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RedHat 9 (Linux i386) - man page for zgebd2 (redhat section l)

ZGEBD2(l)					)					ZGEBD2(l)

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 diago-
	       nal  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 over-
	       written with the lower bidiagonal matrix B; the elements below the first subdiago-
	       nal,  with  the array TAUQ, represent the unitary matrix Q as a product of elemen-
	       tary reflectors, and the elements above the diagonal, with the array TAUP,  repre-
	       sent  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 version 3.0			   15 June 2000 				ZGEBD2(l)


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