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      SUBROUTINE DLABRD( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y,

     $                   LDY )

*

*  -- LAPACK auxiliary routine (version 3.2) --

*  -- LAPACK is a software package provided by Univ. of Tennessee,    --

*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--

*     November 2006

*

*     .. Scalar Arguments ..

      INTEGER            LDA, LDX, LDY, M, N, NB

*     ..

*     .. Array Arguments ..

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

     $                   TAUQ( * ), X( LDX, * ), Y( LDY, * )

*     ..

*

*  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' * 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

*

*  Arguments

*  =========

*

*  M       (input) INTEGER

*          The number of rows in the matrix A.

*

*  N       (input) INTEGER

*          The number of columns in the matrix A.

*

*  NB      (input) INTEGER

*          The number of leading rows and columns of A to be reduced.

*

*  A       (input/output) 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     (input) INTEGER

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

*

*  D       (output) 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       (output) DOUBLE PRECISION array, dimension (NB)

*          The off-diagonal elements of the first NB rows and columns of

*          the reduced matrix.

*

*  TAUQ    (output) DOUBLE PRECISION array dimension (NB)

*          The scalar factors of the elementary reflectors which

*          represent the orthogonal matrix Q. See Further Details.

*

*  TAUP    (output) DOUBLE PRECISION array, dimension (NB)

*          The scalar factors of the elementary reflectors which

*          represent the orthogonal matrix P. See Further Details.

*

*  X       (output) DOUBLE PRECISION array, dimension (LDX,NB)

*          The m-by-nb matrix X required to update the unreduced part

*          of A.

*

*  LDX     (input) INTEGER

*          The leading dimension of the array X. LDX >= M.

*

*  Y       (output) DOUBLE PRECISION array, dimension (LDY,NB)

*          The n-by-nb matrix Y required to update the unreduced part

*          of A.

*

*  LDY     (input) INTEGER

*          The leading dimension of the array Y. LDY >= N.

*

*  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'  and G(i) = I - taup * u * u'

*

*  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' 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' - X*U'.

*

*  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).

*

*  =====================================================================

*

*     .. Parameters ..

      DOUBLE PRECISION   ZERO, ONE

      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0 )

*     ..

*     .. Local Scalars ..

      INTEGER            I

*     ..

*     .. External Subroutines ..

      EXTERNAL           DGEMV, DLARFG, DSCAL

*     ..

*     .. Intrinsic Functions ..

      INTRINSIC          MIN

*     ..

*     .. Executable Statements ..

*

*     Quick return if possible

*

      IF( M.LE.0 .OR. N.LE.0 )

     $   RETURN

*

      IF( M.GE.N ) THEN

*

*        Reduce to upper bidiagonal form

*

         DO 10 I = 1, NB

*

*           Update A(i:m,i)

*

            CALL DGEMV( 'No transpose', M-I+1, I-1, -ONE, A( I, 1 ),

     $                  LDA, Y( I, 1 ), LDY, ONE, A( I, I ), 1 )

            CALL DGEMV( 'No transpose', M-I+1, I-1, -ONE, X( I, 1 ),

     $                  LDX, A( 1, I ), 1, ONE, A( I, I ), 1 )

*

*           Generate reflection Q(i) to annihilate A(i+1:m,i)

*

            CALL DLARFG( M-I+1, A( I, I ), A( MIN( I+1, M ), I ), 1,

     $                   TAUQ( I ) )

            D( I ) = A( I, I )

            IF( I.LT.N ) THEN

               A( I, I ) = ONE

*

*              Compute Y(i+1:n,i)

*

               CALL DGEMV( 'Transpose', M-I+1, N-I, ONE, A( I, I+1 ),

     $                     LDA, A( I, I ), 1, ZERO, Y( I+1, I ), 1 )

               CALL DGEMV( 'Transpose', M-I+1, I-1, ONE, A( I, 1 ), LDA,

     $                     A( I, I ), 1, ZERO, Y( 1, I ), 1 )

               CALL DGEMV( 'No transpose', N-I, I-1, -ONE, Y( I+1, 1 ),

     $                     LDY, Y( 1, I ), 1, ONE, Y( I+1, I ), 1 )

               CALL DGEMV( 'Transpose', M-I+1, I-1, ONE, X( I, 1 ), LDX,

     $                     A( I, I ), 1, ZERO, Y( 1, I ), 1 )

               CALL DGEMV( 'Transpose', I-1, N-I, -ONE, A( 1, I+1 ),

     $                     LDA, Y( 1, I ), 1, ONE, Y( I+1, I ), 1 )

               CALL DSCAL( N-I, TAUQ( I ), Y( I+1, I ), 1 )

*

*              Update A(i,i+1:n)

*

               CALL DGEMV( 'No transpose', N-I, I, -ONE, Y( I+1, 1 ),

     $                     LDY, A( I, 1 ), LDA, ONE, A( I, I+1 ), LDA )

               CALL DGEMV( 'Transpose', I-1, N-I, -ONE, A( 1, I+1 ),

     $                     LDA, X( I, 1 ), LDX, ONE, A( I, I+1 ), LDA )

*

*              Generate reflection P(i) to annihilate A(i,i+2:n)

*

               CALL DLARFG( N-I, A( I, I+1 ), A( I, MIN( I+2, N ) ),

     $                      LDA, TAUP( I ) )

