sgehrd.f

          SUBROUTINE SGEHRD( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO )
    *
    *  -- LAPACK routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          INTEGER            IHI, ILO, INFO, LDA, LWORK, N
    *     ..
    *     .. Array Arguments ..
          REAL              A( LDA, * ), TAU( * ), WORK( * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  SGEHRD reduces a real general matrix A to upper Hessenberg form H by
    *  an orthogonal similarity transformation:  Q' * A * Q = H .
    *
    *  Arguments
    *  =========
    *
    *  N       (input) INTEGER
    *          The order of the matrix A.  N >= 0.
    *
    *  ILO     (input) INTEGER
    *  IHI     (input) INTEGER
    *          It is assumed that A is already upper triangular in rows
    *          and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
    *          set by a previous call to SGEBAL; otherwise they should be
    *          set to 1 and N respectively. See Further Details.
    *          1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
    *
    *  A       (input/output) REAL array, dimension (LDA,N)
    *          On entry, the N-by-N general matrix to be reduced.
    *          On exit, the upper triangle and the first subdiagonal of A
    *          are overwritten with the upper Hessenberg matrix H, and the
    *          elements below the first subdiagonal, with the array TAU,
    *          represent the orthogonal matrix Q as a product of elementary
    *          reflectors. See Further Details.
    *
    *  LDA     (input) INTEGER
    *          The leading dimension of the array A.  LDA >= max(1,N).
    *
    *  TAU     (output) REAL array, dimension (N-1)
    *          The scalar factors of the elementary reflectors (see Further
    *          Details). Elements 1:ILO-1 and IHI:N-1 of TAU are set to
    *          zero.
    *
    *  WORK    (workspace/output) REAL array, dimension (LWORK)
    *          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
    *
    *  LWORK   (input) INTEGER
    *          The length of the array WORK.  LWORK >= max(1,N).
    *          For optimum performance LWORK >= 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 matrix Q is represented as a product of (ihi-ilo) elementary
    *  reflectors
    *
    *     Q = H(ilo) H(ilo+1) . . . H(ihi-1).
    *
    *  Each H(i) has the form
    *
    *     H(i) = I - tau * v * v'
    *
    *  where tau is a real scalar, and v is a real vector with
    *  v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
    *  exit in A(i+2:ihi,i), and tau in TAU(i).
    *
    *  The contents of A are illustrated by the following example, with
    *  n = 7, ilo = 2 and ihi = 6:
    *
    *  on entry,                        on exit,
    *
    *  ( a   a   a   a   a   a   a )    (  a   a   h   h   h   h   a )
    *  (     a   a   a   a   a   a )    (      a   h   h   h   h   a )
    *  (     a   a   a   a   a   a )    (      h   h   h   h   h   h )
    *  (     a   a   a   a   a   a )    (      v2  h   h   h   h   h )
    *  (     a   a   a   a   a   a )    (      v2  v3  h   h   h   h )
    *  (     a   a   a   a   a   a )    (      v2  v3  v4  h   h   h )
    *  (                         a )    (                          a )
    *
    *  where a denotes an element of the original matrix A, h denotes a
    *  modified element of the upper Hessenberg matrix H, and vi denotes an
    *  element of the vector defining H(i).
    *
    *  This file is a slight modification of LAPACK-3.0's SGEHRD
    *  subroutine incorporating improvements proposed by Quintana-Orti and
    *  Van de Geijn (2005).
    *
    *  =====================================================================
    *
    *     .. Parameters ..
          INTEGER            NBMAX, LDT
          PARAMETER          ( NBMAX = 64, LDT = NBMAX+1 )
          REAL              ZERO, ONE
          PARAMETER          ( ZERO = 0.0E+0,
         $                     ONE = 1.0E+0 )
    *     ..
    *     .. Local Scalars ..
          LOGICAL            LQUERY
          INTEGER            I, IB, IINFO, IWS, J, LDWORK, LWKOPT, NB,
         $                   NBMIN, NH, NX
          REAL              EI
    *     ..
    *     .. Local Arrays ..
          REAL              T( LDT, NBMAX )
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           SAXPY, SGEHD2, SGEMM, SLAHR2, SLARFB, STRMM,
         $                   XERBLA
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          MAX, MIN
    *     ..
    *     .. External Functions ..
          INTEGER            ILAENV
          EXTERNAL           ILAENV
    *     ..
    *     .. Executable Statements ..
