f16 Chapter Contents
f16 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_zgbmv (f16sbc)

## 1  Purpose

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

## 2  Specification

 #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)

## 3  Description

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.

## 4  References

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

## 5  Arguments

1:     orderNag_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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2:     transNag_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:     mIntegerInput
On entry: $m$, the number of rows of the matrix $A$.
Constraint: ${\mathbf{m}}\ge 0$.
4:     nIntegerInput
On entry: $n$, the number of columns of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
5:     klIntegerInput
On entry: ${k}_{l}$, the number of subdiagonals within the band of $A$.
Constraint: ${\mathbf{kl}}\ge 0$.
6:     kuIntegerInput
On entry: ${k}_{u}$, the number of superdiagonals within the band of $A$.
Constraint: ${\mathbf{ku}}\ge 0$.
7:     alphaComplexInput
On entry: the scalar $\alpha$.
8:     ab[$\mathit{dim}$]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:     pdabIntegerInput
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:   x[$\mathit{dim}$]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 $x$.
11:   incxIntegerInput
On entry: the increment in the subscripts of x between successive elements of $x$.
Constraint: ${\mathbf{incx}}\ne 0$.
12:   betaComplexInput
On entry: the scalar $\beta$.
13:   y[$\mathit{dim}$]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 $y$.
If ${\mathbf{beta}}=0$, y need not be set.
On exit: the updated vector $y$.
14:   incyIntegerInput
On entry: the increment in the subscripts of y between successive elements of $y$.
Constraint: ${\mathbf{incy}}\ne 0$.
15:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
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: ${\mathbf{pdab}}\ge {\mathbf{kl}}+{\mathbf{ku}}+1$.

## 7  Accuracy

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

Not applicable.

None.

## 10  Example

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

### 10.1  Program Text

Program Text (f16sbce.c)

### 10.2  Program Data

Program Data (f16sbce.d)

### 10.3  Program Results

Program Results (f16sbce.r)