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  == Copyright 2002 The Trustees of Indiana University.

  == Use, modification and distribution is subject to the Boost Software 
  == License, Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at
  == http://www.boost.org/LICENSE_1_0.txt)

  ==  Boost.MultiArray Library
  ==  Authors: Ronald Garcia
  ==           Jeremy Siek
  ==           Andrew Lumsdaine
  ==  See http://www.boost.org/libs/multi_array for documentation.
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<head>
  <title>The Boost Multidimensional Array Library (Boost.MultiArray)</title>
</head>

<body>

<h1>
   <img src="../../../boost.png" alt="boost logo"
    width="277" align="middle" height="86">
   <br>The Boost Multidimensional Array Library 
   <br>(Boost.MultiArray)
</h1>

<h2>Synopsis</h2>

<p>
The Boost Multidimensional Array Library provides a class template for
multidimensional arrays, as well as  semantically equivalent
adaptors for arrays of contiguous data. The classes in this library
implement a common interface, formalized as a generic programming
concept. The interface design is in line with the precedent set by the
C++ Standard Library containers.  Boost MultiArray is a more efficient
and convenient way to express N-dimensional arrays than existing
alternatives (especially the
<tt>std::vector&lt;std::vector&lt;...&gt;&gt;</tt> formulation 
of N-dimensional arrays).  The arrays provided by the library may be
accessed using the familiar syntax of native C++ arrays.  Additional
features, such as resizing, reshaping, and creating views are
available (and described below).  


<h2>Table of Contents</h2>

<ol>
      <li><a href="#sec_introduction">Introduction</a>

      <li><a href="#sec_example">Short Example</a>

      <li><a href="#sec_components">MultiArray Components</a>

      <li><a href="#sec_assignment">Construction and Assignment</a>

      <li><a href="#sec_generators">Array View and Subarray Type Generators</a>

      <li><a href="#sec_dimensions">Specifying Array Dimensions</a>

      <li><a href="#sec_access">Accessing Elements</a>

      <li><a href="#sec_views">Creating Views</a>

      <li><a href="#sec_storage">Storage Ordering</a>

      <li><a href="#sec_base">Setting the Array Base</a>

      <li><a href="#sec_reshape">Changing an Array's Shape</a>

      <li><a href="#sec_resize">Resizing an Array</a>

      <li><a href="#sec_concepts">MultiArray Concept</a>

      <li><a href="#sec_testcases">Test Cases</a>

      <li><a href="#sec_related">Related Work</a>
      <li><a href="#sec_credits">Credits</a>
</ol>  


<a name="sec_introduction"></a>
<h2>Introduction</h2>

<p>
The C++ standard library provides several generic containers, but it
does not provide any multidimensional array types.  The
<tt>std::vector</tt> class template can be used to implement
N-dimensional arrays, for example expressing a 2-dimensional array of
<tt>double</tt> elements using the type
<tt>std::vector&lt;std::vector&lt;double&gt;&gt;</tt>, but the
resulting interface is unwieldy and the memory overhead can be quite
high. Native C++ arrays (i.e. <tt>int arr[2][2][2];</tt>) do not
immediately interoperate well with the C++ Standard Library, and they
also lose information at function call boundaries (specifically the
extent of the last dimension).  Finally, a dynamically allocated
contiguous block of elements can be treated as an array, though this
method requires manual bookkeeping that is error prone and obfuscates
the intent of the programmer.
</p>

<p>
The Boost MultiArray library enhances the C++ standard containers with
versatile multi-dimensional array abstractions. It includes a general
array class template and native array adaptors that support idiomatic
array operations and interoperate with C++ Standard Library containers
and algorithms.  The arrays share a common interface, expressed as a
generic programming in terms of which generic array algorithms can be
implemented.
</p>

<p>
This document is meant to provide an introductory tutorial and user's
guide for the most basic and common usage patterns of MultiArray
components.  The <a href="./reference.html">reference manual</a>
provides more complete and formal documentation of library features.
</p>

