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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dopmtr (f08gg)

## Purpose

nag_lapack_dopmtr (f08gg) multiplies an arbitrary real matrix $C$ by the real orthogonal matrix $Q$ which was determined by nag_lapack_dsptrd (f08ge) when reducing a real symmetric matrix to tridiagonal form.

## Syntax

[ap, c, info] = f08gg(side, uplo, trans, ap, tau, c, 'm', m, 'n', n)
[ap, c, info] = nag_lapack_dopmtr(side, uplo, trans, ap, tau, c, 'm', m, 'n', n)

## Description

nag_lapack_dopmtr (f08gg) is intended to be used after a call to nag_lapack_dsptrd (f08ge), which reduces a real symmetric matrix $A$ to symmetric tridiagonal form $T$ by an orthogonal similarity transformation: $A=QT{Q}^{\mathrm{T}}$. nag_lapack_dsptrd (f08ge) represents the orthogonal matrix $Q$ as a product of elementary reflectors.
This function may be used to form one of the matrix products
 $QC , QTC , CQ ​ or ​ CQT ,$
overwriting the result on $C$ (which may be any real rectangular matrix).
A common application of this function is to transform a matrix $Z$ of eigenvectors of $T$ to the matrix $QZ$ of eigenvectors of $A$.

## References

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

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{side}$ – string (length ≥ 1)
Indicates how $Q$ or ${Q}^{\mathrm{T}}$ is to be applied to $C$.
${\mathbf{side}}=\text{'L'}$
$Q$ or ${Q}^{\mathrm{T}}$ is applied to $C$ from the left.
${\mathbf{side}}=\text{'R'}$
$Q$ or ${Q}^{\mathrm{T}}$ is applied to $C$ from the right.
Constraint: ${\mathbf{side}}=\text{'L'}$ or $\text{'R'}$.
2:     $\mathrm{uplo}$ – string (length ≥ 1)
This must be the same argument uplo as supplied to nag_lapack_dsptrd (f08ge).
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
3:     $\mathrm{trans}$ – string (length ≥ 1)
Indicates whether $Q$ or ${Q}^{\mathrm{T}}$ is to be applied to $C$.
${\mathbf{trans}}=\text{'N'}$
$Q$ is applied to $C$.
${\mathbf{trans}}=\text{'T'}$
${Q}^{\mathrm{T}}$ is applied to $C$.
Constraint: ${\mathbf{trans}}=\text{'N'}$ or $\text{'T'}$.
4:     $\mathrm{ap}\left(:\right)$ – double array
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 ${\mathbf{side}}=\text{'R'}$
Details of the vectors which define the elementary reflectors, as returned by nag_lapack_dsptrd (f08ge).
5:     $\mathrm{tau}\left(:\right)$ – double array
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 ${\mathbf{side}}=\text{'R'}$
Further details of the elementary reflectors, as returned by nag_lapack_dsptrd (f08ge).
6:     $\mathrm{c}\left(\mathit{ldc},:\right)$ – double array
The first dimension of the array c must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
The second dimension of the array c must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The $m$ by $n$ matrix $C$.

### Optional Input Parameters

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

### Output Parameters

1:     $\mathrm{ap}\left(:\right)$ – double array
The dimension of the array ap will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}×\left({\mathbf{m}}+1\right)/2\right)$ 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 ${\mathbf{side}}=\text{'R'}$
Is used as internal workspace prior to being restored and hence is unchanged.
2:     $\mathrm{c}\left(\mathit{ldc},:\right)$ – double array
The first dimension of the array c will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
The second dimension of the array c will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
c stores $QC$ or ${Q}^{\mathrm{T}}C$ or $CQ$ or $C{Q}^{\mathrm{T}}$ as specified by side and trans.
3:     $\mathrm{info}$int64int32nag_int scalar
${\mathbf{info}}=0$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

${\mathbf{info}}=-i$
If ${\mathbf{info}}=-i$, parameter $i$ had an illegal value on entry. The parameters are numbered as follows:
1: side, 2: uplo, 3: trans, 4: m, 5: n, 6: ap, 7: tau, 8: c, 9: ldc, 10: work, 11: 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 $E$ such that
 $E2 = Oε C2 ,$
where $\epsilon$ is the machine precision.

The total number of floating-point operations is approximately $2{m}^{2}n$ if ${\mathbf{side}}=\text{'L'}$ and $2m{n}^{2}$ if ${\mathbf{side}}=\text{'R'}$.
The complex analogue of this function is nag_lapack_zupmtr (f08gu).

## Example

This example computes the two smallest eigenvalues, and the associated eigenvectors, of the matrix $A$, where
 $A = 2.07 3.87 4.20 -1.15 3.87 -0.21 1.87 0.63 4.20 1.87 1.15 2.06 -1.15 0.63 2.06 -1.81 ,$
using packed storage. Here $A$ is symmetric and must first be reduced to tridiagonal form $T$ by nag_lapack_dsptrd (f08ge). The program then calls nag_lapack_dstebz (f08jj) to compute the requested eigenvalues and nag_lapack_dstein (f08jk) to compute the associated eigenvectors of $T$. Finally nag_lapack_dopmtr (f08gg) is called to transform the eigenvectors to those of $A$.
```function f08gg_example

fprintf('f08gg example results\n\n');

% Symmetric matrix A stored in symmetric packed format (Lower)
uplo = 'L';
n = int64(4);
ap = [2.07;     3.87;     4.20;    -1.15;
-0.21;     1.87;     0.63;
1.15;     2.06;
-1.81];

% Reduce A to tridiagonal form
[apf, d, e, tau, info] = f08ge( ...
uplo, n, ap);

% Two smallest eigenvalues
range = 'I';
order = 'B';
vl = 0;
vu = 0;
il = int64(1);
iu = int64(2);
abstol = 0;
[m, nsplit, w, iblock, isplit, info] = ...
f08jj( ...
range, order, vl, vu, il, iu, abstol, d, e);

% Get corresponding eigenvectors of T
[v, ifailv, info] = f08jk( ...
d, e, m, w, iblock, isplit);

% Transform Q*V to get eigenvectors of A
side = 'Left';
trans = 'No transpose';
[~, z, info] = f08gg( ...
side, uplo, trans, apf, tau, v);

% Normalize eigenvectors: largest element positive
for j = 1:m
[~,k] = max(abs(z(:,j)));
if z(k,j) < 0;
z(:,j) = -z(:,j);
end
end

% Display results
fprintf(' Eigenvalues numbered 1 to 2 are:\n   ');
fprintf(' %7.4f',w(1:m));
fprintf('\n\n');

[ifail] = x04ca( ...
'General', ' ', z, 'Corresponding eigenvectors of A');

```
```f08gg example results

Eigenvalues numbered 1 to 2 are:
-5.0034 -1.9987

Corresponding eigenvectors of A
1       2
1   0.5658 -0.2328
2  -0.3478  0.7994
3  -0.4740 -0.4087
4   0.5781  0.3737
```