    *
    *     Test the input parameters
    *
          INFO = 0
          NB = MIN( NBMAX, ILAENV( 1, 'SGEHRD', ' ', N, ILO, IHI, -1 ) )
          LWKOPT = N*NB
          WORK( 1 ) = LWKOPT
          LQUERY = ( LWORK.EQ.-1 )
          IF( N.LT.0 ) THEN
             INFO = -1
          ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN
             INFO = -2
          ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN
             INFO = -3
          ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
             INFO = -5
          ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
             INFO = -8
          END IF
          IF( INFO.NE.0 ) THEN
             CALL XERBLA( 'SGEHRD', -INFO )
             RETURN
          ELSE IF( LQUERY ) THEN
             RETURN
          END IF
    *
    *     Set elements 1:ILO-1 and IHI:N-1 of TAU to zero
    *
          DO 10 I = 1, ILO - 1
             TAU( I ) = ZERO
       10 CONTINUE
          DO 20 I = MAX( 1, IHI ), N - 1
             TAU( I ) = ZERO
       20 CONTINUE
    *
    *     Quick return if possible
    *
          NH = IHI - ILO + 1
          IF( NH.LE.1 ) THEN
             WORK( 1 ) = 1
             RETURN
          END IF
    *
    *     Determine the block size
    *
          NB = MIN( NBMAX, ILAENV( 1, 'SGEHRD', ' ', N, ILO, IHI, -1 ) )
          NBMIN = 2
          IWS = 1
          IF( NB.GT.1 .AND. NB.LT.NH ) THEN
    *
    *        Determine when to cross over from blocked to unblocked code
    *        (last block is always handled by unblocked code)
    *
             NX = MAX( NB, ILAENV( 3, 'SGEHRD', ' ', N, ILO, IHI, -1 ) )
             IF( NX.LT.NH ) THEN
    *
    *           Determine if workspace is large enough for blocked code
    *
                IWS = N*NB
                IF( LWORK.LT.IWS ) THEN
    *
    *              Not enough workspace to use optimal NB:  determine the
    *              minimum value of NB, and reduce NB or force use of
    *              unblocked code
    *
                   NBMIN = MAX( 2, ILAENV( 2, 'SGEHRD', ' ', N, ILO, IHI,
         $                 -1 ) )
                   IF( LWORK.GE.N*NBMIN ) THEN
                      NB = LWORK / N
                   ELSE
                      NB = 1
                   END IF
                END IF
             END IF
          END IF
          LDWORK = N
    *
          IF( NB.LT.NBMIN .OR. NB.GE.NH ) THEN
    *
    *        Use unblocked code below
    *
             I = ILO
    *
          ELSE
    *
    *        Use blocked code
    *
             DO 40 I = ILO, IHI - 1 - NX, NB
                IB = MIN( NB, IHI-I )
    *
    *           Reduce columns i:i+ib-1 to Hessenberg form, returning the
    *           matrices V and T of the block reflector H = I - V*T*V'
    *           which performs the reduction, and also the matrix Y = A*V*T
    *
                CALL SLAHR2( IHI, I, IB, A( 1, I ), LDA, TAU( I ), T, LDT,
         $                   WORK, LDWORK )
    *
    *           Apply the block reflector H to A(1:ihi,i+ib:ihi) from the
    *           right, computing  A := A - Y * V'. V(i+ib,ib-1) must be set
    *           to 1
    *
                EI = A( I+IB, I+IB-1 )
                A( I+IB, I+IB-1 ) = ONE
                CALL SGEMM( 'No transpose', 'Transpose',
         $                  IHI, IHI-I-IB+1,
         $                  IB, -ONE, WORK, LDWORK, A( I+IB, I ), LDA, ONE,
         $                  A( 1, I+IB ), LDA )
                A( I+IB, I+IB-1 ) = EI
    *
    *           Apply the block reflector H to A(1:i,i+1:i+ib-1) from the
    *           right
    *
                CALL STRMM( 'Right', 'Lower', 'Transpose',
         $                  'Unit', I, IB-1,
         $                  ONE, A( I+1, I ), LDA, WORK, LDWORK )
                DO 30 J = 0, IB-2
                   CALL SAXPY( I, -ONE, WORK( LDWORK*J+1 ), 1,
         $                     A( 1, I+J+1 ), 1 )
       30       CONTINUE
    *
    *           Apply the block reflector H to A(i+1:ihi,i+ib:n) from the
    *           left
    *
                CALL SLARFB( 'Left', 'Transpose', 'Forward',
         $                   'Columnwise',
         $                   IHI-I, N-I-IB+1, IB, A( I+1, I ), LDA, T, LDT,
         $                   A( I+1, I+IB ), LDA, WORK, LDWORK )
       40    CONTINUE
          END IF
    *
    *     Use unblocked code to reduce the rest of the matrix
    *
          CALL SGEHD2( N, I, IHI, A, LDA, TAU, WORK, IINFO )
          WORK( 1 ) = IWS
    *
          RETURN
    *
    *     End of SGEHRD
    *
          END