#
The Boost Multidimensional Array Library
(Boost.MultiArray)

## Synopsis

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 `std::vector<std::vector<...>>` 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).

## Table of Contents

1. Introduction
2. Short Example
3. MultiArray Components
4. Construction and Assignment
5. Array View and Subarray Type Generators
6. Specifying Array Dimensions
7. Accessing Elements
8. Creating Views
9. Storage Ordering
10. Setting the Array Base
11. Changing an Array's Shape
12. Resizing an Array
13. MultiArray Concept
14. Test Cases
15. Related Work
16. Credits

## Introduction

The C++ standard library provides several generic containers, but it does not provide any multidimensional array types. The `std::vector` class template can be used to implement N-dimensional arrays, for example expressing a 2-dimensional array of `double` elements using the type `std::vector<std::vector<double>>`, but the resulting interface is unwieldy and the memory overhead can be quite high. Native C++ arrays (i.e. `int arr[2][2][2];`) 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.

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.

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 reference manual provides more complete and formal documentation of library features.

## Short Example

What follows is a brief example of the use of `multi_array`:

>     #include "boost/multi_array.hpp"
>     #include <cassert>

>     int
>     main () {
>       // Create a 3D array that is 3 x 4 x 2
>       typedef boost::multi_array<double, 3> 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;
>     }

## MultiArray Components

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

1. `multi_array`,
2. `multi_array_ref`, and
3. `const_multi_array_ref`

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

`multi_array_ref` adapts an existing array of data to provide the `multi_array` interface. `multi_array_ref` does not own the data passed to it.

`const_multi_array_ref` is similar to `multi_array_ref` but guarantees that the contents of the array are immutable. It can thus wrap pointers of type T const*.

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

## Construction and Assignment

Each of the array types - `multi_array`, `multi_array_ref`, and `const_multi_array_ref` - provides a specialized set of constructors. For further information, consult their reference pages.

All of the non-const array types in this library provide assignment operators`operator=()`. Each of the array types `multi_array`, `multi_array_ref`, `subarray`, and `array_view` can be assigned from any of the others, so long as their shapes match. The const variants, `const_multi_array_ref`, `const_subarray`, and `const_array_view`, 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.

## Array View and Subarray Type Generators

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, `subarray_gen`, `const_subarray_gen`, `array_view_gen`, and `const_array_view_gen` are provided. Thus, the two typedefs in the following example result in the same type:

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

## Specifying Array Dimensions

When creating most of the Boost.MultiArray components, it is necessary to specify both the number of dimensions and the extent of each (`boost::multi_array` 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.

The first method involves passing a Collection of extents to a constructor, most commonly a `boost::array`. 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.

### Example

>       typedef boost::multi_array<double, 3> array_type;
>       boost::array<array_type::index, 3> shape = {{ 3, 4, 2 }};
>       array_type A(shape);

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

### Example

>       typedef boost::multi_array<double, 3> array_type;
>       array_type A(boost::extents[3][4][2]);

## Accessing Elements

The Boost.MultiArray components provide two ways of accessing specific elements within a container. The first uses the traditional C array notation, provided by `operator[]`.

### Example

>       typedef boost::multi_array<double, 3> array_type;
>       array_type A(boost::extents[3][4][2]);
>       A[0][0][0] = 3.14;
>       assert(A[0][0][0] == 3.14);

The second method involves passing a Collection of indices to `operator()`. N indices will be retrieved from the Collection for the N dimensions of the container.

### Example

>       typedef boost::multi_array<double, 3> array_type;
>       array_type A(boost::extents[3][4][2]);
>       boost::array<array_type::index,3> idx = {{0,0,0}};
>       A(idx) = 3.14;
>       assert(A(idx) == 3.14);

This can be useful for writing dimension-independent code, and under some compilers may yield higher performance than `operator[].`

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 `BOOST_DISABLE_ASSERTS` preprocessor macro prior to including multi_array.hpp in your application.

