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Row selections provide views on arbitrary compositions of rows of dense and sparse matrices. These views act as a reference to the selected rows and represent them as another dense or sparse matrix. This reference is valid and can be used in every way any other dense or sparse matrix can be used as long as the matrix containing the rows is not resized or entirely destroyed. The row selection also acts as an alias to the matrix elements in the specified range: Changes made to the rows (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 rows.


Setup of Row Selections

A row selection can be created very conveniently via the rows() function. It can be included via the header file

#include <blaze/math/Rows.h>

The indices of the rows to be selected can be specified either at compile time or at runtime (by means of an initializer list, array or vector):

blaze::DynamicMatrix<double,blaze::rowMajor> A;
// ... Resizing and initialization

// Selecting the rows 4, 6, 8, and 10 (compile time arguments)
auto rs1 = rows<4UL,6UL,8UL,10UL>( A );

// Selecting the rows 3, 2, and 1 (runtime arguments via an initializer list)
const std::initializer_list<size_t> list{ 3UL, 2UL, 1UL };
auto rs2 = rows( A, { 3UL, 2UL, 1UL } );
auto rs3 = rows( A, list );

// Selecting the rows 1, 2, 3, 3, 2, and 1 (runtime arguments via a std::array)
const std::array<size_t> array{ 1UL, 2UL, 3UL, 3UL, 2UL, 1UL };
auto rs4 = rows( A, array );
auto rs5 = rows( A, array.data(), array.size() );

// Selecting the row 4 fives times (runtime arguments via a std::vector)
const std::vector<size_t> vector{ 4UL, 4UL, 4UL, 4UL, 4UL };
auto rs6 = rows( A, vector );
auto rs7 = rows( A, vector.data(), vector.size() );

Note that it is possible to alias the rows of the underlying matrix in any order. Also note that it is possible to use the same index multiple times.

Alternatively it is possible to pass a callable such as a lambda or functor that produces the indices:

blaze::DynamicMatrix<double,blaze::rowMajor> A( 9UL, 18UL );

// Selecting all even rows of the matrix, i.e. selecting the rows 0, 2, 4, 6, and 8
auto rs1 = rows( A, []( size_t i ){ return i*2UL; }, 5UL );

// Selecting all odd rows of the matrix, i.e. selecting the rows 1, 3, 5, and 7
auto rs2 = rows( A, []( size_t i ){ return i*2UL+1UL; }, 4UL );

// Reversing the rows of the matrix, i.e. selecting the rows 8, 7, 6, 5, 4, 3, 2, 1, and 0
auto rs3 = rows( A, [max=A.rows()-1UL]( size_t i ){ return max-i; }, 9UL );

The rows() function returns an expression representing the view on the selected rows. The type of this expression depends on the given 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:

using MatrixType = blaze::DynamicMatrix<int>;
using RowsType = decltype( blaze::rows<3UL,0UL,4UL,8UL>( std::declval<MatrixType>() ) );

The resulting view can be treated as any other dense or sparse matrix, i.e. it can be assigned to, it can be copied from, and it can be used in arithmetic operations. Note, however, that a row selection will always be treated as a row-major matrix, regardless of the storage order of the matrix containing the rows. The view can also be used on both sides of an assignment: It can either be used as an alias to grant write access to specific rows of a matrix primitive on the left-hand side of an assignment or to grant read-access to specific rows of a matrix primitive or expression on the right-hand side of an assignment. The following example demonstrates this in detail:

blaze::DynamicMatrix<double,blaze::rowMajor> A;
blaze::DynamicMatrix<double,blaze::columnMajor> B;
blaze::CompressedMatrix<double,blaze::rowMajor> C;
// ... Resizing and initialization

// Selecting the rows 1, 3, 5, and 7 of A
auto rs = rows( A, { 1UL, 3UL, 5UL, 7UL } );

// Setting rows 1, 3, 5, and 7 of A to row 4 of B
rs = rows( B, { 4UL, 4UL, 4UL, 4UL } );

// Setting the rows 2, 4, 6, and 8 of A to C
rows( A, { 2UL, 4UL, 6UL, 8UL } ) = C;

// Setting the first 4 rows of A to the rows 5, 4, 3, and 2 of C
submatrix( A, 0UL, 0UL, 4UL, A.columns() ) = rows( C, { 5UL, 4UL, 3UL, 2UL } );

