zgebrd.f

          SUBROUTINE ZGEBRD( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK,
         $                   INFO )
    *
    *  -- LAPACK routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          INTEGER            INFO, LDA, LWORK, M, N
    *     ..
    *     .. Array Arguments ..
          DOUBLE PRECISION   D( * ), E( * )
          COMPLEX*16         A( LDA, * ), TAUP( * ), TAUQ( * ), WORK( * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  ZGEBRD reduces a general complex M-by-N matrix A to upper or lower
    *  bidiagonal form B by a unitary transformation: Q**H * 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) COMPLEX*16 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 unitary matrix Q as a product of elementary
    *            reflectors, and the elements above the first superdiagonal,
    *            with the array TAUP, represent the unitary 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 unitary matrix Q as a product of
    *            elementary reflectors, and the elements above the diagonal,
    *            with the array TAUP, represent the unitary 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) COMPLEX*16 array dimension (min(M,N))
    *          The scalar factors of the elementary reflectors which
    *          represent the unitary matrix Q. See Further Details.
    *
    *  TAUP    (output) COMPLEX*16 array, dimension (min(M,N))
    *          The scalar factors of the elementary reflectors which
    *          represent the unitary matrix P. See Further Details.
    *
    *  WORK    (workspace/output) COMPLEX*16 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 complex scalars, and v and u are complex
    *  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 complex scalars, and v and u are complex
    *  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 ..
          COMPLEX*16         ONE
          PARAMETER          ( ONE = ( 1.0D+0, 0.0D+0 ) )
    *     ..
    *     .. Local Scalars ..
          LOGICAL            LQUERY
          INTEGER            I, IINFO, J, LDWRKX, LDWRKY, LWKOPT, MINMN, NB,
         $                   NBMIN, NX
          DOUBLE PRECISION   WS
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           XERBLA, ZGEBD2, ZGEMM, ZLABRD
    *     ..
    *     .. 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, 'ZGEBRD', ' ', 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( 'ZGEBRD', -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, 'ZGEBRD', ' ', 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, 'ZGEBRD', ' ', 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+ib-1 to bidiagonal form and return
    *        the matrices X and Y which are needed to update the unreduced
    *        part of the matrix
    *
             CALL ZLABRD( 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+ib:m,i+ib:n), using
    *        an update of the form  A := A - V*Y' - X*U'
    *
             CALL ZGEMM( 'No transpose', 'Conjugate 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 ZGEMM( '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 ZGEBD2( 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 ZGEBRD
    *
          END