               E( I ) = A( I, I+1 )

               A( I, I+1 ) = ONE

*

*              Compute X(i+1:m,i)

*

               CALL DGEMV( 'No transpose', M-I, N-I, ONE, A( I+1, I+1 ),

     $                     LDA, A( I, I+1 ), LDA, ZERO, X( I+1, I ), 1 )

               CALL DGEMV( 'Transpose', N-I, I, ONE, Y( I+1, 1 ), LDY,

     $                     A( I, I+1 ), LDA, ZERO, X( 1, I ), 1 )

               CALL DGEMV( 'No transpose', M-I, I, -ONE, A( I+1, 1 ),

     $                     LDA, X( 1, I ), 1, ONE, X( I+1, I ), 1 )

               CALL DGEMV( 'No transpose', I-1, N-I, ONE, A( 1, I+1 ),

     $                     LDA, A( I, I+1 ), LDA, ZERO, X( 1, I ), 1 )

               CALL DGEMV( 'No transpose', M-I, I-1, -ONE, X( I+1, 1 ),

     $                     LDX, X( 1, I ), 1, ONE, X( I+1, I ), 1 )

               CALL DSCAL( M-I, TAUP( I ), X( I+1, I ), 1 )

            END IF

   10    CONTINUE

      ELSE

*

*        Reduce to lower bidiagonal form

*

         DO 20 I = 1, NB

*

*           Update A(i,i:n)

*

            CALL DGEMV( 'No transpose', N-I+1, I-1, -ONE, Y( I, 1 ),

     $                  LDY, A( I, 1 ), LDA, ONE, A( I, I ), LDA )

            CALL DGEMV( 'Transpose', I-1, N-I+1, -ONE, A( 1, I ), LDA,

     $                  X( I, 1 ), LDX, ONE, A( I, I ), LDA )

*

*           Generate reflection P(i) to annihilate A(i,i+1:n)

*

            CALL DLARFG( N-I+1, A( I, I ), A( I, MIN( I+1, N ) ), LDA,

     $                   TAUP( I ) )

            D( I ) = A( I, I )

            IF( I.LT.M ) THEN

               A( I, I ) = ONE

*

*              Compute X(i+1:m,i)

*

               CALL DGEMV( 'No transpose', M-I, N-I+1, ONE, A( I+1, I ),

     $                     LDA, A( I, I ), LDA, ZERO, X( I+1, I ), 1 )

               CALL DGEMV( 'Transpose', N-I+1, I-1, ONE, Y( I, 1 ), LDY,

     $                     A( I, I ), LDA, ZERO, X( 1, I ), 1 )

               CALL DGEMV( 'No transpose', M-I, I-1, -ONE, A( I+1, 1 ),

     $                     LDA, X( 1, I ), 1, ONE, X( I+1, I ), 1 )

               CALL DGEMV( 'No transpose', I-1, N-I+1, ONE, A( 1, I ),

     $                     LDA, A( I, I ), LDA, ZERO, X( 1, I ), 1 )

               CALL DGEMV( 'No transpose', M-I, I-1, -ONE, X( I+1, 1 ),

     $                     LDX, X( 1, I ), 1, ONE, X( I+1, I ), 1 )

               CALL DSCAL( M-I, TAUP( I ), X( I+1, I ), 1 )

*

*              Update A(i+1:m,i)

*

               CALL DGEMV( 'No transpose', M-I, I-1, -ONE, A( I+1, 1 ),

     $                     LDA, Y( I, 1 ), LDY, ONE, A( I+1, I ), 1 )

               CALL DGEMV( 'No transpose', M-I, I, -ONE, X( I+1, 1 ),

     $                     LDX, A( 1, I ), 1, ONE, A( I+1, I ), 1 )

*

*              Generate reflection Q(i) to annihilate A(i+2:m,i)

*

               CALL DLARFG( M-I, A( I+1, I ), A( MIN( I+2, M ), I ), 1,

     $                      TAUQ( I ) )

               E( I ) = A( I+1, I )

               A( I+1, I ) = ONE

*

*              Compute Y(i+1:n,i)

*

               CALL DGEMV( 'Transpose', M-I, N-I, ONE, A( I+1, I+1 ),

     $                     LDA, A( I+1, I ), 1, ZERO, Y( I+1, I ), 1 )

               CALL DGEMV( 'Transpose', M-I, I-1, ONE, A( I+1, 1 ), LDA,

     $                     A( I+1, I ), 1, ZERO, Y( 1, I ), 1 )

               CALL DGEMV( 'No transpose', N-I, I-1, -ONE, Y( I+1, 1 ),

     $                     LDY, Y( 1, I ), 1, ONE, Y( I+1, I ), 1 )

               CALL DGEMV( 'Transpose', M-I, I, ONE, X( I+1, 1 ), LDX,

     $                     A( I+1, I ), 1, ZERO, Y( 1, I ), 1 )

               CALL DGEMV( 'Transpose', I, N-I, -ONE, A( 1, I+1 ), LDA,

     $                     Y( 1, I ), 1, ONE, Y( I+1, I ), 1 )

               CALL DSCAL( N-I, TAUQ( I ), Y( I+1, I ), 1 )

            END IF

   20    CONTINUE

      END IF

      RETURN

*

*     End of DLABRD

*

      END