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Blaze 3.9
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Matrix adapter for strictly lower triangular matrices.
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#include <BaseTemplate.h>
Matrix adapter for strictly lower triangular matrices.
The StrictlyLowerMatrix 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 diagonal matrix elements and all matrix elements above the diagonal are 0 (strictly 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 strictly lower triangular matrices:
The storage order of a strictly lower triangular 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 strictly 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 strictly lower matrix will also be a column-major matrix.
A strictly lower triangular 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 strictly lower triangular 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 of the matrix, but not the elements on the diagonal or in the upper part of the matrix. Also, it is only possible to to assign matrices that are strictly lower matrices themselves:
The strictly lower matrix property is also enforced for strictly lower custom matrices: In case the given array of elements does not represent a strictly lower matrix, a std::invalid_argument exception is thrown:
Finally, the strictly lower matrix property is enforced for views (rows, columns, submatrices, ...) on the strictly lower matrix. The following example demonstrates that modifying the elements of an entire row and submatrix of a strictly lower matrix only affects the lower matrix elements:
The next example demonstrates the (compound) assignment to rows/columns and submatrices of strictly lower matrices. Since only lower elements may be modified the matrix to be assigned must be structured such that the strictly lower triangular matrix invariant of the strictly 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 strictly lower matrix this initialization is important since otherwise the strictly lower triangular matrix property of dense strictly lower matrices would not be guaranteed:
It is important to note that dense strictly lower matrices store all elements, including the elements on the diagonal and 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 diagonal and 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 StrictlyLowerMatrix adaptor merely restricts the interface to the matrix type MT and does not change the data layout or the underlying matrix type.
A StrictlyLowerMatrix 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 StrictlyLowerMatrix within arithmetic operations:
Note that it is possible to assign any kind of matrix to a strictly lower matrix. In case the matrix to be assigned is not strictly lower at compile time, a runtime check is performed.
It is also possible to use block-structured strictly lower matrices:
Also in this case the strictly 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 strictly lower triangular matrices whenever and wherever possible. Thus using a strictly lower triangular matrix instead of a general matrix can result in a considerable performance improvement. However, there are also situations when using a strictly lower matrix introduces some overhead. The following examples demonstrate several common situations where strictly lower matrices can positively or negatively impact performance.
When multiplying two matrices, at least one of which is strictly lower triangular, Blaze can exploit the fact that the diagonal and the upper part of the matrix contains only default elements and restrict the algorithm to the lower 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 and medium-sized matrices. Therefore is it highly recommended to use the StrictlyLowerMatrix adaptor when a matrix is known to be strictly 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 strictly lower triangular 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 strictly lower triangular matrix on the right-hand side of an assignment (i.e. for read access), which introduces absolutely no performance penalty, using a strictly lower matrix on the left-hand side of an assignment (i.e. for write access) may introduce additional overhead when it is assigned a matrix, which is not strictly lower triangular at compile time:
When assigning a general, potentially not strictly lower matrix to a strictly lower matrix it is necessary to check at runtime whether the general matrix is strictly lower in order to guarantee the strictly lower triangular property of the strictly lower matrix. In case it turns out to be strictly 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 strictly lower matrices to other strictly lower matrices.
In this context it is especially noteworthy that the addition, subtraction, and multiplication of two strictly lower triangular matrices always results in another strictly lower matrix: