal>
yields the same result as calling 
<literal>a(index_list)</literal>, where <literal>index_list</literal>
is a <ulink url="../../utility/Collection.html">Collection</ulink> containing the values <literal>a1...aN</literal>.
</entry>
	      </row>
	</tbody>
      </tgroup>
    </table>
  </sect2>

<sect2 id="view_types">
      <title>Associated Types for Views</title>
      <para>The following MultiArray  associated 
types define the interface for creating views of existing
MultiArrays. Their interfaces and roles in the
concept are described below.</para>
  
  <sect3 id="index_range">
    <title><literal>index_range</literal></title>

    <para><literal>index_range</literal> objects represent half-open
strided intervals.  They are aggregated (using an
<literal>index_gen</literal> object) and passed to 
a MultiArray's <literal>operator[]</literal>
to create an array view. When creating a view, 
each <literal>index_range</literal> denotes a range of
valid indices along one dimension of a MultiArray.
Elements that are accessed through the set of ranges specified will be 
included in the constructed view. In some cases, an
<literal>index_range</literal> is created without specifying start
or finish values.  In those cases, the object is interpreted to
start at the beginning of a MultiArray dimension
and end at its end.</para>

<para>
<literal>index_range</literal> objects can be constructed and modified 
several ways in order to allow convenient and clear expression of a
range of indices.  To specify ranges, <literal>index_range</literal>
supports a set of constructors, mutating member functions, and a novel 
specification involving inequality operators.  Using inequality
operators,  a half open range [5,10) can be specified as follows:
<programlisting>5 &lt;= index_range() &lt; 10;</programlisting> or
<programlisting>4 &lt; index_range() &lt;= 9;</programlisting> and so on.

The following describes the
<literal>index_range</literal> interface.
</para>

    <table>
      <title>Notation</title>
      <tgroup cols="2">
	<tbody>
	  <row>
	    <entry><literal>i</literal></entry>
	    <entry>An object of type <literal>index_range</literal>.</entry>
	  </row>
	  <row>
	    <entry><literal>idx,idx1,idx2,idx3</literal></entry>
	    <entry>Objects of type <literal>index</literal>.</entry>
	  </row>
	</tbody>
      </tgroup>
    </table>

    <table><title>Associated Types</title>
      <tgroup cols="2">
	<thead>
	  <row>
	    <entry>Type</entry>
	    <entry>Description</entry>
	  </row>
	</thead>
	<tbody>
	  <row>
	    <entry><literal>index</literal></entry>
	    <entry>This is a signed integral type. It is used to
specify the start, finish, and stride values.</entry>
	  </row>
	  <row>
	    <entry><literal>size_type</literal></entry>
	    <entry>This is an unsigned integral type. It is used to
report the size of the range an <literal>index_range</literal> 
represents.</entry>
	  </row>
	</tbody>
      </tgroup>
    </table>
    