<a name="sec_example"></a>
<h2>Short Example</h2>
What follows is a brief example of the use of <tt>multi_array</tt>:

<blockquote>
<pre>
#include "boost/multi_array.hpp"
#include &lt;cassert&gt;

int 
main () {
  // Create a 3D array that is 3 x 4 x 2
  typedef boost::multi_array&lt;double, 3&gt; array_type;
  typedef array_type::index index;
  array_type A(boost::extents[3][4][2]);

  // Assign values to the elements
  int values = 0;
  for(index i = 0; i != 3; ++i) 
    for(index j = 0; j != 4; ++j)
      for(index k = 0; k != 2; ++k)
        A[i][j][k] = values++;

  // Verify values
  int verify = 0;
  for(index i = 0; i != 3; ++i) 
    for(index j = 0; j != 4; ++j)
      for(index k = 0; k != 2; ++k)
        assert(A[i][j][k] == verify++);

  return 0;
}
</pre>
</blockquote>

<a name="sec_components"></a>
<h2>MultiArray Components</h2>

Boost.MultiArray's implementation (boost/multi_array.hpp) provides three user-level class templates:

<ol>
      <li><a href="./reference.html#multi_array"><tt>multi_array</tt></a>, 

      <li><a href="./reference.html#multi_array_ref"><tt>multi_array_ref</tt></a>, and 

      <li><a href="./reference.html#const_multi_array_ref"><tt>const_multi_array_ref</tt></a>  
</ol>

<tt>multi_array</tt> is a container template.  When instantiated, it
allocates space for the number of elements corresponding to the
dimensions specified at construction time.  A <tt>multi_array</tt> may
also be default constructed and resized as needed.

<p>
<tt>multi_array_ref</tt> adapts an existing array of data to provide
the <tt>multi_array</tt> interface. <tt>multi_array_ref</tt> does not own the
data passed to it.

<p>
<tt>const_multi_array_ref</tt> is similar to <tt>multi_array_ref</tt>
but guarantees that the contents of the array are immutable. It can
thus wrap pointers of type <i>T const*</i>. 

<p>
The three components exhibit very similar behavior.  Aside from
constructor parameters, <tt>multi_array</tt> and
<tt>multi_array_ref</tt> export the same interface.
<tt>const_multi_array_ref</tt> provides only the constness-preserving
portions of the <tt>multi_array_ref</tt> interface.

<a name="sec_assignment"></a>
<h2>Construction and Assignment</h2>
<p>Each of the array types - 
<a href="./reference.html#multi_array"><tt>multi_array</tt></a>, 
<a href="./reference.html#multi_array_ref"><tt>multi_array_ref</tt></a>, and
<a href="./reference.html#const_multi_array_ref"><tt>const_multi_array_ref</tt></a> - 
provides a specialized set of constructors.  For further information,
consult their reference pages.

<p>All of the non-const array types in this library provide assignment 
operators<tt>operator=()</tt>. Each of the array types <tt>multi_array</tt>,
      <tt>multi_array_ref</tt>, <tt>subarray</tt>, and
      <tt>array_view</tt> can be assigned from any 
of the others, so long as their shapes match. The 
      const variants, <tt>const_multi_array_ref</tt>, 
<tt>const_subarray</tt>, and <tt>const_array_view</tt>, can be the
      source of a copy to an array with matching shape.
Assignment results in a deep (element by element) copy of the data 
contained within an array.  

<a name="sec_generators"></a>
<h2>Array View and Subarray Type Generators</h2>
In some situations, the use of nested generators for array_view and
subarray types is inconvenient.  For example, inside a
function template parameterized upon array type, the extra
"template" keywords can be obfuscating. More likely though, some
compilers cannot handle templates nested within template parameters.
For this reason the type generators, <tt>subarray_gen</tt>,
<tt>const_subarray_gen</tt>, <tt>array_view_gen</tt>, and
<tt>const_array_view_gen</tt> are provided. Thus, the two typedefs 
in the following example result in the same type:
<blockquote>
<pre>
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;
  // ...
}
</pre>
</blockquote>

<a name="sec_dimensions"></a>
<h2>Specifying Array Dimensions</h2>
When creating most of the Boost.MultiArray components, it is necessary
to specify both the number of dimensions and the extent of each
(<tt>boost::multi_array</tt> also provides a default constructor).
Though the number of dimensions is always specified as a template
parameter, two separate mechanisms have been provided to specify the
extent of each.
<p>The first method involves passing a 
<a href="../../utility/Collection.html">
Collection</a> of extents to a
constructor, most commonly a <tt>boost::array</tt>.  The constructor
will retrieve the beginning iterator from the container and retrieve N 
elements, corresponding to extents for the N dimensions.  This is
useful for writing dimension-independent code.
<h3>Example</h3>
<blockquote>
<pre>
  typedef boost::multi_array&lt;double, 3&gt; array_type;
  boost::array&lt;array_type::index, 3&gt; shape = {{ 3, 4, 2 }};
  array_type A(shape);
</pre>
</blockquote>

<p>The second method involves passing the constructor an <tt>extent_gen</tt>
object, specifying the matrix dimensions. The <tt>extent_gen</tt> type
  is defined in the <tt>multi_array_types</tt> namespace and as a
  member of every array type, but by default, the library constructs a
global <tt>extent_gen</tt> object <tt>boost::extents</tt>.  In case of
concern about memory used by these objects, defining 
<tt>BOOST_MULTI_ARRAY_NO_GENERATORS</tt> before including the library
header inhibits its construction.

<h3>Example</h3>
<blockquote>
<pre>
  typedef boost::multi_array&lt;double, 3&gt; array_type;
  array_type A(boost::extents[3][4][2]);
</pre>
</blockquote>

<a name="sec_access"></a>
<h2>Accessing Elements</h2>
The Boost.MultiArray components provide two ways of accessing
specific elements within a container.  The first uses the traditional
C array notation, provided by <tt>operator[]</tt>.
<h3>Example</h3>
<blockquote>
<pre>
  typedef boost::multi_array&lt;double, 3&gt; array_type;
  array_type A(boost::extents[3][4][2]);
  A[0][0][0] = 3.14;
  assert(A[0][0][0] == 3.14);
</pre>
</blockquote>

<p> The second method involves passing a
<a href="../../utility/Collection.html">
Collection</a> of indices to <tt>operator()</tt>.  N indices will be retrieved
from the Collection for the N dimensions of the container.
<h3>Example</h3>
<blockquote>
<pre>
  typedef boost::multi_array&lt;double, 3&gt; array_type;
  array_type A(boost::extents[3][4][2]);
  boost::array&lt;array_type::index,3&gt; idx = {{0,0,0}};
  A(idx) = 3.14;
  assert(A(idx) == 3.14);
</pre>
</blockquote>
This can be useful for writing dimension-independent code, and under
some compilers may yield higher performance than <tt>operator[].</tt>

<p>
By default, both of the above element access methods 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 <tt>BOOST_DISABLE_ASSERTS</tt> preprocessor macro prior to
including multi_array.hpp in your application.

<a name="sec_views"></a>
<h2>Creating Views</h2>
Boost.MultiArray provides the facilities for creating a sub-view of an 
already existing array component.  It allows you to create a sub-view that
retains the same number of dimensions as the original array or one
that has less dimensions than the original as well.

<p>Sub-view creation occurs by placing a call to operator[], passing
it an <tt>index_gen</tt> type.  The <tt>index_gen</tt> is populated by
passing <tt>index_range</tt> objects to its <tt>operator[]</tt>.
The <tt>index_range</tt> and <tt>index_gen</tt> types are defined in
the <tt>multi_array_types</tt> namespace and as nested members of
every array type.  Similar to <tt>boost::extents</tt>, the library by
default constructs the object <tt>boost::indices</tt>.  You can
suppress this object by
defining <tt>BOOST_MULTI_ARRAY_NO_GENERATORS</tt> before including the
library header. A simple sub-view creation example follows.

