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

DGEBRD(l)					)					DGEBRD(l)

NAME
       DGEBRD  -  reduce a general real M-by-N matrix A to upper or lower bidiagonal form B by an
       orthogonal transformation

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

	   INTEGER	  INFO, LDA, LWORK, M, N

	   DOUBLE	  PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ), TAUQ( * ), WORK( * )

PURPOSE
       DGEBRD reduces a general real M-by-N matrix A to upper or lower bidiagonal form	B  by  an
       orthogonal  transformation: Q**T * 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) DOUBLE PRECISION 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  orthogonal
	       matrix  Q  as a product of elementary reflectors, and the elements above the first
	       superdiagonal, with the array TAUP, represent the orthogonal 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  sub-
	       diagonal,  with	the array TAUQ, represent the orthogonal matrix Q as a product of
	       elementary reflectors, and the elements above the diagonal, with the  array  TAUP,
	       represent the orthogonal matrix P as a product of elementary reflectors.  See Fur-
	       ther 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) DOUBLE PRECISION array dimension (min(M,N))
	       The scalar factors of the elementary reflectors	which  represent  the  orthogonal
	       matrix Q. See Further Details.  TAUP    (output) DOUBLE PRECISION array, dimension
	       (min(M,N)) The scalar factors of the elementary	reflectors  which  represent  the
	       orthogonal  matrix P. See Further Details.  WORK    (workspace/output) DOUBLE PRE-
	       CISION 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 real scalars, and v and u are real 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 real scalars, and v and u are real 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 				DGEBRD(l)


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