# NAG Library Routine Document

## 1Purpose

f08guf (zupmtr) multiplies an arbitrary complex matrix $C$ by the complex unitary matrix $Q$ which was determined by f08gsf (zhptrd) when reducing a complex Hermitian matrix to tridiagonal form.

## 2Specification

Fortran Interface
 Subroutine f08guf ( side, uplo, m, n, ap, tau, c, ldc, work, info)
 Integer, Intent (In) :: m, n, ldc Integer, Intent (Out) :: info Complex (Kind=nag_wp), Intent (In) :: tau(*) Complex (Kind=nag_wp), Intent (Inout) :: ap(*), c(ldc,*), work(*) Character (1), Intent (In) :: side, uplo, trans
#include <nagmk26.h>
 void f08guf_ (const char *side, const char *uplo, const char *trans, const Integer *m, const Integer *n, Complex ap[], const Complex tau[], Complex c[], const Integer *ldc, Complex work[], Integer *info, const Charlen length_side, const Charlen length_uplo, const Charlen length_trans)
The routine may be called by its LAPACK name zupmtr.

## 3Description

f08guf (zupmtr) is intended to be used after a call to f08gsf (zhptrd), which reduces a complex Hermitian matrix $A$ to real symmetric tridiagonal form $T$ by a unitary similarity transformation: $A=QT{Q}^{\mathrm{H}}$. f08gsf (zhptrd) represents the unitary matrix $Q$ as a product of elementary reflectors.
This routine may be used to form one of the matrix products
 $QC , QHC , CQ ​ or ​ CQH ,$
overwriting the result on $C$ (which may be any complex rectangular matrix).
A common application of this routine is to transform a matrix $Z$ of eigenvectors of $T$ to the matrix $QZ$ of eigenvectors of $A$.

## 4References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## 5Arguments

1:     $\mathbf{side}$ – Character(1)Input
On entry: indicates how $Q$ or ${Q}^{\mathrm{H}}$ is to be applied to $C$.
${\mathbf{side}}=\text{'L'}$
$Q$ or ${Q}^{\mathrm{H}}$ is applied to $C$ from the left.
${\mathbf{side}}=\text{'R'}$
$Q$ or ${Q}^{\mathrm{H}}$ is applied to $C$ from the right.
Constraint: ${\mathbf{side}}=\text{'L'}$ or $\text{'R'}$.
2:     $\mathbf{uplo}$ – Character(1)Input
On entry: this must be the same argument uplo as supplied to f08gsf (zhptrd).
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
3:     $\mathbf{trans}$ – Character(1)Input
On entry: indicates whether $Q$ or ${Q}^{\mathrm{H}}$ is to be applied to $C$.
${\mathbf{trans}}=\text{'N'}$
$Q$ is applied to $C$.
${\mathbf{trans}}=\text{'C'}$
${Q}^{\mathrm{H}}$ is applied to $C$.
Constraint: ${\mathbf{trans}}=\text{'N'}$ or $\text{'C'}$.
4:     $\mathbf{m}$ – IntegerInput
On entry: $m$, the number of rows of the matrix $C$; $m$ is also the order of $Q$ if ${\mathbf{side}}=\text{'L'}$.
Constraint: ${\mathbf{m}}\ge 0$.
5:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of columns of the matrix $C$; $n$ is also the order of $Q$ if ${\mathbf{side}}=\text{'R'}$.
Constraint: ${\mathbf{n}}\ge 0$.
6:     $\mathbf{ap}\left(*\right)$ – Complex (Kind=nag_wp) arrayInput/Output
Note: the dimension of the array ap must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}×\left({\mathbf{m}}+1\right)/2\right)$ if ${\mathbf{side}}=\text{'L'}$ and at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$ if ${\mathbf{side}}=\text{'R'}$.
On entry: details of the vectors which define the elementary reflectors, as returned by f08gsf (zhptrd).
On exit: is used as internal workspace prior to being restored and hence is unchanged.
7:     $\mathbf{tau}\left(*\right)$ – Complex (Kind=nag_wp) arrayInput
Note: the dimension of the array tau must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}-1\right)$ if ${\mathbf{side}}=\text{'L'}$ and at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$ if ${\mathbf{side}}=\text{'R'}$.
On entry: further details of the elementary reflectors, as returned by f08gsf (zhptrd).
8:     $\mathbf{c}\left({\mathbf{ldc}},*\right)$ – Complex (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array c must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the $m$ by $n$ matrix $C$.
On exit: c is overwritten by $QC$ or ${Q}^{\mathrm{H}}C$ or $CQ$ or $C{Q}^{\mathrm{H}}$ as specified by side and trans.
9:     $\mathbf{ldc}$ – IntegerInput
On entry: the first dimension of the array c as declared in the (sub)program from which f08guf (zupmtr) is called.
Constraint: ${\mathbf{ldc}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
10:   $\mathbf{work}\left(*\right)$ – Complex (Kind=nag_wp) arrayWorkspace
Note: the dimension of the array work must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ if ${\mathbf{side}}=\text{'L'}$ and at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$ if ${\mathbf{side}}=\text{'R'}$.
11:   $\mathbf{info}$ – IntegerOutput
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

## 6Error Indicators and Warnings

${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

## 7Accuracy

The computed result differs from the exact result by a matrix $E$ such that
 $E2 = Oε C2 ,$
where $\epsilon$ is the machine precision.

## 8Parallelism and Performance

f08guf (zupmtr) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The total number of real floating-point operations is approximately $8{m}^{2}n$ if ${\mathbf{side}}=\text{'L'}$ and $8m{n}^{2}$ if ${\mathbf{side}}=\text{'R'}$.
The real analogue of this routine is f08ggf (dopmtr).

## 10Example

This example computes the two smallest eigenvalues, and the associated eigenvectors, of the matrix $A$, where
 $A = -2.28+0.00i 1.78-2.03i 2.26+0.10i -0.12+2.53i 1.78+2.03i -1.12+0.00i 0.01+0.43i -1.07+0.86i 2.26-0.10i 0.01-0.43i -0.37+0.00i 2.31-0.92i -0.12-2.53i -1.07-0.86i 2.31+0.92i -0.73+0.00i ,$
using packed storage. Here $A$ is Hermitian and must first be reduced to tridiagonal form $T$ by f08gsf (zhptrd). The program then calls f08jjf (dstebz) to compute the requested eigenvalues and f08jxf (zstein) to compute the associated eigenvectors of $T$. Finally f08guf (zupmtr) is called to transform the eigenvectors to those of $A$.

### 10.1Program Text

Program Text (f08gufe.f90)

### 10.2Program Data

Program Data (f08gufe.d)

### 10.3Program Results

Program Results (f08gufe.r)