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