Expression Concepts

Scalar Expression

Description

A Scalar Expression is an expression convertible to a scalar type.

Refinement of

Default Constructible.

Associated types

Public base

scaler_expression<S>

S must be derived from this public base type.

Value type

value_type

The type of the scalar expression.

Notation

S

A type that is a model of Scalar Expression

Definitions

Valid expressions

In addition to the expressions defined in Default Constructible the following expressions must be valid.

Name

Expression

Type requirements

Return type

Evaluation

operator value_type () const

 

value_type

Expression semantics

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

Name Expression Precondition Semantics Postcondition

Evaluation

operator value_type () const

 

  Evaluates the scalar expression.

 

Complexity guarantees

The run-time complexity of the evaluation is specific for the evaluated scalar expression.

Invariants

Models

  • vector_scalar_unary

  • vector_scalar_binary

Vector Expression

Description

A Vector Expression is an expression evaluatable to a vector. Vector Expression provides an Indexed Bidirectional Iterator or an Indexed Random Access Iterator .

Refinement of

Default Constructible.

Associated types

Public base

vector_expression<V>

V must be derived from this public base type.

Value type

value_type

The element type of the vector expression.

Reference type

reference

The return type when accessing an element of a vector expression.
Convertable to a`value_type`.

Const reference type

const_reference

The return type when accessing an element of a constant vector expression.
Convertable to a`value_type`.

Size type

size_type

The index type of the vector expression. Am unsigned integral type used to represent size and index values.
Can represent any nonnegative value of difference_type.

Distance type

difference_type

A signed integral type used to represent the distance between two of the vector expression’s iterators.

Const iterator type

const_iterator

A type of iterator that may be used to examine a vector expression’s elements.

Iterator type

iterator

A type of iterator that may be used to modify a vector expression’s elements.

Const reverse iterator type

const_reverse_iterator

A Reverse Iterator adaptor whose base iterator type is the vector expression’s const iterator type.

Reverse iterator type

reverse_iterator

A Reverse Iterator adaptor whose base iterator type is the vector expression’s iterator type.

Notation

V

A type that is a model of Vector Expression

v, v1, v2

Object of type V

i

Object of a type convertible to size_type

t

Object of a type convertible to value_type

Definitions

Valid expressions

In addition to the expressions defined in Default Constructible the following expressions must be valid.

Name Expression Type requirements Return type

Beginning of range

v.begin ()

 

const_iterator

v.begin ()

v is mutable.

iterator

End of range

v.end ()

 

const_iterator

v.end ()

v is mutable.

iterator

Size

v.size ()

 

size_type

Swap

v1.swap (v2)

v1 and v2 are mutable.

void

Beginning of reverse range

v.rbegin ()

 

const_reverse_iterator

v.rbegin ()

v is mutable.

reverse_iterator

End of reverse range

v.rend ()

 

const_reverse_iterator

v.rend ()

v is mutable.

reverse_iterator

Element access

v (i)

i is convertible to size_type.

Convertible to value_type.

Assignment

v2 = v1

v2 is mutable and v1 is convertible to V.

V &

v2.assign (v1)

v2 is mutable and v1 is convertible to V.

V &

Computed assignment

v2 += v1

v2 is mutable and v1 is convertible to V.

V &

v2.plus_assign (v1)

v2 is mutable and v1 is convertible to V.

V &

v2 -= v1

v2 is mutable and v1 is convertible to V.

V &

v2.minus_assign (v1)

v2 is mutable and v1 is convertible to V.

V &

v *= t

v is mutable and t is convertible to value_type.

V &

Expression semantics

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

Name Expression Precondition Semantics Postcondition

Beginning of range

v.begin ()

 

Returns an iterator pointing to the first element in the vector expression.

v.begin () is either dereferenceable or past-the-end. It is past-the-end if and only if v.size () == 0.

End of range

v.end ()

 

Returns an iterator pointing one past the last element in the vector expression.

v.end () is past-the-end.

Size

v.size ()

 

Returns the size of the vector expression, that is, its number of elements.

v.size () >= 0

Swap

v1.swap (v2)

 

Equivalent to swap (v1, v2).

 

Beginning of reverse range

v.rbegin ()

 

Equivalent to reverse_iterator (v.end ()).

v.rbegin () is either dereferenceable or past-the-end. It is past-the-end if and only if v.size () == 0.

End of reverse range

v.rend ()

 

Equivalent to reverse_iterator (v.begin ()).

v.rend () is past-the-end.

Element access

v (i)

0 ⇐ i < v.size ()

Returns the i-th element of the vector expression.

 

Assignment

v2 = v1

v1.size () == v2.size ()

Assigns every element of the evaluated vector expression v1 to the corresponding element of v2 .

 

v2.assign (v1)

v1.size () == v2.size ()

Assigns every element of v1 to the corresponding element of v2.

 

Computed assignment

v2 += v1

v1.size () == v2.size ()

Adds every element of the evaluated vector expression v1 to the corresponding element of v2.

 

v2.plus_assign (v1)

v1.size () == v2.size ()

Adds every element of v1 to the corresponding element of v2.

 

v2 -= v1

v1.size () == v2.size ()

Subtracts every element of the evaluated vector expression v1 from the corresponding element of v2 .

 

v2.minus_assign (v1)

v1.size () == v2.size ()

Subtracts every element of v1 from the corresponding element of v2.

 

v *= t

 

Multiplies every element of v with t .

 

Complexity guarantees

The run-time complexity of begin () and end () is specific for the evaluated vector expression, typically amortized constant time.

The run-time complexity of size () is constant time.

The run-time complexity of swap () is specific for the evaluated vector expression, typically constant time.

The run-time complexity of rbegin () and rend () is specific for the evaluated vector expression, typically amortized constant time.

The run-time complexity of the element access is specific for the evaluated vector expression, typically amortized constant time for the dense and logarithmic for the sparse case.

The run-time complexity of the arithmetic operations is specific for the evaluated vector expressions, typically linear in the size of the expressions.

Invariants

Valid range

For any vector expression v, [v.begin (), v.end ()) is a valid range.

Completeness

An algorithm that iterates through the range [v.begin (), v.end ()) will pass through every element of v .

Valid reverse range

[v.rbegin (), v.rend ()) is a valid range.

Equivalence of ranges

The distance from v.begin () to v.end () is the same as the distance from v.rbegin () to v.rend ().

Models

  • vector_range;

  • vector_slice

  • matrix_row

  • matrix_column

  • matrix_vector_range

  • matrix_vector_slice

  • vector_unary

  • vector_binary

  • vector_binary_scalar1

  • vector_binary_scalar2

  • matrix_vector_unary1

  • matrix_vector_unary2

  • matrix_vector_binary1

  • matrix_vector_binary2

Matrix Expression

Description

A Matrix Expression is an expression evaluatable to a matrix. Matrix Expression provides an Indexed Bidirectional Column/Row Iterator or an Indexed Random Access Column/Row Iterator .

Refinement of

Default Constructible.

Associated types

immutable types

Public base

matrix_expression<M>

M must be derived from this public base type.

Value type

value_type

The element type of the matrix expression.

Const reference type

const_reference

The return type when accessing an element of a constant matrix expression.
Convertable to a value_type.

Size type

size_type

The index type of the matrix expression. Am unsigned integral type used to represent size and index values.
Can represent any nonnegative value of difference_type.

Distance type

difference_type

A signed integral type used to represent the distance between two of the matrix expression’s iterators.

Const iterator types

const_iterator1

A type of column iterator that may be used to examine a matrix expression’s elements.

const_iterator2

A type of row iterator that may be used to examine a matrix expression’s elements.