<h3>Example</h3>
<blockquote>
<pre>
  // myarray = 2 x 3 x 4

  //
  // array_view dims: [base,bound) (dimension striding default = 1)
  // dim 0: [0,2) 
  // dim 1: [1,3) 
  // dim 2: [0,4) (strided by 2), 
  // 

  typedef boost::multi_array_types::index_range range;
  // OR typedef array_type::index_range range;
  array_type::array_view&lt;3&gt;::type myview =
    myarray[ boost::indices[range(0,2)][range(1,3)][range(0,4,2)] ];

  for (array_type::index i = 0; i != 2; ++i)
    for (array_type::index j = 0; j != 2; ++j)
      for (array_type::index k = 0; k != 2; ++k) 
	assert(myview[i][j][k] == myarray[i][j+1][k*2]);
</pre>
</blockquote>


<p>By passing an integral value to the index_gen, one may create a
subview with fewer dimensions than the original array component (also
called slicing).
<h3>Example</h3>
<blockquote>
<pre>
  // myarray = 2 x 3 x 4

  //
  // array_view dims:
  // [base,stride,bound)
  // [0,1,2), 1, [0,2,4) 
  // 

  typedef boost::multi_array_types::index_range range;
  array_type::index_gen indices;
  array_type::array_view&lt;2&gt;::type myview =
    myarray[ indices[range(0,2)][1][range(0,4,2)] ];

  for (array_type::index i = 0; i != 2; ++i)
    for (array_type::index j = 0; j != 2; ++j)
	assert(myview[i][j] == myarray[i][1][j*2]);
</pre>
</blockquote>

<h3>More on <tt>index_range</tt></h3>
The <tt>index_range</tt> type provides several methods of specifying
ranges for subview generation.  Here are a few range instantiations
that specify the same range.
<h3>Example</h3>
<blockquote>
<pre>
  // [base,stride,bound)
  // [0,2,4) 

  typedef boost::multi_array_types::index_range range;
  range a_range;
  a_range = range(0,4,2);
  a_range = range().start(0).finish(4).stride(2);
  a_range = range().start(0).stride(2).finish(4);
  a_range = 0 &lt;= range().stride(2) &lt; 4;
  a_range = 0 &lt;= range().stride(2) &lt;= 3;
</pre>
</blockquote>

An <tt>index_range</tt> object passed to a slicing operation will
inherit its start and/or finish value from the array being sliced if
you do not supply one.  This conveniently prevents you from having to
know the bounds of the array dimension in certain cases. For example,
the default-constructed range will take the full extent of the
dimension it is used to specify.

<h3>Example</h3>
<blockquote>
<pre>
  typedef boost::multi_array_types::index_range range;
  range a_range;

  // All elements in this dimension
  a_range = range(); 

  // indices i where 3 &lt;= i
  a_range = range().start(3) 
  a_range = 3 &lt;= range();
  a_range = 2 &lt; range();

  // indices i where i &lt; 7
  a_range = range().finish(7)
  a_range = range() &lt; 7;
  a_range = range() &lt;= 6;
</pre>
</blockquote>

The following example slicing operations exhibit some of the
alternatives shown above
<blockquote>
<pre>
    // take all of dimension 1
    // take i &lt; 5 for dimension 2
    // take 4 &lt;= j &lt;= 7 for dimension 3 with stride 2
    myarray[ boost::indices[range()][range() &lt; 5 ][4 &lt;= range().stride(2) &lt;= 7] ];
</pre>
</blockquote>

<a name="sec_storage"></a>
<h2>Storage Ordering</h2>
Each array class provides constructors that accept a storage ordering 
parameter. This is most
useful when interfacing with legacy codes that require an ordering
different from standard C, such as FORTRAN. The possibilities are 
<tt>c_storage_order</tt>, <tt>fortran_storage_order</tt>, and 
<tt>general_storage_order</tt>. 

