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# NAG Toolbox: nag_lapack_dsytrd (f08fe)

## Purpose

nag_lapack_dsytrd (f08fe) reduces a real symmetric matrix to tridiagonal form.

## Syntax

[a, d, e, tau, info] = f08fe(uplo, a, 'n', n)
[a, d, e, tau, info] = nag_lapack_dsytrd(uplo, a, 'n', n)

## Description

nag_lapack_dsytrd (f08fe) reduces a real symmetric matrix $A$ to symmetric tridiagonal form $T$ by an orthogonal similarity transformation: $A=QT{Q}^{\mathrm{T}}$.
The matrix $Q$ is not formed explicitly but is represented as a product of $n-1$ elementary reflectors (see the F08 Chapter Introduction for details). Functions are provided to work with $Q$ in this representation (see Further Comments).

## References

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

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{uplo}$ – string (length ≥ 1)
Indicates whether the upper or lower triangular part of $A$ is stored.
${\mathbf{uplo}}=\text{'U'}$
The upper triangular part of $A$ is stored.
${\mathbf{uplo}}=\text{'L'}$
The lower triangular part of $A$ is stored.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
2:     $\mathrm{a}\left(\mathit{lda},:\right)$ – double array
The first dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The $n$ by $n$ symmetric matrix $A$.
• If ${\mathbf{uplo}}=\text{'U'}$, the upper triangular part of $a$ must be stored and the elements of the array below the diagonal are not referenced.
• If ${\mathbf{uplo}}=\text{'L'}$, the lower triangular part of $a$ must be stored and the elements of the array above the diagonal are not referenced.

### Optional Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
Default: the first dimension of the array a and the second dimension of the array a.
$n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.

### Output Parameters

1:     $\mathrm{a}\left(\mathit{lda},:\right)$ – double array
The first dimension of the array a will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array a will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
a stores the tridiagonal matrix $T$ and details of the orthogonal matrix $Q$ as specified by uplo.
2:     $\mathrm{d}\left(:\right)$ – double array
The dimension of the array d will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The diagonal elements of the tridiagonal matrix $T$.
3:     $\mathrm{e}\left(:\right)$ – double array
The dimension of the array e will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$
The off-diagonal elements of the tridiagonal matrix $T$.
4:     $\mathrm{tau}\left(:\right)$ – double array
The dimension of the array tau will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$
Further details of the orthogonal matrix $Q$.
5:     $\mathrm{info}$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: uplo, 2: n, 3: a, 4: lda, 5: d, 6: e, 7: tau, 8: work, 9: lwork, 10: 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 tridiagonal matrix $T$ is exactly similar to a nearby matrix $\left(A+E\right)$, where
 $E2≤ c n ε A2 ,$
$c\left(n\right)$ is a modestly increasing function of $n$, and $\epsilon$ is the machine precision.
The elements of $T$ themselves may be sensitive to small perturbations in $A$ or to rounding errors in the computation, but this does not affect the stability of the eigenvalues and eigenvectors.

## Further Comments

The total number of floating-point operations is approximately $\frac{4}{3}{n}^{3}$.
To form the orthogonal matrix $Q$ nag_lapack_dsytrd (f08fe) may be followed by a call to nag_lapack_dorgtr (f08ff):
```[a, info] = f08ff(uplo, a, tau);
```
To apply $Q$ to an $n$ by $p$ real matrix $C$ nag_lapack_dsytrd (f08fe) may be followed by a call to nag_lapack_dormtr (f08fg). For example,
```[c, info] = f08fg('Left', uplo, 'No Transpose', a, tau, c);
```
forms the matrix product $QC$.
The complex analogue of this function is nag_lapack_zhetrd (f08fs).

## Example

This example reduces the matrix $A$ to tridiagonal form, where
 $A = 2.07 3.87 4.20 -1.15 3.87 -0.21 1.87 0.63 4.20 1.87 1.15 2.06 -1.15 0.63 2.06 -1.81 .$
```function f08fe_example

fprintf('f08fe example results\n\n');

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];
n = size(a,1);

[~, d, e, tau, info] = f08fe( ...
uplo, a);

fprintf('Diagonal and off-diagonal elements of tridiagonal form\n\n');
fprintf('    i         D           E\n');

for j = 1:n-1
fprintf('%5d%12.5f%12.5f\n', j, d(j), e(j));
end
fprintf('%5d%12.5f\n', n, d(n));

```
```f08fe example results

Diagonal and off-diagonal elements of tridiagonal form

i         D           E
1     2.07000    -5.82575
2     1.47409     2.62405
3    -0.64916     0.91627
4    -1.69493
```

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