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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dorgtr (f08ff)

## Purpose

nag_lapack_dorgtr (f08ff) generates the real orthogonal matrix Q$Q$, which was determined by nag_lapack_dsytrd (f08fe) when reducing a symmetric matrix to tridiagonal form.

## Syntax

[a, info] = f08ff(uplo, a, tau, 'n', n)
[a, info] = nag_lapack_dorgtr(uplo, a, tau, 'n', n)

## Description

nag_lapack_dorgtr (f08ff) is intended to be used after a call to nag_lapack_dsytrd (f08fe), 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_dsytrd (f08fe) represents the orthogonal matrix Q$Q$ as a product of n1$n-1$ elementary reflectors.
This function may be used to generate Q$Q$ explicitly as a square matrix.

## References

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

## Parameters

### Compulsory Input Parameters

1:     uplo – string (length ≥ 1)
This must be the same parameter uplo as supplied to nag_lapack_dsytrd (f08fe).
Constraint: uplo = 'U'${\mathbf{uplo}}=\text{'U'}$ or 'L'$\text{'L'}$.
2:     a(lda, : $:$) – double array
The first dimension of the array a must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
Details of the vectors which define the elementary reflectors, as returned by nag_lapack_dsytrd (f08fe).
3:     tau( : $:$) – double array
Note: the dimension of the array tau must be at least max (1,n1)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
Further details of the elementary reflectors, as returned by nag_lapack_dsytrd (f08fe).

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the array a The second dimension of the array a.
n$n$, the order of the matrix Q$Q$.
Constraint: n0${\mathbf{n}}\ge 0$.

lda work lwork

### Output Parameters

1:     a(lda, : $:$) – double array
The first dimension of the array a will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
ldamax (1,n)$\mathit{lda}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The n$n$ by n$n$ orthogonal matrix Q$Q$.
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: uplo, 2: n, 3: a, 4: lda, 5: tau, 6: work, 7: lwork, 8: 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 matrix Q$Q$ differs from an exactly orthogonal matrix by a matrix E$E$ such that
 ‖E‖2 = O(ε) , $‖E‖2 = O(ε) ,$
where ε$\epsilon$ is the machine precision.

The total number of floating point operations is approximately (4/3)n3$\frac{4}{3}{n}^{3}$.
The complex analogue of this function is nag_lapack_zungtr (f08ft).

## Example

```function nag_lapack_dorgtr_example
uplo = 'L';
a = [2.07, 0, 0, 0;
3.87, -0.21, 0, 0;
4.2, 1.87, 1.15, 0;
-1.15, 0.63, 2.06, -1.81];
[a, d, e, tau, info] = nag_lapack_dsytrd(uplo, a);
[aOut, info] = nag_lapack_dorgtr(uplo, a, tau)
```
```

aOut =

1.0000         0         0         0
0   -0.6643   -0.0400    0.7464
0   -0.7209   -0.2294   -0.6539
0    0.1974   -0.9725    0.1235

info =

0

```
```function f08ff_example
uplo = 'L';
a = [2.07, 0, 0, 0;
3.87, -0.21, 0, 0;
4.2, 1.87, 1.15, 0;
-1.15, 0.63, 2.06, -1.81];
[a, d, e, tau, info] = f08fe(uplo, a);
[aOut, info] = f08ff(uplo, a, tau)
```
```

aOut =

1.0000         0         0         0
0   -0.6643   -0.0400    0.7464
0   -0.7209   -0.2294   -0.6539
0    0.1974   -0.9725    0.1235

info =

0

```