// Rotating the result of the addition between rows 1, 3, 5, and 7 of A and C
B = rows( rs + C, { 2UL, 3UL, 0UL, 1UL } );

Warning: It is the programmer's responsibility to ensure the row selection does not outlive the viewed matrix:

// Creating a row selection on a temporary matrix; results in a dangling reference!
auto rs = rows<2UL,0UL>( DynamicMatrix<int>{ { 1, 2, 3 }, { 4, 5, 6 }, { 7, 8, 9 } } );

Element Access

The elements of a row selection can be directly accessed via the function call operator:

blaze::DynamicMatrix<double,blaze::rowMajor> A;
// ... Resizing and initialization

// Creating a view on the first four rows of A in reverse order
auto rs = rows( A, { 3UL, 2UL, 1UL, 0UL } );

// Setting the element (0,0) of the row selection, which corresponds
// to the element at position (3,0) in matrix A
rs(0,0) = 2.0;

Alternatively, the elements of a row selection can be traversed via (const) iterators. Just as with matrices, in case of non-const row selection, begin() and end() return an iterator, which allows to manipuate the elements, in case of constant row selection an iterator to immutable elements is returned:

blaze::DynamicMatrix<int,blaze::rowMajor> A( 256UL, 512UL );
// ... Resizing and initialization

// Creating a reference to a selection of rows of matrix A
auto rs = rows( A, { 16UL, 32UL, 64UL, 128UL } );

// Traversing the elements of the 0th row via iterators to non-const elements
for( auto it=rs.begin(0); it!=rs.end(0); ++it ) {
   *it = ...;  // OK: Write access to the dense value.
   ... = *it;  // OK: Read access to the dense value.
}

// Traversing the elements of the 1st row via iterators to const elements
for( auto it=rs.cbegin(1); it!=rs.cend(1); ++it ) {
   *it = ...;  // Compilation error: Assignment to the value via iterator-to-const is invalid.
   ... = *it;  // OK: Read access to the dense value.
}
blaze::CompressedMatrix<int,blaze::rowMajor> A( 256UL, 512UL );
// ... Resizing and initialization

// Creating a reference to a selection of rows of matrix A
auto rs = rows( A, { 16UL, 32UL, 64UL, 128UL } );

// Traversing the elements of the 0th row via iterators to non-const elements
for( auto it=rs.begin(0); it!=rs.end(0); ++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 of the 1st row via iterators to const elements
for( auto it=rs.cbegin(1); it!=rs.cend(1); ++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 row selection can be done by several alternative functions. The following example demonstrates all options:

blaze::CompressedMatrix<double,blaze::rowMajor> A( 256UL, 512UL );  // Non-initialized matrix of size 256x512

auto rs = rows( A, { 10UL, 20UL, 30UL, 40UL } );  // View on the rows 10, 20, 30, and 40 of A

// The function call operator provides access to all possible elements of the sparse row
// selection, including the zero elements. In case the function call operator is used to
// access an element that is currently not stored in the sparse row selection, the element
// is inserted into the row selection.
rs(2,4) = 2.0;

// The second operation for inserting elements is the set() function. In case the element is
// not contained in the row selection it is inserted into the row selection, if it is already
// contained in the row selection its value is modified.
rs.set( 2UL, 5UL, -1.2 );

// An alternative for inserting elements into the row selection is the insert() function.
// However, it inserts the element only in case the element is not already contained in the
// row selection.
rs.insert( 2UL, 6UL, 3.7 );

// Just as in the case of sparse matrices, elements can also be inserted via the append()
// function. In case of row selections, append() also requires that the appended element's
// index is strictly larger than the currently largest non-zero index in the according row
// of the row selection and that the according row's capacity is large enough to hold the new
// element. Note however that due to the nature of a row selection, which may be an alias to
// an arbitrary collection of rows, the append() function does not work as efficiently for
// a row selection as it does for a matrix.
rs.reserve( 2UL, 10UL );
rs.append( 2UL, 10UL, -2.1 );

Common Operations

A view on specific rows of a matrix can be used like any other dense or sparse matrix. For instance, the current size of the matrix, i.e. the number of rows or columns can be obtained via the rows() and columns() functions, the current total capacity via the capacity() function, and the number of non-zero elements via the nonZeros() function. However, since row selections are views on specific rows of a matrix, several operations are not possible, such as resizing and swapping:

blaze::DynamicMatrix<int,blaze::rowMajor> A( 42UL, 42UL );
// ... Resizing and initialization