Const reverse iterator types

const_reverse_iterator1

A Reverse Iterator adaptor whose base iterator type is the matrix expression’s const column iterator type.

const_reverse_iterator2

A Reverse Iterator adaptor whose base iterator type is the matrix expression’s const row iterator type.
mutable types

Reference type

reference

The return type when accessing an element of a matrix expression.
Convertable to a value_type.

Iterator types

iterator1

A type of column iterator that may be used to modify a matrix expression’s elements.

iterator2

A type of row iterator that may be used to modify a matrix expression’s elements.

Reverse iterator types

reverse_iterator1

A Reverse Iterator adaptor whose base iterator type is the matrix expression’s column iterator type.

reverse_iterator2

A Reverse Iterator adaptor whose base iterator type is the matrix expression’s row iterator type.

Notation

M

A type that is a model of Matrix Expression

m, m1, m2

Object of type M

i, j

Objects of a type convertible to size_type

t

Object of a type convertible to value_type

Definitions

Valid expressions

In addition to the expressions defined in Default Constructible the following expressions must be valid.

immutable expressions
Name Expression Type requirements Return type

Size

m.size1 ()

 

size_type

m.size2 ()

 

size_type
possibly mutable expressions
Name Expression Type requirements Return type

Beginning of range

m.begin1 ()

 

const_iterator1

m.begin2 ()

 

const_iterator2

m.begin1 ()

m is mutable. 

iterator1

m.begin2 ()

m is mutable.

iterator2

End of range

m.end1 ()

 

const_iterator1

m.end2 ()

 

const_iterator2

m.end1 ()

m is mutable. 

iterator1

m.end2 ()

m is mutable.

iterator2

Swap

m1.swap (m2)

m1 and m2 are mutable. 

void

Beginning of reverse range

m.rbegin1 ()

 

const_reverse_iterator1

m.rbegin2 ()

 

const_reverse_iterator2

m.rbegin1 ()

m is mutable. 

reverse_iterator1

m.rbegin2 ()

m is mutable.

reverse_iterator2

End of reverse range

m.rend1 ()

 

const_reverse_iterator1

m.rend2 ()

 

const_reverse_iterator2

m.rend1 ()

m is mutable.

reverse_iterator1

m.rend2 ()

m is mutable.

reverse_iterator2

Element access

m (i, j)

i and j are convertible to size_type .

Convertible to value_type.

Assignment

m2 = m1

m2 is mutable and m1 is convertible to M.

M &

m2.assign (m1)

m2 is mutable and m1 is convertible to M.

M &

Computed assignment

m2 += m1

m2 is mutable and m1 is convertible to M.

M &

m2.plus_assign (m1)

m2 is mutable and m1 is convertible to M.

M &

m2 -= m1

m2 is mutable and m1 is convertible to M.

M &

m2.minus_assign (m1)

m2 is mutable and m1 is convertible to M.

M &

m *= t

m is mutable and t is convertible to value_type.

M &

Expression semantics

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

Name Expression Precondition Semantics Postcondition

Beginning of range

m.begin1 ()

 

Returns an iterator pointing to the first element in the first column of a matrix expression.

m.begin1 () is either dereferenceable or past-the-end. It is past-the-end if and only if m.size1 () == 0.

m.begin2 ()

 

Returns an iterator pointing to the first element in the first row of a matrix expression.

m.begin2 () is either dereferenceable or past-the-end. It is past-the-end if and only if m.size2 () == 0.

End of range

m.end1 ()

 

Returns an iterator pointing one past the last element in the matrix expression.

m.end1 () is past-the-end.

m.end2 ()

 

Returns an iterator pointing one past the last element in the matrix expression.

m.end2 () is past-the-end.

Size

m.size1 ()

 

Returns the number of rows of the matrix expression.

m.size1 () >= 0

m.size2 ()

 

Returns the number of columns of the matrix expression.

m.size2 () >= 0

Swap

m1.swap (m2)

 

Equivalent to swap (m1, m2).

