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Matrix adapter for lower triangular matrices.
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#include <BaseTemplate.h>
Matrix adapter for lower triangular matrices.
The LowerMatrix 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 the diagonal are 0 (lower triangular matrix). The type of the adapted matrix can be specified via the first template parameter:
The following examples give an impression of several possible lower matrices:
The storage order of a lower 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 lower matrix will also be a row-major matrix. Otherwise, if the adapted matrix is column-major (i.e. is specified as blaze::columnMajor), the lower matrix will also be a column-major matrix.
A lower 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 lower 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 in the lower part or the diagonal of the matrix, but not the elements in the upper part of the matrix. Also, it is only possible to assign matrices that are lower matrices themselves:
The lower matrix property is also enforced for views (rows, columns, submatrices, ...) on the lower matrix. The following example demonstrates that modifying the elements of an entire row and submatrix of a lower matrix only affects the lower and diagonal matrix elements:
The next example demonstrates the (compound) assignment to rows/columns and submatrices of lower matrices. Since only lower and diagonal elements may be modified the matrix to be assigned must be structured such that the lower matrix invariant of the lower 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 lower matrix this initialization is important since otherwise the lower matrix property of dense lower matrices would not be guaranteed:
It is important to note that dense lower matrices store all elements, including the elements in the upper part of the matrix, and therefore don't provide any kind of memory reduction! There are two main reasons for this: First, storing also the upper elements guarantees maximum performance for many algorithms that perform vectorized operations on the lower matrix, which is especially true for small dense matrices. Second, conceptually the LowerMatrix adaptor merely restricts the interface to the matrix type MT and does not change the data layout or the underlying matrix type.
A LowerMatrix 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 LowerMatrix within arithmetic operations:
It is also possible to use block-structured lower matrices:
Also in this case the lower matrix invariant is enforced, i.e. it is not possible to manipulate elements in the upper part of the matrix:
The Blaze library tries to exploit the properties of lower matrices whenever and wherever possible. Thus using a lower triangular matrix instead of a general matrix can result in a considerable performance improvement. However, there are also situations when using a lower triangular matrix introduces some overhead. The following examples demonstrate several common situations where lower matrices can positively or negatively impact performance.
When multiplying two matrices, at least one of which is lower triangular, Blaze can exploit the fact that the upper part of the matrix contains only default elements and restrict the algorithm to the lower and 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. Therefore is it highly recommended to use the LowerMatrix adaptor when a matrix is known to be lower triangular. 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 lower matrix in a matrix/vector multiplication:
In this example, Blaze also exploits the structure of the matrix and approx. halves the runtime of the multiplication. 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 lower triangular matrix on the right-hand side of an assignment (i.e. for read access), which introduces absolutely no performance penalty, using a lower 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 lower triangular at compile time:
When assigning a general, potentially not lower matrix to a lower matrix it is necessary to check whether the matrix is lower at runtime in order to guarantee the lower triangular property of the lower matrix. In case it turns out to be lower triangular, 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 lower matrices to other lower matrices.
In this context it is especially noteworthy that the addition, subtraction, and multiplication of two lower triangular matrices always results in another lower matrix: