zhgeqz.f

famous linear algebra library (LAPACK) ports to windows

      MAXIT = 30*( IHI-ILO+1 )
*
      DO 170 JITER = 1, MAXIT
*
*        Check for too many iterations.
*
         IF( JITER.GT.MAXIT )
     $      GO TO 180
*
*        Split the matrix if possible.
*
*        Two tests:
*           1: H(j,j-1)=0  or  j=ILO
*           2: T(j,j)=0
*
*        Special case: j=ILAST
*
         IF( ILAST.EQ.ILO ) THEN
            GO TO 60
         ELSE
            IF( ABS1( H( ILAST, ILAST-1 ) ).LE.ATOL ) THEN
               H( ILAST, ILAST-1 ) = CZERO
               GO TO 60
            END IF
         END IF
*
         IF( ABS( T( ILAST, ILAST ) ).LE.BTOL ) THEN
            T( ILAST, ILAST ) = CZERO
            GO TO 50
         END IF
*
*        General case: j<ILAST
*
         DO 40 J = ILAST - 1, ILO, -1
*
*           Test 1: for H(j,j-1)=0 or j=ILO
*
            IF( J.EQ.ILO ) THEN
               ILAZRO = .TRUE.
            ELSE
               IF( ABS1( H( J, J-1 ) ).LE.ATOL ) THEN
                  H( J, J-1 ) = CZERO
                  ILAZRO = .TRUE.
               ELSE
                  ILAZRO = .FALSE.
               END IF
            END IF
*
*           Test 2: for T(j,j)=0
*
            IF( ABS( T( J, J ) ).LT.BTOL ) THEN
               T( J, J ) = CZERO
*
*              Test 1a: Check for 2 consecutive small subdiagonals in A
*
               ILAZR2 = .FALSE.
               IF( .NOT.ILAZRO ) THEN
                  IF( ABS1( H( J, J-1 ) )*( ASCALE*ABS1( H( J+1,
     $                J ) ) ).LE.ABS1( H( J, J ) )*( ASCALE*ATOL ) )
     $                ILAZR2 = .TRUE.
               END IF
*
*              If both tests pass (1 & 2), i.e., the leading diagonal
*              element of B in the block is zero, split a 1x1 block off
*              at the top. (I.e., at the J-th row/column) The leading
*              diagonal element of the remainder can also be zero, so
*              this may have to be done repeatedly.
*
               IF( ILAZRO .OR. ILAZR2 ) THEN
                  DO 20 JCH = J, ILAST - 1
                     CTEMP = H( JCH, JCH )
                     CALL ZLARTG( CTEMP, H( JCH+1, JCH ), C, S,
     $                            H( JCH, JCH ) )
                     H( JCH+1, JCH ) = CZERO
                     CALL ZROT( ILASTM-JCH, H( JCH, JCH+1 ), LDH,
     $                          H( JCH+1, JCH+1 ), LDH, C, S )
                     CALL ZROT( ILASTM-JCH, T( JCH, JCH+1 ), LDT,
     $                          T( JCH+1, JCH+1 ), LDT, C, S )
                     IF( ILQ )
     $                  CALL ZROT( N, Q( 1, JCH ), 1, Q( 1, JCH+1 ), 1,
     $                             C, DCONJG( S ) )
                     IF( ILAZR2 )
     $                  H( JCH, JCH-1 ) = H( JCH, JCH-1 )*C
                     ILAZR2 = .FALSE.
