dhgeqz.f

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*           (Single shift only.)
*
            IF( ( DBLE( MAXIT )*SAFMIN )*ABS( A( ILAST-1, ILAST ) ).LT.
     $          ABS( B( ILAST-1, ILAST-1 ) ) ) THEN
               ESHIFT = ESHIFT + A( ILAST-1, ILAST ) /
     $                  B( ILAST-1, ILAST-1 )
            ELSE
               ESHIFT = ESHIFT + ONE / ( SAFMIN*DBLE( MAXIT ) )
            END IF
            S1 = ONE
            WR = ESHIFT
*
*           ------------------- Begin Timing Code ----------------------
            OPST = OPST + DBLE( 4 )
*           -------------------- End Timing Code -----------------------
*
         ELSE
*
*           Shifts based on the generalized eigenvalues of the
*           bottom-right 2x2 block of A and B. The first eigenvalue
*           returned by DLAG2 is the Wilkinson shift (AEP p.512),
*
            CALL DLAG2( A( ILAST-1, ILAST-1 ), LDA,
     $                  B( ILAST-1, ILAST-1 ), LDB, SAFMIN*SAFETY, S1,
     $                  S2, WR, WR2, WI )
*
            TEMP = MAX( S1, SAFMIN*MAX( ONE, ABS( WR ), ABS( WI ) ) )
*
*           ------------------- Begin Timing Code ----------------------
            OPST = OPST + DBLE( 57 )
*           -------------------- End Timing Code -----------------------
*
            IF( WI.NE.ZERO )
     $         GO TO 200
         END IF
*
*        Fiddle with shift to avoid overflow
*
         TEMP = MIN( ASCALE, ONE )*( HALF*SAFMAX )
         IF( S1.GT.TEMP ) THEN
            SCALE = TEMP / S1
         ELSE
            SCALE = ONE
         END IF
*
         TEMP = MIN( BSCALE, ONE )*( HALF*SAFMAX )
         IF( ABS( WR ).GT.TEMP )
     $      SCALE = MIN( SCALE, TEMP / ABS( WR ) )
         S1 = SCALE*S1
         WR = SCALE*WR
*
*        Now check for two consecutive small subdiagonals.
*
         DO 120 J = ILAST - 1, IFIRST + 1, -1
            ISTART = J
            TEMP = ABS( S1*A( J, J-1 ) )
            TEMP2 = ABS( S1*A( J, J )-WR*B( J, J ) )
            TEMPR = MAX( TEMP, TEMP2 )
            IF( TEMPR.LT.ONE .AND. TEMPR.NE.ZERO ) THEN
               TEMP = TEMP / TEMPR
               TEMP2 = TEMP2 / TEMPR
            END IF
            IF( ABS( ( ASCALE*A( J+1, J ) )*TEMP ).LE.( ASCALE*ATOL )*
     $          TEMP2 )GO TO 130
  120    CONTINUE
*
         ISTART = IFIRST
  130    CONTINUE
*
*        Do an implicit single-shift QZ sweep.
*
*        Initial Q
*
         TEMP = S1*A( ISTART, ISTART ) - WR*B( ISTART, ISTART )
         TEMP2 = S1*A( ISTART+1, ISTART )
         CALL DLARTG( TEMP, TEMP2, C, S, TEMPR )
*
*        Sweep
*
         DO 190 J = ISTART, ILAST - 1
            IF( J.GT.ISTART ) THEN
               TEMP = A( J, J-1 )
               CALL DLARTG( TEMP, A( J+1, J-1 ), C, S, A( J, J-1 ) )
               A( J+1, J-1 ) = ZERO
            END IF
*
            DO 140 JC = J, ILASTM
               TEMP = C*A( J, JC ) + S*A( J+1, JC )
               A( J+1, JC ) = -S*A( J, JC ) + C*A( J+1, JC )
               A( J, JC ) = TEMP
               TEMP2 = C*B( J, JC ) + S*B( J+1, JC )
               B( J+1, JC ) = -S*B( J, JC ) + C*B( J+1, JC )
               B( J, JC ) = TEMP2
  140       CONTINUE
            IF( ILQ ) THEN
               DO 150 JR = 1, N
                  TEMP = C*Q( JR, J ) + S*Q( JR, J+1 )
                  Q( JR, J+1 ) = -S*Q( JR, J ) + C*Q( JR, J+1 )
                  Q( JR, J ) = TEMP
  150          CONTINUE
            END IF
*
            TEMP = B( J+1, J+1 )
            CALL DLARTG( TEMP, B( J+1, J ), C, S, B( J+1, J+1 ) )
            B( J+1, J ) = ZERO
*
            DO 160 JR = IFRSTM, MIN( J+2, ILAST )
               TEMP = C*A( JR, J+1 ) + S*A( JR, J )
               A( JR, J ) = -S*A( JR, J+1 ) + C*A( JR, J )
               A( JR, J+1 ) = TEMP
