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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zgebrd (f08ks)

## Purpose

nag_lapack_zgebrd (f08ks) reduces a complex m$m$ by n$n$ 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$m$ by n$n$ matrix A$A$ to real bidiagonal form B$B$ by a unitary transformation: A = QBPH$A=QB{P}^{\mathrm{H}}$, where Q$Q$ and PH${P}^{\mathrm{H}}$ are unitary matrices of order m$m$ and n$n$ respectively.
If mn$m\ge n$, the reduction is given by:
A = Q
 ( B1 ) 0
PH = Q1 B1 PH ,
$A =Q B1 0 PH = Q1 B1 PH ,$
where B1${B}_{1}$ is a real n$n$ by n$n$ upper bidiagonal matrix and Q1${Q}_{1}$ consists of the first n$n$ columns of Q$Q$.
If m < n$m, the reduction is given by
A = Q
 ( B1 0 )
PH = Q B1 P1H ,
$A =Q B1 0 PH = Q B1 P1H ,$
where B1${B}_{1}$ is a real m$m$ by m$m$ lower bidiagonal matrix and P1H${P}_{1}^{\mathrm{H}}$ consists of the first m$m$ rows of PH${P}^{\mathrm{H}}$.
The unitary matrices Q$Q$ and P$P$ 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$Q$ and P$P$ in this representation (see Section [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:     a(lda, : $:$) – complex array
The first dimension of the array a 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 A$A$.

### Optional Input Parameters

1:     m – int64int32nag_int scalar
Default: The first dimension of the array a.
m$m$, the number of rows of the matrix A$A$.
Constraint: m0${\mathbf{m}}\ge 0$.
2:     n – int64int32nag_int scalar
Default: The second dimension of the array a.
n$n$, the number of columns of the matrix A$A$.
Constraint: n0${\mathbf{n}}\ge 0$.

lda work lwork

### Output Parameters

1:     a(lda, : $:$) – complex array
The first dimension of the array a 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)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
ldamax (1,m)$\mathit{lda}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
If mn$m\ge n$, the diagonal and first superdiagonal store the upper bidiagonal matrix B$B$, elements below the diagonal store details of the unitary matrix Q$Q$ and elements above the first superdiagonal store details of the unitary matrix P$P$.
If m < n$m, the diagonal and first subdiagonal store the lower bidiagonal matrix B$B$, elements below the first subdiagonal store details of the unitary matrix Q$Q$ and elements above the diagonal store details of the unitary matrix P$P$.
2:     d( : $:$) – double array
Note: the dimension of the array d must be at least max (1,min (m,n))$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$.
The diagonal elements of the bidiagonal matrix B$B$.
3:     e( : $:$) – double array
Note: the dimension of the array e must be at least max (1,min (m,n)1)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)-1\right)$.
The off-diagonal elements of the bidiagonal matrix B$B$.
4:     tauq( : $:$) – complex array
Note: the dimension of the array tauq must be at least max (1,min (m,n))$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$.
Further details of the unitary matrix Q$Q$.
5:     taup( : $:$) – complex array
Note: the dimension of the array taup must be at least max (1,min (m,n))$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$.
Further details of the unitary matrix P$P$.
6:     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: 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$B$ satisfies QBPH = A + E$QB{P}^{\mathrm{H}}=A+E$, where
 ‖E‖2 ≤ c (n) ε ‖A‖2 , $‖E‖2 ≤ c (n) ε ‖A‖2 ,$
c(n)$c\left(n\right)$ is a modestly increasing function of n$n$, and ε$\epsilon$ is the machine precision.
The elements of B$B$ themselves may be sensitive to small perturbations in A$A$ or to rounding errors in the computation, but this does not affect the stability of the singular values and vectors.

The total number of real floating point operations is approximately 16n2(3mn) / 3$16{n}^{2}\left(3m-n\right)/3$ if mn$m\ge n$ or 16m2(3nm) / 3$16{m}^{2}\left(3n-m\right)/3$ if m < n$m.
If mn$m\gg n$, it can be more efficient to first call nag_lapack_zgeqrf (f08as) to perform a QR$QR$ factorization of A$A$, and then to call nag_lapack_zgebrd (f08ks) to reduce the factor R$R$ to bidiagonal form. This requires approximately 8n2(m + n)$8{n}^{2}\left(m+n\right)$ floating point operations.
If mn$m\ll n$, it can be more efficient to first call nag_lapack_zgelqf (f08av) to perform an LQ$LQ$ factorization of A$A$, and then to call nag_lapack_zgebrd (f08ks) to reduce the factor L$L$ to bidiagonal form. This requires approximately 8m2(m + n)$8{m}^{2}\left(m+n\right)$ operations.
To form the unitary matrices PH${P}^{\mathrm{H}}$ and/or Q$Q$ nag_lapack_zgebrd (f08ks) may be followed by calls to nag_lapack_zungbr (f08kt):
to form the m$m$ by m$m$ unitary matrix Q$Q$
```[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$n$ by n$n$ unitary matrix PH${P}^{\mathrm{H}}$
```[a, info] = f08kt('P', m, a, taup);
```
but note that the first dimension of the array a, specified by the parameter lda, must be at least n, which may be larger than was required by nag_lapack_zgebrd (f08ks).
To apply Q$Q$ or P$P$ to a complex rectangular matrix C$C$, 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

