System upgrade: RLTFR8
============================

If the number of second members to be solved simultaneously is greater than 4, a block method implemented by routine RLBFR8 is used, otherwise MLTDRA is called for each second member.

**Data**: these are the pointers from MLTPRE:

:math:`\mathit{ADRESS}`,

:math:`\mathit{GLOBAL}`,

:math:`\mathit{SUPND}`,

:math:`\mathit{LGSN}`,

:math:`\mathit{ANC}`,

:math:`\mathit{NOUV}`,

:math:`\mathit{SEQ}`,

:math:`\mathit{LGBLOC}`,

:math:`\mathit{NCBLOC}`,

:math:`\mathit{DECAL}`.

And the factored matrix from MULFR8

:math:`\mathit{FACTOL}`:. VALF (and :math:`\mathit{FACTOU}`:. WALF in non-symmetric)

The second members:

:math:`\mathit{XSOL}(\mathrm{1 }\mathrm{:}\mathrm{\ne },\mathrm{1 }\mathrm{:}\mathit{NBSOL})`

**Results**: They are found in the table that contained the second members: :math:`\mathit{XSOL}`.




MLTDRA
------

This routine performs, for a single second member, the descent, the division by the term diagonal, and the ascent successively. In the descent, we use the matrix-vector product :math:`\mathit{DGEMV}` from Blas libraries, beyond a certain threshold. Below we optimize "by hand" in routine SSPMVB.

(This optimization is a vectorization written for CRAY, so it could be reviewed or even removed! )




RLBFR8
------

In the presence of several second members, the calls to DGEMV mentioned in the preceding paragraph are replaced by calls to DGEMM, matrix*matrix product, on matrix blocks of order :math:`\mathit{NB}`. We thus aim at the performance provided by these level 3 Blas, as in numerical factorization.

The routines called are:


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+ .. image:: images/10000000000000F2000000EF626AF42A426F7D59.png                                                              +
|    :width: 2.1453in                                                                                                         |
+    :height: 2.3327in                                                                                                        +
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**Figure 5 Descent-ascent by blocks**