F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF07BSF (ZGBTRS)

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

F07BSF (ZGBTRS) solves a complex band system of linear equations with multiple right-hand sides,
 $AX=B , ATX=B or AHX=B ,$
where $A$ has been factorized by F07BRF (ZGBTRF).

## 2  Specification

 SUBROUTINE F07BSF ( TRANS, N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, INFO)
 INTEGER N, KL, KU, NRHS, LDAB, IPIV(*), LDB, INFO COMPLEX (KIND=nag_wp) AB(LDAB,*), B(LDB,*) CHARACTER(1) TRANS
The routine may be called by its LAPACK name zgbtrs.

## 3  Description

F07BSF (ZGBTRS) is used to solve a complex band system of linear equations $AX=B$, ${A}^{\mathrm{T}}X=B$ or ${A}^{\mathrm{H}}X=B$, the routine must be preceded by a call to F07BRF (ZGBTRF) 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 ${\mathbf{TRANS}}=\text{'T'}$, the solution is computed by solving ${U}^{\mathrm{T}}Y=B$ and then ${L}^{\mathrm{T}}{P}^{\mathrm{T}}X=Y$.
If ${\mathbf{TRANS}}=\text{'C'}$, the solution is computed by solving ${U}^{\mathrm{H}}Y=B$ and then ${L}^{\mathrm{H}}{P}^{\mathrm{T}}X=Y$.

## 4  References

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

## 5  Parameters

1:     TRANS – CHARACTER(1)Input
On entry: indicates the form of the equations.
${\mathbf{TRANS}}=\text{'N'}$
$AX=B$ is solved for $X$.
${\mathbf{TRANS}}=\text{'T'}$
${A}^{\mathrm{T}}X=B$ is solved for $X$.
${\mathbf{TRANS}}=\text{'C'}$
${A}^{\mathrm{H}}X=B$ is solved for $X$.
Constraint: ${\mathbf{TRANS}}=\text{'N'}$, $\text{'T'}$ or $\text{'C'}$.
2:     N – INTEGERInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 0$.
3:     KL – INTEGERInput
On entry: ${k}_{l}$, the number of subdiagonals within the band of the matrix $A$.
Constraint: ${\mathbf{KL}}\ge 0$.
4:     KU – INTEGERInput
On entry: ${k}_{u}$, the number of superdiagonals within the band of the matrix $A$.
Constraint: ${\mathbf{KU}}\ge 0$.
5:     NRHS – INTEGERInput
On entry: $r$, the number of right-hand sides.
Constraint: ${\mathbf{NRHS}}\ge 0$.
6:     AB(LDAB,$*$) – COMPLEX (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 F07BRF (ZGBTRF).
7:     LDAB – INTEGERInput
On entry: the first dimension of the array AB as declared in the (sub)program from which F07BSF (ZGBTRS) is called.
Constraint: ${\mathbf{LDAB}}\ge 2×{\mathbf{KL}}+{\mathbf{KU}}+1$.
8:     IPIV($*$) – 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 F07BRF (ZGBTRF).
9:     B(LDB,$*$) – 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 $r$ 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 F07BSF (ZGBTRS) 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.

## 7  Accuracy

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
 $E≤ckεLU ,$
$c\left(k\right)$ 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 $\mathrm{cond}\left({A}^{\mathrm{H}}\right)$ (which is the same as $\mathrm{cond}\left({A}^{\mathrm{T}}\right)$) can be much larger (or smaller) than $\mathrm{cond}\left(A\right)$.
Forward and backward error bounds can be computed by calling F07BVF (ZGBRFS), and an estimate for ${\kappa }_{\infty }\left(A\right)$ can be obtained by calling F07BUF (ZGBCON) with ${\mathbf{NORM}}=\text{'I'}$.

## 8  Further Comments

The total number of real floating point operations is approximately $8n\left(2{k}_{l}+{k}_{u}\right)r$, assuming $n\gg {k}_{l}$ and $n\gg {k}_{u}$.
This routine may be followed by a call to F07BVF (ZGBRFS) to refine the solution and return an error estimate.
The real analogue of this routine is F07BEF (DGBTRS).

## 9  Example

This example solves the system of equations $AX=B$, where
 $A= -1.65+2.26i -2.05-0.85i 0.97-2.84i 0.00+0.00i 0.00+6.30i -1.48-1.75i -3.99+4.01i 0.59-0.48i 0.00+0.00i -0.77+2.83i -1.06+1.94i 3.33-1.04i 0.00+0.00i 0.00+0.00i 4.48-1.09i -0.46-1.72i$
and
 $B= -1.06+21.50i 12.85+02.84i -22.72-53.90i -70.22+21.57i 28.24-38.60i -20.70-31.23i -34.56+16.73i 26.01+31.97i .$
Here $A$ is nonsymmetric and is treated as a band matrix, which must first be factorized by F07BRF (ZGBTRF).

### 9.1  Program Text

Program Text (f07bsfe.f90)

### 9.2  Program Data

Program Data (f07bsfe.d)

### 9.3  Program Results

Program Results (f07bsfe.r)