<?xml version="1.0" encoding="utf-8"?>
<!DOCTYPE library PUBLIC "-//Boost//DTD BoostBook XML V1.0//EN"
  "http://www.boost.org/tools/boostbook/dtd/boostbook.dtd" [
 <!ENTITY concepts SYSTEM "MultiArray.xml">
 <!ENTITY multi_array SYSTEM "multi_array.xml">
 <!ENTITY multi_array_ref SYSTEM "multi_array_ref.xml">
 <!ENTITY const_multi_array_ref SYSTEM "const_multi_array_ref.xml">
]>
<library name="MultiArray" dirname="multi_array" id="multi_array"
   xmlns:xi="http://www.w3.org/2001/XInclude"
   last-revision="$Date$">
  <libraryinfo>
    <author>
      <firstname>Ronald</firstname>
      <surname>Garcia</surname>
      <affiliation>
	<orgname>Indiana University</orgname>
	<orgdiv>Open Systems Lab</orgdiv>
      </affiliation>
    </author>
    <orgname>BOOST</orgname>
    <copyright>
      <year>2002</year>
      <holder>The Trustees of Indiana University</holder>
    </copyright>
       <librarypurpose>Multidimensional containers and adaptors for
       arrays of contiguous data</librarypurpose>
       <librarycategory name="category:math"/>
       <librarycategory name="category:containers"/>
  </libraryinfo>

<title>Boost.MultiArray Reference Manual</title>


<para>Boost.MultiArray is composed of several components.
The MultiArray concept defines a generic interface to multidimensional
containers.
<literal>multi_array</literal> is a general purpose container class
that models MultiArray. <literal>multi_array_ref</literal>
and <literal>const_multi_array_ref</literal> are adapter
classes. Using them, 
you can manipulate any block of contiguous data as though it were a
<literal>multi_array</literal>.
<literal>const_multi_array_ref</literal> differs from
<literal>multi_array_ref</literal> in that its elements cannot
be modified through its interface. Finally, several auxiliary classes are used
to create and specialize arrays and some global objects are defined as
part of the library interface.</para>

<sect1 id="synopsis">
<title>Library Synopsis</title>
    <para>To use Boost.MultiArray, you must include the header 
<filename>boost/multi_array.hpp</filename> in your source. This file
brings the following declarations into scope:</para>
<programlisting>
<![CDATA[namespace boost {
  
  namespace multi_array_types {
    typedef *unspecified* index;
    typedef *unspecified* size_type;
    typedef *unspecified* difference_type;
    typedef *unspecified* index_range;
    typedef *unspecified* extent_range;
    typedef *unspecified* index_gen;
    typedef *unspecified* extent_gen;
  }

  template <typename ValueType, 
            std::size_t NumDims, 
            typename Allocator = std::allocator<ValueType> >
  class multi_array;

  template <typename ValueType, 
            std::size_t NumDims>
  class multi_array_ref;

  template <typename ValueType, 
            std::size_t NumDims> 
  class const_multi_array_ref;

  multi_array_types::extent_gen extents;
  multi_array_types::index_gen  indices;

  template <typename Array, int N> class subarray_gen;
  template <typename Array, int N> class const_subarray_gen;
  template <typename Array, int N> class array_view_gen;
  template <typename Array, int N> class const_array_view_gen;

  class c_storage_order; 
  class fortran_storage_order;
  template <std::size_t NumDims> class general_storage_order;

}]]>
</programlisting>
</sect1>

&concepts;

<sect1 id="array_types">
<title>Array Components</title>
<para>
Boost.MultiArray defines an array class,
<literal>multi_array</literal>, and two adapter classes,
<literal>multi_array_ref</literal> and 
<literal>const_multi_array_ref</literal>. The three classes model 
MultiArray and so they share a lot of functionality.
<literal>multi_array_ref</literal> differs from
<literal>multi_array</literal> in that the
<literal>multi_array</literal> manages its own memory, while
<literal>multi_array_ref</literal> is passed a block of memory that it
expects to be externally managed.
<literal>const_multi_array_ref</literal> differs from
<literal>multi_array_ref</literal> in that the underlying elements it
adapts cannot be modified through its interface, though some array
properties, including the array shape and index bases, can be altered.
Functionality the classes have in common is described
below.
</para>

<formalpara>
<title>Note: Preconditions, Effects, and Implementation</title>
<para>
Throughout the following sections, small pieces of C++ code are
used to specify constraints such as preconditions, effects, and
postconditions.  These do not necessarily describe the underlying
implementation of array components; rather, they describe the 
expected input to and
behavior of the specified operations.  Failure to meet
preconditions results in undefined behavior. Not all effects
(i.e. copy constructors, etc.) must be mimicked exactly.  The code
snippets for effects intend to capture the essence of the described
operation. 
</para>
</formalpara>

<formalpara>
<title>Queries</title>

<variablelist>
<varlistentry>
<term><programlisting>element* data();
const element* data() const;</programlisting></term>
<listitem>
<para>This returns a pointer to the beginning of the
contiguous block that contains the array's data. If all dimensions of
the array are 0-indexed and stored in ascending order, this is 
equivalent to <literal>origin()</literal>. Note that
<literal>const_multi_array_ref</literal> only provides the const
version of this function.
</para>
</listitem>
</varlistentry>

<varlistentry>
<term><programlisting>element* origin();
const element* origin() const;</programlisting></term>
<listitem>
<para>This returns the origin element of the
<literal>multi_array</literal>. Note that
<literal>const_multi_array_ref</literal> only provides the const
version of this function. (Required by MultiArray)
</para>
</listitem>
</varlistentry>

<varlistentry>
<term><function>const index* index_bases();</function></term>
<listitem>
<para>This returns the index bases for the
<literal>multi_array</literal>. (Required by MultiArray)
</para>
</listitem>
</varlistentry>

<varlistentry>
<term><function>const index* strides();</function></term>
<listitem>
<para>This returns the strides for the
<literal>multi_array</literal>. (Required by MultiArray)
</para>
</listitem>
</varlistentry>

<varlistentry>
<term><function>const size_type* shape();</function></term>
<listitem>
<para>This returns the shape of the
<literal>multi_array</literal>. (Required by MultiArray)
</para>
</listitem>
</varlistentry>
</variablelist>

</formalpara>

<formalpara>
<title>Comparators</title>
<variablelist>
<varlistentry>
<term><programlisting><![CDATA[
bool operator==(const *array-type*& rhs);
bool operator!=(const *array-type*& rhs);
bool operator<(const *array-type*& rhs);
bool operator>(const *array-type*& rhs);
bool operator>=(const *array-type*& rhs);
bool operator<=(const *array-type*& rhs);]]></programlisting></term>

<listitem>
<para>Each comparator executes a lexicographical compare over
the value types of the two arrays.
(Required by MultiArray)
</para>
<formalpara>
<title>Preconditions</title>
<para><literal>element</literal> must support the
comparator corresponding to that called on
<literal>multi_array</literal>.</para>
</formalpara>

<formalpara>
<title>Complexity</title>
<para>O(<literal>num_elements()</literal>).</para>
</formalpara>

</listitem>
</varlistentry>

</variablelist>
</formalpara>

<formalpara>
<title>Modifiers</title>

<variablelist>

<varlistentry>
<term>
<programlisting>
<![CDATA[
template <typename SizeList>
void reshape(const SizeList& sizes)
]]>
</programlisting>
</term>

<listitem>
<para>This changes the shape of the <literal>multi_array</literal>.  The
number of elements and the index bases remain the same, but the number
of values at each level of the nested container hierarchy may
change.</para>

<formalpara><title><literal>SizeList</literal> Requirements</title>
<para><literal>SizeList</literal> must model
<ulink url="../../utility/Collection.html">Collection</ulink>.</para>
</formalpara>

<formalpara><title>Preconditions</title>
<para>
<programlisting>
<![CDATA[std::accumulate(sizes.begin(),sizes.end(),size_type(1),std::times<size_type>()) == this->num_elements();
sizes.size() == NumDims;]]>
</programlisting></para>
</formalpara>


<formalpara><title>Postconditions</title>
<para>
<literal>std::equal(sizes.begin(),sizes.end(),this->shape) == true;</literal>
</para>
</formalpara>
</listitem>
</varlistentry>

<varlistentry>
<term>
<programlisting>
<![CDATA[
template <typename BaseList>
void reindex(const BaseList& values);
]]>
</programlisting>
</term>
<listitem>
<para>This changes the index bases of the <literal>multi_array</literal> to
correspond to the the values in <literal>values</literal>.</para>

<formalpara>
<title><literal>BaseList</literal> Requirements</title>
<para><literal>BaseList</literal> must model
<ulink url="../../utility/Collection.html">Collection</ulink>.</para>
</formalpara>

<formalpara>
<title>Preconditions</title>
<para><literal>values.size() == NumDims;</literal></para>
</formalpara>


<formalpara>
<title>Postconditions</title>
<para><literal>std::equal(values.begin(),values.end(),this->index_bases());
</literal></para>
</formalpara>
</listitem>
</varlistentry>

<varlistentry>
<term>
<programlisting>
<![CDATA[
void reindex(index value);
]]>
</programlisting>
</term>
<listitem>
<para>This changes the index bases of all dimensions of the
<literal>multi_array</literal> to <literal>value</literal>.</para>

<formalpara>
<title>Postconditions</title>
<para>
<programlisting>
<![CDATA[
std::count_if(this->index_bases(),this->index_bases()+this->num_dimensions(),
              std::bind_2nd(std::equal_to<index>(),value)) == 
              this->num_dimensions();
]]>
</programlisting>
</para>
</formalpara>
</listitem>
</varlistentry>

</variablelist>
</formalpara>

&multi_array;
&multi_array_ref;
&const_multi_array_ref;

</sect1>


<sect1 id="auxiliary">
    <title>Auxiliary Components</title>

<sect2 id="multi_array_types">
<title><literal>multi_array_types</literal></title>

<programlisting>
<![CDATA[namespace multi_array_types {
  typedef *unspecified* index;
  typedef *unspecified* size_type;
  typedef *unspecified* difference_type;
  typedef *unspecified* index_range;
  typedef *unspecified* extent_range;
  typedef *unspecified* index_gen;
  typedef *unspecified* extent_gen;
}]]>
</programlisting>

<para>Namespace <literal>multi_array_types</literal> defines types
associated with <literal>multi_array</literal>,
<literal>multi_array_ref</literal>, and
<literal>const_multi_array_ref</literal> that are not
dependent upon template parameters.  These types find common use with
all Boost.Multiarray components.  They are defined
in a namespace from which they can be accessed conveniently.
With the exception of <literal>extent_gen</literal> and 
<literal>extent_range</literal>, these types fulfill the roles of the
same name required by MultiArray and are described in its
concept definition.  <literal>extent_gen</literal> and
<literal>extent_range</literal> are described below.
</para>
</sect2>


<sect2 id="extent_range">
    <title><classname>extent_range</classname></title>

<para><classname>extent_range</classname> objects define half open
intervals.  They provide shape and index base information to
<literal>multi_array</literal>, <literal>multi_array_ref</literal>,
 and <literal>const_multi_array_ref</literal> constructors.
<classname>extent_range</classname>s are passed in
aggregate to an array constructor (see
<classname>extent_gen</classname> for more details).
</para>

<formalpara>
	<title>Synopsis</title>
<programlisting><![CDATA[
class extent_range {
public:
  typedef multi_array_types::index      index;
  typedef multi_array_types::size_type  size_type;

  // Structors
  extent_range(index start, index finish);
  extent_range(index finish);
  ~extent_range();

  // Queries
  index start();
  index finish();
  size_type size();
};]]></programlisting>
</formalpara>

      <formalpara>
	<title>Model Of</title>
	<para>DefaultConstructible,CopyConstructible</para>
      </formalpara>