  160       CONTINUE
            DO 170 JR = IFRSTM, J
               TEMP = C*B( JR, J+1 ) + S*B( JR, J )
               B( JR, J ) = -S*B( JR, J+1 ) + C*B( JR, J )
               B( JR, J+1 ) = TEMP
  170       CONTINUE
            IF( ILZ ) THEN
               DO 180 JR = 1, N
                  TEMP = C*Z( JR, J+1 ) + S*Z( JR, J )
                  Z( JR, J ) = -S*Z( JR, J+1 ) + C*Z( JR, J )
                  Z( JR, J+1 ) = TEMP
  180          CONTINUE
            END IF
  190    CONTINUE
*
*        --------------------- Begin Timing Code -----------------------
         OPST = OPST + DBLE( 6+( ILAST-ISTART )*
     $          ( 8+14+36+12*( ILASTM-IFRSTM )+6*( NQ+NZ ) ) )
*        ---------------------- End Timing Code ------------------------
*
         GO TO 350
*
*        Use Francis double-shift
*
*        Note: the Francis double-shift should work with real shifts,
*              but only if the block is at least 3x3.
*              This code may break if this point is reached with
*              a 2x2 block with real eigenvalues.
*
  200    CONTINUE
         IF( IFIRST+1.EQ.ILAST ) THEN
*
*           Special case -- 2x2 block with complex eigenvectors
*
*           Step 1: Standardize, that is, rotate so that
*
*                       ( B11  0  )
*                   B = (         )  with B11 non-negative.
*                       (  0  B22 )
*
            CALL DLASV2( B( ILAST-1, ILAST-1 ), B( ILAST-1, ILAST ),
     $                   B( ILAST, ILAST ), B22, B11, SR, CR, SL, CL )
*
            IF( B11.LT.ZERO ) THEN
               CR = -CR
               SR = -SR
               B11 = -B11
               B22 = -B22
            END IF
*
            CALL DROT( ILASTM+1-IFIRST, A( ILAST-1, ILAST-1 ), LDA,
     $                 A( ILAST, ILAST-1 ), LDA, CL, SL )
            CALL DROT( ILAST+1-IFRSTM, A( IFRSTM, ILAST-1 ), 1,
     $                 A( IFRSTM, ILAST ), 1, CR, SR )
*
            IF( ILAST.LT.ILASTM )
     $         CALL DROT( ILASTM-ILAST, B( ILAST-1, ILAST+1 ), LDB,
     $                    B( ILAST, ILAST+1 ), LDA, CL, SL )
            IF( IFRSTM.LT.ILAST-1 )
     $         CALL DROT( IFIRST-IFRSTM, B( IFRSTM, ILAST-1 ), 1,
     $                    B( IFRSTM, ILAST ), 1, CR, SR )
*
            IF( ILQ )
     $         CALL DROT( N, Q( 1, ILAST-1 ), 1, Q( 1, ILAST ), 1, CL,
     $                    SL )
            IF( ILZ )
     $         CALL DROT( N, Z( 1, ILAST-1 ), 1, Z( 1, ILAST ), 1, CR,
     $                    SR )
*
            B( ILAST-1, ILAST-1 ) = B11
            B( ILAST-1, ILAST ) = ZERO
            B( ILAST, ILAST-1 ) = ZERO
            B( ILAST, ILAST ) = B22
*
*           If B22 is negative, negate column ILAST
*
            IF( B22.LT.ZERO ) THEN
               DO 210 J = IFRSTM, ILAST
                  A( J, ILAST ) = -A( J, ILAST )
                  B( J, ILAST ) = -B( J, ILAST )
  210          CONTINUE
*
               IF( ILZ ) THEN
                  DO 220 J = 1, N
                     Z( J, ILAST ) = -Z( J, ILAST )
  220             CONTINUE
               END IF
            END IF
*
*           Step 2: Compute ALPHAR, ALPHAI, and BETA (see refs.)
*
*           Recompute shift
*
            CALL DLAG2( A( ILAST-1, ILAST-1 ), LDA,
     $                  B( ILAST-1, ILAST-1 ), LDB, SAFMIN*SAFETY, S1,
     $                  TEMP, WR, TEMP2, WI )
*
*           ------------------- Begin Timing Code ----------------------
            OPST = OPST + DBLE( 103+12*( ILASTM+ILAST-IFIRST-IFRSTM )+6*
     $             ( NQ+NZ ) )
*           -------------------- End Timing Code -----------------------
*
*           If standardization has perturbed the shift onto real line,
*           do another (real single-shift) QR step.