// Creating a view on the rows 8, 16, 24, and 32 of matrix A
auto rs = rows( A, { 8UL, 16UL, 24UL, 32UL } );

rs.rows();      // Returns the number of rows of the row selection
rs.columns();   // Returns the number of columns of the row selection
rs.capacity();  // Returns the capacity of the row selection
rs.nonZeros();  // Returns the number of non-zero elements contained in the row selection

rs.resize( 10UL, 8UL );  // Compilation error: Cannot resize a row selection

auto rs2 = rows( A, 9UL, 17UL, 25UL, 33UL );
swap( rs, rs2 );  // Compilation error: Swap operation not allowed

Arithmetic Operations

Both dense and sparse row selections can be used in all arithmetic operations that any other dense or sparse matrix can be used in. The following example gives an impression of the use of dense row selctions within arithmetic operations. All operations (addition, subtraction, multiplication, scaling, ...) can be performed on all possible combinations of dense and sparse matrices with fitting element types:

blaze::DynamicMatrix<double,blaze::rowMajor> D1, D2, D3;
blaze::CompressedMatrix<double,blaze::rowMajor> S1, S2;

blaze::CompressedVector<double,blaze::columnVector> a, b;

// ... Resizing and initialization

std::initializer_list<size_t> indices1{ 0UL, 3UL, 6UL,  9UL, 12UL, 15UL, 18UL, 21UL };
std::initializer_list<size_t> indices2{ 1UL, 4UL, 7UL, 10UL, 13UL, 16UL, 19UL, 22UL };
std::initializer_list<size_t> indices3{ 2UL, 5UL, 8UL, 11UL, 14UL, 17UL, 20UL, 23UL };

auto rs = rows( D1, indices1 );  // Selecting the every third row of D1 in the range [0..21]

rs = D2;                    // Dense matrix assignment to the selected rows
rows( D1, indices2 ) = S1;  // Sparse matrix assignment to the selected rows

D3 = rs + D2;                    // Dense matrix/dense matrix addition
S2 = S1 - rows( D1, indices2 );  // Sparse matrix/dense matrix subtraction
D2 = rs % rows( D1, indices3 );  // Dense matrix/dense matrix Schur product
D2 = rows( D1, indices2 ) * D1;  // Dense matrix/dense matrix multiplication

rows( D1, indices2 ) *= 2.0;      // In-place scaling of the second selection of rows
D2 = rows( D1, indices3 ) * 2.0;  // Scaling of the elements in the third selection of rows
D2 = 2.0 * rows( D1, indices3 );  // Scaling of the elements in the third selection of rows

rows( D1, indices1 ) += D2;  // Addition assignment
rows( D1, indices2 ) -= S1;  // Subtraction assignment
rows( D1, indices3 ) %= rs;  // Schur product assignment

a = rows( D1, indices1 ) * b;  // Dense matrix/sparse vector multiplication

Row Selections on Column-Major Matrices

Especially noteworthy is that row selections 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::columnMajor> A( 64UL, 32UL );
// ... Resizing and initialization

// Creating a reference to the 1st and 3rd row of a column-major matrix A
auto rs = rows( A, { 1UL, 3UL } );

// Traversing row 0 of the selection, which corresponds to the 1st row of matrix A
for( auto it=rs.begin( 0UL ); it!=rs.end( 0UL ); ++it ) {
   // ...
}

However, please note that creating a row selection on a matrix stored in a column-major fashion can result in a considerable performance decrease in comparison to a row selection on a matrix with row-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 column-major matrices
blaze::DynamicMatrix<double,blaze::columnMajor> A( 128UL, 128UL );
blaze::DynamicMatrix<double,blaze::columnMajor> B( 128UL, 128UL );
// ... Resizing and initialization

// The computation of the 15th, 30th, and 45th row of the multiplication between A and B ...
blaze::DynamicMatrix<double,blaze::rowMajor> x = rows( A * B, { 15UL, 30UL, 45UL } );

// ... is essentially the same as the following computation, which multiplies
// the 15th, 30th, and 45th row of the column-major matrix A with B.
blaze::DynamicMatrix<double,blaze::rowMajor> x = rows( A, { 15UL, 30UL, 45UL } ) * B;

Although Blaze performs the resulting matrix/matrix multiplication as efficiently as possible using a row-major storage order for matrix A would result in a more efficient evaluation.


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