HLIBpro
2.6

The foundation of HLIBpro is provided by the BLAS and LAPACK software packages. Especially the advanced functions of LAPACK, e.g. singular value decomposition, play a crucial role in the Harithmetics. To simplify the usage of these functions and to reduce the potential for errors, an extra layer was implemented between the Harithmetics and the BLAS/LAPACK functions. The BLAS module provides this layer.
All data classes in the BLAS namespace share special properties with respect to the memory management of the underlying data:
Since the matrix and vector classes were implemented with BLAS/LAPACK in mind, some restrictions apply:
For the following code, an
is assumed.
Due to performance issues on multicore systems the memory management of the BLAS classes was changed in HLIBpro. Please read the following carefully to prevent programming errors.
Both vector and matrix classes in the BLAS namespace store a pointer to some memory block. This memory may be either owned or referenced by the corresponding object. If the pointer only references some memory block owned by another object, no reference counting is done. Hence, if the owning object is freed, the reference will be invalid.
A major cause of such problems are temporary objects, e.g. some auxilliary vectors or matrices inside a function, which are assigned to some return value:
Here, x
will by default only reference t
. To make the
new owner of the data in t
, one has to use the move semantics in C++:
Now, t
will be empty and x
owns the data of t
.
If the result is the return value of the function, the compiler will automatically choose the move semantics:
A range is an index set in the BLAS namespace.
Here, an index set containing \(\left\{0,1,\ldots,10\right\}\) is defined. Please note, that the index interval is closed on both ends, e.g. the first and the last index is given.
In addition to a standard index set, you may also define a stride, e.g. a distance between elements in the index set:
This defines the index set \(\left\{10,12,14,\ldots,20\right\}\)
For convenience, a constructor for an index set containing just one element is also available:
yields \(\left\{21\right\}\).
Furthermore, the special range Range::all indicates, that the whole range shall be used:
Here, all columns of M
are used in M2
but only the rows 5 to 10.
Usually, a BLAS vector is intialised with the corresponding size as the constructor argument:
creates a double vector of size 10. Leaving out the size argument, intialises the vector with size 0:
Providing another vector in the constructor, creates a new vector pointing to the same data as the argument vector:
Changing z
also changes x!
With the usage of range objects, parts of vectors can be represented as another vector:
Here, x2
addresses the second half of the coefficients of x1 whereas x3
contains each second coefficient of x2
and by that, the coefficients 5, 7 and 9 of x1
.
By default, the policy for the copy constructors is to use a reference to the given data. You may change that, such that a copy of the data is created, e.g. a new memory block is allocated:
Now, x3
does not address any coefficients in x1
or x2
.
Single coefficients in vectors are accessed using the () operator:
The length of a vector is accessible via the length()
function:
Most of the BLAS vector functions are available in HLIBpro and some, which have no representation in the BLAS package.
An examples, involving filling, scaling vectors, forming linear combinations and computing dot products:
As mentioned before, you can not mix vectors of different value type, e.g.
since BLAS/LAPACK do not support such mixing
For a complete overview of the algebra functions refer to the BLAS module.
Constructing matrices follows the same scheme as vectors, with the only difference being the additional dimension:
Addressing subblocks of a given matrix is done by providing Range objects:
Here, M2
references the last 10 rows of M1
. M3
also shares these coefficients but further restricts the elements to each second column of M1
. Hence
will affect the same memory position.
The standard copy policy for matrices, i.e. copy_reference
, can be changed by the last, optional parameter to the above constructor:
This will create a 10ร10 matrix with coefficients copied from matrix M2
.
As for vectors, the arithmetic functionality is implemented as functions and not as special C++ operators for objects of type Matrix.
๐ง๐ซ๐จ๐ก๐๐๐ provides different <emph>matrix view</emph> for accessing matrices, e.g. transposed and hermitian view. A view object will not change or copy the data of a matrix but changes for instance the way the data is access, e.g. interchanging the indices. For a given matrix
The transposed view of M
is defined by
and the adjoint view by
These views may be used for arithmetic functions, albeit not all functions, to simplify the usage and to enhance the performance of these functions.
The most common form of matrixvector arithmetic is the matrix vector multiplication in the general form
\[ y := \alpha A x + \beta y \]
:
Using matrix views allows the multiplication with the transposed or adjoint matrix:
If M
is a triangular matrix, the multiplication is performed in place, i.e. the single argument vector is modified:
\[ x := A \cdot x \]
The exact shape of the matrix is defined by additional arguments of type tri_type_t
(lower_triangular
or upper_triangular
) and diag_type_t
(unit_diag
or general_diag
). The second argument describes the diagonal which is either identical to 1 or not:
Also available is the rank1 update of a matrix
\[ A := A + \alpha a \cdot b^H \]
:
For triangular matrices \(A\), solving equation systems \(Ax=b\) is supported via:
Here, upon input x
contains \(b\)b and is overwritten with the result \(x\).
The BLAS namespace defines common functions for single matrices and between matrices. Single matrix functions are
Different matrix norms are available:
Other functions, involving more matrices are:
Inversion and factorisation functions are:
It is also possible to solve matrix equations of the form
\[ A X = B \]
with known \(A\) and \(b\):
X
will initially contain the matrix \(B\). Furthermore, the content of the matrix A
will be destroyed during computation.
If \(A\) is a triangular matrix, the equation may take the more general forms:
\[ A X = \alpha B \quad \textrm{or} \quad X A = \alpha B \]
The corresponding side from which to apply \(A\) to \(X\) is defined by a parameter (form_left
or form_right
):
Eigenvalues and singular values and vectors of a matrix can also be computed:
If M
is symmetric, selected eigen values may be computed: