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

     $                   INFO )

*

*  -- LAPACK 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            INFO, LDA, LWORK, M, N

*     ..

*     .. Array Arguments ..

      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 diagonal 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 subdiagonal, 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 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) 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 PRECISION array, dimension (MAX(1,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).

*

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

*

*     .. Parameters ..

      DOUBLE PRECISION   ONE

      PARAMETER          ( ONE = 1.0D+0 )

*     ..

*     .. Local Scalars ..

      LOGICAL            LQUERY

      INTEGER            I, IINFO, J, LDWRKX, LDWRKY, LWKOPT, MINMN, NB,

     $                   NBMIN, NX

      DOUBLE PRECISION   WS

*     ..

*     .. External Subroutines ..

      EXTERNAL           DGEBD2, DGEMM, DLABRD, XERBLA

*     ..

*     .. Intrinsic Functions ..

      INTRINSIC          DBLE, MAX, MIN

*     ..

*     .. External Functions ..

      INTEGER            ILAENV

      EXTERNAL           ILAENV

*     ..

*     .. Executable Statements ..

*

*     Test the input parameters

*

      INFO = 0

      NB = MAX( 1, ILAENV( 1, 'DGEBRD', ' ', M, N, -1, -1 ) )

      LWKOPT = ( M+N )*NB

      WORK( 1 ) = DBLE( LWKOPT )

      LQUERY = ( LWORK.EQ.-1 )

      IF( M.LT.0 ) THEN

         INFO = -1

      ELSE IF( N.LT.0 ) THEN

         INFO = -2

      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN

         INFO = -4

      ELSE IF( LWORK.LT.MAX( 1, M, N ) .AND. .NOT.LQUERY ) THEN

         INFO = -10

      END IF

      IF( INFO.LT.0 ) THEN

         CALL XERBLA( 'DGEBRD', -INFO )

         RETURN

      ELSE IF( LQUERY ) THEN

         RETURN

      END IF

*

*     Quick return if possible

*

      MINMN = MIN( M, N )

      IF( MINMN.EQ.0 ) THEN

         WORK( 1 ) = 1

         RETURN

      END IF

*

      WS = MAX( M, N )

      LDWRKX = M

      LDWRKY = N

*

      IF( NB.GT.1 .AND. NB.LT.MINMN ) THEN

*

*        Set the crossover point NX.

*

         NX = MAX( NB, ILAENV( 3, 'DGEBRD', ' ', M, N, -1, -1 ) )

*

*        Determine when to switch from blocked to unblocked code.

*

         IF( NX.LT.MINMN ) THEN

            WS = ( M+N )*NB

            IF( LWORK.LT.WS ) THEN

*

*              Not enough work space for the optimal NB, consider using

*              a smaller block size.

*

               NBMIN = ILAENV( 2, 'DGEBRD', ' ', M, N, -1, -1 )

               IF( LWORK.GE.( M+N )*NBMIN ) THEN

                  NB = LWORK / ( M+N )

               ELSE

                  NB = 1

                  NX = MINMN

               END IF

            END IF

         END IF

      ELSE

         NX = MINMN

      END IF

*

      DO 30 I = 1, MINMN - NX, NB

*

*        Reduce rows and columns i:i+nb-1 to bidiagonal form and return

*        the matrices X and Y which are needed to update the unreduced

*        part of the matrix

*

         CALL DLABRD( M-I+1, N-I+1, NB, A( I, I ), LDA, D( I ), E( I ),

     $                TAUQ( I ), TAUP( I ), WORK, LDWRKX,

     $                WORK( LDWRKX*NB+1 ), LDWRKY )

*

*        Update the trailing submatrix A(i+nb:m,i+nb:n), using an update

*        of the form  A := A - V*Y' - X*U'

*

         CALL DGEMM( 'No transpose', 'Transpose', M-I-NB+1, N-I-NB+1,

     $               NB, -ONE, A( I+NB, I ), LDA,

     $               WORK( LDWRKX*NB+NB+1 ), LDWRKY, ONE,

     $               A( I+NB, I+NB ), LDA )

         CALL DGEMM( 'No transpose', 'No transpose', M-I-NB+1, N-I-NB+1,

     $               NB, -ONE, WORK( NB+1 ), LDWRKX, A( I, I+NB ), LDA,

     $               ONE, A( I+NB, I+NB ), LDA )

*

*        Copy diagonal and off-diagonal elements of B back into A

*

         IF( M.GE.N ) THEN

            DO 10 J = I, I + NB - 1

               A( J, J ) = D( J )

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

   10       CONTINUE

         ELSE

            DO 20 J = I, I + NB - 1

               A( J, J ) = D( J )

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

   20       CONTINUE

         END IF

   30 CONTINUE

*

*     Use unblocked code to reduce the remainder of the matrix

*

      CALL DGEBD2( M-I+1, N-I+1, A( I, I ), LDA, D( I ), E( I ),

     $             TAUQ( I ), TAUP( I ), WORK, IINFO )

      WORK( 1 ) = WS

      RETURN

*

*     End of DGEBRD

*

      END