claein.f

          SUBROUTINE CLAEIN( RIGHTV, NOINIT, N, H, LDH, W, V, B, LDB, RWORK,
         $                   EPS3, SMLNUM, INFO )
    *
    *  -- LAPACK auxiliary routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          LOGICAL            NOINIT, RIGHTV
          INTEGER            INFO, LDB, LDH, N
          REAL               EPS3, SMLNUM
          COMPLEX            W
    *     ..
    *     .. Array Arguments ..
          REAL               RWORK( * )
          COMPLEX            B( LDB, * ), H( LDH, * ), V( * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  CLAEIN uses inverse iteration to find a right or left eigenvector
    *  corresponding to the eigenvalue W of a complex upper Hessenberg
    *  matrix H.
    *
    *  Arguments
    *  =========
    *
    *  RIGHTV   (input) LOGICAL
    *          = .TRUE. : compute right eigenvector;
    *          = .FALSE.: compute left eigenvector.
    *
    *  NOINIT   (input) LOGICAL
    *          = .TRUE. : no initial vector supplied in V
    *          = .FALSE.: initial vector supplied in V.
    *
    *  N       (input) INTEGER
    *          The order of the matrix H.  N >= 0.
    *
    *  H       (input) COMPLEX array, dimension (LDH,N)
    *          The upper Hessenberg matrix H.
    *
    *  LDH     (input) INTEGER
    *          The leading dimension of the array H.  LDH >= max(1,N).
    *
    *  W       (input) COMPLEX
    *          The eigenvalue of H whose corresponding right or left
    *          eigenvector is to be computed.
    *
    *  V       (input/output) COMPLEX array, dimension (N)
    *          On entry, if NOINIT = .FALSE., V must contain a starting
    *          vector for inverse iteration; otherwise V need not be set.
    *          On exit, V contains the computed eigenvector, normalized so
    *          that the component of largest magnitude has magnitude 1; here
    *          the magnitude of a complex number (x,y) is taken to be
    *          |x| + |y|.
    *
    *  B       (workspace) COMPLEX array, dimension (LDB,N)
    *
    *  LDB     (input) INTEGER
    *          The leading dimension of the array B.  LDB >= max(1,N).
    *
    *  RWORK   (workspace) REAL array, dimension (N)
    *
    *  EPS3    (input) REAL
    *          A small machine-dependent value which is used to perturb
    *          close eigenvalues, and to replace zero pivots.
    *
    *  SMLNUM  (input) REAL
    *          A machine-dependent value close to the underflow threshold.
    *
    *  INFO    (output) INTEGER
    *          = 0:  successful exit
    *          = 1:  inverse iteration did not converge; V is set to the
    *                last iterate.
    *
    *  =====================================================================
    *
    *     .. Parameters ..
          REAL               ONE, TENTH
          PARAMETER          ( ONE = 1.0E+0, TENTH = 1.0E-1 )
          COMPLEX            ZERO
          PARAMETER          ( ZERO = ( 0.0E+0, 0.0E+0 ) )
    *     ..
    *     .. Local Scalars ..
          CHARACTER          NORMIN, TRANS
          INTEGER            I, IERR, ITS, J
          REAL               GROWTO, NRMSML, ROOTN, RTEMP, SCALE, VNORM
          COMPLEX            CDUM, EI, EJ, TEMP, X
    *     ..
    *     .. External Functions ..
          INTEGER            ICAMAX
          REAL               SCASUM, SCNRM2
          COMPLEX            CLADIV
          EXTERNAL           ICAMAX, SCASUM, SCNRM2, CLADIV
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           CLATRS, CSSCAL
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          ABS, AIMAG, MAX, REAL, SQRT
    *     ..
    *     .. Statement Functions ..
          REAL               CABS1
    *     ..
    *     .. Statement Function definitions ..
          CABS1( CDUM ) = ABS( REAL( CDUM ) ) + ABS( AIMAG( CDUM ) )
    *     ..
    *     .. Executable Statements ..
    *
          INFO = 0
    *
    *     GROWTO is the threshold used in the acceptance test for an
    *     eigenvector.
    *
          ROOTN = SQRT( REAL( N ) )
          GROWTO = TENTH / ROOTN
          NRMSML = MAX( ONE, EPS3*ROOTN )*SMLNUM
    *
    *     Form B = H - W*I (except that the subdiagonal elements are not
    *     stored).