<formalpara><title>Methods and Types</title>
<variablelist>
<varlistentry>
<term><function>extent_range(index start, index finish)</function></term>
<listitem>
<para>  This constructor defines the half open interval
<literal>[start,finish)</literal>. The expression
<literal>finish</literal> must be greater than <literal>start</literal>.
</para>
</listitem>
</varlistentry>

<varlistentry><term><function>extent_range(index finish)</function></term>
<listitem>
<para>This constructor defines the half open interval
<literal>[0,finish)</literal>. The value of <literal>finish</literal>
must be positive.</para> 
</listitem>
</varlistentry>

<varlistentry><term><function>index start()</function></term>
<listitem>
<para>This function returns the first index represented by the range</para>
</listitem>
</varlistentry>

<varlistentry><term><function>index finish()</function></term>
<listitem>
<para>This function returns the upper boundary value of the half-open
interval.  Note that the range does not include this value.</para>
</listitem>
</varlistentry>

<varlistentry>
<term><function>size_type size()</function></term>
<listitem>
<para>This function returns the size of the specified range. It is
equivalent to <literal>finish()-start()</literal>.</para>
</listitem>
</varlistentry>

</variablelist>
</formalpara>
</sect2>

<sect2 id="extent_gen">
    <title><classname>extent_gen</classname></title>
    <para>The <classname>extent_gen</classname> class defines an
interface for aggregating array shape and indexing information to be
passed to a <literal>multi_array</literal>, 
<literal>multi_array_ref</literal>, or <literal>const_multi_array_ref</literal>
constructor. Its interface mimics 
 the syntax used to declare built-in array types
in C++. For example, while a 3-dimensional array of 
<classname>int</classname> values in C++ would be
declared as:
<programlisting>int A[3][4][5],</programlisting>
a similar <classname>multi_array</classname> would be declared:
<programlisting>multi_array&lt;int,3&gt; A(extents[3][4][5]).</programlisting>
</para>

<formalpara><title>Synopsis</title>
<programlisting><![CDATA[
template <std::size_t NumRanges>
class *implementation_defined* {
public:
  typedef multi_array_types::index index;
  typedef multi_array_types::size_type size_type;

  template <std::size_t NumRanges> class gen_type;

  gen_type<NumRanges+1>::type  operator[](const range& a_range) const;
  gen_type<NumRanges+1>::type  operator[](index idx) const;
};

typedef *implementation_defined*<0> extent_gen;
]]></programlisting>
</formalpara>

<formalpara><title>Methods and Types</title>
<variablelist>
<varlistentry>
<term><function>template gen_type&lt;Ranges&gt;::type</function></term>
<listitem>
<para>This type generator is used to specify the result of 
<literal>Ranges</literal> chained calls to
<literal>extent_gen::operator[].</literal> The types
<classname>extent_gen</classname> and
<classname>gen_type&lt;0&gt;::type</classname> are the same.</para>
</listitem>
</varlistentry>

<varlistentry>
<term><function>gen_type&lt;NumRanges+1&gt;::type  
operator[](const extent_range&amp; a_range) const;</function></term>
<listitem>
<para>This function returns a new object containing all previous
<classname>extent_range</classname> objects in addition to
<literal>a_range.</literal> <classname>extent_range</classname>
objects are aggregated by chained calls to
<function>operator[]</function>.</para>
</listitem>
</varlistentry>

<varlistentry>
<term><function>gen_type&lt;NumRanges+1&gt;::type
operator[](index idx) const;</function></term>
<listitem>
<para>This function returns a new object containing all previous
<classname>extent_range</classname> objects in addition to
<literal>extent_range(0,idx).</literal> This function gives the array
constructors a similar syntax to traditional C multidimensional array
declaration.</para>
</listitem>
</varlistentry>

</variablelist>
</formalpara>      
</sect2>
  
<sect2>
    <title>Global Objects</title>
    <para>For syntactic convenience, Boost.MultiArray defines two 
global objects as part of its
interface.  These objects play the role of object generators;
expressions involving them create other objects of interest.
</para>

