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dlabrd.f(3)				      LAPACK				      dlabrd.f(3)

NAME
       dlabrd.f -

SYNOPSIS
   Functions/Subroutines
       subroutine dlabrd (M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y, LDY)
	   DLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.

Function/Subroutine Documentation
   subroutine dlabrd (integerM, integerN, integerNB, double precision, dimension( lda, * )A,
       integerLDA, double precision, dimension( * )D, double precision, dimension( * )E, double
       precision, dimension( * )TAUQ, double precision, dimension( * )TAUP, double precision,
       dimension( ldx, * )X, integerLDX, double precision, dimension( ldy, * )Y, integerLDY)
       DLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.

       Purpose:

	    DLABRD reduces the first NB rows and columns of a real general
	    m by n matrix A to upper or lower bidiagonal form by an orthogonal
	    transformation Q**T * A * P, and returns the matrices X and Y which
	    are needed to apply the transformation to the unreduced part of A.

	    If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower
	    bidiagonal form.

	    This is an auxiliary routine called by DGEBRD

       Parameters:
	   M

		     M is INTEGER
		     The number of rows in the matrix A.

	   N

		     N is INTEGER
		     The number of columns in the matrix A.

	   NB

		     NB is INTEGER
		     The number of leading rows and columns of A to be reduced.

	   A

		     A is DOUBLE PRECISION array, dimension (LDA,N)
		     On entry, the m by n general matrix to be reduced.
		     On exit, the first NB rows and columns of the matrix are
		     overwritten; the rest of the array is unchanged.
		     If m >= n, elements on and below the diagonal in the first NB
		       columns, with the array TAUQ, represent the orthogonal
		       matrix Q as a product of elementary reflectors; and
		       elements above the diagonal in the first NB rows, with the
		       array TAUP, represent the orthogonal matrix P as a product
		       of elementary reflectors.
		     If m < n, elements below the diagonal in the first NB
		       columns, with the array TAUQ, represent the orthogonal
		       matrix Q as a product of elementary reflectors, and
		       elements on and above the diagonal in the first NB rows,
		       with the array TAUP, represent the orthogonal matrix P as
		       a product of elementary reflectors.
		     See Further Details.

	   LDA

		     LDA is INTEGER
		     The leading dimension of the array A.  LDA >= max(1,M).

	   D

		     D is DOUBLE PRECISION array, dimension (NB)
		     The diagonal elements of the first NB rows and columns of
		     the reduced matrix.  D(i) = A(i,i).

	   E

		     E is DOUBLE PRECISION array, dimension (NB)
		     The off-diagonal elements of the first NB rows and columns of
		     the reduced matrix.

	   TAUQ

		     TAUQ is DOUBLE PRECISION array dimension (NB)
		     The scalar factors of the elementary reflectors which
		     represent the orthogonal matrix Q. See Further Details.

	   TAUP

		     TAUP is DOUBLE PRECISION array, dimension (NB)
		     The scalar factors of the elementary reflectors which
		     represent the orthogonal matrix P. See Further Details.

	   X

		     X is DOUBLE PRECISION array, dimension (LDX,NB)
		     The m-by-nb matrix X required to update the unreduced part
		     of A.

	   LDX

		     LDX is INTEGER
		     The leading dimension of the array X. LDX >= max(1,M).

	   Y

		     Y is DOUBLE PRECISION array, dimension (LDY,NB)
		     The n-by-nb matrix Y required to update the unreduced part
		     of A.

	   LDY

		     LDY is INTEGER
		     The leading dimension of the array Y. LDY >= max(1,N).

       Author:
	   Univ. of Tennessee

	   Univ. of California Berkeley

	   Univ. of Colorado Denver

	   NAG Ltd.

       Date:
	   September 2012

       Further Details:

	     The matrices Q and P are represented as products of elementary
	     reflectors:

		Q = H(1) H(2) . . . H(nb)  and	P = G(1) G(2) . . . G(nb)

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

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

	     where tauq and taup are real scalars, and v and u are real vectors.

	     If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in
	     A(i:m,i); u(1:i) = 0, u(i+1) = 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).

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

	     The elements of the vectors v and u together form the m-by-nb matrix
	     V and the nb-by-n matrix U**T which are needed, with X and Y, to apply
	     the transformation to the unreduced part of the matrix, using a block
	     update of the form:  A := A - V*Y**T - X*U**T.

	     The contents of A on exit are illustrated by the following examples
	     with nb = 2:

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

	       (  1   1   u1  u1  u1 )		 (  1	u1  u1	u1  u1	u1 )
	       (  v1  1   1   u2  u2 )		 (  1	1   u2	u2  u2	u2 )
	       (  v1  v2  a   a   a  )		 (  v1	1   a	a   a	a  )
	       (  v1  v2  a   a   a  )		 (  v1	v2  a	a   a	a  )
	       (  v1  v2  a   a   a  )		 (  v1	v2  a	a   a	a  )
	       (  v1  v2  a   a   a  )

	     where a denotes an element of the original matrix which is unchanged,
	     vi denotes an element of the vector defining H(i), and ui an element
	     of the vector defining G(i).

       Definition at line 210 of file dlabrd.f.

Author
       Generated automatically by Doxygen for LAPACK from the source code.

Version 3.4.2				 Tue Sep 25 2012			      dlabrd.f(3)
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