slabrd.f(3)							      LAPACK							       slabrd.f(3)

NAME

       slabrd.f -

SYNOPSIS

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

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

       Purpose:

	    SLABRD 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 SGEBRD

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

	   TAUQ

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

	   TAUP

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

	   X

		     X is REAL 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 REAL 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 slabrd.f.

Author
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Version 3.4.2							  Tue Sep 25 2012						       slabrd.f(3)