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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zunmtr (f08fu)

## Purpose

nag_lapack_zunmtr (f08fu) multiplies an arbitrary complex matrix C$C$ by the complex unitary matrix Q$Q$ which was determined by nag_lapack_zhetrd (f08fs) when reducing a complex Hermitian matrix to tridiagonal form.

## Syntax

[c, info] = f08fu(side, uplo, trans, a, tau, c, 'm', m, 'n', n)
[c, info] = nag_lapack_zunmtr(side, uplo, trans, a, tau, c, 'm', m, 'n', n)

## Description

nag_lapack_zunmtr (f08fu) is intended to be used after a call to nag_lapack_zhetrd (f08fs), which reduces a complex Hermitian matrix A$A$ to real symmetric tridiagonal form T$T$ by a unitary similarity transformation: A = QTQH$A=QT{Q}^{\mathrm{H}}$. nag_lapack_zhetrd (f08fs) represents the unitary matrix Q$Q$ as a product of elementary reflectors.
This function may be used to form one of the matrix products
 QC , QHC , CQ ​ or ​ CQH , $QC , QHC , CQ ​ or ​ CQH ,$
overwriting the result on C$C$ (which may be any complex rectangular matrix).
A common application of this function is to transform a matrix Z$Z$ of eigenvectors of T$T$ to the matrix QZ$\mathit{QZ}$ of eigenvectors of A$A$.

## References

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

## Parameters

### Compulsory Input Parameters

1:     side – string (length ≥ 1)
Indicates how Q$Q$ or QH${Q}^{\mathrm{H}}$ is to be applied to C$C$.
side = 'L'${\mathbf{side}}=\text{'L'}$
Q$Q$ or QH${Q}^{\mathrm{H}}$ is applied to C$C$ from the left.
side = 'R'${\mathbf{side}}=\text{'R'}$
Q$Q$ or QH${Q}^{\mathrm{H}}$ is applied to C$C$ from the right.
Constraint: side = 'L'${\mathbf{side}}=\text{'L'}$ or 'R'$\text{'R'}$.
2:     uplo – string (length ≥ 1)
This must be the same parameter uplo as supplied to nag_lapack_zhetrd (f08fs).
Constraint: uplo = 'U'${\mathbf{uplo}}=\text{'U'}$ or 'L'$\text{'L'}$.
3:     trans – string (length ≥ 1)
Indicates whether Q$Q$ or QH${Q}^{\mathrm{H}}$ is to be applied to C$C$.
trans = 'N'${\mathbf{trans}}=\text{'N'}$
Q$Q$ is applied to C$C$.
trans = 'C'${\mathbf{trans}}=\text{'C'}$
QH${Q}^{\mathrm{H}}$ is applied to C$C$.
Constraint: trans = 'N'${\mathbf{trans}}=\text{'N'}$ or 'C'$\text{'C'}$.
4:     a(lda, : $:$) – complex array
The first dimension, lda, of the array a must satisfy
• if side = 'L'${\mathbf{side}}=\text{'L'}$, lda max (1,m) $\mathit{lda}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
• if side = 'R'${\mathbf{side}}=\text{'R'}$, lda max (1,n) $\mathit{lda}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array must be at least max (1,m)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$ if side = 'L'${\mathbf{side}}=\text{'L'}$ and at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ if side = 'R'${\mathbf{side}}=\text{'R'}$
Details of the vectors which define the elementary reflectors, as returned by nag_lapack_zhetrd (f08fs).
5:     tau( : $:$) – complex array
Note: the dimension of the array tau must be at least max (1,m1)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}-1\right)$ if side = 'L'${\mathbf{side}}=\text{'L'}$ and at least max (1,n1)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$ if side = 'R'${\mathbf{side}}=\text{'R'}$.
Further details of the elementary reflectors, as returned by nag_lapack_zhetrd (f08fs).
6:     c(ldc, : $:$) – complex array
The first dimension of the array c must be at least max (1,m)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$
The second dimension of the array must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The m$m$ by n$n$ matrix C$C$.

### Optional Input Parameters

1:     m – int64int32nag_int scalar
Default: The first dimension of the array c.
m$m$, the number of rows of the matrix C$C$; m$m$ is also the order of Q$Q$ if side = 'L'${\mathbf{side}}=\text{'L'}$.
Constraint: m0${\mathbf{m}}\ge 0$.
2:     n – int64int32nag_int scalar
Default: The second dimension of the array c.
n$n$, the number of columns of the matrix C$C$; n$n$ is also the order of Q$Q$ if side = 'R'${\mathbf{side}}=\text{'R'}$.
Constraint: n0${\mathbf{n}}\ge 0$.

