f16 Chapter Contents
f16 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_ztpmv (f16shc)

## 1  Purpose

nag_ztpmv (f16shc) performs matrix-vector multiplication for a complex triangular matrix stored in packed form.

## 2  Specification

 #include #include
 void nag_ztpmv (Nag_OrderType order, Nag_UploType uplo, Nag_TransType trans, Nag_DiagType diag, Integer n, Complex alpha, const Complex ap[], Complex x[], Integer incx, NagError *fail)

## 3  Description

nag_ztpmv (f16shc) performs one of the matrix-vector operations
 $x←αAx , x←αATx or x←αAHx ,$
where $A$ is an $n$ by $n$ complex triangular matrix, stored in packed form, $x$ is an $n$-element complex vector and $\alpha$ is a complex scalar.

## 4  References

The BLAS Technical Forum Standard (2001) http://www.netlib.org/blas/blast-forum

## 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 Nag_ColMajor.
2:     uploNag_UploTypeInput
On entry: specifies whether $A$ is upper or lower triangular.
${\mathbf{uplo}}=\mathrm{Nag_Upper}$
$A$ is upper triangular.
${\mathbf{uplo}}=\mathrm{Nag_Lower}$
$A$ is lower triangular.
Constraint: ${\mathbf{uplo}}=\mathrm{Nag_Upper}$ or $\mathrm{Nag_Lower}$.
3:     transNag_TransTypeInput
On entry: specifies the operation to be performed.
${\mathbf{trans}}=\mathrm{Nag_NoTrans}$
$x←\alpha Ax$.
${\mathbf{trans}}=\mathrm{Nag_Trans}$
$x←\alpha {A}^{\mathrm{T}}x$.
${\mathbf{trans}}=\mathrm{Nag_ConjTrans}$
$x←\alpha {A}^{\mathrm{H}}x$.
Constraint: ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$, $\mathrm{Nag_Trans}$ or $\mathrm{Nag_ConjTrans}$.
4:     diagNag_DiagTypeInput
On entry: specifies whether $A$ has nonunit or unit diagonal elements.
${\mathbf{diag}}=\mathrm{Nag_NonUnitDiag}$
The diagonal elements are stored explicitly.
${\mathbf{diag}}=\mathrm{Nag_UnitDiag}$
The diagonal elements are assumed to be $1$ and are not referenced.
Constraint: ${\mathbf{diag}}=\mathrm{Nag_NonUnitDiag}$ or $\mathrm{Nag_UnitDiag}$.
5:     nIntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
6:     alphaComplexInput
On entry: the scalar $\alpha$.
7:     ap[$\mathit{dim}$]const ComplexInput
Note: the dimension, dim, of the array ap must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$.
On entry: the $n$ by $n$ triangular matrix $A$, packed by rows or columns.
The storage of elements ${A}_{ij}$ depends on the order and uplo arguments as follows:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Upper}$,
${A}_{ij}$ is stored in ${\mathbf{ap}}\left[\left(j-1\right)×j/2+i-1\right]$, for $i\le j$;
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Lower}$,
${A}_{ij}$ is stored in ${\mathbf{ap}}\left[\left(2n-j\right)×\left(j-1\right)/2+i-1\right]$, for $i\ge j$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Upper}$,
${A}_{ij}$ is stored in ${\mathbf{ap}}\left[\left(2n-i\right)×\left(i-1\right)/2+j-1\right]$, for $i\le j$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Lower}$,
${A}_{ij}$ is stored in ${\mathbf{ap}}\left[\left(i-1\right)×i/2+j-1\right]$, for $i\ge j$.
If ${\mathbf{diag}}=\mathrm{Nag_UnitDiag}$, the diagonal elements of $\mathrm{AP}$ are assumed to be $1$, and are not referenced; the same storage scheme is used whether ${\mathbf{diag}}=\mathrm{Nag_NonUnitDiag}$ or ${\mathbf{diag}}=\mathrm{Nag_UnitDiag}$.
8:     x[$\mathit{dim}$]ComplexInput/Output
Note: the dimension, dim, of the array x must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,1+\left({\mathbf{n}}-1\right)\left|{\mathbf{incx}}\right|\right)$.
On entry: the right-hand side vector $b$.
On exit: the solution vector $x$.
9:     incxIntegerInput
On entry: the increment in the subscripts of x between successive elements of $x$.
Constraint: ${\mathbf{incx}}\ne 0$.
10:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

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, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.

## 7  Accuracy

The BLAS standard requires accurate implementations which avoid unnecessary over/underflow (see Section 2.7 of The BLAS Technical Forum Standard (2001)).

None.

## 9  Example

This example computes the matrix-vector product
 $y=αAx$
where
 $A = 1.0+1.0i 0.0+0.0i 0.0+0.0i 0.0+0.0i 2.0+1.0i 2.0+2.0i 0.0+0.0i 0.0+0.0i 3.0+1.0i 3.0+2.0i 3.0+3.0i 0.0+0.0i 4.0+1.0i 4.0+2.0i 4.0+3.0i 4.0+4.0i ,$
 $x = 1.0+0.0i 0.0-1.0i -1.0+0.0i 0.0+1.0i$
and
 $α=1.0+0.0i .$

### 9.1  Program Text

Program Text (f16shce.c)

### 9.2  Program Data

Program Data (f16shce.d)

### 9.3  Program Results

Program Results (f16shce.r)