HLIBpro
2.7

For unsymmetric matrices, 𝖧𝖫𝖨𝖡𝗉𝗋𝗈 implements LU and LDU factorisation, i.e. the matrix \( A \) is factorised into
\[ A = LU \quad \textrm{or} \quad A = LDU \]
with a lower triangular, unit diagonal matrix \( L \), a upper triangular matrix \( U \) and, in the case of LDU a diagonal matrix \( D \) (in which case \( U \) is also unit diagonal).
Both matrix factorisation are available in two modes:
In the case of pointwise factorisation, the possibility of a breakdown is high because pivoting is only possible to a very small degree for Hmatrices and therefore not implemented in 𝖧𝖫𝖨𝖡𝗉𝗋𝗈. Therefore, blockwise factorisation is the default factorisation mode, since it can only break down in the case of singular diagonal blocks (in which case some remedy is avaibable, see below).
The factorisation methods are implemented in the classes TLU
and TLDU
. Factorising \( A \) using default parameters can be performed as
Here, a blockwise accuracy of \( 10^{4} \) were used during LU. By replacing TLU
with TLDU
, the corresponding factorisation method is used. Furthermore, for both variants, functional forms are available, called lu and ldu:
Matrix factorisation is performed in place, i.e. the individual factors are stored within \( A \), thereby overwriting the original content. That also means, that \( A \) can no longer be used as a standard matrix object, e.g. for matrix vector multiplication, since for that, the content needs special interpretation, e.g. first multiply with upper triangular part, then with lower triangular.
Special classes are avaiable in 𝖧𝖫𝖨𝖡𝗉𝗋𝗈 to perform such interpretation, namely:
TLUMatrix
, TLDUMatrix:
compute matrix vector product \( LU x = y \) and \( LDU x = y \) respectively,TLUInvMatrix
, TLDUInvMatrix:
compute matrix vector product \( (LU)^{1} x = y \) and \( (LDU))^{1} x = y \) respectively.Since the corresponding objects do not provide standard matrix functionality, they are representatives of the linear operator class TLinearOperator
. Linear operators only provide evaluation of the matrixvector product, which itself is a limited version of the standard TMatrix matrixvector multiplication since only updates of the destination vector are possible, e.g. either
\[ y = A \cdot x \]
or
\[ y = y + \alpha A \cdot x \]
However, also the transposed or hermitian evaluation are supported.
TLUMatrix
(TLDUMatrix
) and TLUInvMatrix
(TLDUInvMatrix
) objects may be created explicitely or by using the corresponding functions of TLU
(TLDU
):
So far, matrix factorisation was performed with default parameters. Options for matrix factorisation are provided in the form of a fac_options_t object. With it, you may select pointwise factorisation, e.g.:
Also, a progress meter may be assigned for the factorisation:
If \( A \) is symmetric ( \( A = A^T \)) or hermitian ( \( A = A^H \)), 𝖧𝖫𝖨𝖡𝗉𝗋𝗈 provides the Cholesky (LL) and the LDL factorisation. These are implemented in the classes TLL
and TLDL:
or available in functional form:
TLU
and TLDU
will not work for symmetric/hermitian matrices as only the lower triangular part is stored and therefore, will throw an exception if called with such a matrix.Evaluation of a factorised \( A \) is provided by TLLMatrix
and TLDLMatrix
, whereas evaluation of \( A^1 \) is implemented in TLLInvMatrix
and TLDLInvMatrix
. Please note, that the matrix format, e.g. whether symmetric or hermitian, is lost during factorisation but crucial for matrix evaluation and therefore, has to be provided while constructing these matrix objects:
TLDL
uses blockwise factorisation but can be switched to pointwise mode using fac_options_t objects as described above. Also, factorisation modifications/stabilisations may be activated for LDL factorisation. Neither of these is supported for Cholesky factorisation as this always used pointwise mode (blockwise is not possible). It is therefore not as stable as LDL factorisation.
Instead of chosing the factorisation technique by yourself and handling the creation of the corresponding linear operators, this can be simplified by using the functions factorise
and factorise_inv:
This works for unsymmetric and symmetric/hermitian matrices. Furthermore, options to the factorisation may be provided as an additional argument.