```function nag_lapack_zgebrd_example
a = [ 0.96 - 0.81i,  -0.03 + 0.96i,  -0.91 + 2.06i,  -0.05 + 0.41i;
-0.98 + 1.98i,  -1.2 + 0.19i,  -0.66 + 0.42i, ...
-0.81 + 0.56i;
0.62 - 0.46i,  1.01 + 0.02i,  0.63 - 0.17i,  -1.11 + 0.6i;
-0.37 + 0.38i,  0.19 - 0.54i,  -0.98 - 0.36i,  0.22 - 0.2i;
0.83 + 0.51i,  0.2 + 0.01i,  -0.17 - 0.46i,  1.47 + 1.59i;
1.08 - 0.28i,  0.2 - 0.12i,  -0.07 + 1.23i,  0.26 + 0.26i];
[aOut, d, e, tauq, taup, info] = nag_lapack_zgebrd(a)
```
```

aOut =

-3.0870 + 0.0000i   2.1126 + 0.0000i   0.0543 + 0.4543i   0.3757 + 0.1070i
-0.3270 + 0.4238i   2.0660 + 0.0000i   1.2628 + 0.0000i   0.0283 + 0.1650i
0.1692 - 0.0798i  -0.2585 - 0.0137i   1.8731 + 0.0000i  -1.6126 + 0.0000i
-0.1060 + 0.0727i   0.0582 + 0.0068i  -0.3219 + 0.3404i   2.0022 + 0.0000i
0.1729 + 0.1606i   0.0884 - 0.1430i  -0.4052 - 0.2475i   0.2871 + 0.1826i
0.2699 - 0.0152i  -0.0551 - 0.1065i   0.2172 + 0.2910i   0.5596 - 0.0569i

d =

-3.0870
2.0660
1.8731
2.0022

e =

2.1126
1.2628
-1.6126

tauq =

1.3110 - 0.2624i
1.7965 - 0.0234i
1.2420 - 0.1807i
1.0144 + 0.6225i

taup =

1.2312 - 0.5404i
1.2623 - 0.9286i
1.7829 - 0.6221i
0.0000 + 0.0000i

info =

0

```
```function f08ks_example
a = [ 0.96 - 0.81i,  -0.03 + 0.96i,  -0.91 + 2.06i,  -0.05 + 0.41i;
-0.98 + 1.98i,  -1.2 + 0.19i,  -0.66 + 0.42i, ...
-0.81 + 0.56i;
0.62 - 0.46i,  1.01 + 0.02i,  0.63 - 0.17i,  -1.11 + 0.6i;
-0.37 + 0.38i,  0.19 - 0.54i,  -0.98 - 0.36i,  0.22 - 0.2i;
0.83 + 0.51i,  0.2 + 0.01i,  -0.17 - 0.46i,  1.47 + 1.59i;
1.08 - 0.28i,  0.2 - 0.12i,  -0.07 + 1.23i,  0.26 + 0.26i];
[aOut, d, e, tauq, taup, info] = f08ks(a)
```
```

aOut =

-3.0870 + 0.0000i   2.1126 + 0.0000i   0.0543 + 0.4543i   0.3757 + 0.1070i
-0.3270 + 0.4238i   2.0660 + 0.0000i   1.2628 + 0.0000i   0.0283 + 0.1650i
0.1692 - 0.0798i  -0.2585 - 0.0137i   1.8731 + 0.0000i  -1.6126 + 0.0000i
-0.1060 + 0.0727i   0.0582 + 0.0068i  -0.3219 + 0.3404i   2.0022 + 0.0000i
0.1729 + 0.1606i   0.0884 - 0.1430i  -0.4052 - 0.2475i   0.2871 + 0.1826i
0.2699 - 0.0152i  -0.0551 - 0.1065i   0.2172 + 0.2910i   0.5596 - 0.0569i

d =

-3.0870
2.0660
1.8731
2.0022

e =

2.1126
1.2628
-1.6126

tauq =

1.3110 - 0.2624i
1.7965 - 0.0234i
1.2420 - 0.1807i
1.0144 + 0.6225i

taup =

1.2312 - 0.5404i
1.2623 - 0.9286i
1.7829 - 0.6221i
0.0000 + 0.0000i

info =

0

```