NAG Library Routine Document

f07vsf (ztbtrs)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:     $\mathbf{uplo}$ – 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:     $\mathbf{trans}$ – 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:     $\mathbf{diag}$ – 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:     $\mathbf{n}$ – IntegerInput

On entry: $n$, the order of the matrix $A$.

Constraint: $\mathbf{n}\ge 0$.

5:     $\mathbf{kd}$ – 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:     $\mathbf{nrhs}$ – IntegerInput

On entry: $r$, the number of right-hand sides.

Constraint: $\mathbf{nrhs}\ge 0$.

7:     $\mathbf{ab}\left(\mathbf{ldab},*\right)$ – 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:     $\mathbf{ldab}$ – 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:     $\mathbf{b}\left(\mathbf{ldb},*\right)$ – 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:   $\mathbf{ldb}$ – 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:   $\mathbf{info}$ – IntegerOutput

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

6

Error Indicators and Warnings

$\mathbf{info}<0$

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

$\mathbf{info}>0$

Element $〈\mathit{\text{value}}〉$ of the diagonal is exactly zero. $A$ is singular and the solution has not been computed.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (f07vsfe.f90)

10.2

Program Data

Program Data (f07vsfe.d)

10.3

Program Results

Program Results (f07vsfe.r)