    <para> Under some circumstances, the two global objects may be
considered excessive overhead.  Their construction can be prevented by
defining the preprocessor symbol
<literal>BOOST_MULTI_ARRAY_NO_GENERATORS</literal> before including
<filename>boost/multi_array.hpp.</filename></para>

<sect3 id="extents">
<title><literal>extents</literal></title>

<programlisting>
<![CDATA[namespace boost {
  multi_array_base::extent_gen extents;
}]]>
</programlisting>

    <para>Boost.MultiArray's array classes use the
<literal>extents</literal> global object to specify 
array shape during their construction. 
For example,
a 3 by 3 by 3 <classname>multi_array</classname> is constructed as follows:
<programlisting>multi_array&lt;int,3&gt; A(extents[3][3][3]);</programlisting>
The same array could also be created by explicitly declaring an <literal>extent_gen</literal> 
object locally,, but the global object makes this declaration unnecessary.  
</para>
</sect3>

<sect3 id="indices">
<title><literal>indices</literal></title>

<programlisting>
<![CDATA[namespace boost {
  multi_array_base::index_gen  indices;
}]]>
</programlisting>

      <para>The MultiArray concept specifies an
<literal>index_gen</literal> associated type that is used to
create views.
<literal>indices</literal> is a global object that serves the role of
<literal>index_gen</literal> for all array components provided by this
library and their associated subarrays and views. 
</para>
<para>For example, using the <literal>indices</literal> object,
a view of an array <literal>A</literal> is constructed as follows:
<programlisting>
A[indices[index_range(0,5)][2][index_range(2,4)]];
</programlisting>
</para>
</sect3>
</sect2>

<sect2 id="generators">
<title>View and SubArray Generators</title>
<para>
Boost.MultiArray provides traits classes, <literal>subarray_gen</literal>,
<literal>const_subarray_gen</literal>,
<literal>array_view_gen</literal>,
and <literal>const_array_view_gen</literal>, for naming of
array associated types within function templates.  
In general this is no more convenient to use than the nested 
type generators, but the library author found that some C++ compilers do not 
properly handle templates nested within function template parameter types. 
These generators constitute a workaround for this deficit.  
The following code snippet illustrates
the correspondence between the <literal>array_view_gen</literal>
traits class and the <literal>array_view</literal> type associated to
an array:

<programlisting>
template &lt;typename Array&gt;
void my_function() {
  typedef typename Array::template array_view&lt;3&gt;::type view1_t;
  typedef typename boost::array_view_gen&lt;Array,3&gt;::type view2_t;
  // ...
}
</programlisting>

In the above example, <literal>view1_t</literal> and
<literal>view2_t</literal> have the same type.
</para>
</sect2>


<sect2 id="memory_layout">
<title>Memory Layout Specifiers</title>
<para>
While a multidimensional array represents a hierarchy of containers of
elements, at some point the elements must be laid out in
memory.  As a result, a single multidimensional array 
can be represented in memory more than one way.
</para>

<para>For example, consider the two dimensional array shown below in
matrix notation:

<graphic fileref="matrix.gif"/>

Here is how the above array is expressed in C++:
<programlisting>
int a[3][4] = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 };
</programlisting>
This is an example of row-major storage, where elements of each row
are stored contiguously.  

While C++ transparently handles accessing elements of an array, you
can also manage the array and its indexing manually.  One way that 
this may be expressed in memory is as follows:
<programlisting>
int a[] = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 };
int s[] = { 4, 1 };
</programlisting>

With the latter declaration of <literal>a</literal> and 
strides <literal>s</literal>, element <literal>a(i,j)</literal>
of the array can be
accessed using the expression 
<programlisting>*a+i*s[0]+j*s[1]</programlisting>.
</para>

<para>The same two dimensional array could be laid out by column as follows:

<programlisting>
int a[] = { 0, 4, 8, 1, 5, 9, 2, 6, 10, 3, 7, 11 };
int s[] = { 3, 1 };
</programlisting>
Notice that the strides here are different. As a result,
The expression given above to access values will work with this pair
of data and strides as well.
</para>

