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Blaze
3.6
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Matrix adapter for strictly upper triangular matrices.
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#include <BaseTemplate.h>
Matrix adapter for strictly upper triangular matrices.
The StrictlyUpperMatrix 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 below the diagonal are 0 (strictly upper 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 upper triangular matrices:
The storage order of a strictly upper 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 upper 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 upper matrix will also be a column-major matrix.
A strictly upper 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 upper 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 upper part of the matrix, but not the elements on the diagonal or in the lower part of the matrix. Also, it is only possible to assign matrices that are strictly upper triangular matrices themselves:
The strictly upper matrix property is also enforced for strictly upper custom matrices: In case the given array of elements does not represent a strictly upper matrix, a std::invalid_argument exception is thrown:
Finally, the strictly upper matrix property is enforced for views (rows, columns, submatrices, ...) on the strictly upper matrix. The following example demonstrates that modifying the elements of an entire row and submatrix of a strictly upper matrix only affects the upper matrix elements:
The next example demonstrates the (compound) assignment to rows/columns and submatrices of strictly upper matrices. Since only upper elements may be modified the matrix to be assigned must be structured such that the strictly upper triangular matrix invariant of the strictly upper 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 upper matrix this initialization is important since otherwise the strictly upper triangular matrix property of dense strictly upper matrices would not be guaranteed:
It is important to note that dense strictly upper matrices store all elements, including the elements on the diagonal and in the lower 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 lower elements guarantees maximum performance for many algorithms that perform vectorized operations on the upper matrix, which is especially true for small dense matrices. Second, conceptually the StrictlyUpperMatrix adaptor merely restricts the interface to the matrix type MT and does not change the data layout or the underlying matrix type.
An StrictlyUpperMatrix 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 StrictlyUpperMatrix within arithmetic operations:
Note that it is possible to assign any kind of matrix to a strictly upper matrix. In case the matrix to be assigned is not strictly upper at compile time, a runtime check is performed.
It is also possible to use block-structured strictly upper matrices:
Also in this case the strictly upper matrix invariant is enforced, i.e. it is not possible to manipulate elements in the lower part of the matrix:
The Blaze library tries to exploit the properties of strictly upper triangular matrices whenever and wherever possible. Thus using a strictly upper triangular matrix instead of a general matrix can result in a considerable performance improvement. However, there are also situations when using a strictly upper matrix introduces some overhead. The following examples demonstrate several common situations where strictly upper matrices can positively or negatively impact performance.
When multiplying two matrices, at least one of which is strictly upper triangular, Blaze can exploit the fact that the diagonal and the lower part of the matrix contains only default elements and restrict the algorithm to the upper 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 StrictlyUpperMatrix adaptor when a matrix is known to be strictly upper 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 upper 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 upper triangular matrix on the right-hand side of an assignment (i.e. for read access), which introduces absolutely no performance penalty, using a strictly upper 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 upper triangular at compile time:
When assigning a general, potentially not strictly upper matrix to a strictly upper matrix it is necessary to check whether the general matrix is strictly upper at runtime in order to guarantee the strictly upper triangular property of the strictly upper matrix. In case it turns out to be strictly upper 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 upper matrices to other strictly upper matrices.
In this context it is especially noteworthy that the addition, subtraction, and multiplication of two strictly upper triangular matrices always results in another strictly upper matrix: