NAG Library Routine Document

F07BVF (ZGBRFS)

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     TRANS – CHARACTER(1)Input

On entry: indicates the form of the linear equations for which $X$ is the computed solution as follows:

$\mathbf{TRANS}=\text{'N'}$

The linear equations are of the form $AX=B$.

$\mathbf{TRANS}=\text{'T'}$

The linear equations are of the form $A^{\mathrm{T}}X=B$.

$\mathbf{TRANS}=\text{'C'}$

The linear equations are of the form $A^{\mathrm{H}}X=B$.

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 $\max \left(1,\mathbf{N}\right)$.

On entry: the original $n$ by $n$ band matrix $A$ as supplied to F07BRF (ZGBTRF).

The matrix is stored in rows $1$ to $k_l+k_u+1$, more precisely, the element $A_{ij}$ must be stored in

$$\mathbf{AB}\left(k_u+1+i-j,j\right)\text{\hspace{1em} for }\max \left(1,j-k_u\right)\le i\le \min \left(n,j+k_l\right)\text{.}$$

See Section 8 in F07BNF (ZGBSV) for further details.

7:     LDAB – INTEGERInput

On entry: the first dimension of the array AB as declared in the (sub)program from which F07BVF (ZGBRFS) is called.

Constraint: $\mathbf{LDAB}\ge \mathbf{KL}+\mathbf{KU}+1$.

8:     AFB(LDAFB,$*$) – COMPLEX (KIND=nag_wp) arrayInput

Note: the second dimension of the array AFB must be at least $\max \left(1,\mathbf{N}\right)$.

On entry: the $LU$ factorization of $A$, as returned by F07BRF (ZGBTRF).

9:     LDAFB – INTEGERInput

On entry: the first dimension of the array AFB as declared in the (sub)program from which F07BVF (ZGBRFS) is called.

Constraint: $\mathbf{LDAFB}\ge 2\times \mathbf{KL}+\mathbf{KU}+1$.

10:   IPIV($*$) – INTEGER arrayInput

Note: the dimension of the array IPIV must be at least $\max \left(1,\mathbf{N}\right)$.

On entry: the pivot indices, as returned by F07BRF (ZGBTRF).

11:   B(LDB,$*$) – COMPLEX (KIND=nag_wp) arrayInput

Note: the second dimension of the array B must be at least $\max \left(1,\mathbf{NRHS}\right)$.

On entry: the $n$ by $r$ right-hand side matrix $B$.

12:   LDB – INTEGERInput

On entry: the first dimension of the array B as declared in the (sub)program from which F07BVF (ZGBRFS) is called.

Constraint: $\mathbf{LDB}\ge \max \left(1,\mathbf{N}\right)$.

13:   X(LDX,$*$) – COMPLEX (KIND=nag_wp) arrayInput/Output

Note: the second dimension of the array X must be at least $\max \left(1,\mathbf{NRHS}\right)$.

On entry: the $n$ by $r$ solution matrix $X$, as returned by F07BSF (ZGBTRS).

On exit: the improved solution matrix $X$.

14:   LDX – INTEGERInput

On entry: the first dimension of the array X as declared in the (sub)program from which F07BVF (ZGBRFS) is called.

Constraint: $\mathbf{LDX}\ge \max \left(1,\mathbf{N}\right)$.

15:   FERR(NRHS) – REAL (KIND=nag_wp) arrayOutput

On exit: $\mathbf{FERR}\left(\mathit{j}\right)$ contains an estimated error bound for the $\mathit{j}$th solution vector, that is, the $\mathit{j}$th column of $X$, for $\mathit{j}=1,2,\dots ,r$.

16:   BERR(NRHS) – REAL (KIND=nag_wp) arrayOutput

On exit: $\mathbf{BERR}\left(\mathit{j}\right)$ contains the component-wise backward error bound $\beta$ for the $\mathit{j}$th solution vector, that is, the $\mathit{j}$th column of $X$, for $\mathit{j}=1,2,\dots ,r$.

17:   WORK($2\times \mathbf{N}$) – COMPLEX (KIND=nag_wp) arrayWorkspace

18:   RWORK(N) – REAL (KIND=nag_wp) arrayWorkspace

19:   INFO – INTEGEROutput

On exit: $\mathbf{INFO}=0$ unless the routine detects an error (see Section 6).

6  Error Indicators and Warnings

$\mathbf{INFO}<0$

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

8  Further Comments

9  Example

9.1  Program Text

Program Text (f07bvfe.f90)

9.2  Program Data

Program Data (f07bvfe.d)

9.3  Program Results

Program Results (f07bvfe.r)