clahqr.f

来自「famous linear algebra library (LAPACK) p」· F 代码 · 共 470 行 · 第 1/2 页

*     subdiagonal element has become negligible.
*
      L = ILO
      DO 130 ITS = 0, ITMAX
*
*        Look for a single small subdiagonal element.
*
         DO 40 K = I, L + 1, -1
            IF( CABS1( H( K, K-1 ) ).LE.SMLNUM )
     $         GO TO 50
            TST = CABS1( H( K-1, K-1 ) ) + CABS1( H( K, K ) )
            IF( TST.EQ.ZERO ) THEN
               IF( K-2.GE.ILO )
     $            TST = TST + ABS( REAL( H( K-1, K-2 ) ) )
               IF( K+1.LE.IHI )
     $            TST = TST + ABS( REAL( H( K+1, K ) ) )
            END IF
*           ==== The following is a conservative small subdiagonal
*           .    deflation criterion due to Ahues & Tisseur (LAWN 122,
*           .    1997). It has better mathematical foundation and
*           .    improves accuracy in some examples.  ====
            IF( ABS( REAL( H( K, K-1 ) ) ).LE.ULP*TST ) THEN
               AB = MAX( CABS1( H( K, K-1 ) ), CABS1( H( K-1, K ) ) )
               BA = MIN( CABS1( H( K, K-1 ) ), CABS1( H( K-1, K ) ) )
               AA = MAX( CABS1( H( K, K ) ),
     $              CABS1( H( K-1, K-1 )-H( K, K ) ) )
               BB = MIN( CABS1( H( K, K ) ),
     $              CABS1( H( K-1, K-1 )-H( K, K ) ) )
               S = AA + AB
               IF( BA*( AB / S ).LE.MAX( SMLNUM,
     $             ULP*( BB*( AA / S ) ) ) )GO TO 50
            END IF
   40    CONTINUE
   50    CONTINUE
         L = K
         IF( L.GT.ILO ) THEN
*
*           H(L,L-1) is negligible
*
            H( L, L-1 ) = ZERO
         END IF
*
*        Exit from loop if a submatrix of order 1 has split off.
*
         IF( L.GE.I )
     $      GO TO 140
*
*        Now the active submatrix is in rows and columns L to I. If
*        eigenvalues only are being computed, only the active submatrix
*        need be transformed.
*
         IF( .NOT.WANTT ) THEN
            I1 = L
            I2 = I
         END IF
*
         IF( ITS.EQ.10 .OR. ITS.EQ.20 ) THEN
*
*           Exceptional shift.
*
            S = DAT1*ABS( REAL( H( I, I-1 ) ) )
            T = S + H( I, I )
         ELSE
*
*           Wilkinson's shift.
*
            T = H( I, I )
            U = SQRT( H( I-1, I ) )*SQRT( H( I, I-1 ) )
            S = CABS1( U )
            IF( S.NE.RZERO ) THEN
               X = HALF*( H( I-1, I-1 )-T )
               SX = CABS1( X )
               S = MAX( S, CABS1( X ) )
               Y = S*SQRT( ( X / S )**2+( U / S )**2 )
               IF( SX.GT.RZERO ) THEN
                  IF( REAL( X / SX )*REAL( Y )+AIMAG( X / SX )*
     $                AIMAG( Y ).LT.RZERO )Y = -Y
               END IF
               T = T - U*CLADIV( U, ( X+Y ) )
            END IF
         END IF
*
*        Look for two consecutive small subdiagonal elements.
*
         DO 60 M = I - 1, L + 1, -1
*
*           Determine the effect of starting the single-shift QR
*           iteration at row M, and see if this would make H(M,M-1)
*           negligible.
*
            H11 = H( M, M )
            H22 = H( M+1, M+1 )
            H11S = H11 - T
            H21 = H( M+1, M )
            S = CABS1( H11S ) + ABS( H21 )
            H11S = H11S / S
            H21 = H21 / S
            V( 1 ) = H11S
            V( 2 ) = H21
            H10 = H( M, M-1 )
            IF( ABS( H10 )*ABS( H21 ).LE.ULP*
     $          ( CABS1( H11S )*( CABS1( H11 )+CABS1( H22 ) ) ) )
     $          GO TO 70
   60    CONTINUE
         H11 = H( L, L )
         H22 = H( L+1, L+1 )
         H11S = H11 - T
         H21 = H( L+1, L )
         S = CABS1( H11S ) + ABS( H21 )
         H11S = H11S / S
         H21 = H21 / S
         V( 1 ) = H11S
         V( 2 ) = H21
   70    CONTINUE
*
*        Single-shift QR step
*
         DO 120 K = M, I - 1
*
*           The first iteration of this loop determines a reflection G
*           from the vector V and applies it from left and right to H,
*           thus creating a nonzero bulge below the subdiagonal.
*
*           Each subsequent iteration determines a reflection G to
*           restore the Hessenberg form in the (K-1)th column, and thus
*           chases the bulge one step toward the bottom of the active
*           submatrix.
*
*           V(2) is always real before the call to CLARFG, and hence
*           after the call T2 ( = T1*V(2) ) is also real.
*
            IF( K.GT.M )
     $         CALL CCOPY( 2, H( K, K-1 ), 1, V, 1 )
            CALL CLARFG( 2, V( 1 ), V( 2 ), 1, T1 )
            IF( K.GT.M ) THEN
               H( K, K-1 ) = V( 1 )
               H( K+1, K-1 ) = ZERO
            END IF
            V2 = V( 2 )
            T2 = REAL( T1*V2 )
*
*           Apply G from the left to transform the rows of the matrix
*           in columns K to I2.
*
            DO 80 J = K, I2
               SUM = CONJG( T1 )*H( K, J ) + T2*H( K+1, J )
               H( K, J ) = H( K, J ) - SUM
               H( K+1, J ) = H( K+1, J ) - SUM*V2
   80       CONTINUE
*
*           Apply G from the right to transform the columns of the
*           matrix in rows I1 to min(K+2,I).
*
            DO 90 J = I1, MIN( K+2, I )
               SUM = T1*H( J, K ) + T2*H( J, K+1 )
               H( J, K ) = H( J, K ) - SUM
               H( J, K+1 ) = H( J, K+1 ) - SUM*CONJG( V2 )
   90       CONTINUE
*
            IF( WANTZ ) THEN
*
*              Accumulate transformations in the matrix Z
*
               DO 100 J = ILOZ, IHIZ
                  SUM = T1*Z( J, K ) + T2*Z( J, K+1 )
                  Z( J, K ) = Z( J, K ) - SUM
                  Z( J, K+1 ) = Z( J, K+1 ) - SUM*CONJG( V2 )
  100          CONTINUE
            END IF
*
            IF( K.EQ.M .AND. M.GT.L ) THEN
*
*              If the QR step was started at row M > L because two
*              consecutive small subdiagonals were found, then extra
*              scaling must be performed to ensure that H(M,M-1) remains
*              real.
*
               TEMP = ONE - T1
               TEMP = TEMP / ABS( TEMP )
               H( M+1, M ) = H( M+1, M )*CONJG( TEMP )
               IF( M+2.LE.I )
     $            H( M+2, M+1 ) = H( M+2, M+1 )*TEMP
               DO 110 J = M, I
                  IF( J.NE.M+1 ) THEN
                     IF( I2.GT.J )
     $                  CALL CSCAL( I2-J, TEMP, H( J, J+1 ), LDH )
                     CALL CSCAL( J-I1, CONJG( TEMP ), H( I1, J ), 1 )
                     IF( WANTZ ) THEN
                        CALL CSCAL( NZ, CONJG( TEMP ), Z( ILOZ, J ), 1 )
                     END IF
                  END IF
  110          CONTINUE
            END IF
  120    CONTINUE
*
*        Ensure that H(I,I-1) is real.
*
         TEMP = H( I, I-1 )
         IF( AIMAG( TEMP ).NE.RZERO ) THEN
            RTEMP = ABS( TEMP )
            H( I, I-1 ) = RTEMP
            TEMP = TEMP / RTEMP
            IF( I2.GT.I )
     $         CALL CSCAL( I2-I, CONJG( TEMP ), H( I, I+1 ), LDH )
            CALL CSCAL( I-I1, TEMP, H( I1, I ), 1 )
            IF( WANTZ ) THEN
               CALL CSCAL( NZ, TEMP, Z( ILOZ, I ), 1 )
            END IF
         END IF
*
  130 CONTINUE
*
*     Failure to converge in remaining number of iterations
*
      INFO = I
      RETURN
*
  140 CONTINUE
*
*     H(I,I-1) is negligible: one eigenvalue has converged.
*
      W( I ) = H( I, I )
*
*     return to start of the main loop with new value of I.
*
      I = L - 1
      GO TO 30
*
  150 CONTINUE
      RETURN
*
*     End of CLAHQR
*
      END