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    1:       SUBROUTINE DGEHRD( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO )
    2: *
    3: *  -- LAPACK routine (version 3.2.1)                                  --
    4: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
    5: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
    6: *  -- April 2009                                                      --
    7: *
    8: *     .. Scalar Arguments ..
    9:       INTEGER            IHI, ILO, INFO, LDA, LWORK, N
   10: *     ..
   11: *     .. Array Arguments ..
   12:       DOUBLE PRECISION  A( LDA, * ), TAU( * ), WORK( * )
   13: *     ..
   14: *
   15: *  Purpose
   16: *  =======
   17: *
   18: *  DGEHRD reduces a real general matrix A to upper Hessenberg form H by
   19: *  an orthogonal similarity transformation:  Q' * A * Q = H .
   20: *
   21: *  Arguments
   22: *  =========
   23: *
   24: *  N       (input) INTEGER
   25: *          The order of the matrix A.  N >= 0.
   26: *
   27: *  ILO     (input) INTEGER
   28: *  IHI     (input) INTEGER
   29: *          It is assumed that A is already upper triangular in rows
   30: *          and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
   31: *          set by a previous call to DGEBAL; otherwise they should be
   32: *          set to 1 and N respectively. See Further Details.
   33: *          1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
   34: *
   35: *  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
   36: *          On entry, the N-by-N general matrix to be reduced.
   37: *          On exit, the upper triangle and the first subdiagonal of A
   38: *          are overwritten with the upper Hessenberg matrix H, and the
   39: *          elements below the first subdiagonal, with the array TAU,
   40: *          represent the orthogonal matrix Q as a product of elementary
   41: *          reflectors. See Further Details.
   42: *
   43: *  LDA     (input) INTEGER
   44: *          The leading dimension of the array A.  LDA >= max(1,N).
   45: *
   46: *  TAU     (output) DOUBLE PRECISION array, dimension (N-1)
   47: *          The scalar factors of the elementary reflectors (see Further
   48: *          Details). Elements 1:ILO-1 and IHI:N-1 of TAU are set to
   49: *          zero.
   50: *
   51: *  WORK    (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
   52: *          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
   53: *
   54: *  LWORK   (input) INTEGER
   55: *          The length of the array WORK.  LWORK >= max(1,N).
   56: *          For optimum performance LWORK >= N*NB, where NB is the
   57: *          optimal blocksize.
   58: *
   59: *          If LWORK = -1, then a workspace query is assumed; the routine
   60: *          only calculates the optimal size of the WORK array, returns
   61: *          this value as the first entry of the WORK array, and no error
   62: *          message related to LWORK is issued by XERBLA.
   63: *
   64: *  INFO    (output) INTEGER
   65: *          = 0:  successful exit
   66: *          < 0:  if INFO = -i, the i-th argument had an illegal value.
   67: *
   68: *  Further Details
   69: *  ===============
   70: *
   71: *  The matrix Q is represented as a product of (ihi-ilo) elementary
   72: *  reflectors
   73: *
   74: *     Q = H(ilo) H(ilo+1) . . . H(ihi-1).
   75: *
   76: *  Each H(i) has the form
   77: *
   78: *     H(i) = I - tau * v * v'
   79: *
   80: *  where tau is a real scalar, and v is a real vector with
   81: *  v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
   82: *  exit in A(i+2:ihi,i), and tau in TAU(i).
   83: *
   84: *  The contents of A are illustrated by the following example, with
   85: *  n = 7, ilo = 2 and ihi = 6:
   86: *
   87: *  on entry,                        on exit,
   88: *
   89: *  ( a   a   a   a   a   a   a )    (  a   a   h   h   h   h   a )
   90: *  (     a   a   a   a   a   a )    (      a   h   h   h   h   a )
   91: *  (     a   a   a   a   a   a )    (      h   h   h   h   h   h )
   92: *  (     a   a   a   a   a   a )    (      v2  h   h   h   h   h )
   93: *  (     a   a   a   a   a   a )    (      v2  v3  h   h   h   h )
   94: *  (     a   a   a   a   a   a )    (      v2  v3  v4  h   h   h )
   95: *  (                         a )    (                          a )
   96: *
   97: *  where a denotes an element of the original matrix A, h denotes a
   98: *  modified element of the upper Hessenberg matrix H, and vi denotes an
   99: *  element of the vector defining H(i).