                     IF( ABS1( T( JCH+1, JCH+1 ) ).GE.BTOL ) THEN
                        IF( JCH+1.GE.ILAST ) THEN
                           GO TO 60
                        ELSE
                           IFIRST = JCH + 1
                           GO TO 70
                        END IF
                     END IF
                     T( JCH+1, JCH+1 ) = CZERO
   20             CONTINUE
                  GO TO 50
               ELSE
*
*                 Only test 2 passed -- chase the zero to T(ILAST,ILAST)
*                 Then process as in the case T(ILAST,ILAST)=0
*
                  DO 30 JCH = J, ILAST - 1
                     CTEMP = T( JCH, JCH+1 )
                     CALL ZLARTG( CTEMP, T( JCH+1, JCH+1 ), C, S,
     $                            T( JCH, JCH+1 ) )
                     T( JCH+1, JCH+1 ) = CZERO
                     IF( JCH.LT.ILASTM-1 )
     $                  CALL ZROT( ILASTM-JCH-1, T( JCH, JCH+2 ), LDT,
     $                             T( JCH+1, JCH+2 ), LDT, C, S )
                     CALL ZROT( ILASTM-JCH+2, H( JCH, JCH-1 ), LDH,
     $                          H( JCH+1, JCH-1 ), LDH, C, S )
                     IF( ILQ )
     $                  CALL ZROT( N, Q( 1, JCH ), 1, Q( 1, JCH+1 ), 1,
     $                             C, DCONJG( S ) )
                     CTEMP = H( JCH+1, JCH )
                     CALL ZLARTG( CTEMP, H( JCH+1, JCH-1 ), C, S,
     $                            H( JCH+1, JCH ) )
                     H( JCH+1, JCH-1 ) = CZERO
                     CALL ZROT( JCH+1-IFRSTM, H( IFRSTM, JCH ), 1,
     $                          H( IFRSTM, JCH-1 ), 1, C, S )
                     CALL ZROT( JCH-IFRSTM, T( IFRSTM, JCH ), 1,
     $                          T( IFRSTM, JCH-1 ), 1, C, S )
                     IF( ILZ )
     $                  CALL ZROT( N, Z( 1, JCH ), 1, Z( 1, JCH-1 ), 1,
     $                             C, S )
   30             CONTINUE
                  GO TO 50
               END IF
            ELSE IF( ILAZRO ) THEN
*
*              Only test 1 passed -- work on J:ILAST
*
               IFIRST = J
               GO TO 70
            END IF
*
*           Neither test passed -- try next J
*
   40    CONTINUE
*
*        (Drop-through is "impossible")
*
         INFO = 2*N + 1
         GO TO 210
*
*        T(ILAST,ILAST)=0 -- clear H(ILAST,ILAST-1) to split off a
*        1x1 block.
*
   50    CONTINUE
         CTEMP = H( ILAST, ILAST )
         CALL ZLARTG( CTEMP, H( ILAST, ILAST-1 ), C, S,
     $                H( ILAST, ILAST ) )
         H( ILAST, ILAST-1 ) = CZERO
         CALL ZROT( ILAST-IFRSTM, H( IFRSTM, ILAST ), 1,
     $              H( IFRSTM, ILAST-1 ), 1, C, S )
         CALL ZROT( ILAST-IFRSTM, T( IFRSTM, ILAST ), 1,
     $              T( IFRSTM, ILAST-1 ), 1, C, S )
         IF( ILZ )
     $      CALL ZROT( N, Z( 1, ILAST ), 1, Z( 1, ILAST-1 ), 1, C, S )
*
*        H(ILAST,ILAST-1)=0 -- Standardize B, set ALPHA and BETA
*
   60    CONTINUE
         ABSB = ABS( T( ILAST, ILAST ) )
         IF( ABSB.GT.SAFMIN ) THEN
            SIGNBC = DCONJG( T( ILAST, ILAST ) / ABSB )
            T( ILAST, ILAST ) = ABSB
            IF( ILSCHR ) THEN
               CALL ZSCAL( ILAST-IFRSTM, SIGNBC, T( IFRSTM, ILAST ), 1 )
               CALL ZSCAL( ILAST+1-IFRSTM, SIGNBC, H( IFRSTM, ILAST ),
     $                     1 )
            ELSE
               H( ILAST, ILAST ) = H( ILAST, ILAST )*SIGNBC
            END IF
            IF( ILZ )
     $         CALL ZSCAL( N, SIGNBC, Z( 1, ILAST ), 1 )
         ELSE
            T( ILAST, ILAST ) = CZERO
         END IF
         ALPHA( ILAST ) = H( ILAST, ILAST )
         BETA( ILAST ) = T( ILAST, ILAST )
*
*        Go to next block -- exit if finished.
