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`
Beginning of range	`v.begin ()`	`v` is mutable.	`iterator`
End of range	`v.end ()`		`const_iterator`
End of range	`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`
Beginning of reverse range	`v.rbegin ()`	`v` is mutable.	`reverse_iterator`
End of reverse range	`v.rend ()`		`const_reverse_iterator`
End of reverse range	`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 &`
Assignment	`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` .
Assignment	`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

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.
Reference type	`reference`	The return type when accessing an element of a matrix expression. Convertable to a`value_type`.
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 iterator types	`const_iterator2`	A type of row iterator that may be used to examine a matrix expression's elements.
Iterator types	`iterator1`	A type of column iterator that may be used to modify a matrix expression's elements.
Iterator types	`iterator2`	A type of row iterator that may be used to modify 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 iterator types	`const_reverse_iterator2`	A Reverse Iterator adaptor whose base iterator type is the matrix expression's const row iterator type.
Reverse iterator types	`reverse_iterator1`	A Reverse Iterator adaptor whose base iterator type is the matrix expression's column iterator type.
Reverse iterator types	`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.

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`
Size	`m.size1 ()`		`size_type`
Size	`m.size2 ()`		`size_type`
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 &`
Assignment	`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`.
Beginning of range	`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.
End of range	`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`
Size	`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`.
Beginning of reverse range	`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.
End of reverse range	`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` .
Assignment	`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
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