  100: *
  101: *  This file is a slight modification of LAPACK-3.0's DGEHRD
  102: *  subroutine incorporating improvements proposed by Quintana-Orti and
  103: *  Van de Geijn (2006). (See DLAHR2.)
  104: *
  105: *  =====================================================================
  106: *
  107: *     .. Parameters ..
  108:       INTEGER            NBMAX, LDT
  109:       PARAMETER          ( NBMAX = 64, LDT = NBMAX+1 )
  110:       DOUBLE PRECISION  ZERO, ONE
  111:       PARAMETER          ( ZERO = 0.0D+0, 
  112:      $                     ONE = 1.0D+0 )
  113: *     ..
  114: *     .. Local Scalars ..
  115:       LOGICAL            LQUERY
  116:       INTEGER            I, IB, IINFO, IWS, J, LDWORK, LWKOPT, NB,
  117:      $                   NBMIN, NH, NX
  118:       DOUBLE PRECISION  EI
  119: *     ..
  120: *     .. Local Arrays ..
  121:       DOUBLE PRECISION  T( LDT, NBMAX )
  122: *     ..
  123: *     .. External Subroutines ..
  124:       EXTERNAL           DAXPY, DGEHD2, DGEMM, DLAHR2, DLARFB, DTRMM,
  125:      $                   XERBLA
  126: *     ..
  127: *     .. Intrinsic Functions ..
  128:       INTRINSIC          MAX, MIN
  129: *     ..
  130: *     .. External Functions ..
  131:       INTEGER            ILAENV
  132:       EXTERNAL           ILAENV
  133: *     ..
  134: *     .. Executable Statements ..
  135: *
  136: *     Test the input parameters
  137: *
  138:       INFO = 0
  139:       NB = MIN( NBMAX, ILAENV( 1, 'DGEHRD', ' ', N, ILO, IHI, -1 ) )
  140:       LWKOPT = N*NB
  141:       WORK( 1 ) = LWKOPT
  142:       LQUERY = ( LWORK.EQ.-1 )
  143:       IF( N.LT.0 ) THEN
  144:          INFO = -1
  145:       ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN
  146:          INFO = -2
  147:       ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN
  148:          INFO = -3
  149:       ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
  150:          INFO = -5
  151:       ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
  152:          INFO = -8
  153:       END IF
  154:       IF( INFO.NE.0 ) THEN
  155:          CALL XERBLA( 'DGEHRD', -INFO )
  156:          RETURN
  157:       ELSE IF( LQUERY ) THEN
  158:          RETURN
  159:       END IF
  160: *
  161: *     Set elements 1:ILO-1 and IHI:N-1 of TAU to zero
  162: *
  163:       DO 10 I = 1, ILO - 1
  164:          TAU( I ) = ZERO
  165:    10 CONTINUE
  166:       DO 20 I = MAX( 1, IHI ), N - 1
  167:          TAU( I ) = ZERO
  168:    20 CONTINUE
  169: *
  170: *     Quick return if possible
  171: *
  172:       NH = IHI - ILO + 1
  173:       IF( NH.LE.1 ) THEN
  174:          WORK( 1 ) = 1
  175:          RETURN
  176:       END IF
  177: *
  178: *     Determine the block size
  179: *
  180:       NB = MIN( NBMAX, ILAENV( 1, 'DGEHRD', ' ', N, ILO, IHI, -1 ) )
  181:       NBMIN = 2
  182:       IWS = 1
  183:       IF( NB.GT.1 .AND. NB.LT.NH ) THEN
  184: *
  185: *        Determine when to cross over from blocked to unblocked code
  186: *        (last block is always handled by unblocked code)
  187: *
  188:          NX = MAX( NB, ILAENV( 3, 'DGEHRD', ' ', N, ILO, IHI, -1 ) )
  189:          IF( NX.LT.NH ) THEN