    <table><title>Valid Expressions</title>
      <tgroup cols="3">
	<thead>
	  <row>
	    <entry>Expression</entry>
	    <entry>Return type</entry>
	    <entry>Semantics</entry>
	  </row>
	</thead>
	<tbody>
	  <row>
	    <entry><literal>index_range(idx1,idx2,idx3)</literal></entry>
	    <entry><literal>index_range</literal></entry>
	    <entry>This constructs an <literal>index_range</literal>
	    representing the interval <literal>[idx1,idx2)</literal>
 with stride <literal>idx3</literal>.</entry>
	  </row>
	  <row>
	    <entry><literal>index_range(idx1,idx2)</literal></entry>
	    <entry><literal>index_range</literal></entry>
	    <entry>This constructs an <literal>index_range</literal>
	    representing the interval <literal>[idx1,idx2)</literal>
 with unit stride. It is equivalent to
	    <literal>index_range(idx1,idx2,1)</literal>.</entry>
	  </row>
	  <row>
	    <entry><literal>index_range()</literal></entry>
	    <entry><literal>index_range</literal></entry>
	    <entry>This construct an <literal>index_range</literal>
with unspecified start and finish values.</entry>
	  </row>
	  <row>
	    <entry><literal>i.start(idx1)</literal></entry> 
	    <entry><literal>index&amp;</literal></entry>
	    <entry>This sets the start index of <literal>i</literal> to
	    <literal>idx</literal>.</entry>
	  </row>
	  <row>
	    <entry><literal>i.finish(idx)</literal></entry>
	    <entry><literal>index&amp;</literal></entry>
	    <entry>This sets the finish index of <literal>i</literal> to 
            <literal>idx</literal>.</entry>
	  </row>
	  <row>
	    <entry><literal>i.stride(idx)</literal></entry>
	    <entry><literal>index&amp;</literal></entry>
	    <entry>This sets the stride length of <literal>i</literal> to
            <literal>idx</literal>.</entry>
	  </row>
	  <row>
	    <entry><literal>i.start()</literal></entry>
	    <entry><literal>index</literal></entry>
	    <entry>This returns the start index of <literal>i</literal>.</entry>
	  </row>
	  <row>
	    <entry><literal>i.finish()</literal></entry>
	    <entry><literal>index</literal></entry>
	    <entry>This returns the finish index of <literal>i</literal>.</entry>
	  </row>
	  <row>
	    <entry><literal>i.stride()</literal></entry>
	    <entry><literal>index</literal></entry>
	    <entry>This returns the stride length of <literal>i</literal>.</entry>
	  </row>
	  <row>
	    <entry><literal>i.get_start(idx)</literal></entry> 
	    <entry><literal>index</literal></entry>
	    <entry>If <literal>i</literal> specifies a start
value, this is equivalent to <literal>i.start()</literal>. Otherwise it
returns <literal>idx</literal>.</entry>
	  </row>
	  <row>
	    <entry><literal>i.get_finish(idx)</literal></entry>
	    <entry><literal>index</literal></entry>
	    <entry>If <literal>i</literal> specifies a finish
value, this is equivalent to <literal>i.finish()</literal>. Otherwise it
returns <literal>idx</literal>.</entry>
	  </row>
	  <row>
	    <entry><literal>i.size(idx)</literal></entry>
	    <entry><literal>size_type</literal></entry>
	    <entry>If <literal>i</literal> specifies a both finish and
start values, this is equivalent to
<literal>(i.finish()-i.start())/i.stride()</literal>. Otherwise it
returns <literal>idx</literal>.</entry>
	  </row>
	  <row>
	    <entry><literal>i &lt; idx</literal></entry>
	    <entry><literal>index</literal></entry>
	    <entry>This is another syntax for specifying the finish
value. This notation does not include 
<literal>idx</literal> in the range of valid indices. It is equivalent to 
<literal>index_range(r.start(), idx, r.stride())</literal></entry>
	  </row>
	  <row>
	    <entry><literal>i &lt;= idx</literal></entry>
	    <entry><literal>index</literal></entry>
	    <entry>This is another syntax for specifying the finish
value. This notation includes 
<literal>idx</literal> in the range of valid indices. It is equivalent to 
<literal>index_range(r.start(), idx + 1, r.stride())</literal></entry>
	  </row>
	  <row>
	    <entry><literal>idx &lt; i</literal></entry>
	    <entry><literal>index</literal></entry>
	    <entry>This is another syntax for specifying the start
value. This notation does not include 
<literal>idx</literal> in the range of valid indices. It is equivalent to 
<literal>index_range(idx + 1, i.finish(), i.stride())</literal>.</entry>
	  </row>
	  <row>
	    <entry><literal>idx &lt;= i</literal></entry>
	    <entry><literal>index</literal></entry>
	    <entry>This is another syntax for specifying the start
value. This notation includes
<literal>idx1</literal> in the range of valid indices. It is equivalent to 
<literal>index_range(idx, i.finish(), i.stride())</literal>.</entry>
	  </row>
	  <row>
	    <entry><literal>i + idx</literal></entry>
	    <entry><literal>index</literal></entry>
	    <entry>This expression shifts the start and finish values
of <literal>i</literal> up by <literal>idx</literal>. It is equivalent to 
<literal>index_range(r.start()+idx1, r.finish()+idx, r.stride())</literal></entry>
	  </row>
	  <row>
	    <entry><literal>i - idx</literal></entry>
	    <entry><literal>index</literal></entry>
	    <entry>This expression shifts the start and finish values
of <literal>i</literal> up by <literal>idx</literal>. It is equivalent to 
<literal>index_range(r.start()-idx1, r.finish()-idx, r.stride())</literal></entry>
	  </row>
	</tbody>
      </tgroup>
    </table>
  </sect3>