## Creating Views

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.

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

### Example

>       // 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<3>::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]);

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).

### Example

>       // 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<2>::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]);

### More on `index_range`

The `index_range` type provides several methods of specifying ranges for subview generation. Here are a few range instantiations that specify the same range.

### Example

>       // [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 <= range().stride(2) < 4;
>       a_range = 0 <= range().stride(2) <= 3;

An `index_range` 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.

### Example

>       typedef boost::multi_array_types::index_range range;
>       range a_range;

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

>       // indices i where 3 <= i
>       a_range = range().start(3)
>       a_range = 3 <= range();
>       a_range = 2 < range();

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

The following example slicing operations exhibit some of the alternatives shown above

>         // take all of dimension 1
>         // take i < 5 for dimension 2
>         // take 4 <= j <= 7 for dimension 3 with stride 2
>         myarray[ boost::indices[range()][range() < 5 ][4 <= range().stride(2) <= 7] ];

## Storage Ordering

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 `c_storage_order`, `fortran_storage_order`, and `general_storage_order`.

`c_storage_order`, 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.

`fortran_storage_order` 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.

### Example

>       typedef boost::multi_array<double,3> array_type;
>       array_type A(boost::extents[3][4][2],boost::fortran_storage_order());
>       call_fortran_function(A.data());

`general_storage_order` allows one to customize both the order in which dimensions are stored in memory and whether dimensions are stored in ascending or descending order.

### Example

>       typedef boost::general_storage_order<3> storage;
>       typedef boost::multi_array<int,3> 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));

## Setting The Array Base

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 `extent_range` type, to the `extent_gen` constructor in order to set the base value.

### Example

>       typedef boost::multi_array<double, 3> 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)]);

An alternative is to first construct the array normally then reset the bases. To set all bases to the same value, use the `reindex` member function, passing it a single new index value.

### Example

>       typedef boost::multi_array<double, 3> array_type;

>       array_type::extent_gen extents;

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

An alternative is to set each base separately using the `reindex` member function, passing it a Collection of index bases.

### Example

>       typedef boost::multi_array<double, 3> 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<array_type::index,ndims> bases = {{0, 1, -1}};
>       A.reindex(bases);

## Changing an Array's Shape

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.

### Example

>       typedef boost::multi_array<double, 3> array_type;

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

Note that reshaping an array does not affect the indexing.

## Resizing an Array

The `boost::multi_array` 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.

### Example

>       typedef boost::multi_array<int, 3> 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.

## MultiArray Concept

Boost.MultiArray defines and uses the MultiArray concept. It specifies an interface for N-dimensional containers.

## Test Cases

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 here.

## Related Work

boost::array and std::vector are one-dimensional containers of user data. Both manage their own memory. `std::valarray` is a low-level C++ Standard Library component meant to provide portable high performance for numerical applications. Blitz++ is an array library developed by Todd Veldhuizen. It uses advanced C++ techniques to provide near-Fortran performance for array-based numerical applications. array_traits is a beta library, formerly distributed with Boost, that provides a means to create iterators over native C++ arrays. This library is analogous to boost::array in that it augments C style N-dimensional arrays, as `boost::array` does for C one-dimensional arrays.

## Credits

- Ronald Garcia is the primary author of the library.
- Jeremy Siek helped with the library and provided a sounding board for ideas, advice, and assistance porting to Microsoft Visual C++.
- Giovanni Bavestrelli provided an early implementation of an N-dimensional array which inspired feedback from the Boost mailing list members. Some design decisions in this work were based upon this implementation and the comments it elicited.
- Todd Veldhuizen wrote Blitz++, which inspired some aspects of this design. In addition, he supplied feedback on the design and implementation of the library.
- Jeremiah Willcock provided feedback on the implementation and design of the library and some suggestions for features.
- Beman Dawes helped immensely with porting the library to Microsoft Windows compilers.

Last modified: Tue Feb 7 17:15:50 EST 2006