# NAG Library Routine Document

## 1Purpose

f08ksf (zgebrd) reduces a complex $m$ by $n$ matrix to bidiagonal form.

## 2Specification

Fortran Interface
 Subroutine f08ksf ( m, n, a, lda, d, e, tauq, taup, work, 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))
#include nagmk26.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)
The routine may be called by its LAPACK name zgebrd.

## 3Description

f08ksf (zgebrd) 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). Routines are provided to work with $Q$ and $P$ in this representation (see Section 9).

## 4References

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

## 5Arguments

1:     $\mathbf{m}$ – IntegerInput
On entry: $m$, the number of rows of the matrix $A$.
Constraint: ${\mathbf{m}}\ge 0$.
2:     $\mathbf{n}$ – IntegerInput
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) arrayInput/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, 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:     $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08ksf (zgebrd) 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) arrayOutput
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) arrayOutput
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 off-diagonal elements of the bidiagonal matrix $B$.
7:     $\mathbf{tauq}\left(*\right)$ – Complex (Kind=nag_wp) arrayOutput
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) arrayOutput
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) arrayWorkspace
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:   $\mathbf{lwork}$ – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08ksf (zgebrd) 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, ${\mathbf{lwork}}\ge \left({\mathbf{m}}+{\mathbf{n}}\right)×\mathit{nb}$, 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:   $\mathbf{info}$ – IntegerOutput
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

## 6Error 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.

## 7Accuracy

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.

## 8Parallelism and Performance

f08ksf (zgebrd) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08ksf (zgebrd) 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 implementation-specific information.

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 f08asf (zgeqrf) to perform a $QR$ factorization of $A$, and then to call f08ksf (zgebrd) 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 f08avf (zgelqf) to perform an $LQ$ factorization of $A$, and then to call f08ksf (zgebrd) 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 $Q$ f08ksf (zgebrd) may be followed by calls to f08ktf (zungbr):
to form the $m$ by $m$ unitary matrix $Q$
```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 (zgebrd);
to form the $n$ by $n$ unitary matrix ${P}^{\mathrm{H}}$
```Call zungbr('P',n,n,m,a,lda,taup,work,lwork,info)
```
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 f08ksf (zgebrd).
To apply $Q$ or $P$ to a complex rectangular matrix $C$, f08ksf (zgebrd) may be followed by a call to f08kuf (zunmbr).
The real analogue of this routine is f08kef (dgebrd).

## 10Example

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

### 10.1Program Text

Program Text (f08ksfe.f90)

### 10.2Program Data

Program Data (f08ksfe.d)

### 10.3Program Results

Program Results (f08ksfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017