NAG Library Routine Document
F04BBF
1 Purpose
F04BBF computes the solution to a real system of linear equations $AX=B$, where $A$ is an $n$ by $n$ band matrix, with ${k}_{l}$ subdiagonals and ${k}_{u}$ superdiagonals, 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
SUBROUTINE F04BBF ( 
N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, RCOND, ERRBND, IFAIL) 
INTEGER 
N, KL, KU, NRHS, LDAB, IPIV(N), LDB, IFAIL 
REAL (KIND=nag_wp) 
AB(LDAB,*), B(LDB,*), RCOND, ERRBND 

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 the product of permutation matrices and unit lower triangular matrices with ${k}_{l}$ subdiagonals, and $U$ is upper triangular with $\left({k}_{l}+{k}_{u}\right)$ superdiagonals. 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: $\mathrm{KL}$ – INTEGERInput

On entry: the number of subdiagonals ${k}_{l}$, within the band of $A$.
Constraint:
${\mathbf{KL}}\ge 0$.
 3: $\mathrm{KU}$ – INTEGERInput

On entry: the number of superdiagonals ${k}_{u}$, within the band of $A$.
Constraint:
${\mathbf{KU}}\ge 0$.
 4: $\mathrm{NRHS}$ – INTEGERInput

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

Note: the second dimension of the array
AB
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
On entry: the
$n$ by
$n$ matrix
$A$.
The matrix is stored in rows
${k}_{l}+1$ to
$2{k}_{l}+{k}_{u}+1$; the first
${k}_{l}$ rows need not be set, more precisely, the element
${A}_{ij}$ must be stored in
See
Section 9 for further details.
On exit: if
${\mathbf{IFAIL}}\ge {\mathbf{0}}$,
AB is overwritten by details of the factorization.
The upper triangular band matrix $U$, with ${k}_{l}+{k}_{u}$ superdiagonals, is stored in rows $1$ to ${k}_{l}+{k}_{u}+1$ of the array, and the multipliers used to form the matrix $L$ are stored in rows ${k}_{l}+{k}_{u}+2$ to $2{k}_{l}+{k}_{u}+1$.
 6: $\mathrm{LDAB}$ – INTEGERInput

On entry: the first dimension of the array
AB as declared in the (sub)program from which F04BBF is called.
Constraint:
${\mathbf{LDAB}}\ge 2\times {\mathbf{KL}}+{\mathbf{KU}}+1$.
 7: $\mathrm{IPIV}\left({\mathbf{N}}\right)$ – 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.
 8: $\mathrm{B}\left({\mathbf{LDB}},*\right)$ – REAL (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 $X$.
 9: $\mathrm{LDB}$ – INTEGERInput

On entry: the first dimension of the array
B as declared in the (sub)program from which F04BBF is called.
Constraint:
${\mathbf{LDB}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
 10: $\mathrm{RCOND}$ – 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 ${\mathbf{RCOND}}=1/\left({\Vert A\Vert}_{1}{\Vert {A}^{1}\Vert}_{1}\right)$.
 11: $\mathrm{ERRBND}$ – 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.
 12: $\mathrm{IFAIL}$ – 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. F04BBF 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
F04BBF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
F04BBF 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 integer allocatable memory required is
N, and the real allocatable memory required is
$3\times {\mathbf{N}}$. In this case the factorization and the solution
$X$ have been computed, but
RCOND and
ERRBND have not been computed.
The band storage scheme for the array
AB is illustrated by the following example, when
$n=6$,
${k}_{l}=1$, and
${k}_{u}=2$.
Storage of the band matrix
$A$ in the array
AB:
Array elements marked $*$ need not be set and are not referenced by the routine. Array elements marked + need not be set, but are defined on exit from the routine and contain the elements
${u}_{14}$,
${u}_{25}$ and
${u}_{36}$.
The total number of floatingpoint operations required to solve the equations $AX=B$ depends upon the pivoting required, but if $n\gg {k}_{l}+{k}_{u}$ then it is approximately bounded by $\mathit{O}\left(n{k}_{l}\left({k}_{l}+{k}_{u}\right)\right)$ for the factorization and $\mathit{O}\left(n\left(2{k}_{l}+{k}_{u}\right)r\right)$ for the solution following the factorization. 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 complex analogue of F04BBF is
F04CBF.
10 Example
This example solves the equations
where
$A$ is the band matrix
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 (f04bbfe.f90)
10.2 Program Data
Program Data (f04bbfe.d)
10.3 Program Results
Program Results (f04bbfe.r)