*
         ILAST = ILAST - 1
         IF( ILAST.LT.ILO )
     $      GO TO 190
*
*        Reset counters
*
         IITER = 0
         ESHIFT = CZERO
         IF( .NOT.ILSCHR ) THEN
            ILASTM = ILAST
            IF( IFRSTM.GT.ILAST )
     $         IFRSTM = ILO
         END IF
         GO TO 160
*
*        QZ step
*
*        This iteration only involves rows/columns IFIRST:ILAST.  We
*        assume IFIRST < ILAST, and that the diagonal of B is non-zero.
*
   70    CONTINUE
         IITER = IITER + 1
         IF( .NOT.ILSCHR ) THEN
            IFRSTM = IFIRST
         END IF
*
*        Compute the Shift.
*
*        At this point, IFIRST < ILAST, and the diagonal elements of
*        T(IFIRST:ILAST,IFIRST,ILAST) are larger than BTOL (in
*        magnitude)
*
         IF( ( IITER / 10 )*10.NE.IITER ) THEN
*
*           The Wilkinson shift (AEP p.512), i.e., the eigenvalue of
*           the bottom-right 2x2 block of A inv(B) which is nearest to
*           the bottom-right element.
*
*           We factor B as U*D, where U has unit diagonals, and
*           compute (A*inv(D))*inv(U).
*
            U12 = ( BSCALE*T( ILAST-1, ILAST ) ) /
     $            ( BSCALE*T( ILAST, ILAST ) )
            AD11 = ( ASCALE*H( ILAST-1, ILAST-1 ) ) /
     $             ( BSCALE*T( ILAST-1, ILAST-1 ) )
            AD21 = ( ASCALE*H( ILAST, ILAST-1 ) ) /
     $             ( BSCALE*T( ILAST-1, ILAST-1 ) )
            AD12 = ( ASCALE*H( ILAST-1, ILAST ) ) /
     $             ( BSCALE*T( ILAST, ILAST ) )
            AD22 = ( ASCALE*H( ILAST, ILAST ) ) /
     $             ( BSCALE*T( ILAST, ILAST ) )
            ABI22 = AD22 - U12*AD21
*
            T1 = HALF*( AD11+ABI22 )
            RTDISC = SQRT( T1**2+AD12*AD21-AD11*AD22 )
            TEMP = DBLE( T1-ABI22 )*DBLE( RTDISC ) +
     $             DIMAG( T1-ABI22 )*DIMAG( RTDISC )
            IF( TEMP.LE.ZERO ) THEN
               SHIFT = T1 + RTDISC
            ELSE
               SHIFT = T1 - RTDISC
            END IF
         ELSE
*
*           Exceptional shift.  Chosen for no particularly good reason.
*
            ESHIFT = ESHIFT + DCONJG( ( ASCALE*H( ILAST-1, ILAST ) ) /
     $               ( BSCALE*T( ILAST-1, ILAST-1 ) ) )
            SHIFT = ESHIFT
         END IF
*
*        Now check for two consecutive small subdiagonals.
*
         DO 80 J = ILAST - 1, IFIRST + 1, -1
            ISTART = J
            CTEMP = ASCALE*H( J, J ) - SHIFT*( BSCALE*T( J, J ) )
            TEMP = ABS1( CTEMP )
            TEMP2 = ASCALE*ABS1( H( J+1, J ) )
            TEMPR = MAX( TEMP, TEMP2 )
            IF( TEMPR.LT.ONE .AND. TEMPR.NE.ZERO ) THEN
               TEMP = TEMP / TEMPR
               TEMP2 = TEMP2 / TEMPR
            END IF
            IF( ABS1( H( J, J-1 ) )*TEMP2.LE.TEMP*ATOL )
     $         GO TO 90
   80    CONTINUE
*
         ISTART = IFIRST
         CTEMP = ASCALE*H( IFIRST, IFIRST ) -
     $           SHIFT*( BSCALE*T( IFIRST, IFIRST ) )
   90    CONTINUE
*
*        Do an implicit-shift QZ sweep.
