NAG Library Routine Document
F04CAF
1 Purpose
F04CAF computes the solution to a complex system of linear equations $AX=B$, where $A$ is an $n$ by $n$ matrix and $X$ and $B$ are $n$ by $r$ matrices. An estimate of the condition number of $A$ and an error bound for the computed solution are also returned.
2 Specification
INTEGER 
N, NRHS, LDA, IPIV(N), LDB, IFAIL 
REAL (KIND=nag_wp) 
RCOND, ERRBND 
COMPLEX (KIND=nag_wp) 
A(LDA,*), B(LDB,*) 

3 Description
The $LU$ decomposition with partial pivoting and row interchanges is used to factor $A$ as $A=PLU$, where $P$ is a permutation matrix, $L$ is unit lower triangular, and $U$ is upper triangular. The factored form of $A$ is then used to solve the system of equations $AX=B$.
4 References
Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999)
LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia
http://www.netlib.org/lapack/lug
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia
5 Parameters
 1: $\mathrm{N}$ – INTEGERInput

On entry: the number of linear equations $n$, i.e., the order of the matrix $A$.
Constraint:
${\mathbf{N}}\ge 0$.
 2: $\mathrm{NRHS}$ – INTEGERInput

On entry: the number of righthand sides $r$, i.e., the number of columns of the matrix $B$.
Constraint:
${\mathbf{NRHS}}\ge 0$.
 3: $\mathrm{A}\left({\mathbf{LDA}},*\right)$ – COMPLEX (KIND=nag_wp) arrayInput/Output

Note: the second dimension of the array
A
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
On entry: the $n$ by $n$ coefficient matrix $A$.
On exit: if ${\mathbf{IFAIL}}\ge {\mathbf{0}}$, the factors $L$ and $U$ from the factorization $A=PLU$. The unit diagonal elements of $L$ are not stored.
 4: $\mathrm{LDA}$ – INTEGERInput

On entry: the first dimension of the array
A as declared in the (sub)program from which F04CAF is called.
Constraint:
${\mathbf{LDA}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
 5: $\mathrm{IPIV}\left({\mathbf{N}}\right)$ – INTEGER arrayOutput

On exit: if ${\mathbf{IFAIL}}\ge {\mathbf{0}}$, the pivot indices that define the permutation matrix $P$; at the $i$th step row $i$ of the matrix was interchanged with row ${\mathbf{IPIV}}\left(i\right)$. ${\mathbf{IPIV}}\left(i\right)=i$ indicates a row interchange was not required.
 6: $\mathrm{B}\left({\mathbf{LDB}},*\right)$ – COMPLEX (KIND=nag_wp) arrayInput/Output

Note: the second dimension of the array
B
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{NRHS}}\right)$.
On entry: the $n$ by $r$ matrix of righthand sides $B$.
On exit: if ${\mathbf{IFAIL}}={\mathbf{0}}$ or ${\mathbf{N}+{\mathbf{1}}}$, the $n$ by $r$ solution matrix $X$.
 7: $\mathrm{LDB}$ – INTEGERInput

On entry: the first dimension of the array
B as declared in the (sub)program from which F04CAF is called.
Constraint:
${\mathbf{LDB}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
 8: $\mathrm{RCOND}$ – REAL (KIND=nag_wp)Output

On exit: if no constraints are violated, an estimate of the reciprocal of the condition number of the matrix $A$, computed as ${\mathbf{RCOND}}=1/\left({\Vert A\Vert}_{1}{\Vert {A}^{1}\Vert}_{1}\right)$.
 9: $\mathrm{ERRBND}$ – REAL (KIND=nag_wp)Output

On exit: if
${\mathbf{IFAIL}}={\mathbf{0}}$ or
${\mathbf{N}+{\mathbf{1}}}$, an estimate of the forward error bound for a computed solution
$\hat{x}$, such that
${\Vert \hat{x}x\Vert}_{1}/{\Vert x\Vert}_{1}\le {\mathbf{ERRBND}}$, where
$\hat{x}$ is a column of the computed solution returned in the array
B and
$x$ is the corresponding column of the exact solution
$X$. If
RCOND is less than
machine precision, then
ERRBND is returned as unity.
 10: $\mathrm{IFAIL}$ – INTEGERInput/Output

On entry:
IFAIL must be set to
$0$,
$1\text{ or}1$. If you are unfamiliar with this parameter you should refer to
Section 3.3 in the Essential Introduction 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 parameter, 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
${\mathbf{IFAIL}}={\mathbf{0}}$ or
${{\mathbf{1}}}$, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Errors or warnings detected by the routine:
 ${\mathbf{IFAIL}}<0\hspace{0.17em}\text{and}\hspace{0.17em}{\mathbf{IFAIL}}\ne 999$

If ${\mathbf{IFAIL}}=i$, the $i$th argument had an illegal value.
 ${\mathbf{IFAIL}}<0\hspace{0.17em}\text{and}\hspace{0.17em}{\mathbf{IFAIL}}\le {\mathbf{N}}$

If ${\mathbf{IFAIL}}=i$, ${u}_{ii}$ is exactly zero. The factorization has been completed, but the factor $U$ is exactly singular, so the solution could not be computed.
 ${\mathbf{IFAIL}}={\mathbf{N}}+1$

RCOND is less than
machine precision, so that the matrix
$A$ is numerically singular. A solution to the equations
$AX=B$ has nevertheless been computed.
 ${\mathbf{IFAIL}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
See
Section 3.8 in the Essential Introduction for further information.
 ${\mathbf{IFAIL}}=399$
Your licence key may have expired or may not have been installed correctly.
See
Section 3.7 in the Essential Introduction for further information.
 ${\mathbf{IFAIL}}=999$
Dynamic memory allocation failed.
See
Section 3.6 in the Essential Introduction for further information.
7 Accuracy
The computed solution for a single righthand side,
$\hat{x}$, satisfies an equation of the form
where
and
$\epsilon $ is the
machine precision. An approximate error bound for the computed solution is given by
where
$\kappa \left(A\right)={\Vert {A}^{1}\Vert}_{1}{\Vert A\Vert}_{1}$, the condition number of
$A$ with respect to the solution of the linear equations. F04CAF uses the approximation
${\Vert E\Vert}_{1}=\epsilon {\Vert A\Vert}_{1}$ to estimate
ERRBND. See Section 4.4 of
Anderson et al. (1999) for further details.
8 Parallelism and Performance
F04CAF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
F04CAF makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the
Users' Note for your implementation for any additional implementationspecific information.
The
complex
allocatable memory required is
$2\times {\mathbf{N}}$, and the real allocatable memory required is
$2\times {\mathbf{N}}$. In this case the factorization and the solution
$X$ have been computed, but
RCOND and
ERRBND have not been computed.
The total number of floatingpoint operations required to solve the equations $AX=B$ is proportional to $\left(\frac{2}{3}{n}^{3}+{n}^{2}r\right)$. The condition number estimation typically requires between four and five solves and never more than eleven solves, following the factorization.
In practice the condition number estimator is very reliable, but it can underestimate the true condition number; see Section 15.3 of
Higham (2002) for further details.
The real analogue of F04CAF is
F04BAF.
10 Example
This example solves the equations
where
and
An estimate of the condition number of $A$ and an approximate error bound for the computed solutions are also printed.
10.1 Program Text
Program Text (f04cafe.f90)
10.2 Program Data
Program Data (f04cafe.d)
10.3 Program Results
Program Results (f04cafe.r)