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Just as rows provide a view on a specific row of a matrix, columns provide views on a specific column of a dense or sparse matrix. As such, columns act as a reference to a specific column. This reference is valid an can be used in every way any other column vector can be used as long as the matrix containing the column is not resized or entirely destroyed. Changes made to the elements (e.g. modifying values, inserting or erasing elements) are immediately visible in the matrix and changes made via the matrix are immediately visible in the column.
A reference to a dense or sparse column can be created very conveniently via the column()
function. It can be included via the header file
The column index must be in the range from , where
N
is the total number of columns of the matrix, and can be specified both at compile time or at runtime:
The column()
function returns an expression representing the column view. The type of this expression depends on the given column arguments, primarily the type of the matrix and the compile time arguments. If the type is required, it can be determined via the decltype
specifier:
The resulting view can be treated as any other column vector, i.e. it can be assigned to, it can be copied from, and it can be used in arithmetic operations. The reference can also be used on both sides of an assignment: The column can either be used as an alias to grant write access to a specific column of a matrix primitive on the left-hand side of an assignment or to grant read-access to a specific column of a matrix primitive or expression on the right-hand side of an assignment. The following example demonstrates this in detail:
The elements of a column can be directly accessed with the subscript operator.
The numbering of the column elements is
where N is the number of rows of the referenced matrix. Alternatively, the elements of a column can be traversed via iterators. Just as with vectors, in case of non-const columns, begin()
and end()
return an iterator, which allows to manipulate the elements, in case of constant columns an iterator to immutable elements is returned:
Inserting/accessing elements in a sparse column can be done by several alternative functions. The following example demonstrates all options:
A column view can be used like any other column vector. This means that with only a few exceptions all Vector Operations and Arithmetic Operations can be used. For instance, the current number of elements can be obtained via the size()
function, the current capacity via the capacity()
function, and the number of non-zero elements via the nonZeros()
function. However, since columns are references to specific columns of a matrix, several operations are not possible on views, such as resizing and swapping. The following example shows this by means of a dense column view:
Both dense and sparse columns can be used in all arithmetic operations that any other dense or sparse column vector can be used in. The following example gives an impression of the use of dense columns within arithmetic operations. All operations (addition, subtraction, multiplication, scaling, ...) can be performed on all possible combinations of dense and sparse columns with fitting element types:
Especially noteworthy is that column views can be created for both row-major and column-major matrices. Whereas the interface of a row-major matrix only allows to traverse a row directly and the interface of a column-major matrix only allows to traverse a column, via views it is possible to traverse a row of a column-major matrix or a column of a row-major matrix. For instance:
However, please note that creating a column view on a matrix stored in a row-major fashion can result in a considerable performance decrease in comparison to a column view on a matrix with column-major storage format. This is due to the non-contiguous storage of the matrix elements. Therefore care has to be taken in the choice of the most suitable storage order:
Although Blaze performs the resulting matrix/vector multiplication as efficiently as possible using a column-major storage order for matrix B
would result in a more efficient evaluation.
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