  190: *
  191: *           Determine if workspace is large enough for blocked code
  192: *
  193:             IWS = N*NB
  194:             IF( LWORK.LT.IWS ) THEN
  195: *
  196: *              Not enough workspace to use optimal NB:  determine the
  197: *              minimum value of NB, and reduce NB or force use of
  198: *              unblocked code
  199: *
  200:                NBMIN = MAX( 2, ILAENV( 2, 'DGEHRD', ' ', N, ILO, IHI,
  201:      $                 -1 ) )
  202:                IF( LWORK.GE.N*NBMIN ) THEN
  203:                   NB = LWORK / N
  204:                ELSE
  205:                   NB = 1
  206:                END IF
  207:             END IF
  208:          END IF
  209:       END IF
  210:       LDWORK = N
  211: *
  212:       IF( NB.LT.NBMIN .OR. NB.GE.NH ) THEN
  213: *
  214: *        Use unblocked code below
  215: *
  216:          I = ILO
  217: *
  218:       ELSE
  219: *
  220: *        Use blocked code
  221: *
  222:          DO 40 I = ILO, IHI - 1 - NX, NB
  223:             IB = MIN( NB, IHI-I )
  224: *
  225: *           Reduce columns i:i+ib-1 to Hessenberg form, returning the
  226: *           matrices V and T of the block reflector H = I - V*T*V'
  227: *           which performs the reduction, and also the matrix Y = A*V*T
  228: *
  229:             CALL DLAHR2( IHI, I, IB, A( 1, I ), LDA, TAU( I ), T, LDT,
  230:      $                   WORK, LDWORK )
  231: *
  232: *           Apply the block reflector H to A(1:ihi,i+ib:ihi) from the
  233: *           right, computing  A := A - Y * V'. V(i+ib,ib-1) must be set
  234: *           to 1
  235: *
  236:             EI = A( I+IB, I+IB-1 )
  237:             A( I+IB, I+IB-1 ) = ONE
  238:             CALL DGEMM( 'No transpose', 'Transpose', 
  239:      $                  IHI, IHI-I-IB+1,
  240:      $                  IB, -ONE, WORK, LDWORK, A( I+IB, I ), LDA, ONE,
  241:      $                  A( 1, I+IB ), LDA )
  242:             A( I+IB, I+IB-1 ) = EI
  243: *
  244: *           Apply the block reflector H to A(1:i,i+1:i+ib-1) from the
  245: *           right
  246: *
  247:             CALL DTRMM( 'Right', 'Lower', 'Transpose',
  248:      $                  'Unit', I, IB-1,
  249:      $                  ONE, A( I+1, I ), LDA, WORK, LDWORK )
  250:             DO 30 J = 0, IB-2
  251:                CALL DAXPY( I, -ONE, WORK( LDWORK*J+1 ), 1,
  252:      $                     A( 1, I+J+1 ), 1 )
  253:    30       CONTINUE
  254: *
  255: *           Apply the block reflector H to A(i+1:ihi,i+ib:n) from the
  256: *           left
  257: *
  258:             CALL DLARFB( 'Left', 'Transpose', 'Forward',
  259:      $                   'Columnwise',
  260:      $                   IHI-I, N-I-IB+1, IB, A( I+1, I ), LDA, T, LDT,
  261:      $                   A( I+1, I+IB ), LDA, WORK, LDWORK )
  262:    40    CONTINUE
  263:       END IF
  264: *
  265: *     Use unblocked code to reduce the rest of the matrix
  266: *
  267:       CALL DGEHD2( N, I, IHI, A, LDA, TAU, WORK, IINFO )
  268:       WORK( 1 ) = IWS
  269: *
  270:       RETURN
  271: *
  272: *     End of DGEHRD
  273: *
  274:       END