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# NAG Toolbox: nag_lapack_zgebrd (f08ks)

## Purpose

nag_lapack_zgebrd (f08ks) reduces a complex $m$ by $n$ matrix to bidiagonal form.

## Syntax

[a, d, e, tauq, taup, info] = f08ks(a, 'm', m, 'n', n)
[a, d, e, tauq, taup, info] = nag_lapack_zgebrd(a, 'm', m, 'n', n)

## Description

nag_lapack_zgebrd (f08ks) reduces a complex $m$ by $n$ matrix $A$ to real bidiagonal form $B$ by a unitary transformation: $A=QB{P}^{\mathrm{H}}$, where $Q$ and ${P}^{\mathrm{H}}$ are unitary matrices of order $m$ and $n$ respectively.
If $m\ge n$, the reduction is given by:
 $A =Q B1 0 PH = Q1 B1 PH ,$
where ${B}_{1}$ is a real $n$ by $n$ upper bidiagonal matrix and ${Q}_{1}$ consists of the first $n$ columns of $Q$.
If $m, the reduction is given by
 $A =Q B1 0 PH = Q B1 P1H ,$
where ${B}_{1}$ is a real $m$ by $m$ lower bidiagonal matrix and ${P}_{1}^{\mathrm{H}}$ consists of the first $m$ rows of ${P}^{\mathrm{H}}$.
The unitary matrices $Q$ and $P$ are not formed explicitly but are represented as products of elementary reflectors (see the F08 Chapter Introduction for details). Functions are provided to work with $Q$ and $P$ 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:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex array
The first dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
The second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The $m$ by $n$ matrix $A$.

### Optional Input Parameters

1:     $\mathrm{m}$int64int32nag_int scalar
Default: the first dimension of the array a.
$m$, the number of rows of the matrix $A$.
Constraint: ${\mathbf{m}}\ge 0$.
2:     $\mathrm{n}$int64int32nag_int scalar
Default: the second dimension of the array a.
$n$, the number of columns of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.

### Output Parameters

1:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex array
The first dimension of the array a will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
The second dimension of the array a will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
If $m\ge n$, the diagonal and first superdiagonal store the upper bidiagonal matrix $B$, elements below the diagonal store details of the unitary matrix $Q$ and elements above the first superdiagonal store details of the unitary matrix $P$.
If $m, the diagonal and first subdiagonal store the lower bidiagonal matrix $B$, elements below the first subdiagonal store details of the unitary matrix $Q$ and elements above the diagonal store details of the unitary matrix $P$.
2:     $\mathrm{d}\left(:\right)$ – double array
The dimension of the array d will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$
The diagonal elements of the bidiagonal matrix $B$.
3:     $\mathrm{e}\left(:\right)$ – double array
The dimension of the array e will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)-1\right)$
The off-diagonal elements of the bidiagonal matrix $B$.
4:     $\mathrm{tauq}\left(:\right)$ – complex array
The dimension of the array tauq will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$
Further details of the unitary matrix $Q$.
5:     $\mathrm{taup}\left(:\right)$ – complex array
The dimension of the array taup will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$
Further details of the unitary matrix $P$.
6:     $\mathrm{info}$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: m, 2: n, 3: a, 4: lda, 5: d, 6: e, 7: tauq, 8: taup, 9: work, 10: lwork, 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 bidiagonal form $B$ satisfies $QB{P}^{\mathrm{H}}=A+E$, 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 $B$ 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 singular values and vectors.

## Further Comments

The total number of real floating-point operations is approximately $16{n}^{2}\left(3m-n\right)/3$ if $m\ge n$ or $16{m}^{2}\left(3n-m\right)/3$ if $m.
If $m\gg n$, it can be more efficient to first call nag_lapack_zgeqrf (f08as) to perform a $QR$ factorization of $A$, and then to call nag_lapack_zgebrd (f08ks) to reduce the factor $R$ to bidiagonal form. This requires approximately $8{n}^{2}\left(m+n\right)$ floating-point operations.
If $m\ll n$, it can be more efficient to first call nag_lapack_zgelqf (f08av) to perform an $LQ$ factorization of $A$, and then to call nag_lapack_zgebrd (f08ks) to reduce the factor $L$ to bidiagonal form. This requires approximately $8{m}^{2}\left(m+n\right)$ operations.
To form the unitary matrices ${P}^{\mathrm{H}}$ and/or $Q$ nag_lapack_zgebrd (f08ks) may be followed by calls to nag_lapack_zungbr (f08kt):
to form the $m$ by $m$ unitary matrix $Q$
```[a, info] = f08kt('Q', n, a, tauq);
```
but note that the second dimension of the array a must be at least m, which may be larger than was required by nag_lapack_zgebrd (f08ks);
to form the $n$ by $n$ unitary matrix ${P}^{\mathrm{H}}$
```[a, info] = f08kt('P', m, a, taup);
```
but note that the first dimension of the array a, specified by the argument lda, must be at least n, which may be larger than was required by nag_lapack_zgebrd (f08ks).
To apply $Q$ or $P$ to a complex rectangular matrix $C$, nag_lapack_zgebrd (f08ks) may be followed by a call to nag_lapack_zunmbr (f08ku).
The real analogue of this function is nag_lapack_dgebrd (f08ke).

## Example

This example reduces the matrix $A$ to bidiagonal form, where
 $A = 0.96-0.81i -0.03+0.96i -0.91+2.06i -0.05+0.41i -0.98+1.98i -1.20+0.19i -0.66+0.42i -0.81+0.56i 0.62-0.46i 1.01+0.02i 0.63-0.17i -1.11+0.60i -0.37+0.38i 0.19-0.54i -0.98-0.36i 0.22-0.20i 0.83+0.51i 0.20+0.01i -0.17-0.46i 1.47+1.59i 1.08-0.28i 0.20-0.12i -0.07+1.23i 0.26+0.26i .$
```function f08ks_example

fprintf('f08ks example results\n\n');

a = [ 0.96 - 0.81i, -0.03 + 0.96i, -0.91 + 2.06i, -0.05 + 0.41i;
-0.98 + 1.98i, -1.20 + 0.19i, -0.66 + 0.42i, -0.81 + 0.56i;
0.62 - 0.46i,  1.01 + 0.02i,  0.63 - 0.17i, -1.11 + 0.60i;
-0.37 + 0.38i,  0.19 - 0.54i, -0.98 - 0.36i,  0.22 - 0.20i;
0.83 + 0.51i,  0.20 + 0.01i, -0.17 - 0.46i,  1.47 + 1.59i;
1.08 - 0.28i,  0.20 - 0.12i, -0.07 + 1.23i,  0.26 + 0.26i];

% Reduce general complex matrix to real bidiagonal form A = ZBP^H
[B, d, e, tauq, taup, info] = f08ks(a);

fprintf(' Bidiagonal matrix B\n   Main diagonal  ');
fprintf(' %7.3f',d);
fprintf('\n   super-diagonal ');
fprintf(' %7.3f',e);
fprintf('\n');

```
```f08ks example results

Bidiagonal matrix B
Main diagonal    -3.087   2.066   1.873   2.002
super-diagonal    2.113   1.263  -1.613
```

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