 

Beginning of reverse range

m.rbegin1 ()

 

Equivalent to reverse_iterator1 (m.end1 ()).

m.rbegin1 () is either dereferenceable or past-the-end. It is past-the-end if and only if m.size1 () == 0.

m.rbegin2 ()

 

Equivalent to reverse_iterator2 (m.end2 ()).

m.rbegin2 () is either dereferenceable or past-the-end. It is past-the-end if and only if m.size2 () == 0.

End of reverse range

m.rend1 ()

 

Equivalent to reverse_iterator1 (m.begin1 ()).

m.rend1 () is past-the-end.

m.rend2 ()

 

Equivalent to reverse_iterator2 (m.begin2 ()).

m.rend2 () is past-the-end.

Element access

m (i, j)

0 ⇐ i < m.size1 () and 0 ⇐ j < m.size2 ()

Returns the j-th element of the i-th row of the matrix expression.

 

Assignment

m2 = m1

m1.size1 () == m2.size1 () and ` m1.size2 () == m2.size2 ()`

Assigns every element of the evaluated matrix expression m1 to the corresponding element of m2 .

 

m2.assign (m1)

m1.size1 () == m2.size1 () and ` m1.size2 () == m2.size2 ()`

Assigns every element of m1 to the corresponding element of m2.

 

Computed assignment

m2 += m1

m1.size1 () == m2.size1 () and ` m1.size2 () == m2.size2 ()`

Adds every element of the evaluated matrix expression m1 to the corresponding element of m2.

 

m2.plus_assign (m1)

m1.size1 () == m2.size1 () and ` m1.size2 () == m2.size2 ()`

Adds every element of m1 to the corresponding element of m2.

 

m2 -= m1

m1.size1 () == m2.size1 () and ` m1.size2 () == m2.size2 ()`

Subtracts every element of the evaluated matrix expression m1 from the corresponding element of m2 .

 

m2.minus_assign (m1)

m1.size1 () == m2.size1 () and ` m1.size2 () == m2.size2 ()`

Subtracts every element of m1 from the corresponding element of m2.

 

m *= t

 

Multiplies every element of m with t .

 

Complexity guarantees

The run-time complexity of begin1 (), begin2 () , end1 () and end2 () is specific for the evaluated matrix expression.

The run-time complexity of size1 () and size2 () is constant time.

The run-time complexity of swap () is specific for the evaluated matrix expression, typically constant time.

The run-time complexity of rbegin1 (), rbegin2 () , rend1 () and rend2 () is specific for the evaluated matrix expression.

The run-time complexity of the element access is specific for the evaluated matrix expression, typically amortized constant time for the dense and logarithmic for the sparse case.

The run-time complexity of the arithmetic operations is specific for the evaluated matrix expressions, typically quadratic in the size of the proxies.

Invariants

Valid range

For any matrix expression m, [m.begin1 (), m.end1 ()) and [m.begin2 (), m.end2 ()) are valid ranges.

Completeness

An algorithm that iterates through the range [m.begin1 (), m.end1 ()) will pass through every row of m , an algorithm that iterates through the range [m.begin2 (), m.end2 ()) will pass through every column of m .

Valid reverse range

[m.rbegin1 (), m.rend1 ()) and [m.rbegin2 (), m.rend2 ()) are valid ranges.

Equivalence of ranges

The distance from m.begin1 () to m.end1 () is the same as the distance from m.rbegin1 () to m.rend1 () and the distance from m.begin2 () to m.end2 () is the same as the distance from m.rbegin2 () to m.rend2 ().

Models

  • matrix_range

  • matrix_slice;

  • triangular_adaptor

  • symmetric_adaptor

  • banded_adaptor

  • vector_matrix_binary

  • matrix_unary1

  • matrix_unary2

  • matrix_binary

  • matrix_binary_scalar1

  • matrix_binary_scalar2

  • matrix_matrix_binary

Copyright (©) 2000-2002 Joerg Walter, Mathias Koch
Copyright (©) 2021 Shikhar Vashistha
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 ).