<p><tt>c_storage_order</tt>, which is the default, will store elements
in memory in the same order as a C array would, that is, the
dimensions are stored from last to first.

<p><tt>fortran_storage_order</tt> will store elements in memory in the same order
as FORTRAN would: from the first dimension to
the last. Note that with use of this parameter, the array
indices will remain zero-based.
<h3>Example</h3>
<blockquote>
<pre>
  typedef boost::multi_array&lt;double,3&gt; array_type;
  array_type A(boost::extents[3][4][2],boost::fortran_storage_order); 
  call_fortran_function(A.data());
</pre>
</blockquote>

<p><tt>general_storage_order</tt> allows one to customize both the order in
which dimensions are stored in memory and whether dimensions are
stored in ascending or descending order.
<h3>Example</h3>
<blockquote>
<pre>
  typedef boost::general_storage_order&lt;3&gt; storage;
  typedef boost::multi_array&lt;int,3&gt; array_type;
 
  // Store last dimension, then first, then middle
  array_type::size_type ordering[] = {2,0,1};

  // Store the first dimension(dimension 0) in descending order 
  bool ascending[] = {false,true,true};

  array_type A(extents[3][4][2],storage(ordering,ascending)); 
</pre>
</blockquote>


<a name="sec_base"></a>
<h2>Setting The Array Base</h2>
In some situations, it may be inconvenient or awkward to use an 
array that is zero-based.
the Boost.MultiArray components provide two facilities for changing the
bases of an array.  One may specify a pair of range values, with
the <tt>extent_range</tt> type, to
the <tt>extent_gen</tt> constructor in order to set the base value. 

<h3>Example</h3>
<blockquote>
<pre>
  typedef boost::multi_array&lt;double, 3&gt; array_type;
  typedef boost::multi_array_types::extent_range range;
  // OR typedef array_type::extent_range range;

  array_type::extent_gen extents;
 
  // dimension 0: 0-based
  // dimension 1: 1-based
  // dimension 2: -1 - based
  array_type A(extents[2][range(1,4)][range(-1,3)]);
</pre>
</blockquote>

<p>
An alternative is to first construct the array normally then 
reset the bases.  To set all bases to the same value, use the
<tt>reindex</tt> member function, passing it a single new index value.
<h3>Example</h3>
<blockquote>
<pre>
  typedef boost::multi_array&lt;double, 3&gt; array_type;

  array_type::extent_gen extents;
 
  array_type A(extents[2][3][4]);
  // change to 1-based
  A.reindex(1)
</pre>
</blockquote>

<p>
An alternative is to set each base separately using the
<tt>reindex</tt> member function, passing it a Collection of index bases.
<h3>Example</h3>
<blockquote>
<pre>
  typedef boost::multi_array&lt;double, 3&gt; array_type;

  array_type::extent_gen extents;
 
  // dimension 0: 0-based
  // dimension 1: 1-based
  // dimension 2: (-1)-based
  array_type A(extents[2][3][4]);
  boost::array&lt;array_type::index,ndims&gt; bases = {{0, 1, -1}};       
  A.reindex(bases);
</pre>
</blockquote>


<a name="sec_reshape"></a>
<h2>Changing an Array's Shape</h2>
The Boost.MultiArray arrays provide a reshape operation.  While the
number of dimensions must remain the same, the shape of the array may 
change so long as the total number of 
elements contained remains the same.
<h3>Example</h3>
<blockquote>
<pre>
  typedef boost::multi_array&lt;double, 3&gt; array_type;

  array_type::extent_gen extents;
  array_type A(extents[2][3][4]);
  boost::array&lt;array_type::index,ndims&gt; dims = {{4, 3, 2}};       
  A.reshape(dims);
</pre>
</blockquote>

<p>
Note that reshaping an array does not affect the indexing.

