F11DPF solves a system of complex linear equations involving the incomplete
$LU$ preconditioning matrix generated by
F11DNF.
F11DPF solves a system of complex linear equations
according to the value of the parameter
TRANS, where the matrix
$M=PLDUQ$ corresponds to an incomplete
$LU$ decomposition of a complex sparse matrix stored in coordinate storage (CS) format (see
Section 2.1.1 in the F11 Chapter Introduction), as generated by
F11DNF.
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 F11DPF through the matrix
which is an
N by
N sparse matrix, stored in CS format, as returned by
F11DNF. The permutation matrices
$P$ and
$Q$ are returned from
F11DNF via the arrays
IPIVP and
IPIVQ.
It is envisaged that a common use of F11DPF will be to carry out the preconditioning step required in the application of
F11BSF to sparse complex linear systems. F11DPF is used for this purpose by the Black Box routine
F11DQF.
F11DPF may also be used in combination with
F11DNF to solve a sparse system of complex linear equations directly (see
Section 8.5 in F11DNF). This use of F11DPF is illustrated 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 F11DPF is proportional to the value of
NNZC returned from
F11DNF.
It is expected that a common use of F11DPF will be to carry out the preconditioning step required in the application of
F11BSF to sparse complex linear systems. In this situation F11DPF 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 to set
${\mathbf{CHECK}}=\text{'N'}$ for all subsequent calls.
This example reads in a complex sparse non-Hermitian matrix
$A$ and a vector
$y$. It then calls
F11DNF, with
${\mathbf{LFILL}}=-1$ and
${\mathbf{DTOL}}=0.0$, to compute the
complete
$LU$ decomposition
Finally it calls F11DPF to solve the system