# Container Concepts

## Vector

#### Description

A Vector describes common aspects of dense, packed and sparse vectors.

#### Refinement of

DefaultConstructible, Vector Expression [1].

#### Associated types

In addition to the types defined by Vector Expression

Public base | vector_container<V> | V must be derived from this public base type.
Storage array | V::array_type | Dense Vector ONLY. The type of underlying storage array used to store the elements. The array_type must model the Storage concept.

#### Notation

`V` | A type that is a model of Vector
`v` | Objects of type `V`
`n, i` | Objects of a type convertible to `size_type`
`t` | Object of a type convertible to `value_type`
`p` | Object of a type convertible to `bool`

#### Definitions

#### Valid expressions

In addition to the expressions defined in DefaultConstructible, Vector Expression the following expressions must be valid.

Name | Expression | Type requirements | Return type
--- | --- | --- | ---
Sizing constructor | `V v (n)` |  | `V`
Insert | `v.insert_element (i, t)` | `v` is mutable. | `void`
Erase | `v.erase_element (i)` | `v` is mutable. | `void`
Clear | `v.clear ()` | `v` is mutable. | `void`
Resize | `v.resize (n)` `v.resize (n, p)` | `v` is mutable. | `void`
Storage | `v.data()` | `v` is mutable and Dense. | `array_type&` if `v` is mutable, `const array_type&` otherwise

#### Expression semantics

Semantics of an expression is defined only where it differs from, or is not defined in Vector Expression .

Name | Expression | Precondition | Semantics | Postcondition
--- | --- | --- | --- | ---
Sizing constructor | `V v (n)` | `n >= 0` | Allocates a vector of`n` elements. | `v.size () == n`.
Element access [2] | `v[n]` | `0<n>v.size()` | returns the n-th element in v |
Insert | `v.insert_element (i, t)` | `0 <= i < v.size ()`. | Inserts an element at `v (i)` with value `t`. The storage requirement of the Vector may be increased. | `v (i)` is equal to `t`.
Erase | `v.erase_element (i)` | `0 <= i < v.size ()` | Destroys the element as `v (i)` and replaces it with the default `value_type ()`. The storage requirement of the Vector may be decreased. | `v (i)` is equal to `value_type ()`.
Clear | `v.clear ()` |  | Equivalent to `for (i = 0; i < v.size (); ++ i)` `v.erase_element (i);` |
Resize | `v.resize (n) v.resize (n, p)` |  | Reallocates the vector so that it can hold `n` elements. Erases or appends elements in order to bring the vector to the prescribed size. Appended elements copies of `value_type()`. When `p == false` then existing elements are not preserved and elements will not appended as normal. Instead the vector is in the same state as that after an equivalent sizing constructor. | `v.size () == n`.
Storage | `v.data()` |  | Returns a reference to the underlying dense storage. |

#### Complexity guarantees

The run-time complexity of the sizing constructor is linear in the vector's size.

The run-time complexity of insert_element and erase_element is specific for the Vector model and it depends on increases/decreases in storage requirements.

The run-time complexity of resize is linear in the vector's size.

#### Invariants

#### Models

- `vector`, `bounded_vector`, `c_vector`
- `unit_vector`, `zero_vector`, `scalar_vector`
- `mapped_vector;`, `compressed_vector`, `coordinate_vector`

#### Notes

[1] As a user you need not care about `Vector` being a refinement of the VectorExpression. Being a refinement of the VectorExpression is only important for the template-expression engine but not the user.

[2] The `operator[]` is added purely for convenience and compatibility with the `std::vector`. In uBLAS however, generally `operator()` is used for indexing because this can be used for both vectors and matrices.

## Matrix

#### Description

A Matrix describes common aspects of dense, packed and sparse matrices.

#### Refinement of

DefaultConstructible, Matrix Expression [1] .

#### Associated types

In addition to the types defined by Matrix Expression

Public base | matrix_container<M> | M must be derived from this public base type.
Storage array | M::array_type | Dense Matrix ONLY. The type of underlying storage array used to store the elements. The array_type must model the Storage concept.

#### Notation

`M` | A type that is a model of Matrix
`m` | Objects of type `M`
`n1, n2, i, j` | Objects of a type convertible to `size_type`
`t` | Object of a type convertible to `value_type`
`p` | Object of a type convertible to `bool`

#### Definitions

#### Valid expressions

In addition to the expressions defined in Matrix Expression the following expressions must be valid.

Name | Expression | Type requirements | Return type
--- | --- | --- | ---
Sizing constructor | `M m (n1, n2)` |  | `M`
Insert | `m.insert_element (i, j, t)` | `m` is mutable. | `void`
Erase | `m.erase_element (i, j)` | `m` is mutable. | `void`
Clear | `m.clear ()` | `m` is mutable. | `void`
Resize | `m.resize (n1, n2)` `m.resize (n1, n2, p)` | `m` is mutable. | `void`
Storage | `m.data()` | `m` is mutable and Dense. | `array_type&` if `m` is mutable, `const array_type&` otherwise

#### Expression semantics

Semantics of an expression is defined only where it differs from, or is not defined in Matrix Expression .

Name | Expression | Precondition | Semantics | Postcondition
--- | --- | --- | --- | ---
Sizing constructor | `M m (n1, n2)` | `n1 >= 0` and `n2 >= 0` | Allocates a matrix of `n1` rows and `n2` columns. | `m.size1 () == n1` and `m.size2 () == n2`.
Insert | `m.insert_element (i, j, t)` | `0 <= i < m.size1 ()`, `0 <= j < m.size2 ()`. | Inserts an element at `m (i, j)` with value `t`. The storage requirement of the Matrix may be increased. | `m (i, j)` is equal to `t`.
Erase | `m.erase_element (i, j)` | `0 <= i < m.size1 ()`and `0 <= j < m.size2` | Destroys the element as `m (i, j)` and replaces it with the default `value_type ()`. The storage requirement of the Matrix may be decreased. | `m (i, j)` is equal to `value_type ()`.
Clear | `m.clear ()` |  | Equivalent to `for (i = 0; i < m.size1 (); ++ i)` `for (j = 0; j < m.size2 (); ++ j)` `m.erase_element (i, j);` |
Resize | `m.resize (n1, n2) m.resize (n1, n2, p)` |  | Reallocate the matrix so that it can hold `n1` rows and `n2` columns. Erases or appends elements in order to bring the matrix to the prescribed size. Appended elements are `value_type()` copies. When `p == false` then existing elements are not preserved and elements will not appended as normal. Instead the matrix is in the same state as that after an equivalent sizing constructor. | `m.size1 () == n1` and `m.size2 () == n2`.
Storage | `m.data()` |  | Returns a reference to the underlying dense storage. |

#### Complexity guarantees

The run-time complexity of the sizing constructor is quadratic in the matrix's size.

The run-time complexity of insert_element and erase_element is specific for the Matrix model and it depends on increases/decreases in storage requirements.

The run-time complexity of resize is quadratic in the matrix's size.

#### Invariants

#### Models

- `matrix`, `bounded_matrix`, `c_matrix`
- `identity_matrix` , `zero_matrix` , `scalar_matrix`
- `triangular_matrix` , `symmetric_matrix` , `banded_matrix`
- `mapped_matrix` , `compressed_matrix` , `coordinate_matrix`

#### Notes

[1] As a user you need not care about `Matrix` being a refinement of the MatrixExpression. Being a refinement of the MatrixExpression is only important for the template-expression engine but not the user.

## Tensor

#### Description

A Tensor describes common aspects of dense multidimensional arrays.

#### Refinement of

DefaultConstructible, Tensor Expression [1] .

#### Associated types

In addition to the types defined by Tensor Expression

Public base | `tensor_container<tensor_t>` | `tensor_t` must be derived from this public base type.
Storage array | `tensor_t::array_type``` | Dense tensor ONLY. The type of underlying storage array used to store the elements. The array_type must model the Storage concept.

#### Notation

`tensor_t` | A type that is a model of Tensor
`t` | Objects of type `tensor_t`
`n1, n2, np, m1, m2, mq` | Dimension objects of a type convertible to `size_type`
`i1, i2, ip, j, k` | Index objects of a type convertible to `size_type`
`v` | Object of a type convertible to `value_type`

#### Definitions

#### Valid expressions

In addition to the expressions defined in Tensor Expression the following expressions must be valid.

Name | Expression | Type requirements | Return type
--- | --- | --- | ---
Sizing constructor | `T t(n1, n2, ..., np)` |  | `T`
Write | `t.at(i1, i2, ..., ip)` | `t` is mutable. | `void`
Read | `t.at(i1, i2, ..., ip)` | `t` is mutable. | `v`
Clear | `t.clear ()` | `t` is mutable. | `void`
Resize | `t.resize(m1, m2, ... , mq)` | `t` is mutable. | `void`
Storage | `t.data()` | `t` is mutable and dense. | `pointer` if `t` is mutable, `const_pointer` otherwise

#### Expression semantics

Semantics of an expression is defined only where it differs from, or is not defined in Tensor Expression .

Name | Expression | Precondition | Semantics | Postcondition
--- | --- | --- | --- | ---
Sizing constructor | `T t(n1, n2, ..., np)` | $n_r \geq 1$ for $1\leq 1 \leq p $ | Allocates a `p`-order tensor with dimension extents $n_1,n_2,\dots,n_p$. | `t.size(r)==nr` for $1\leq r \leq p$.
Write | `t.at(i1,i2,...,ip)=v` | $0 \leq i_r < n_r$ for $1 \leq r \leq p$. | Writes an element at multi-index position $i_1,i_2,\dots,i_p$ with value `v`. | `t(i1,i2,...,ip)` is equal to `v`.
Read | `v=t.at(i1,i2,...,ip)` | $0 \leq i_r < n_r$ for $1 \leq r \leq p$. | Reads the element at multi-index position $(i_1,i2_,\dots,i_p)$ and returns a value `v`. | `t(i1,i2,...,ip)` is equal to `v`.
Clear | `t.clear()` |  | Removes all elements from the container. |
Resize | `t.resize(m1, m2, ..., mq)` | $m_r \geq 1$ for $1\leq 1 \leq q $ | Reallocate the matrix so that it can hold $m_1\times m_2\times \cdots \times m_q$ elements. Erases or appends elements in order to bring the matrix to the prescribed size. Appended elements are `value_type()` copies. | `t.size(r) == mr` for $1\leq r \leq q$.
Storage | `m.data()` |  | Returns a reference to the underlying dense storage. |

#### Complexity guarantees

The run-time complexity of contructor is linear in the tensor's size $n_1 \times n_2 \times \cdots \times n_p$.

The run-time complexity of `write()` and `read()` is linear in the order of the tensor.

The run-time complexity of resize is at most linear in the tensor's size $m_1 \times m_2 \times \cdots \times n_q$.

#### Invariants

#### Models

- `tensor`

#### Notes

[1] As a user you need not care about `Tensor` being a refinement of the TensorExpression. Being a refinement of the TensorExpression is only important for the template-expression engine but not the user.

Copyright (©) 2000-2002 Joerg Walter, Mathias Koch
Copyright (©) 2018 Cem Bassoy
Use, modification and distribution are 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 ).