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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$C$ by the real orthogonal matrix Q$Q$ 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$A$ to symmetric tridiagonal form T$T$ by an orthogonal similarity transformation: A = QTQT$A=QT{Q}^{\mathrm{T}}$. nag_lapack_dsptrd (f08ge) represents the orthogonal matrix Q$Q$ as a product of elementary reflectors.
This function may be used to form one of the matrix products
 QC , QTC , CQ ​ or ​ CQT , $QC , QTC , CQ ​ or ​ CQT ,$
overwriting the result on C$C$ (which may be any real rectangular matrix).
A common application of this function is to transform a matrix Z$Z$ of eigenvectors of T$T$ to the matrix QZ$QZ$ 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 QT${Q}^{\mathrm{T}}$ is to be applied to C$C$.
side = 'L'${\mathbf{side}}=\text{'L'}$
Q$Q$ or QT${Q}^{\mathrm{T}}$ is applied to C$C$ from the left.
side = 'R'${\mathbf{side}}=\text{'R'}$
Q$Q$ or QT${Q}^{\mathrm{T}}$ 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_dsptrd (f08ge).
Constraint: uplo = 'U'${\mathbf{uplo}}=\text{'U'}$ or 'L'$\text{'L'}$.
3:     trans – string (length ≥ 1)
Indicates whether Q$Q$ or QT${Q}^{\mathrm{T}}$ is to be applied to C$C$.
trans = 'N'${\mathbf{trans}}=\text{'N'}$
Q$Q$ is applied to C$C$.
trans = 'T'${\mathbf{trans}}=\text{'T'}$
QT${Q}^{\mathrm{T}}$ is applied to C$C$.
Constraint: trans = 'N'${\mathbf{trans}}=\text{'N'}$ or 'T'$\text{'T'}$.
4:     ap( : $:$) – double array
Note: the dimension of the array ap must be at least max (1, m × (m + 1) / 2 ) $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}×\left({\mathbf{m}}+1\right)/2\right)$ if side = 'L'${\mathbf{side}}=\text{'L'}$ and at least max (1, n × (n + 1) / 2 ) $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$ if side = 'R'${\mathbf{side}}=\text{'R'}$.
Details of the vectors which define the elementary reflectors, as returned by nag_lapack_dsptrd (f08ge).
5:     tau( : $:$) – double 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_dsptrd (f08ge).
6:     c(ldc, : $:$) – double 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${\mathbf{n}}\ge 0$.

ldc work

### Output Parameters

1:     ap( : $:$) – double array
Note: the dimension of the array ap must be at least max (1, m × (m + 1) / 2 ) $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}×\left({\mathbf{m}}+1\right)/2\right)$ if side = 'L'${\mathbf{side}}=\text{'L'}$ and at least max (1, n × (n + 1) / 2 ) $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$ if side = 'R'${\mathbf{side}}=\text{'R'}$.
Is used as internal workspace prior to being restored and hence is unchanged.
2:     c(ldc, : $:$) – double 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 QTC${Q}^{\mathrm{T}}C$ or CQ$CQ$ or CQT$C{Q}^{\mathrm{T}}$ as specified by side and trans.
3:     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: 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$E$ 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 floating point operations is approximately 2m2n$2{m}^{2}n$ if side = 'L'${\mathbf{side}}=\text{'L'}$ and 2mn2$2m{n}^{2}$ if side = 'R'${\mathbf{side}}=\text{'R'}$.
The complex analogue of this function is nag_lapack_zupmtr (f08gu).

## Example

```function nag_lapack_dopmtr_example
side = 'Left';
uplo = 'L';
trans = 'No transpose';
ap = [2.07;
-5.825753170191817;
0.4331793442217867;
-0.1186086299654892;
1.474093708197552;
2.624045178795586;
0.8062881532775791;
-0.6491595075457843;
0.9162727563219193;
-1.694934200651768];
tau = [1.664291789738249;
1.212047324162142;
0];
c = [0.5657591788223874, -0.2328424308031574;
0.6869179572505918, -0.1626170961491636;
-0.4395889372131648, -0.3017273343882724;
0.1217449705930083, 0.9101102670229791];
[apOut, cOut, info] = nag_lapack_dopmtr(side, uplo, trans, ap, tau, c)
```
```

apOut =

2.0700
-5.8258
0.4332
-0.1186
1.4741
2.6240
0.8063
-0.6492
0.9163
-1.6949

cOut =

0.5658   -0.2328
-0.3478    0.7994
-0.4740   -0.4087
0.5781    0.3737

info =

0

```
```function f08gg_example
side = 'Left';
uplo = 'L';
trans = 'No transpose';
ap = [2.07;
-5.825753170191817;
0.4331793442217867;
-0.1186086299654892;
1.474093708197552;
2.624045178795586;
0.8062881532775791;
-0.6491595075457843;
0.9162727563219193;
-1.694934200651768];
tau = [1.664291789738249;
1.212047324162142;
0];
c = [0.5657591788223874, -0.2328424308031574;
0.6869179572505918, -0.1626170961491636;
-0.4395889372131648, -0.3017273343882724;
0.1217449705930083, 0.9101102670229791];
[apOut, cOut, info] = f08gg(side, uplo, trans, ap, tau, c)
```
```

apOut =

2.0700
-5.8258
0.4332
-0.1186
1.4741
2.6240
0.8063
-0.6492
0.9163
-1.6949

cOut =

0.5658   -0.2328
-0.3478    0.7994
-0.4740   -0.4087
0.5781    0.3737

info =

0

```