F07VEF (DTBTRS) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF07VEF (DTBTRS)

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

F07VEF (DTBTRS) solves a real triangular band system of linear equations with multiple right-hand sides, $AX=B$ or ${A}^{\mathrm{T}}X=B$.

## 2  Specification

 SUBROUTINE F07VEF ( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, B, LDB, INFO)
 INTEGER N, KD, NRHS, LDAB, LDB, INFO REAL (KIND=nag_wp) AB(LDAB,*), B(LDB,*) CHARACTER(1) UPLO, TRANS, DIAG
The routine may be called by its LAPACK name dtbtrs.

## 3  Description

F07VEF (DTBTRS) solves a real triangular band system of linear equations $AX=B$ or ${A}^{\mathrm{T}}X=B$.

## 4  References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Higham N J (1989) The accuracy of solutions to triangular systems SIAM J. Numer. Anal. 26 1252–1265

## 5  Parameters

1:     UPLO – CHARACTER(1)Input
On entry: specifies whether $A$ is upper or lower triangular.
${\mathbf{UPLO}}=\text{'U'}$
$A$ is upper triangular.
${\mathbf{UPLO}}=\text{'L'}$
$A$ is lower triangular.
Constraint: ${\mathbf{UPLO}}=\text{'U'}$ or $\text{'L'}$.
2:     TRANS – CHARACTER(1)Input
On entry: indicates the form of the equations.
${\mathbf{TRANS}}=\text{'N'}$
The equations are of the form $AX=B$.
${\mathbf{TRANS}}=\text{'T'}$ or $\text{'C'}$
The equations are of the form ${A}^{\mathrm{T}}X=B$.
Constraint: ${\mathbf{TRANS}}=\text{'N'}$, $\text{'T'}$ or $\text{'C'}$.
3:     DIAG – CHARACTER(1)Input
On entry: indicates whether $A$ is a nonunit or unit triangular matrix.
${\mathbf{DIAG}}=\text{'N'}$
$A$ is a nonunit triangular matrix.
${\mathbf{DIAG}}=\text{'U'}$
$A$ is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be $1$.
Constraint: ${\mathbf{DIAG}}=\text{'N'}$ or $\text{'U'}$.
4:     N – INTEGERInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 0$.
5:     KD – INTEGERInput
On entry: ${k}_{d}$, the number of superdiagonals of the matrix $A$ if ${\mathbf{UPLO}}=\text{'U'}$, or the number of subdiagonals if ${\mathbf{UPLO}}=\text{'L'}$.
Constraint: ${\mathbf{KD}}\ge 0$.
6:     NRHS – INTEGERInput
On entry: $r$, the number of right-hand sides.
Constraint: ${\mathbf{NRHS}}\ge 0$.
7:     AB(LDAB,$*$) – 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 $n$ by $n$ triangular band matrix $A$.
The matrix is stored in rows $1$ to ${k}_{d}+1$, more precisely,
• if ${\mathbf{UPLO}}=\text{'U'}$, the elements of the upper triangle of $A$ within the band must be stored with element ${A}_{ij}$ in ${\mathbf{AB}}\left({k}_{d}+1+i-j,j\right)\text{​ for ​}\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,j-{k}_{d}\right)\le i\le j$;
• if ${\mathbf{UPLO}}=\text{'L'}$, the elements of the lower triangle of $A$ within the band must be stored with element ${A}_{ij}$ in ${\mathbf{AB}}\left(1+i-j,j\right)\text{​ for ​}j\le i\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,j+{k}_{d}\right)\text{.}$
If ${\mathbf{DIAG}}=\text{'U'}$, the diagonal elements of $A$ are assumed to be $1$, and are not referenced.
8:     LDAB – INTEGERInput
On entry: the first dimension of the array AB as declared in the (sub)program from which F07VEF (DTBTRS) is called.
Constraint: ${\mathbf{LDAB}}\ge {\mathbf{KD}}+1$.
9:     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 $r$ right-hand side matrix $B$.
On exit: the $n$ by $r$ solution matrix $X$.
10:   LDB – INTEGERInput
On entry: the first dimension of the array B as declared in the (sub)program from which F07VEF (DTBTRS) is called.
Constraint: ${\mathbf{LDB}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
11:   INFO – INTEGEROutput
On exit: ${\mathbf{INFO}}=0$ unless the routine detects an error (see Section 6).

## 6  Error Indicators and Warnings

Errors or warnings detected by the routine:
${\mathbf{INFO}}<0$
If ${\mathbf{INFO}}=-i$, the $i$th parameter had an illegal value. An explanatory message is output, and execution of the program is terminated.
${\mathbf{INFO}}>0$
If ${\mathbf{INFO}}=i$, $a\left(i,i\right)$ is exactly zero; $A$ is singular and the solution has not been computed.

## 7  Accuracy

The solutions of triangular systems of equations are usually computed to high accuracy. See Higham (1989).
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
 $E≤ckεA ,$
$c\left(k\right)$ is a modest linear function of $k$, and $\epsilon$ is the machine precision.
If $\stackrel{^}{x}$ is the true solution, then the computed solution $x$ satisfies a forward error bound of the form
 $x-x^∞ x∞ ≤ckcondA,xε , provided ckcondA,xε<1 ,$
where $\mathrm{cond}\left(A,x\right)={‖\left|{A}^{-1}\right|\left|A\right|\left|x\right|‖}_{\infty }/{‖x‖}_{\infty }$.
Note that $\mathrm{cond}\left(A,x\right)\le \mathrm{cond}\left(A\right)={‖\left|{A}^{-1}\right|\left|A\right|‖}_{\infty }\le {\kappa }_{\infty }\left(A\right)$; $\mathrm{cond}\left(A,x\right)$ can be much smaller than $\mathrm{cond}\left(A\right)$ and it is also possible for $\mathrm{cond}\left({A}^{\mathrm{T}}\right)$ to be much larger (or smaller) than $\mathrm{cond}\left(A\right)$.
Forward and backward error bounds can be computed by calling F07VHF (DTBRFS), and an estimate for ${\kappa }_{\infty }\left(A\right)$ can be obtained by calling F07VGF (DTBCON) with ${\mathbf{NORM}}=\text{'I'}$.

## 8  Further Comments

The total number of floating point operations is approximately $2nkr$ if $k\ll n$.
The complex analogue of this routine is F07VSF (ZTBTRS).

## 9  Example

This example solves the system of equations $AX=B$, where
 $A= -4.16 0.00 0.00 0.00 -2.25 4.78 0.00 0.00 0.00 5.86 6.32 0.00 0.00 0.00 -4.82 0.16 and B= -16.64 -4.16 -13.78 -16.59 13.10 -4.94 -14.14 -9.96 .$
Here $A$ is treated as a lower triangular band matrix with one subdiagonal.

### 9.1  Program Text

Program Text (f07vefe.f90)

### 9.2  Program Data

Program Data (f07vefe.d)

### 9.3  Program Results

Program Results (f07vefe.r)

F07VEF (DTBTRS) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual