![]() |
Matrix adapter for diagonal matrices.
More...
#include <BaseTemplate.h>
Matrix adapter for diagonal matrices.
The DiagonalMatrix class template is an adapter for existing dense and sparse matrix types. It inherits the properties and the interface of the given matrix type MT and extends it by enforcing the additional invariant that all matrix elements above and below the diagonal are 0 (diagonal matrix). The type of the adapted matrix can be specified via the first template parameter:
The following examples give an impression of several possible diagonal matrices:
The storage order of a diagonal matrix is depending on the storage order of the adapted matrix type MT. In case the adapted matrix is stored in a row-wise fashion (i.e. is specified as blaze::rowMajor), the diagonal matrix will also be a row-major matrix. Otherwise, if the adapted matrix is column-major (i.e. is specified as blaze::columnMajor), the diagonal matrix will also be a column-major matrix.
A diagonal matrix is used exactly like a matrix of the underlying, adapted matrix type MT. It also provides (nearly) the same interface as the underlying matrix type. However, there are some important exceptions resulting from the diagonal matrix constraint:
In case a resizable matrix is used (as for instance blaze::HybridMatrix, blaze::DynamicMatrix, or blaze::CompressedMatrix), this means that the according constructors, the resize()
and the extend()
functions only expect a single parameter, which specifies both the number of rows and columns, instead of two (one for the number of rows and one for the number of columns):
In case a matrix with a fixed size is used (as for instance blaze::StaticMatrix), the number of rows and number of columns must be specified equally:
This means that it is only allowed to modify elements on the the diagonal of the matrix, but not the elements in the lower or upper part of the matrix. Also, it is only possible to assign matrices that are diagonal matrices themselves:
The diagonal matrix property is also enforced for diagonal custom matrices: In case the given array of elements does not represent a diagonal matrix, a std::invalid_argument exception is thrown:
Finally, the diagonal matrix property is enforced for views (rows, columns, submatrices, ...) on the diagonal matrix. The following example demonstrates that modifying the elements of an entire row and submatrix of a diagonal matrix only affects the diagonal matrix elements:
The next example demonstrates the (compound) assignment to rows/columns and submatrices of diagonal matrices. Since only diagonal elements may be modified the matrix to be assigned must be structured such that the diagonal matrix invariant of the diagonal matrix is preserved. Otherwise a std::invalid_argument exception is thrown:
Although this results in a small loss of efficiency during the creation of a dense diagonal matrix this initialization is important since otherwise the diagonal matrix property of dense diagonal matrices would not be guaranteed:
It is very important to note that dense diagonal matrices store all elements, including the non-diagonal elements, and therefore don't provide any kind of memory reduction! There are two main reasons for this: First, storing also the non-diagonal elements guarantees maximum performance for many algorithms that perform vectorized operations on the diagonal matrix, which is especially true for small dense matrices. Second, conceptually the DiagonalMatrix adaptor merely restricts the interface to the matrix type MT and does not change the data layout or the underlying matrix type. Thus, in order to achieve the perfect combination of performance and memory consumption it is recommended to use dense matrices for small diagonal matrices and sparse matrices for large diagonal matrices:
A DiagonalMatrix matrix can participate in numerical operations in any way any other dense or sparse matrix can participate. It can also be combined with any other dense or sparse vector or matrix. The following code example gives an impression of the use of DiagonalMatrix within arithmetic operations:
Note that it is possible to assign any kind of matrix to a diagonal matrix. In case the matrix to be assigned is not diagonal at compile time, a runtime check is performed.
It is also possible to use block-structured diagonal matrices:
Also in this case the diagonal matrix invariant is enforced, i.e. it is not possible to manipulate elements in the lower and upper part of the matrix:
The Blaze library tries to exploit the properties of diagonal matrices whenever and wherever possible. In fact, diagonal matrices come with several special kernels and additionally profit from all optimizations for symmetric and triangular matrices. Thus using a diagonal matrix instead of a general matrix can result in a considerable performance improvement. However, there are also situations when using a diagonal triangular matrix introduces some overhead. The following examples demonstrate several common situations where diagonal matrices can positively or negatively impact performance.
When multiplying two matrices, at least one of which is diagonal, Blaze can exploit the fact that the lower and upper part of the matrix contains only default elements and restrict the algorithm to the diagonal elements. The following example demonstrates this by means of a dense matrix/dense matrix multiplication:
In comparison to a general matrix multiplication, the performance advantage is significant, especially for large matrices. In this particular case, the multiplication performs similarly to a matrix addition since the complexity is reduced from to
. Therefore is it highly recommended to use the DiagonalMatrix adaptor when a matrix is known to be diagonal. Note however that the performance advantage is most pronounced for dense matrices and much less so for sparse matrices.
A similar performance improvement can be gained when using a diagonal matrix in a matrix/vector multiplication:
In this example, Blaze also exploits the structure of the matrix and performs similarly to a vector addition. Also in case of matrix/vector multiplications the performance improvement is most pronounced for dense matrices and much less so for sparse matrices.
In contrast to using a diagonal matrix on the right-hand side of an assignment (i.e. for read access), which introduces absolutely no performance penalty, using a diagonal matrix on the left-hand side of an assignment (i.e. for write access) may introduce additional overhead when it is assigned a general matrix, which is not diagonal at compile time:
When assigning a general, potentially not diagonal matrix to a diagonal matrix it is necessary to check whether the matrix is diagonal at runtime in order to guarantee the diagonal property of the diagonal matrix. In case it turns out to be diagonal, it is assigned as efficiently as possible, if it is not, an exception is thrown. In order to prevent this runtime overhead it is therefore generally advisable to assign diagonal matrices to other diagonal matrices.
In this context it is especially noteworthy that the addition, subtraction, and multiplication of two diagonal matrices always results in another diagonal matrix: