# Wiki

# blaze / Columns

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.

## Setup of Columns

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

```
#include <blaze/math/Column.h>
```

The column index must be in the range from `[0..N-1]`

, where `N`

is the total number of columns of the matrix, and can be specified both at compile time or at runtime:

```
blaze::DynamicMatrix<double,blaze::columnMajor> A;
// ... Resizing and initialization
// Creating a reference to the 1st column of matrix A (compile time index)
auto col1 = column<1UL>( A );
// Creating a reference to the 2nd column of matrix A (runtime index)
auto col2 = column( A, 2UL );
```

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 `decltype`

or via the `ColumnExprTrait`

class template:

```
using MatrixType = blaze::DynamicMatrix<int>;
using ColumnType1 = decltype( blaze::column<1UL>( std::declval<MatrixType>() ) );
using ColumnType2 = blaze::ColumnExprTrait<MatrixType,1UL>::Type;
```

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:

```
blaze::DynamicVector<double,blaze::columnVector> x;
blaze::CompressedVector<double,blaze::columnVector> y;
blaze::DynamicMatrix<double,blaze::columnMajor> A, B;
blaze::CompressedMatrix<double,blaze::columnMajor> C, D;
// ... Resizing and initialization
// Setting the 1st column of matrix A to x
auto col1 = column( A, 1UL );
col1 = x;
// Setting the 4th column of matrix B to y
column( B, 4UL ) = y;
// Setting x to the 2nd column of the result of the matrix multiplication
x = column( A * B, 2UL );
// Setting y to the 2nd column of the result of the sparse matrix multiplication
y = column( C * D, 2UL );
```

## Element Access

The elements of a column can be directly accessed with the subscript operator. The indices to access a column are zero-based:

```
blaze::DynamicMatrix<double,blaze::columnMajor> A;
// ... Resizing and initialization
// Creating a view on the 4th column of matrix A
auto col4 = column( A, 4UL );
// Setting the 1st element of the dense column, which corresponds
// to the 1st element in the 4th column of matrix A
col4[1] = 2.0;
```

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:

```
blaze::DynamicMatrix<int,blaze::columnMajor> A( 128UL, 256UL );
// ... Resizing and initialization
// Creating a reference to the 31st column of matrix A
auto col31 = column( A, 31UL );
// Traversing the elements via iterators to non-const elements
for( auto it=col31.begin(); it!=col31.end(); ++it ) {
*it = ...; // OK; Write access to the dense column value
... = *it; // OK: Read access to the dense column value.
}
// Traversing the elements via iterators to const elements
for( auto it=col31.cbegin(); it!=col31.cend(); ++it ) {
*it = ...; // Compilation error: Assignment to the value via iterator-to-const is invalid.
... = *it; // OK: Read access to the dense column value.
}
```

```
blaze::CompressedMatrix<int,blaze::columnMajor> A( 128UL, 256UL );
// ... Resizing and initialization
// Creating a reference to the 31st column of matrix A
auto col31 = column( A, 31UL );
// Traversing the elements via iterators to non-const elements
for( auto it=col31.begin(); it!=col31.end(); ++it ) {
it->value() = ...; // OK: Write access to the value of the non-zero element.
... = it->value(); // OK: Read access to the value of the non-zero element.
it->index() = ...; // Compilation error: The index of a non-zero element cannot be changed.
... = it->index(); // OK: Read access to the index of the sparse element.
}
// Traversing the elements via iterators to const elements
for( auto it=col31.cbegin(); it!=col31.cend(); ++it ) {
it->value() = ...; // Compilation error: Assignment to the value via iterator-to-const is invalid.
... = it->value(); // OK: Read access to the value of the non-zero element.
it->index() = ...; // Compilation error: The index of a non-zero element cannot be changed.
... = it->index(); // OK: Read access to the index of the sparse element.
}
```

## Element Insertion

Inserting/accessing elements in a sparse column can be done by several alternative functions. The following example demonstrates all options:

```
blaze::CompressedMatrix<double,blaze::columnMajor> A( 100UL, 10UL ); // Non-initialized 100x10 matrix
auto col0( column( A, 0UL ) ); // Reference to the 0th column of A
// The subscript operator provides access to all possible elements of the sparse column,
// including the zero elements. In case the subscript operator is used to access an element
// that is currently not stored in the sparse column, the element is inserted into the column.
col0[42] = 2.0;
// The second operation for inserting elements is the set() function. In case the element
// is not contained in the column it is inserted into the column, if it is already contained
// in the column its value is modified.
col0.set( 45UL, -1.2 );
// An alternative for inserting elements into the column is the insert() function. However,
// it inserts the element only in case the element is not already contained in the column.
col0.insert( 50UL, 3.7 );
// A very efficient way to add new elements to a sparse column is the append() function.
// Note that append() requires that the appended element's index is strictly larger than
// the currently largest non-zero index of the column and that the column's capacity is
// large enough to hold the new element.
col0.reserve( 10UL );
col0.append( 51UL, -2.1 );
```

## Common Operations

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:

```
blaze::DynamicMatrix<int,blaze::columnMajor> A( 42UL, 42UL );
// ... Resizing and initialization
// Creating a reference to the 2nd column of matrix A
auto col2 = column( A, 2UL );
col2.size(); // Returns the number of elements in the column
col2.capacity(); // Returns the capacity of the column
col2.nonZeros(); // Returns the number of non-zero elements contained in the column
col2.resize( 84UL ); // Compilation error: Cannot resize a single column of a matrix
auto col3 = column( A, 3UL );
swap( col2, col3 ); // Compilation error: Swap operation not allowed
```

## Arithmetic Operations

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:

```
blaze::DynamicVector<double,blaze::columnVector> a( 2UL, 2.0 ), b;
blaze::CompressedVector<double,blaze::columnVector> c( 2UL );
c[1] = 3.0;
blaze::DynamicMatrix<double,blaze::columnMajor> A( 2UL, 4UL ); // Non-initialized 2x4 matrix
auto col0( column( A, 0UL ) ); // Reference to the 0th column of A
col0[0] = 0.0; // Manual initialization of the 0th column of A
col0[1] = 0.0;
column( A, 1UL ) = 1.0; // Homogeneous initialization of the 1st column of A
column( A, 2UL ) = a; // Dense vector initialization of the 2nd column of A
column( A, 3UL ) = c; // Sparse vector initialization of the 3rd column of A
b = col0 + a; // Dense vector/dense vector addition
b = c + column( A, 1UL ); // Sparse vector/dense vector addition
b = col0 * column( A, 2UL ); // Component-wise vector multiplication
column( A, 1UL ) *= 2.0; // In-place scaling of the 1st column
b = column( A, 1UL ) * 2.0; // Scaling of the 1st column
b = 2.0 * column( A, 1UL ); // Scaling of the 1st column
column( A, 2UL ) += a; // Addition assignment
column( A, 2UL ) -= c; // Subtraction assignment
column( A, 2UL ) *= column( A, 0UL ); // Multiplication assignment
double scalar = trans( c ) * column( A, 1UL ); // Scalar/dot/inner product between two vectors
A = column( A, 1UL ) * trans( c ); // Outer product between two vectors
```

## Views on Matrices with Non-Fitting Storage Order

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:

```
blaze::DynamicMatrix<int,blaze::rowMajor> A( 64UL, 32UL );
// ... Resizing and initialization
// Creating a reference to the 1st column of a column-major matrix A
auto col1 = column( A, 1UL );
for( auto it=col1.begin(); it!=col1.end(); ++it ) {
// ...
}
```

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:

```
// Setup of two row-major matrices
blaze::DynamicMatrix<double,blaze::rowMajor> A( 128UL, 128UL );
blaze::DynamicMatrix<double,blaze::rowMajor> B( 128UL, 128UL );
// ... Resizing and initialization
// The computation of the 15th column of the multiplication between A and B ...
blaze::DynamicVector<double,blaze::columnVector> x = column( A * B, 15UL );
// ... is essentially the same as the following computation, which multiplies
// A with the 15th column of the row-major matrix B.
blaze::DynamicVector<double,blaze::columnVector> x = A * column( B, 15UL );
```

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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