F06 Chapter Contents
F06 Chapter Introduction
NAG Library Manual

NAG Library Routine DocumentF06SKF (ZTBSV)

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

F06SKF (ZTBSV) solves a complex triangular banded system of equations with a single right hand side.

2  Specification

 SUBROUTINE F06SKF ( UPLO, TRANS, DIAG, N, K, A, LDA, X, INCX)
 INTEGER N, K, LDA, INCX COMPLEX (KIND=nag_wp) A(LDA,*), X(*) CHARACTER(1) UPLO, TRANS, DIAG
The routine may be called by its BLAS name ztbsv.

3  Description

F06SKF (ZTBSV) performs one of the matrix-vector operations
 $x←A-1x , x←A-Tx or x←A-Hx ,$
where $A$ is an $n$ by $n$ complex triangular band matrix with $k$ subdiagonals or superdiagonals, and $x$ is an $n$-element complex vector. ${A}^{-\mathrm{T}}$ denotes ${\left({A}^{\mathrm{T}}\right)}^{-1}$ or equivalently ${\left({A}^{-1}\right)}^{\mathrm{T}}$; ${A}^{-\mathrm{H}}$ denotes ${\left({A}^{\mathrm{H}}\right)}^{-1}$ or equivalently ${\left({A}^{-1}\right)}^{\mathrm{H}}$.
No test for singularity or near-singularity of $A$ is included in this routine. Such tests must be performed before calling this routine.

None.

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.
${\mathbf{TRANS}}=\text{'N'}$
$x←{A}^{-1}x$.
${\mathbf{TRANS}}=\text{'T'}$
$x←{A}^{-\mathrm{T}}x$.
${\mathbf{TRANS}}=\text{'C'}$
$x←{A}^{-\mathrm{H}}x$.
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 ${\mathbf{A}}\left(k+1+i-j,j\right)\text{​ for ​}\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,j-k\right)\le i\le j$;
• if ${\mathbf{UPLO}}=\text{'L'}$, the elements of the lower triangle of $A$ within the band must be stored with element ${A}_{ij}$ in ${\mathbf{A}}\left(1+i-j,j\right)\text{​ for ​}j\le i\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,j+k\right)\text{.}$
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 F06SKF (ZTBSV) 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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,1+\left({\mathbf{N}}-1\right)×\left|{\mathbf{INCX}}\right|\right)$.
On entry: the vector $x$.
If ${\mathbf{INCX}}>0$, ${x}_{\mathit{i}}$ must be stored in ${\mathbf{X}}\left(1+\left(\mathit{i}–1\right)×{\mathbf{INCX}}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{N}}$.
If ${\mathbf{INCX}}<0$, ${x}_{\mathit{i}}$ must be stored in ${\mathbf{X}}\left(1–\left({\mathbf{N}}–\mathit{i}\right)×{\mathbf{INCX}}\right)$, 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: ${\mathbf{INCX}}\ne 0$.

None.

Not applicable.