# NAG Library Function Document

## 1Purpose

nag_zgbmv (f16sbc) performs matrix-vector multiplication for a complex band matrix.

## 2Specification

 #include #include
 void nag_zgbmv (Nag_OrderType order, Nag_TransType trans, Integer m, Integer n, Integer kl, Integer ku, Complex alpha, const Complex ab[], Integer pdab, const Complex x[], Integer incx, Complex beta, Complex y[], Integer incy, NagError *fail)

## 3Description

nag_zgbmv (f16sbc) performs one of the matrix-vector operations
 $y←αAx+βy, y←αATx+βy or y←αAHx+βy$
where $A$ is an $m$ by $n$ complex band matrix with ${k}_{l}$ subdiagonals and ${k}_{u}$ superdiagonals, $x$ and $y$ are complex vectors, and $\alpha$ and $\beta$ are complex scalars.
If $m=0$ or $n=0$, no operation is performed.

## 4References

Basic Linear Algebra Subprograms Technical (BLAST) Forum (2001) Basic Linear Algebra Subprograms Technical (BLAST) Forum Standard University of Tennessee, Knoxville, Tennessee http://www.netlib.org/blas/blast-forum/blas-report.pdf

## 5Arguments

1:    $\mathbf{order}$Nag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2:    $\mathbf{trans}$Nag_TransTypeInput
On entry: specifies the operation to be performed.
${\mathbf{trans}}=\mathrm{Nag_NoTrans}$
$y←\alpha Ax+\beta y$.
${\mathbf{trans}}=\mathrm{Nag_Trans}$
$y←\alpha {A}^{\mathrm{T}}x+\beta y$.
${\mathbf{trans}}=\mathrm{Nag_ConjTrans}$
$y←\alpha {A}^{\mathrm{H}}x+\beta y$.
Constraint: ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$, $\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$.
3:    $\mathbf{m}$IntegerInput
On entry: $m$, the number of rows of the matrix $A$.
Constraint: ${\mathbf{m}}\ge 0$.
4:    $\mathbf{n}$IntegerInput
On entry: $n$, the number of columns of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
5:    $\mathbf{kl}$IntegerInput
On entry: ${k}_{l}$, the number of subdiagonals within the band of $A$.
Constraint: ${\mathbf{kl}}\ge 0$.
6:    $\mathbf{ku}$IntegerInput
On entry: ${k}_{u}$, the number of superdiagonals within the band of $A$.
Constraint: ${\mathbf{ku}}\ge 0$.
7:    $\mathbf{alpha}$ComplexInput
On entry: the scalar $\alpha$.
8:    $\mathbf{ab}\left[\mathit{dim}\right]$const ComplexInput
Note: the dimension, dim, of the array ab must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdab}}×{\mathbf{n}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}×{\mathbf{pdab}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the $m$ by $n$ band matrix $A$.
This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements ${A}_{ij}$, for row $i=1,\dots ,m$ and column $j=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,i-{k}_{l}\right),\dots ,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,i+{k}_{u}\right)$, depends on the order argument as follows:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${A}_{ij}$ is stored as ${\mathbf{ab}}\left[\left(j-1\right)×{\mathbf{pdab}}+{\mathbf{ku}}+i-j\right]$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${A}_{ij}$ is stored as ${\mathbf{ab}}\left[\left(i-1\right)×{\mathbf{pdab}}+{\mathbf{kl}}+j-i\right]$.
9:    $\mathbf{pdab}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix $A$ in the array ab.
Constraint: ${\mathbf{pdab}}\ge {\mathbf{kl}}+{\mathbf{ku}}+1$.
10:  $\mathbf{x}\left[\mathit{dim}\right]$const ComplexInput
Note: the dimension, dim, of the array x must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,1+\left({\mathbf{n}}-1\right)\left|{\mathbf{incx}}\right|\right)$ when ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,1+\left({\mathbf{m}}-1\right)\left|{\mathbf{incx}}\right|\right)$ when ${\mathbf{trans}}=\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$.
On entry: the vector $x$.
If ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$, then $x$ is an $n$-element vector.
• If ${\mathbf{incx}}>0$, ${x}_{\mathit{i}}$ must be stored in ${\mathbf{x}}\left[\left(\mathit{i}-1\right)×{\mathbf{incx}}\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
• If ${\mathbf{incx}}<0$, ${x}_{\mathit{i}}$ must be stored in ${\mathbf{x}}\left[\left({\mathbf{n}}-\mathit{i}\right)×\left|{\mathbf{incx}}\right|\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
• Intermediate elements of x are not referenced. If ${\mathbf{n}}=0$, x is not referenced and may be NULL.
Otherwise, $x$ is an $m$-element vector.
• If ${\mathbf{incx}}>0$, ${x}_{\mathit{i}}$ must be stored in ${\mathbf{x}}\left[\left(\mathit{i}-1\right)×{\mathbf{incx}}\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{m}}$.
• If ${\mathbf{incx}}<0$, ${x}_{\mathit{i}}$ must be stored in ${\mathbf{x}}\left[\left({\mathbf{m}}-\mathit{i}\right)×\left|{\mathbf{incx}}\right|\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{m}}$.
• Intermediate elements of x are not referenced. If ${\mathbf{m}}=0$, x is not referenced and may be NULL.
11:  $\mathbf{incx}$IntegerInput
On entry: the increment in the subscripts of x between successive elements of $x$.
Constraint: ${\mathbf{incx}}\ne 0$.
12:  $\mathbf{beta}$ComplexInput
On entry: the scalar $\beta$.
13:  $\mathbf{y}\left[\mathit{dim}\right]$ComplexInput/Output
Note: the dimension, dim, of the array y must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,1+\left({\mathbf{m}}-1\right)\left|{\mathbf{incy}}\right|\right)$ when ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,1+\left({\mathbf{n}}-1\right)\left|{\mathbf{incy}}\right|\right)$ when ${\mathbf{trans}}=\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$.
On entry: the vector $y$. See x for details of storage.
If ${\mathbf{beta}}=0$, y need not be set.
On exit: the updated vector $y$.
14:  $\mathbf{incy}$IntegerInput
On entry: the increment in the subscripts of y between successive elements of $y$.
Constraint: ${\mathbf{incy}}\ne 0$.
15:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{incx}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{incx}}\ne 0$.
On entry, ${\mathbf{incy}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{incy}}\ne 0$.
On entry, ${\mathbf{kl}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{kl}}\ge 0$.
On entry, ${\mathbf{ku}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ku}}\ge 0$.
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}\ge 0$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
NE_INT_3
On entry, ${\mathbf{pdab}}=〈\mathit{\text{value}}〉$, ${\mathbf{kl}}=〈\mathit{\text{value}}〉$, ${\mathbf{ku}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdab}}\ge {\mathbf{kl}}+{\mathbf{ku}}+1$.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