*
            IF( WI.EQ.ZERO )
     $         GO TO 350
            S1INV = ONE / S1
*
*           Do EISPACK (QZVAL) computation of alpha and beta
*
            A11 = A( ILAST-1, ILAST-1 )
            A21 = A( ILAST, ILAST-1 )
            A12 = A( ILAST-1, ILAST )
            A22 = A( ILAST, ILAST )
*
*           Compute complex Givens rotation on right
*           (Assume some element of C = (sA - wB) > unfl )
*                            __
*           (sA - wB) ( CZ   -SZ )
*                     ( SZ    CZ )
*
            C11R = S1*A11 - WR*B11
            C11I = -WI*B11
            C12 = S1*A12
            C21 = S1*A21
            C22R = S1*A22 - WR*B22
            C22I = -WI*B22
*
            IF( ABS( C11R )+ABS( C11I )+ABS( C12 ).GT.ABS( C21 )+
     $          ABS( C22R )+ABS( C22I ) ) THEN
               T = DLAPY3( C12, C11R, C11I )
               CZ = C12 / T
               SZR = -C11R / T
               SZI = -C11I / T
            ELSE
               CZ = DLAPY2( C22R, C22I )
               IF( CZ.LE.SAFMIN ) THEN
                  CZ = ZERO
                  SZR = ONE
                  SZI = ZERO
               ELSE
                  TEMPR = C22R / CZ
                  TEMPI = C22I / CZ
                  T = DLAPY2( CZ, C21 )
                  CZ = CZ / T
                  SZR = -C21*TEMPR / T
                  SZI = C21*TEMPI / T
               END IF
            END IF
*
*           Compute Givens rotation on left
*
*           (  CQ   SQ )
*           (  __      )  A or B
*           ( -SQ   CQ )
*
            AN = ABS( A11 ) + ABS( A12 ) + ABS( A21 ) + ABS( A22 )
            BN = ABS( B11 ) + ABS( B22 )
            WABS = ABS( WR ) + ABS( WI )
            IF( S1*AN.GT.WABS*BN ) THEN
               CQ = CZ*B11
               SQR = SZR*B22
               SQI = -SZI*B22
            ELSE
               A1R = CZ*A11 + SZR*A12
               A1I = SZI*A12
               A2R = CZ*A21 + SZR*A22
               A2I = SZI*A22
               CQ = DLAPY2( A1R, A1I )
               IF( CQ.LE.SAFMIN ) THEN
                  CQ = ZERO
                  SQR = ONE
                  SQI = ZERO
               ELSE
                  TEMPR = A1R / CQ
                  TEMPI = A1I / CQ
                  SQR = TEMPR*A2R + TEMPI*A2I
                  SQI = TEMPI*A2R - TEMPR*A2I
               END IF
            END IF
            T = DLAPY3( CQ, SQR, SQI )
            CQ = CQ / T
            SQR = SQR / T
            SQI = SQI / T
*
*           Compute diagonal elements of QBZ
*
            TEMPR = SQR*SZR - SQI*SZI
            TEMPI = SQR*SZI + SQI*SZR
            B1R = CQ*CZ*B11 + TEMPR*B22
            B1I = TEMPI*B22
            B1A = DLAPY2( B1R, B1I )
            B2R = CQ*CZ*B22 + TEMPR*B11
            B2I = -TEMPI*B11
            B2A = DLAPY2( B2R, B2I )
*
*           Normalize so beta > 0, and Im( alpha1 ) > 0
*
            BETA( ILAST-1 ) = B1A
            BETA( ILAST ) = B2A
            ALPHAR( ILAST-1 ) = ( WR*B1A )*S1INV
            ALPHAI( ILAST-1 ) = ( WI*B1A )*S1INV
            ALPHAR( ILAST ) = ( WR*B2A )*S1INV
            ALPHAI( ILAST ) = -( WI*B2A )*S1INV
*
*           ------------------- Begin Timing Code ----------------------
            OPST = OPST + DBLE( 93 )
*           -------------------- End Timing Code -----------------------
*
*           Step 3: Go to next block -- exit if finished.
*
            ILAST = IFIRST - 1
            IF( ILAST.LT.ILO )
     $         GO TO 380
*
*           Reset counters
*
            IITER = 0
            ESHIFT = ZERO
            IF( .NOT.ILSCHR ) THEN
               ILASTM = ILAST
               IF( IFRSTM.GT.ILAST )
     $            IFRSTM = ILO