    *
          DO 20 J = 1, N
             DO 10 I = 1, J - 1
                B( I, J ) = H( I, J )
       10    CONTINUE
             B( J, J ) = H( J, J ) - W
       20 CONTINUE
    *
          IF( NOINIT ) THEN
    *
    *        Initialize V.
    *
             DO 30 I = 1, N
                V( I ) = EPS3
       30    CONTINUE
          ELSE
    *
    *        Scale supplied initial vector.
    *
             VNORM = SCNRM2( N, V, 1 )
             CALL CSSCAL( N, ( EPS3*ROOTN ) / MAX( VNORM, NRMSML ), V, 1 )
          END IF
    *
          IF( RIGHTV ) THEN
    *
    *        LU decomposition with partial pivoting of B, replacing zero
    *        pivots by EPS3.
    *
             DO 60 I = 1, N - 1
                EI = H( I+1, I )
                IF( CABS1( B( I, I ) ).LT.CABS1( EI ) ) THEN
    *
    *              Interchange rows and eliminate.
    *
                   X = CLADIV( B( I, I ), EI )
                   B( I, I ) = EI
                   DO 40 J = I + 1, N
                      TEMP = B( I+1, J )
                      B( I+1, J ) = B( I, J ) - X*TEMP
                      B( I, J ) = TEMP
       40          CONTINUE
                ELSE
    *
    *              Eliminate without interchange.
    *
                   IF( B( I, I ).EQ.ZERO )
         $            B( I, I ) = EPS3
                   X = CLADIV( EI, B( I, I ) )
                   IF( X.NE.ZERO ) THEN
                      DO 50 J = I + 1, N
                         B( I+1, J ) = B( I+1, J ) - X*B( I, J )
       50             CONTINUE
                   END IF
                END IF
       60    CONTINUE
             IF( B( N, N ).EQ.ZERO )
         $      B( N, N ) = EPS3
    *
             TRANS = 'N'
    *
          ELSE
    *
    *        UL decomposition with partial pivoting of B, replacing zero
    *        pivots by EPS3.
    *
             DO 90 J = N, 2, -1
                EJ = H( J, J-1 )
                IF( CABS1( B( J, J ) ).LT.CABS1( EJ ) ) THEN
    *
    *              Interchange columns and eliminate.
    *
                   X = CLADIV( B( J, J ), EJ )
                   B( J, J ) = EJ
                   DO 70 I = 1, J - 1
                      TEMP = B( I, J-1 )
                      B( I, J-1 ) = B( I, J ) - X*TEMP
                      B( I, J ) = TEMP
       70          CONTINUE
                ELSE
    *
    *              Eliminate without interchange.
    *
                   IF( B( J, J ).EQ.ZERO )
         $            B( J, J ) = EPS3
                   X = CLADIV( EJ, B( J, J ) )
                   IF( X.NE.ZERO ) THEN
                      DO 80 I = 1, J - 1
                         B( I, J-1 ) = B( I, J-1 ) - X*B( I, J )
       80             CONTINUE
                   END IF
                END IF
       90    CONTINUE
             IF( B( 1, 1 ).EQ.ZERO )
         $      B( 1, 1 ) = EPS3
    *
             TRANS = 'C'
    *
          END IF
    *
          NORMIN = 'N'
          DO 110 ITS = 1, N
    *
    *        Solve U*x = scale*v for a right eigenvector
    *          or U'*x = scale*v for a left eigenvector,
    *        overwriting x on v.
    *
             CALL CLATRS( 'Upper', TRANS, 'Nonunit', NORMIN, N, B, LDB, V,
         $                SCALE, RWORK, IERR )
             NORMIN = 'Y'
    *
    *        Test for sufficient growth in the norm of v.
    *
             VNORM = SCASUM( N, V, 1 )
             IF( VNORM.GE.GROWTO*SCALE )
         $      GO TO 120
    *
    *        Choose new orthogonal starting vector and try again.
    *
             RTEMP = EPS3 / ( ROOTN+ONE )
             V( 1 ) = EPS3
             DO 100 I = 2, N
                V( I ) = RTEMP
      100    CONTINUE
             V( N-ITS+1 ) = V( N-ITS+1 ) - EPS3*ROOTN
      110 CONTINUE
    *
    *     Failure to find eigenvector in N iterations.
    *
          INFO = 1
    *
      120 CONTINUE
    *
    *     Normalize eigenvector.
    *
          I = ICAMAX( N, V, 1 )
          CALL CSSCAL( N, ONE / CABS1( V( I ) ), V, 1 )
    *
          RETURN
    *
    *     End of CLAEIN
    *
          END