# NAG Library Routine Document

## 1Purpose

f07bef (dgbtrs) solves a real band system of linear equations with multiple right-hand sides,
 $AX=B or ATX=B ,$
where $A$ has been factorized by f07bdf (dgbtrf).

## 2Specification

Fortran Interface
 Subroutine f07bef ( n, kl, ku, nrhs, ab, ldab, ipiv, b, ldb, info)
 Integer, Intent (In) :: n, kl, ku, nrhs, ldab, ipiv(*), ldb Integer, Intent (Out) :: info Real (Kind=nag_wp), Intent (In) :: ab(ldab,*) Real (Kind=nag_wp), Intent (Inout) :: b(ldb,*) Character (1), Intent (In) :: trans
#include <nagmk26.h>
 void f07bef_ (const char *trans, const Integer *n, const Integer *kl, const Integer *ku, const Integer *nrhs, const double ab[], const Integer *ldab, const Integer ipiv[], double b[], const Integer *ldb, Integer *info, const Charlen length_trans)
The routine may be called by its LAPACK name dgbtrs.

## 3Description

f07bef (dgbtrs) is used to solve a real band system of linear equations $AX=B$ or ${A}^{\mathrm{T}}X=B$, the routine must be preceded by a call to f07bdf (dgbtrf) which computes the $LU$ factorization of $A$ as $A=PLU$. The solution is computed by forward and backward substitution.
If ${\mathbf{trans}}=\text{'N'}$, the solution is computed by solving $PLY=B$ and then $UX=Y$.
If ${\mathbf{trans}}=\text{'T'}$ or $\text{'C'}$, the solution is computed by solving ${U}^{\mathrm{T}}Y=B$ and then ${L}^{\mathrm{T}}{P}^{\mathrm{T}}X=Y$.

## 4References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## 5Arguments

1:     $\mathbf{trans}$ – Character(1)Input
On entry: indicates the form of the equations.
${\mathbf{trans}}=\text{'N'}$
$AX=B$ is solved for $X$.
${\mathbf{trans}}=\text{'T'}$ or $\text{'C'}$
${A}^{\mathrm{T}}X=B$ is solved for $X$.
Constraint: ${\mathbf{trans}}=\text{'N'}$, $\text{'T'}$ or $\text{'C'}$.
2:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3:     $\mathbf{kl}$ – IntegerInput
On entry: ${k}_{l}$, the number of subdiagonals within the band of the matrix $A$.
Constraint: ${\mathbf{kl}}\ge 0$.
4:     $\mathbf{ku}$ – IntegerInput
On entry: ${k}_{u}$, the number of superdiagonals within the band of the matrix $A$.
Constraint: ${\mathbf{ku}}\ge 0$.
5:     $\mathbf{nrhs}$ – IntegerInput
On entry: $r$, the number of right-hand sides.
Constraint: ${\mathbf{nrhs}}\ge 0$.
6:     $\mathbf{ab}\left({\mathbf{ldab}},*\right)$ – Real (Kind=nag_wp) arrayInput
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 $LU$ factorization of $A$, as returned by f07bdf (dgbtrf).
7:     $\mathbf{ldab}$ – IntegerInput
On entry: the first dimension of the array ab as declared in the (sub)program from which f07bef (dgbtrs) is called.
Constraint: ${\mathbf{ldab}}\ge 2×{\mathbf{kl}}+{\mathbf{ku}}+1$.
8:     $\mathbf{ipiv}\left(*\right)$ – Integer arrayInput
Note: the dimension of the array ipiv must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the pivot indices, as returned by f07bdf (dgbtrf).
9:     $\mathbf{b}\left({\mathbf{ldb}},*\right)$ – 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 $r$ right-hand side matrix $B$.
On exit: the $n$ by $r$ solution matrix $X$.
10:   $\mathbf{ldb}$ – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f07bef (dgbtrs) is called.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
11:   $\mathbf{info}$ – IntegerOutput
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

## 6Error Indicators and Warnings

${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

## 7Accuracy

For each right-hand side vector $b$, the computed solution $x$ is the exact solution of a perturbed system of equations $\left(A+E\right)x=b$, where
 $E≤ckεPLU ,$
$c\left(k\right)$ is a modest linear function of $k={k}_{l}+{k}_{u}+1$, and $\epsilon$ is the machine precision. This assumes $k\ll n$.
If $\stackrel{^}{x}$ is the true solution, then the computed solution $x$ satisfies a forward error bound of the form
 $x-x^∞ x∞ ≤ckcondA,xε$
where $\mathrm{cond}\left(A,x\right)={‖\left|{A}^{-1}\right|\left|A\right|\left|x\right|‖}_{\infty }/{‖x‖}_{\infty }\le \mathrm{cond}\left(A\right)={‖\left|{A}^{-1}\right|\left|A\right|‖}_{\infty }\le {\kappa }_{\infty }\left(A\right)$.
Note that $\mathrm{cond}\left(A,x\right)$ can be much smaller than $\mathrm{cond}\left(A\right)$, and $\mathrm{cond}\left({A}^{\mathrm{T}}\right)$ can be much larger (or smaller) than $\mathrm{cond}\left(A\right)$.
Forward and backward error bounds can be computed by calling f07bhf (dgbrfs), and an estimate for ${\kappa }_{\infty }\left(A\right)$ can be obtained by calling f07bgf (dgbcon) with ${\mathbf{norm}}=\text{'I'}$.

## 8Parallelism and Performance

f07bef (dgbtrs) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07bef (dgbtrs) 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 implementation-specific information.

The total number of floating-point operations is approximately $2n\left(2{k}_{l}+{k}_{u}\right)r$, assuming $n\gg {k}_{l}$ and $n\gg {k}_{u}$.
This routine may be followed by a call to f07bhf (dgbrfs) to refine the solution and return an error estimate.
The complex analogue of this routine is f07bsf (zgbtrs).

## 10Example

This example solves the system of equations $AX=B$, where
 $A= -0.23 2.54 -3.66 0.00 -6.98 2.46 -2.73 -2.13 0.00 2.56 2.46 4.07 0.00 0.00 -4.78 -3.82 and B= 4.42 -36.01 27.13 -31.67 -6.14 -1.16 10.50 -25.82 .$
Here $A$ is nonsymmetric and is treated as a band matrix, which must first be factorized by f07bdf (dgbtrf).

### 10.1Program Text

Program Text (f07befe.f90)

### 10.2Program Data

Program Data (f07befe.d)

### 10.3Program Results

Program Results (f07befe.r)