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

ZGEBRD(l)					)					ZGEBRD(l)

NAME
       ZGEBRD - reduce a general complex M-by-N matrix A to upper or lower bidiagonal form B by a
       unitary transformation

SYNOPSIS
       SUBROUTINE ZGEBRD( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK, INFO )

	   INTEGER	  INFO, LDA, LWORK, M, N

	   DOUBLE	  PRECISION D( * ), E( * )

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

PURPOSE
       ZGEBRD reduces a general complex M-by-N matrix A to upper or lower bidiagonal form B by	a
       unitary	transformation:  Q**H * 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/output) COMPLEX*16 array, dimension
	       (LWORK) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

       LWORK   (input) INTEGER
	       The length of the array WORK.  LWORK >= max(1,M,N).  For optimum performance LWORK
	       >= (M+N)*NB, where NB is the optimal blocksize.

	       If  LWORK = -1, then 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 by XERBLA.

       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, and 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 				ZGEBRD(l)


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