*
*        Initial Q
*
         CTEMP2 = ASCALE*H( ISTART+1, ISTART )
         CALL ZLARTG( CTEMP, CTEMP2, C, S, CTEMP3 )
*
*        Sweep
*
         DO 150 J = ISTART, ILAST - 1
            IF( J.GT.ISTART ) THEN
               CTEMP = H( J, J-1 )
               CALL ZLARTG( CTEMP, H( J+1, J-1 ), C, S, H( J, J-1 ) )
               H( J+1, J-1 ) = CZERO
            END IF
*
            DO 100 JC = J, ILASTM
               CTEMP = C*H( J, JC ) + S*H( J+1, JC )
               H( J+1, JC ) = -DCONJG( S )*H( J, JC ) + C*H( J+1, JC )
               H( J, JC ) = CTEMP
               CTEMP2 = C*T( J, JC ) + S*T( J+1, JC )
               T( J+1, JC ) = -DCONJG( S )*T( J, JC ) + C*T( J+1, JC )
               T( J, JC ) = CTEMP2
  100       CONTINUE
            IF( ILQ ) THEN
               DO 110 JR = 1, N
                  CTEMP = C*Q( JR, J ) + DCONJG( S )*Q( JR, J+1 )
                  Q( JR, J+1 ) = -S*Q( JR, J ) + C*Q( JR, J+1 )
                  Q( JR, J ) = CTEMP
  110          CONTINUE
            END IF
*
            CTEMP = T( J+1, J+1 )
            CALL ZLARTG( CTEMP, T( J+1, J ), C, S, T( J+1, J+1 ) )
            T( J+1, J ) = CZERO
*
            DO 120 JR = IFRSTM, MIN( J+2, ILAST )
               CTEMP = C*H( JR, J+1 ) + S*H( JR, J )
               H( JR, J ) = -DCONJG( S )*H( JR, J+1 ) + C*H( JR, J )
               H( JR, J+1 ) = CTEMP
  120       CONTINUE
            DO 130 JR = IFRSTM, J
               CTEMP = C*T( JR, J+1 ) + S*T( JR, J )
               T( JR, J ) = -DCONJG( S )*T( JR, J+1 ) + C*T( JR, J )
               T( JR, J+1 ) = CTEMP
  130       CONTINUE
            IF( ILZ ) THEN
               DO 140 JR = 1, N
                  CTEMP = C*Z( JR, J+1 ) + S*Z( JR, J )
                  Z( JR, J ) = -DCONJG( S )*Z( JR, J+1 ) + C*Z( JR, J )
                  Z( JR, J+1 ) = CTEMP
  140          CONTINUE
            END IF
  150    CONTINUE
*
  160    CONTINUE
*
  170 CONTINUE
*
*     Drop-through = non-convergence
*
  180 CONTINUE
      INFO = ILAST
      GO TO 210
*
*     Successful completion of all QZ steps
*
  190 CONTINUE
*
*     Set Eigenvalues 1:ILO-1
*
      DO 200 J = 1, ILO - 1
         ABSB = ABS( T( J, J ) )
         IF( ABSB.GT.SAFMIN ) THEN
            SIGNBC = DCONJG( T( J, J ) / ABSB )
            T( J, J ) = ABSB
            IF( ILSCHR ) THEN
               CALL ZSCAL( J-1, SIGNBC, T( 1, J ), 1 )
               CALL ZSCAL( J, SIGNBC, H( 1, J ), 1 )
            ELSE
               H( J, J ) = H( J, J )*SIGNBC
            END IF
            IF( ILZ )
     $         CALL ZSCAL( N, SIGNBC, Z( 1, J ), 1 )
         ELSE
            T( J, J ) = CZERO
         END IF
         ALPHA( J ) = H( J, J )
         BETA( J ) = T( J, J )
  200 CONTINUE
*
*     Normal Termination
*
      INFO = 0
*
*     Exit (other than argument error) -- return optimal workspace size
*
  210 CONTINUE
      WORK( 1 ) = DCMPLX( N )
      RETURN
*
*     End of ZHGEQZ
*
      END