NAG FL Interface
f08ksf (zgebrd)
1
Purpose
f08ksf reduces a complex $m$ by $n$ matrix to bidiagonal form.
2
Specification
Fortran Interface
Subroutine f08ksf ( 
m, n, a, lda, d, e, tauq, taup, work, lwork, info) 
Integer, Intent (In) 
:: 
m, n, lda, lwork 
Integer, Intent (Out) 
:: 
info 
Real (Kind=nag_wp), Intent (Inout) 
:: 
d(*), e(*) 
Complex (Kind=nag_wp), Intent (Inout) 
:: 
a(lda,*), tauq(*), taup(*) 
Complex (Kind=nag_wp), Intent (Out) 
:: 
work(max(1,lwork)) 

C Header Interface
#include <nag.h>
void 
f08ksf_ (const Integer *m, const Integer *n, Complex a[], const Integer *lda, double d[], double e[], Complex tauq[], Complex taup[], Complex work[], const Integer *lwork, Integer *info) 

C++ Header Interface
#include <nag.h> extern "C" {
void 
f08ksf_ (const Integer &m, const Integer &n, Complex a[], const Integer &lda, double d[], double e[], Complex tauq[], Complex taup[], Complex work[], const Integer &lwork, Integer &info) 
}

The routine may be called by the names f08ksf, nagf_lapackeig_zgebrd or its LAPACK name zgebrd.
3
Description
f08ksf 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:
where
${B}_{1}$ is a real
$n$ by
$n$ upper bidiagonal matrix and
${Q}_{1}$ consists of the first
$n$ columns of
$Q$.
If
$m<n$, the reduction is given by
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). Routines are provided to work with
$Q$ and
$P$ in this representation (see
Section 9).
4
References
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5
Arguments

1:
$\mathbf{m}$ – Integer
Input

On entry: $m$, the number of rows of the matrix $A$.
Constraint:
${\mathbf{m}}\ge 0$.

2:
$\mathbf{n}$ – Integer
Input

On entry: $n$, the number of columns of the matrix $A$.
Constraint:
${\mathbf{n}}\ge 0$.

3:
$\mathbf{a}\left({\mathbf{lda}},*\right)$ – Complex (Kind=nag_wp) array
Input/Output

Note: the second dimension of the array
a
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the $m$ by $n$ matrix $A$.
On exit: if
$m\ge n$, the diagonal and first superdiagonal are overwritten by the upper bidiagonal matrix
$B$, elements below the diagonal are overwritten by details of the unitary matrix
$Q$ and elements above the first superdiagonal are overwritten by details of the unitary matrix
$P$.
If $m<n$, the diagonal and first subdiagonal are overwritten by the lower bidiagonal matrix $B$, elements below the first subdiagonal are overwritten by details of the unitary matrix $Q$ and elements above the diagonal are overwritten by details of the unitary matrix $P$.

4:
$\mathbf{lda}$ – Integer
Input

On entry: the first dimension of the array
a as declared in the (sub)program from which
f08ksf is called.
Constraint:
${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.

5:
$\mathbf{d}\left(*\right)$ – Real (Kind=nag_wp) array
Output

Note: the dimension of the array
d
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$.
On exit: the diagonal elements of the bidiagonal matrix $B$.

6:
$\mathbf{e}\left(*\right)$ – Real (Kind=nag_wp) array
Output

Note: the dimension of the array
e
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)1\right)$.
On exit: the offdiagonal elements of the bidiagonal matrix $B$.

7:
$\mathbf{tauq}\left(*\right)$ – Complex (Kind=nag_wp) array
Output

Note: the dimension of the array
tauq
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$.
On exit: further details of the unitary matrix $Q$.

8:
$\mathbf{taup}\left(*\right)$ – Complex (Kind=nag_wp) array
Output

Note: the dimension of the array
taup
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$.
On exit: further details of the unitary matrix $P$.

9:
$\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lwork}}\right)\right)$ – Complex (Kind=nag_wp) array
Workspace

On exit: if
${\mathbf{info}}={\mathbf{0}}$, the real part of
${\mathbf{work}}\left(1\right)$ contains the minimum value of
lwork required for optimal performance.

10:
$\mathbf{lwork}$ – Integer
Input

On entry: the dimension of the array
work as declared in the (sub)program from which
f08ksf is called.
If
${\mathbf{lwork}}=1$, a workspace query is assumed; the routine only calculates the optimal size of the
work array, returns this value as the first entry of the
work array, and no error message related to
lwork is issued.
Suggested value:
for optimal performance, ${\mathbf{lwork}}\ge \left({\mathbf{m}}+{\mathbf{n}}\right)\times \mathit{nb}$, where $\mathit{nb}$ is the optimal block size.
Constraint:
${\mathbf{lwork}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}},{\mathbf{n}}\right)$ or ${\mathbf{lwork}}=1$.

11:
$\mathbf{info}$ – Integer
Output

On exit:
${\mathbf{info}}=0$ unless the routine detects an error (see
Section 6).
6
Error Indicators and Warnings
 ${\mathbf{info}}<0$
If ${\mathbf{info}}=i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
7
Accuracy
The computed bidiagonal form
$B$ satisfies
$QB{P}^{\mathrm{H}}=A+E$, where
$c\left(n\right)$ 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.
8
Parallelism and Performance
f08ksf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08ksf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the
Users' Note for your implementation for any additional implementationspecific information.
The total number of real floatingpoint operations is approximately $16{n}^{2}\left(3mn\right)/3$ if $m\ge n$ or $16{m}^{2}\left(3nm\right)/3$ if $m<n$.
If
$m\gg n$, it can be more efficient to first call
f08asf to perform a
$QR$ factorization of
$A$, and then to call
f08ksf to reduce the factor
$R$ to bidiagonal form. This requires approximately
$8{n}^{2}\left(m+n\right)$ floatingpoint operations.
If
$m\ll n$, it can be more efficient to first call
f08avf to perform an
$LQ$ factorization of
$A$, and then to call
f08ksf to reduce the factor
$L$ to bidiagonal form. This requires approximately
$8{m}^{2}\left(m+n\right)$ operations.
To form the
$m$ by
$m$ unitary matrix
$Q$ f08ksf may be followed by calls to
f08ktf
. For example
Call zungbr('Q',m,m,n,a,lda,tauq,work,lwork,info)
but note that the second dimension of the array
a must be at least
m, which may be larger than was required by
f08ksf.
To form the
$n$ by
$n$ unitary matrix
${P}^{\mathrm{H}}$ another call to
f08kff may be made
. For example
Call zungbr('P',n,n,m,a,lda,taup,work,lwork,info)
but note that the first dimension of the array
a must be at least
n, which may be larger than was required by
f08ksf.
To apply
$Q$ or
$P$ to a complex rectangular matrix
$C$,
f08ksf may be followed by a call to
f08kuf.
The real analogue of this routine is
f08kef.
10
Example
This example reduces the matrix
$A$ to bidiagonal form, where
10.1
Program Text
10.2
Program Data
10.3
Program Results