The BLAS standard requires accurate implementations which avoid unnecessary over/underflow (see Section 2.7 of Basic Linear Algebra Subprograms Technical (BLAST) Forum (2001)).

## 8Parallelism and Performance

nag_zgbmv (f16sbc) is not threaded in any implementation.

None.

## 10Example

This example computes the matrix-vector product
 $y=αAx+βy$
where
 $A = 1.0+1.0i 1.0+2.0i 0.0+0.0i 0.0+0.0i 2.0+1.0i 2.0+2.0i 2.0+3.0i 0.0+0.0i 3.0+1.0i 3.0+2.0i 3.0+3.0i 3.0+4.0i 0.0+0.0i 4.0+2.0i 4.0+3.0i 4.0+4.0i 0.0+0.0i 0.0+0.0i 5.0+3.0i 5.0+4.0i 0.0+0.0i 0.0+0.0i 0.0+0.0i 6.0+4.0i ,$
 $x = 1.0-1.0i 2.0-2.0i 3.0-3.0i 4.0-4.0i ,$
 $y = -3.5+00.0i -11.5+01.0i -27.5+03.0i -29.0+07.5i -25.5+10.0i -14.5+10.0i ,$
 $α=1.0+0.0i and β=2.0+0.0i .$

### 10.1Program Text

Program Text (f16sbce.c)

### 10.2Program Data

Program Data (f16sbce.d)

### 10.3Program Results

Program Results (f16sbce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017