### Input Parameters Omitted from the MATLAB Interface

lda ldc work lwork

### Output Parameters

1:     c(ldc, : $:$) – complex array
The first dimension of the array c will be max (1,m)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$
The second dimension of the array will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
ldcmax (1,m)$\mathit{ldc}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
c stores QC$QC$ or QHC${Q}^{\mathrm{H}}C$ or CQ$CQ$ or CQH$C{Q}^{\mathrm{H}}$ as specified by side and trans.
2:     info – int64int32nag_int scalar
info = 0${\mathbf{info}}=0$ unless the function detects an error (see Section [Error Indicators and Warnings]).

## Error Indicators and Warnings

info = i${\mathbf{info}}=-i$
If info = i${\mathbf{info}}=-i$, parameter i$i$ had an illegal value on entry. The parameters are numbered as follows:
1: side, 2: uplo, 3: trans, 4: m, 5: n, 6: a, 7: lda, 8: tau, 9: c, 10: ldc, 11: work, 12: lwork, 13: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.

## Accuracy

The computed result differs from the exact result by a matrix E$E$ such that
 ‖E‖2 = O(ε) ‖C‖2 , $‖E‖2 = O(ε) ‖C‖2 ,$
where ε$\epsilon$ is the machine precision.

The total number of real floating point operations is approximately 8m2n$8{m}^{2}n$ if side = 'L'${\mathbf{side}}=\text{'L'}$ and 8mn2$8m{n}^{2}$ if side = 'R'${\mathbf{side}}=\text{'R'}$.
The real analogue of this function is nag_lapack_dormtr (f08fg).

## Example

```function nag_lapack_zunmtr_example
side = 'Left';
uplo = 'L';
trans = 'No transpose';
a = [complex(-2.28),  0 + 0i,  0 + 0i,  0 + 0i;
1.78 + 2.03i,  -1.12 + 0i,  0 + 0i,  0 + 0i;
2.26 - 0.1i,  0.01 - 0.43i,  -0.37 + 0i,  0 + 0i;
-0.12 - 2.53i,  -1.07 - 0.86i,  2.31 + 0.92i,  -0.73 + 0i];
range = 'I';
order = 'B';
vl = 0;
vu = 0;
il = int64(1);
iu = int64(2);
abstol = 0;
[a, d, e, tau, info] = nag_lapack_zhetrd(uplo, a);
[m, nsplit, w, iblock, isplit, info] = ...
nag_lapack_dstebz(range, order, vl, vu, il, iu, abstol, d, e);
[c, ifailv, info] = nag_lapack_zstein(d, e, m, w, iblock, isplit);
[cOut, info] = nag_lapack_zunmtr(side, uplo, trans, a, tau, c)
```
```

cOut =

0.7299 + 0.0000i  -0.2595 + 0.0000i
-0.1663 - 0.2061i   0.5969 + 0.4214i
-0.4165 - 0.1417i  -0.2965 - 0.1507i
0.1743 + 0.4162i   0.3482 + 0.4085i

info =

0

```
```function f08fu_example
side = 'Left';
uplo = 'L';
trans = 'No transpose';
a = [complex(-2.28),  0 + 0i,  0 + 0i,  0 + 0i;
1.78 + 2.03i,  -1.12 + 0i,  0 + 0i,  0 + 0i;
2.26 - 0.1i,  0.01 - 0.43i,  -0.37 + 0i,  0 + 0i;
-0.12 - 2.53i,  -1.07 - 0.86i,  2.31 + 0.92i,  -0.73 + 0i];
range = 'I';
order = 'B';
vl = 0;
vu = 0;
il = int64(1);
iu = int64(2);
abstol = 0;
[a, d, e, tau, info] = f08fs(uplo, a);
[m, nsplit, w, iblock, isplit, info] = ...
f08jj(range, order, vl, vu, il, iu, abstol, d, e);
[c, ifailv, info] = f08jx(d, e, m, w, iblock, isplit);
[cOut, info] = f08fu(side, uplo, trans, a, tau, c)
```
```

cOut =

0.7299 + 0.0000i  -0.2595 + 0.0000i
-0.1663 - 0.2061i   0.5969 + 0.4214i
-0.4165 - 0.1417i  -0.2965 - 0.1507i
0.1743 + 0.4162i   0.3482 + 0.4085i

info =

0

```