      <sect3 id="index_gen">
	<title><literal>index_gen</literal></title>
	<para> <literal>index_gen</literal> aggregates 
<literal>index_range</literal> objects in order to specify view
parameters.  Chained calls to <literal>operator[]</literal> store
range and dimension information used to 
instantiate a new view into a MultiArray.
</para>
	<table>
	  <title>Notation</title>
	  <tgroup cols="2">
	    <tbody>
	      <row>
		<entry><literal>Dims,Ranges</literal></entry>
		<entry>Unsigned integral values.</entry>
	      </row>
	      <row>
		<entry><literal>x</literal></entry>
		<entry>An object of type 
<literal>template gen_type&lt;Dims,Ranges&gt;::type</literal>.</entry>
	      </row>
	      <row>
		<entry><literal>i</literal></entry>
		<entry>An object of type 
<literal>index_range</literal>.</entry>
	      </row>
	      <row>
		<entry><literal>idx</literal></entry>
		<entry>Objects of type <literal>index</literal>.</entry>
	      </row>
	    </tbody>
	  </tgroup>
	</table>
	
    <table><title>Associated Types</title>
      <tgroup cols="2">
	<thead>
	  <row>
	    <entry>Type</entry>
	    <entry>Description</entry>
	  </row>
	</thead>
	<tbody>
	  <row>
	    <entry><literal>index</literal></entry>
	    <entry>This is a signed integral type. It is used to
specify degenerate dimensions.</entry>
	  </row>
	  <row>
	    <entry><literal>size_type</literal></entry>
	    <entry>This is an unsigned integral type. It is used to
report the size of the range an <literal>index_range</literal> 
represents.</entry>
	  </row>
	      <row>
		<entry>
<literal>template gen_type::&lt;Dims,Ranges&gt;::type</literal></entry>
	<entry>This type generator names the result of 
<literal>Dims</literal> chained calls to
<literal>index_gen::operator[]</literal>.  The
<literal>Ranges</literal> parameter is determined by the number of
degenerate ranges specified (i.e. calls to
<literal>operator[](index)</literal>). Note that  
<classname>index_gen</classname> and
<classname>gen_type&lt;0,0&gt;::type</classname> are the same type.</entry>
	      </row>
	</tbody>
      </tgroup>
    </table>
    
      


    <table><title>Valid Expressions</title>
      <tgroup cols="3">
	<thead>
	  <row>
	    <entry>Expression</entry>
	    <entry>Return type</entry>
	    <entry>Semantics</entry>
	  </row>
	</thead>
	<tbody>
	  <row>
	    <entry><literal>index_gen()</literal></entry>
	    <entry><literal>gen_type&lt;0,0&gt;::type</literal></entry>
	    <entry>This constructs an <literal>index_gen</literal>
object. This object can then be used to generate tuples of
<literal>index_range</literal> values.</entry>
	  </row>

	      <row>
		<entry><literal>x[i]</literal></entry>
		<entry><literal>gen_type&lt;Dims+1,Ranges+1&gt;::type</literal>
</entry>
		<entry>Returns a new object containing all previous
<classname>index_range</classname> objects in addition to
<literal>i.</literal> Chained calls to
<function>operator[]</function> are the means by which
<classname>index_range</classname> objects are aggregated.</entry>
	      </row>
	      <row>
		<entry><literal>x[idx]</literal></entry>
		<entry><literal>gen_type&lt;Dims,Ranges+1&gt;::type</literal>
</entry>
	<entry>Returns a new object containing all previous
<classname>index_range</classname> objects in addition to a degenerate
range, <literal>index_range(idx,idx).</literal> Note that this is NOT
equivalent to <literal>x[index_range(idx,idx)].</literal>, which will
return an object of type
<literal>gen_type&lt;Dims+1,Ranges+1&gt;::type</literal>.
</entry>
	      </row>
	    </tbody>
	  </tgroup>
	</table>
      </sect3>
      
    </sect2>

<sect2>
<title>Models</title>

    <itemizedlist>
      <listitem> <literal>multi_array</literal> </listitem>
      <listitem> <literal>multi_array_ref</literal> </listitem>
      <listitem> <literal>const_multi_array_ref</literal> </listitem>
      <listitem> 
	<literal>template array_view&lt;Dims&gt;::type</literal> 
      </listitem>
      <listitem> 
	<literal>template const_array_view&lt;Dims&gt;::type</literal> 
      </listitem>
      <listitem>
	<literal>template subarray&lt;Dims&gt;::type</literal>
      </listitem>
      <listitem>
	<literal>template const_subarray&lt;Dims&gt;::type</literal>
      </listitem>
    </itemizedlist>
</sect2>

</sect1>