F04 Chapter Contents
F04 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF04BBF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 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:     N – INTEGERInput
On entry: the number of linear equations $n$, i.e., the order of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 0$.
2:     KL – INTEGERInput
On entry: the number of subdiagonals ${k}_{l}$, within the band of $A$.
Constraint: ${\mathbf{KL}}\ge 0$.
3:     KU – INTEGERInput
On entry: the number of superdiagonals ${k}_{u}$, within the band of $A$.
Constraint: ${\mathbf{KU}}\ge 0$.
4:     NRHS – INTEGERInput
On entry: the number of right-hand sides $r$, i.e., the number of columns of the matrix $B$.
Constraint: ${\mathbf{NRHS}}\ge 0$.
5:     AB(LDAB,$*$) – 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
 $ABkl+ku+1+i-jj=Aij for ​max1,j-ku≤i≤minn,j+kl.$
See Section 8 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:     LDAB – INTEGERInput
On entry: the first dimension of the array AB as declared in the (sub)program from which F04BBF is called.
Constraint: ${\mathbf{LDAB}}\ge 2×{\mathbf{KL}}+{\mathbf{KU}}+1$.
7:     IPIV(N) – 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:     B(LDB,$*$) – 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 right-hand sides $B$.
On exit: if ${\mathbf{IFAIL}}={\mathbf{0}}$ or $\mathbf{N}+{\mathbf{1}}$, the $n$ by $r$ solution matrix $X$.
9:     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:   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 ${\mathbf{RCOND}}=1/\left({‖A‖}_{1}{‖{A}^{-1}‖}_{1}\right)$.
11:   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 $\stackrel{^}{x}$, such that ${‖\stackrel{^}{x}-x‖}_{1}/{‖x‖}_{1}\le {\mathbf{ERRBND}}$, where $\stackrel{^}{x}$ 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:   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:
If ${\mathbf{IFAIL}}=-i$, the $i$th argument had an illegal value.
${\mathbf{IFAIL}}=-999$
Allocation of memory failed. The integer allocatable memory required is N, and the real allocatable memory required is $3×{\mathbf{N}}$. In this case the factorization and the solution $X$ have been computed, but RCOND and ERRBND have not been computed.
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.

## 7  Accuracy

The computed solution for a single right-hand side, $\stackrel{^}{x}$, satisfies an equation of the form
 $A+E x^=b,$
where
 $E1 = Oε A1$
and $\epsilon$ is the machine precision. An approximate error bound for the computed solution is given by
 $x^-x1 x1 ≤ κA E1 A1 ,$
where $\kappa \left(A\right)={‖{A}^{-1}‖}_{1}{‖A‖}_{1}$, the condition number of $A$ with respect to the solution of the linear equations. F04BBF uses the approximation ${‖E‖}_{1}=\epsilon {‖A‖}_{1}$ to estimate ERRBND. See Section 4.4 of Anderson et al. (1999) for further details.

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:
 $* * * + + + * * a13 a24 a35 a46 * a12 a23 a34 a45 a56 a11 a22 a33 a44 a55 a66 a21 a32 a43 a54 a65 *$
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 floating point 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.

## 9  Example

This example solves the equations
 $AX=B,$
where $A$ is the band matrix
 $A= -0.23 2.54 -3.66 0 -6.98 2.46 -2.73 -2.13 0 2.56 2.46 4.07 0 0 -4.78 -3.82 and B= 4.42 -36.01 27.13 -31.67 -6.14 -1.16 10.50 -25.82 .$
An estimate of the condition number of $A$ and an approximate error bound for the computed solutions are also printed.

### 9.1  Program Text

Program Text (f04bbfe.f90)

### 9.2  Program Data

Program Data (f04bbfe.d)

### 9.3  Program Results

Program Results (f04bbfe.r)