F11DBF solves a system of linear equations involving the incomplete
$LU$ preconditioning matrix generated by
F11DAF.
F11DBF solves a system of linear equations
according to the value of the parameter
TRANS, where the matrix
$M=PLDUQ$, corresponds to an incomplete
$LU$ decomposition of a sparse matrix stored in coordinate storage (CS) format (see
Section 2.1.1 in the F11 Chapter Introduction), as generated by
F11DAF.
In the above decomposition
$L$ is a lower triangular sparse matrix with unit diagonal elements,
$D$ is a diagonal matrix,
$U$ is an upper triangular sparse matrix with unit diagonal elements and,
$P$ and
$Q$ are permutation matrices.
$L$,
$D$ and
$U$ are supplied to F11DBF through the matrix
which is an
N by
N sparse matrix, stored in CS format, as returned by
F11DAF. The permutation matrices
$P$ and
$Q$ are returned from
F11DAF via the arrays
IPIVP and
IPIVQ.
It is envisaged that a common use of F11DBF will be to carry out the preconditioning step required in the application of
F11BEF to sparse linear systems. F11DBF is used for this purpose by the Black Box routine
F11DCF.
F11DBF may also be used in combination with
F11DAF to solve a sparse system of linear equations directly (see
Section 8.5 in F11DAF). This use of F11DBF is demonstrated in
Section 9.
None.
If on entry
${\mathbf{IFAIL}}={\mathbf{0}}$ or
${-{\mathbf{1}}}$, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
If
${\mathbf{TRANS}}=\text{'N'}$ the computed solution
$x$ is the exact solution of a perturbed system of equations
$\left(M+\delta M\right)x=y$, where
$c\left(n\right)$ is a modest linear function of
$n$, and
$\epsilon $ is the
machine precision. An equivalent result holds when
${\mathbf{TRANS}}=\text{'T'}$.
The time taken for a call to F11DBF is proportional to the value of
NNZC returned from
F11DAF.
It is expected that a common use of F11DBF will be to carry out the preconditioning step required in the application of
F11BEF to sparse linear systems. In this situation F11DBF is likely to be called many times with the same matrix
$M$. In the interests of both reliability and efficiency, you are recommended to set
${\mathbf{CHECK}}=\text{'C'}$ for the first of such calls, and for all subsequent calls set
${\mathbf{CHECK}}=\text{'N'}$.
This example reads in a sparse nonsymmetric matrix
$A$ and a vector
$y$. It then calls
F11DAF, with
${\mathbf{LFILL}}=-1$ and
${\mathbf{DTOL}}=0.0$, to compute the
complete
$LU$ decomposition
Finally it calls F11DBF to solve the system