NAG Library Routine Document
f11dpf (complex_gen_precon_ilu_solve)
1
Purpose
f11dpf solves a system of complex linear equations involving the incomplete
$LU$ preconditioning matrix generated by
f11dnf.
2
Specification
Fortran Interface
Subroutine f11dpf ( 
trans, n, a, la, irow, icol, ipivp, ipivq, istr, idiag, check, y, x, ifail) 
Integer, Intent (In)  ::  n, la, irow(la), icol(la), istr(n+1), idiag(n)  Integer, Intent (Inout)  ::  ipivp(n), ipivq(n), ifail  Complex (Kind=nag_wp), Intent (In)  ::  a(la), y(n)  Complex (Kind=nag_wp), Intent (Out)  ::  x(n)  Character (1), Intent (In)  ::  trans, check 

C Header Interface
#include nagmk26.h
void 
f11dpf_ (const char *trans, const Integer *n, const Complex a[], const Integer *la, const Integer irow[], const Integer icol[], Integer ipivp[], Integer ipivq[], const Integer istr[], const Integer idiag[], const char *check, const Complex y[], Complex x[], Integer *ifail, const Charlen length_trans, const Charlen length_check) 

3
Description
f11dpf solves a system of complex linear equations
according to the value of the argument
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 9.5 in
f11dnf). This use of
f11dpf is illustrated in
Section 10.
4
References
None.
5
Arguments
 1: $\mathbf{trans}$ – Character(1)Input

On entry: specifies whether or not the matrix
$M$ is transposed.
 ${\mathbf{trans}}=\text{'N'}$
 $Mx=y$ is solved.
 ${\mathbf{trans}}=\text{'T'}$
 ${M}^{\mathrm{T}}x=y$ is solved.
Constraint:
${\mathbf{trans}}=\text{'N'}$ or $\text{'T'}$.
 2: $\mathbf{n}$ – IntegerInput

On entry:
$n$, the order of the matrix
$M$. This
must be the same value as was supplied in the preceding call to
f11dnf.
Constraint:
${\mathbf{n}}\ge 1$.
 3: $\mathbf{a}\left({\mathbf{la}}\right)$ – Complex (Kind=nag_wp) arrayInput

On entry: the values returned in the array
a by a previous call to
f11dnf.
 4: $\mathbf{la}$ – IntegerInput

On entry: the dimension of the arrays
a,
irow and
icol as declared in the (sub)program from which
f11dpf is called. This
must be the same value supplied in the preceding call to
f11dnf.
 5: $\mathbf{irow}\left({\mathbf{la}}\right)$ – Integer arrayInput
 6: $\mathbf{icol}\left({\mathbf{la}}\right)$ – Integer arrayInput
 7: $\mathbf{ipivp}\left({\mathbf{n}}\right)$ – Integer arrayInput
 8: $\mathbf{ipivq}\left({\mathbf{n}}\right)$ – Integer arrayInput
 9: $\mathbf{istr}\left({\mathbf{n}}+1\right)$ – Integer arrayInput
 10: $\mathbf{idiag}\left({\mathbf{n}}\right)$ – Integer arrayInput

On entry: the values returned in arrays
irow,
icol,
ipivp,
ipivq,
istr and
idiag by a previous call to
f11dnf.
 11: $\mathbf{check}$ – Character(1)Input

On entry: specifies whether or not the CS representation of the matrix
$M$ should be checked.
 ${\mathbf{check}}=\text{'C'}$
 Checks are carried on the values of n, irow, icol, ipivp, ipivq, istr and idiag.
 ${\mathbf{check}}=\text{'N'}$
 None of these checks are carried out.
Constraint:
${\mathbf{check}}=\text{'C'}$ or $\text{'N'}$.
 12: $\mathbf{y}\left({\mathbf{n}}\right)$ – Complex (Kind=nag_wp) arrayInput

On entry: the righthand side vector $y$.
 13: $\mathbf{x}\left({\mathbf{n}}\right)$ – Complex (Kind=nag_wp) arrayOutput

On exit: the solution vector $x$.
 14: $\mathbf{ifail}$ – IntegerInput/Output

On entry:
ifail must be set to
$0$,
$1\text{ or}1$. If you are unfamiliar with this argument you should refer to
Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
$1\text{ or}1$ is recommended. If the output of error messages is undesirable, then the value
$1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is
$0$.
When the value $\mathbf{1}\text{ or}\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit:
${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see
Section 6).
6
Error Indicators and Warnings
If on entry
${\mathbf{ifail}}=0$ or
$1$, explanatory error messages are output on the current error message unit (as defined by
x04aaf).
Errors or warnings detected by the routine:
 ${\mathbf{ifail}}=1$

On entry,  ${\mathbf{trans}}\ne \text{'N'}$ or $\text{'T'}$, 
or  ${\mathbf{check}}\ne \text{'C'}$ or $\text{'N'}$. 
 ${\mathbf{ifail}}=2$

On entry,  ${\mathbf{n}}<1$. 
 ${\mathbf{ifail}}=3$

On entry, the CS representation of the preconditioning matrix
$M$ is invalid. Further details are given in the error message. Check that the call to
f11dpf has been preceded by a valid call to
f11dnf and that the arrays
a,
irow,
icol,
ipivp,
ipivq,
istr and
idiag have not been corrupted between the two calls.
 ${\mathbf{ifail}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
See
Section 3.9 in How to Use the NAG Library and its Documentation for further information.
 ${\mathbf{ifail}}=399$
Your licence key may have expired or may not have been installed correctly.
See
Section 3.8 in How to Use the NAG Library and its Documentation for further information.
 ${\mathbf{ifail}}=999$
Dynamic memory allocation failed.
See
Section 3.7 in How to Use the NAG Library and its Documentation for further information.
7
Accuracy
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
${\mathbf{trans}}=\text{'T'}$.
8
Parallelism and Performance
f11dpf is not threaded in any implementation.
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.
10
Example
This example reads in a complex sparse nonHermitian 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
10.1
Program Text
Program Text (f11dpfe.f90)
10.2
Program Data
Program Data (f11dpfe.d)
10.3
Program Results
Program Results (f11dpfe.r)