<para>In addition to dimension order, it is also possible to
store any dimension in descending order. For example, returning to the 
first example, the first dimension of the example array, the 
rows,  could be stored in 
reverse, resulting in the following:

<programlisting>
int data[] = { 8, 9, 10, 11, 4, 5, 6, 7, 0, 1, 2, 3 };
int *a = data + 8;
int s[] = { -4, 1 };
</programlisting>

Note that in this example <literal>a</literal> must be explicitly set
to the origin. In the previous examples, the
first element stored in memory was the origin; here this is no longer
the case. 
</para>

<para>
Alternatively, the second dimension, or the columns, could be reversed
and the rows stored in ascending order:

<programlisting>
int data[] = { 3, 2, 1, 0,  7, 6, 5, 4, 11, 10, 9, 8 };
int *a = data + 3;
int s[] = { 4, -1 };
</programlisting>
</para>

<para>
Finally, both dimensions could be stored in descending order:

<programlisting>
int data[] = {11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0};
int *a = data + 11;
int s[] = { -4, -1 };
</programlisting>
<literal>
</literal>
</para>

<para>
All of the above arrays are equivalent. The expression
given above for <literal>a(i,j)</literal> will yield the same value
regardless of the memory layout.

Boost.MultiArray arrays can be created with customized storage
parameters as described above. Thus, existing data can be adapted
(with <literal>multi_array_ref</literal> or
<literal>const_multi_array_ref</literal>) as suited to the array
abstraction.  A common usage of this feature would be to wrap arrays
that must interoperate with Fortran routines so they can be
manipulated naturally at both the C++ and Fortran levels. The
following sections describe the Boost.MultiArray components used to
specify memory layout.
</para>

<sect3 id="c_storage_order">
<title><literal>c_storage_order</literal></title>
<programlisting>
<![CDATA[class c_storage_order {
  c_storage_order();
};]]>
</programlisting>

<para><literal>c_storage_order</literal> is used to specify that an
array should store its elements using the same layout as that used by
primitive C++ multidimensional arrays, that is, from last dimension
to first. This is the default storage order for the arrays provided by
this library.</para>
</sect3>

<sect3 id="fortran_storage_order">
<title><literal>fortran_storage_order</literal></title>
<programlisting>
<![CDATA[class fortran_storage_order {
  fortran_storage_order();
};]]>
</programlisting>

<para><literal>fortran_storage_order</literal> is used to specify that
an array should store its elements using the same memory layout as a
Fortran multidimensional array would, that is, from first dimension to
last.</para>
</sect3>

<sect3 id="general_storage_order">
<title><literal>general_storage_order</literal></title>
<programlisting>
<![CDATA[template <std::size_t NumDims> 
class general_storage_order {

  template <typename OrderingIter, typename AscendingIter>
  general_storage_order(OrderingIter ordering, AscendingIter ascending);
};]]>
</programlisting>

<para><literal>general_storage_order</literal> allows the user to
specify an arbitrary memory layout for the contents of an array.  The
constructed object is passed to the array constructor in order to
specify storage order.</para>

<para>
<literal>OrderingIter</literal> and <literal>AscendingIter</literal>
must model the <literal>InputIterator</literal> concept.  Both
iterators must refer to a range of <literal>NumDims</literal>
elements.  <literal>AscendingIter</literal> points to objects
convertible to <literal>bool</literal>.  A value of
<literal>true</literal> means that a dimension is stored in ascending
order while <literal>false</literal> means that a dimension is stored
in descending order.  <literal>OrderingIter</literal> specifies the
order in which dimensions are stored.
</para>  

</sect3>
</sect2>

<sect2 id="range_checking">
<title>Range Checking</title>
<para>
By default, the array access methods <literal>operator()</literal> and
<literal>operator[]</literal> perform range
checking.  If a supplied index is out of the range defined for an
array, an assertion will abort the program.  To disable range
checking (for performance reasons in production releases), define
the <literal>BOOST_DISABLE_ASSERTS</literal> preprocessor macro prior to
including multi_array.hpp in an application.
</para>

</sect2>
</sect1>


</library>