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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

```