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LAPACK-3

man page of LAPACK-3

LAPACK-3: reduces a complex general m by n matrix A to upper or lower real bidiagonal form B by a unitary transformation

```NAME
LAPACK-3  - reduces 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
overwritten 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.

FURTHER DETAILS
The matrices Q and P are represented as products of elementary
reflectors:
If m >= n,
Q = H(1) H(2) . . . H(n)  and  P = G(1) 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
vectors; 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,
Q = H(1) H(2) . . . H(m-1)  and  P = G(1) 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).

LAPACK routine (version 3.2)     March 2011                       LAPACK-3(3)
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