<a name="sec_resize"></a>
<h2>Resizing an Array</h2>

The <tt>boost::multi_array</tt> class provides an element-preserving
resize operation.  The number of dimensions must remain the same, but
the extent of each dimension may be increased and decreased as
desired.  When an array is made strictly larger, the existing elements
will be preserved by copying them into the new underlying memory and
subsequently destructing the elements in the old underlying memory.
Any new elements in the array are default constructed.  However, if
the new array size shrinks some of the dimensions, some elements will
no longer be available.

<h3>Example</h3>
<blockquote>
<pre>
  typedef boost::multi_array&lt;int, 3&gt; array_type;

  array_type::extent_gen extents;
  array_type A(extents[3][3][3]);
  A[0][0][0] = 4;
  A[2][2][2] = 5;
  A.resize(extents[2][3][4]);
  assert(A[0][0][0] == 4);
  // A[2][2][2] is no longer valid.
</pre>
</blockquote>


<a name="sec_concepts"></a>
<h2>MultiArray Concept</h2>
Boost.MultiArray defines and uses the 
<a href="./reference.html#MultiArray">MultiArray</a>
concept.  It specifies an interface for N-dimensional containers.

<a name="sec_testcases"></a>
<h2>Test Cases</h2>
Boost.MultiArray comes with a suite of test cases meant to exercise
the features and semantics of the library.  A description of the test
cases can be found <a href="./test_cases.html">here</a>.

<a name="sec_related"></a>
<h2>Related Work</h2>

<a href="../../array/index.html">boost::array</a>
 and <a href="http://www.sgi.com/tech/stl/Vector.html">std::vector</a> are
    one-dimensional containers of user data.  Both manage their own
    memory. <tt>std::valarray</tt> is a low-level 
    C++ Standard Library component
    meant to provide portable high performance for numerical applications.
<a href="http://www.oonumerics.org/blitz/">Blitz++</a> is 
    an array library developed by Todd
    Veldhuizen. It uses
    advanced C++ techniques to provide near-Fortran performance for
    array-based numerical applications.
    <b>array_traits</b> is a beta library, formerly distributed with
    Boost, that provides a means to create iterators over native C++
    arrays.

This library is analogous to 
<a href="../../array/index.html">boost::array</a> in that it augments C style N-dimensional
arrays, as <tt>boost::array</tt> does for C one-dimensional arrays.


<a name="sec_credits"></a>
<h2>Credits</h2>
<ul>
      
      <li><a href="mailto:garcia@osl.iu.edu">Ronald Garcia</a> 
      is the primary author of the library.

      <li><a href="http://www.boost.org/people/jeremy_siek.htm">Jeremy Siek</a>
      helped with the library and provided a sounding board for ideas,
       advice, and assistance porting to Microsoft Visual C++.

      <li><a href="mailto:gbavestrelli@yahoo.com">Giovanni Bavestrelli</a>
      provided an early implementation of an
      N-dimensional array which inspired feedback from the 
      <a href="http://www.boost.org/">Boost</a> mailing list
      members. Some design decisions in this work were based upon this
      implementation and the comments it elicited.

      <li><a href="mailto:tveldhui@acm.org">Todd Veldhuizen</a> wrote
      <a href="http://oonumerics.org/blitz/">Blitz++</a>, which
      inspired some aspects of this design. In addition, he supplied 
      feedback on the design and implementation of the library. 

      <li><a href="mailto:jewillco@osl.iu.edu">Jeremiah Willcock</a>
      provided feedback on the implementation and design of the
      library and some suggestions for features.

      <li><a href="mailto:bdawes@acm.org">Beman Dawes</a>
      helped immensely with porting the library to Microsoft Windows
      compilers.
</ul>

<hr>

<address>
<a href="mailto:garcia@.cs.indiana.edu">Ronald Garcia</a>
</address>
<!-- Created: Fri Jun 29 10:53:07 EST 2001 -->
<!-- hhmts start -->Last modified: Tue Feb  7 17:15:50 EST 2006 <!-- hhmts end -->

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