NAG Library Routine Document

F06SGF (ZTBMV)

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: specifies the operation to be performed.

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

$x\leftarrow Ax$.

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

$x\leftarrow A^{\mathrm{T}}x$.

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

$x\leftarrow A^{\mathrm{H}}x$.

Constraint: $\mathbf{TRANS}=\text{'N'}$, $\text{'T'}$ or $\text{'C'}$.

3:     DIAG – CHARACTER(1)Input

On entry: specifies whether $A$ has nonunit or unit diagonal elements.

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

The diagonal elements are stored explicitly.

$\mathbf{DIAG}=\text{'U'}$

The diagonal elements are assumed to be $1$, and are not referenced.

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:     K – INTEGERInput

On entry: $k$, the number of subdiagonals or superdiagonals of the matrix $A$.

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

6:     A(LDA,$*$) – COMPLEX (KIND=nag_wp) arrayInput

Note: the second dimension of the array A must be at least $\mathbf{N}$.

On entry: the $n$ by $n$ triangular band matrix $A$ 

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

If $\mathbf{DIAG}=\text{'U'}$, the diagonal elements of $A$ are assumed to be $1$, and are not referenced.

7:     LDA – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which F06SGF (ZTBMV) is called.

Constraint: $\mathbf{LDA}\ge \mathbf{K}+1$.

8:     X($*$) – COMPLEX (KIND=nag_wp) arrayInput/Output

Note: the dimension of the array X must be at least $\max \left(1,1+\left(\mathbf{N}-1\right)\times \left|\mathbf{INCX}\right|\right)$.

On entry: the vector $x$.

If $\mathbf{INCX}>0$, $x_{\mathit{i}}$ must be stored in $\mathbf{X}\left(1+\left(\mathit{i}–1\right)\times \mathbf{INCX}\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{N}$.

If $\mathbf{INCX}<0$, $x_{\mathit{i}}$ must be stored in $\mathbf{X}\left(1–\left(\mathbf{N}–\mathit{i}\right)\times \mathbf{INCX}\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{N}$.

On exit: the updated vector $x$ stored in the array elements used to supply the original vector $x$.

9:     INCX – INTEGERInput

On entry: the increment in the subscripts of X between successive elements of $x$.

Constraint: $\mathbf{INCX}\ne 0$.

6  Error Indicators and Warnings

7  Accuracy

8  Further Comments

9  Example