F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF08FUF (ZUNMTR)

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

F08FUF (ZUNMTR) multiplies an arbitrary complex matrix $C$ by the complex unitary matrix $Q$ which was determined by F08FSF (ZHETRD) when reducing a complex Hermitian matrix to tridiagonal form.

## 2  Specification

 SUBROUTINE F08FUF ( SIDE, UPLO, TRANS, M, N, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
 INTEGER M, N, LDA, LDC, LWORK, INFO COMPLEX (KIND=nag_wp) A(LDA,*), TAU(*), C(LDC,*), WORK(max(1,LWORK)) CHARACTER(1) SIDE, UPLO, TRANS
The routine may be called by its LAPACK name zunmtr.

## 3  Description

F08FUF (ZUNMTR) is intended to be used after a call to F08FSF (ZHETRD), which reduces a complex Hermitian matrix $A$ to real symmetric tridiagonal form $T$ by a unitary similarity transformation: $A=QT{Q}^{\mathrm{H}}$. F08FSF (ZHETRD) 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 $\mathit{QZ}$ of eigenvectors of $A$.

## 4  References

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

## 5  Parameters

1:     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:     UPLO – CHARACTER(1)Input
On entry: this must be the same parameter UPLO as supplied to F08FSF (ZHETRD).
Constraint: ${\mathbf{UPLO}}=\text{'U'}$ or $\text{'L'}$.
3:     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:     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:     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:     A(LDA,$*$) – COMPLEX (KIND=nag_wp) arrayInput
Note: the second dimension of the array A must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{M}}\right)$ if ${\mathbf{SIDE}}=\text{'L'}$ and at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$ if ${\mathbf{SIDE}}=\text{'R'}$.
On entry: details of the vectors which define the elementary reflectors, as returned by F08FSF (ZHETRD).
7:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F08FUF (ZUNMTR) is called.
Constraints:
• if ${\mathbf{SIDE}}=\text{'L'}$, ${\mathbf{LDA}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{M}}\right)$;
• if ${\mathbf{SIDE}}=\text{'R'}$, ${\mathbf{LDA}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
8:     TAU($*$) – 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 F08FSF (ZHETRD).
9:     C(LDC,$*$) – 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.
10:   LDC – INTEGERInput
On entry: the first dimension of the array C as declared in the (sub)program from which F08FUF (ZUNMTR) is called.
Constraint: ${\mathbf{LDC}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{M}}\right)$.
11:   WORK($\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{LWORK}}\right)$) – COMPLEX (KIND=nag_wp) arrayWorkspace
On exit: if ${\mathbf{INFO}}={\mathbf{0}}$, the real part of ${\mathbf{WORK}}\left(1\right)$ contains the minimum value of LWORK required for optimal performance.
12:   LWORK – INTEGERInput
On entry: the dimension of the array WORK as declared in the (sub)program from which F08FUF (ZUNMTR) is called.
If ${\mathbf{LWORK}}=-1$, a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued.
Suggested value: for optimal performance, ${\mathbf{LWORK}}\ge {\mathbf{N}}×\mathit{nb}$ if ${\mathbf{SIDE}}=\text{'L'}$ and at least ${\mathbf{M}}×\mathit{nb}$ if ${\mathbf{SIDE}}=\text{'R'}$, where $\mathit{nb}$ is the optimal block size.
Constraints:
• if ${\mathbf{SIDE}}=\text{'L'}$, ${\mathbf{LWORK}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$ or ${\mathbf{LWORK}}=-1$;
• if ${\mathbf{SIDE}}=\text{'R'}$, ${\mathbf{LWORK}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{M}}\right)$ or ${\mathbf{LWORK}}=-1$.
13:   INFO – INTEGEROutput
On exit: ${\mathbf{INFO}}=0$ unless the routine detects an error (see Section 6).

## 6  Error Indicators and Warnings

Errors or warnings detected by the routine:
${\mathbf{INFO}}<0$
If ${\mathbf{INFO}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

## 7  Accuracy

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

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 F08FGF (DORMTR).

## 9  Example

This example computes the two smallest eigenvalues, and the associated eigenvectors, of the matrix $A$, where
 $A = -2.28+0.00i 1.78-2.03i 2.26+0.10i -0.12+2.53i 1.78+2.03i -1.12+0.00i 0.01+0.43i -1.07+0.86i 2.26-0.10i 0.01-0.43i -0.37+0.00i 2.31-0.92i -0.12-2.53i -1.07-0.86i 2.31+0.92i -0.73+0.00i .$
Here $A$ is Hermitian and must first be reduced to tridiagonal form $T$ by F08FSF (ZHETRD). The program then calls F08JJF (DSTEBZ) to compute the requested eigenvalues and F08JXF (ZSTEIN) to compute the associated eigenvectors of $T$. Finally F08FUF (ZUNMTR) is called to transform the eigenvectors to those of $A$.

### 9.1  Program Text

Program Text (f08fufe.f90)

### 9.2  Program Data

Program Data (f08fufe.d)

### 9.3  Program Results

Program Results (f08fufe.r)