NAG Library Routine Document

F07VSF (ZTBTRS)

1  Purpose

2  Specification

3  Description

4  References

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 $AX=B$.

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

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

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

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

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,$*$) – 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 $n$ 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 $\mathbf{AB}\left(k_d+1+i-j,j\right)\text{ for }\max \left(1,j-k_d\right)\le i\le j$;
• if $\mathbf{UPLO}=\text{'L'}$, the elements of the lower triangle of $A$ within the band must be stored with element $A_{ij}$ in $\mathbf{AB}\left(1+i-j,j\right)\text{ for }j\le i\le \min \left(n,j+k_d\right)\text{.}$

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 F07VSF (ZTBTRS) is called.

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

9:     B(LDB,$*$) – COMPLEX (KIND=nag_wp) arrayInput/Output

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 $B$.

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 F07VSF (ZTBTRS) is called.

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

11:   INFO – INTEGEROutput

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

6  Error Indicators and Warnings

$\mathbf{INFO}<0$

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

8  Further Comments

9  Example

9.1  Program Text

Program Text (f07vsfe.f90)

9.2  Program Data

Program Data (f07vsfe.d)

9.3  Program Results

Program Results (f07vsfe.r)