# Ubuntu Feisty 7.04 manual page repository

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### NAME

```        ZGEBD2  -  reduce  a  complex general m by n matrix A to upper or lower
real bidiagonal form B by a unitary transformation
```

### SYNOPSIS

```        SUBROUTINE ZGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO )

INTEGER        INFO, LDA, M, N

DOUBLE         PRECISION D( * ), E( * )

COMPLEX*16     A( LDA, * ), TAUP( * ), TAUQ( * ), WORK( * )
```

### PURPOSE

```        ZGEBD2 reduces a complex general m by n matrix A to upper or lower real
bidiagonal form B by a unitary transformation: Q’ * A * P = B.  If m >=
n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
```

### ARGUMENTS

```        M       (input) INTEGER
The number of rows in the matrix A.  M >= 0.

N       (input) INTEGER
The number of columns in the matrix A.  N >= 0.

A       (input/output) COMPLEX*16 array, dimension (LDA,N)
On entry, the m by n general matrix to be reduced.  On exit, if
m  >= n, the diagonal and the first superdiagonal are overwrit‐
ten with the upper bidiagonal matrix B; the elements below  the
diagonal,  with  the array TAUQ, represent the unitary matrix Q
as a product of elementary reflectors, and the  elements  above
the  first  superdiagonal,  with  the array TAUP, represent the
unitary matrix P as a product of elementary reflectors; if m  <
n,  the diagonal and the first subdiagonal are overwritten with
the lower bidiagonal matrix B; the  elements  below  the  first
subdiagonal,  with the array TAUQ, represent the unitary matrix
Q as a product of elementary reflectors, and the elements above
the diagonal, with the array TAUP, represent the unitary matrix
P as a product of elementary reflectors.  See Further  Details.
LDA      (input)  INTEGER The leading dimension of the array A.
LDA >= max(1,M).

D       (output) DOUBLE PRECISION array, dimension (min(M,N))
The diagonal elements  of  the  bidiagonal  matrix  B:  D(i)  =
A(i,i).

E       (output) DOUBLE PRECISION array, dimension (min(M,N)-1)
The  off-diagonal  elements of the bidiagonal matrix B: if m >=
n, E(i) = A(i,i+1) for i =  1,2,...,n-1;  if  m  <  n,  E(i)  =
A(i+1,i) for i = 1,2,...,m-1.

TAUQ    (output) COMPLEX*16 array dimension (min(M,N))
The scalar factors of the elementary reflectors which represent
the unitary matrix Q. See Further  Details.   TAUP     (output)
COMPLEX*16  array,  dimension  (min(M,N)) The scalar factors of
the elementary reflectors which represent the unitary matrix P.
See  Further  Details.   WORK     (workspace) COMPLEX*16 array,
dimension (max(M,N))

INFO    (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
The matrices Q and P are represented as products of elementary  reflec‐
tors:

If m >= n,

G(2) . . . G(n-1)

Each H(i) and G(i) has the form:

H(i) = I - tauq * v * v’  and G(i) = I - taup * u * u’

where  tauq  and taup are complex scalars, and v and u are complex vec‐
tors; v(1:i-1) = 0, v(i) =  1,  and  v(i+1:m)  is  stored  on  exit  in
A(i+1:m,i);  u(1:i)  = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in
A(i,i+2:n); tauq is stored in TAUQ(i) and taup in TAUP(i).

If m < n,

G(2) . . . G(m)

Each H(i) and G(i) has the form:

H(i) = I - tauq * v * v’  and G(i) = I - taup * u * u’

where tauq and taup are complex scalars, v and u are  complex  vectors;
v(1:i)  =  0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit  in  A(i,i+1:n);
tauq is stored in TAUQ(i) and taup in TAUP(i).

The contents of A on exit are illustrated by the following examples:

m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):

(  d   e   u1  u1  u1 )           (  d   u1  u1  u1  u1  u1 )
(  v1  d   e   u2  u2 )           (  e   d   u2  u2  u2  u2 )
(  v1  v2  d   e   u3 )           (  v1  e   d   u3  u3  u3 )
(  v1  v2  v3  d   e  )           (  v1  v2  e   d   u4  u4 )
(  v1  v2  v3  v4  d  )           (  v1  v2  v3  e   d   u5 )
(  v1  v2  v3  v4  v5 )

where  d  and  e  denote  diagonal  and  off-diagonal elements of B, vi
denotes an element of the vector defining H(i), and ui  an  element  of
the vector defining G(i).

```
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