Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dgebrd (f08ke)

## Purpose

nag_lapack_dgebrd (f08ke) reduces a real m$m$ by n$n$ matrix to bidiagonal form.

## Syntax

[a, d, e, tauq, taup, info] = f08ke(a, 'm', m, 'n', n)
[a, d, e, tauq, taup, info] = nag_lapack_dgebrd(a, 'm', m, 'n', n)

## Description

nag_lapack_dgebrd (f08ke) reduces a real m$m$ by n$n$ matrix A$A$ to bidiagonal form B$B$ by an orthogonal transformation: A = QBPT$A=QB{P}^{\mathrm{T}}$, where Q$Q$ and PT${P}^{\mathrm{T}}$ are orthogonal matrices of order m$m$ and n$n$ respectively.
If mn$m\ge n$, the reduction is given by:
A = Q
 ( B1 ) 0
PT = Q1 B1 PT ,
$A =Q B1 0 PT = Q1 B1 PT ,$
where B1${B}_{1}$ is an 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 )
PT = Q B1 P1T ,
$A =Q B1 0 PT = Q B1 P1T ,$
where B1${B}_{1}$ is an m$m$ by m$m$ lower bidiagonal matrix and P1T${P}_{1}^{\mathrm{T}}$ consists of the first m$m$ rows of PT${P}^{\mathrm{T}}$.
The orthogonal 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, : $:$) – double 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, : $:$) – double 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 orthogonal matrix Q$Q$ and elements above the first superdiagonal store details of the orthogonal matrix P$P$.
If m < n$m, the diagonal and first subdiagonal store the lower bidiagonal matrix B$B$, elements below the first subdiagonal store details of the orthogonal matrix Q$Q$ and elements above the diagonal store details of the orthogonal 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( : $:$) – double 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 orthogonal matrix Q$Q$.
5:     taup( : $:$) – double 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 orthogonal 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 QBPT = A + E$QB{P}^{\mathrm{T}}=A+E$, 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 floating point operations is approximately (4/3)n2(3mn)$\frac{4}{3}{n}^{2}\left(3m-n\right)$ if mn$m\ge n$ or (4/3)m2(3nm)$\frac{4}{3}{m}^{2}\left(3n-m\right)$ if m < n$m.
If mn$m\gg n$, it can be more efficient to first call nag_lapack_dgeqrf (f08ae) to perform a QR$QR$ factorization of A$A$, and then to call nag_lapack_dgebrd (f08ke) to reduce the factor R$R$ to bidiagonal form. This requires approximately 2n2(m + n)$2{n}^{2}\left(m+n\right)$ floating point operations.
If mn$m\ll n$, it can be more efficient to first call nag_lapack_dgelqf (f08ah) to perform an LQ$LQ$ factorization of A$A$, and then to call nag_lapack_dgebrd (f08ke) to reduce the factor L$L$ to bidiagonal form. This requires approximately 2m2(m + n)$2{m}^{2}\left(m+n\right)$ operations.
To form the orthogonal matrices PT${P}^{\mathrm{T}}$ and/or Q$Q$ nag_lapack_dgebrd (f08ke) may be followed by calls to nag_lapack_dorgbr (f08kf):
to form the m$m$ by m$m$ orthogonal matrix Q$Q$
```[a, info] = f08kf('Q', k, 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_dgebrd (f08ke);
to form the n$n$ by n$n$ orthogonal matrix PT${P}^{\mathrm{T}}$
```[a, info] = f08kf('P', k, 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_dgebrd (f08ke).
To apply Q$Q$ or P$P$ to a real rectangular matrix C$C$, nag_lapack_dgebrd (f08ke) may be followed by a call to nag_lapack_dormbr (f08kg).
The complex analogue of this function is nag_lapack_zgebrd (f08ks).

## Example

```function nag_lapack_dgebrd_example
a = [-0.57, -1.28, -0.39, 0.25;
-1.93, 1.08, -0.31, -2.14;
2.3, 0.24, 0.4, -0.35;
-1.93, 0.64, -0.66, 0.08;
0.15, 0.3, 0.15, -2.13;
-0.02, 1.03, -1.43, 0.5];
[aOut, d, e, tauq, taup, info] = nag_lapack_dgebrd(a)
```
```

aOut =

3.6177    1.2587   -0.4668   -0.4110
0.4609    2.4161    1.5262   -0.2095
-0.5492    0.1219   -1.9213   -1.1895
0.4609    0.0770    0.0557   -1.4265
-0.0358    0.3309   -0.1248   -0.4048
0.0048    0.2796    0.8322    0.2196

d =

3.6177
2.4161
-1.9213
-1.4265

e =

1.2587
1.5262
-1.1895

tauq =

1.1576
1.6550
1.1687
1.6501

taup =

1.4422
1.9159
0
0

info =

0

```
```function f08ke_example
a = [-0.57, -1.28, -0.39, 0.25;
-1.93, 1.08, -0.31, -2.14;
2.3, 0.24, 0.4, -0.35;
-1.93, 0.64, -0.66, 0.08;
0.15, 0.3, 0.15, -2.13;
-0.02, 1.03, -1.43, 0.5];
[aOut, d, e, tauq, taup, info] = f08ke(a)
```
```

aOut =

3.6177    1.2587   -0.4668   -0.4110
0.4609    2.4161    1.5262   -0.2095
-0.5492    0.1219   -1.9213   -1.1895
0.4609    0.0770    0.0557   -1.4265
-0.0358    0.3309   -0.1248   -0.4048
0.0048    0.2796    0.8322    0.2196

d =

3.6177
2.4161
-1.9213
-1.4265

e =

1.2587
1.5262
-1.1895

tauq =

1.1576
1.6550
1.1687
1.6501

taup =

1.4422
1.9159
0
0

info =

0

```