slanv2.f

          SUBROUTINE SLANV2( A, B, C, D, RT1R, RT1I, RT2R, RT2I, CS, SN )
    *
    *  -- LAPACK driver routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          REAL               A, B, C, CS, D, RT1I, RT1R, RT2I, RT2R, SN
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  SLANV2 computes the Schur factorization of a real 2-by-2 nonsymmetric
    *  matrix in standard form:
    *
    *       [ A  B ] = [ CS -SN ] [ AA  BB ] [ CS  SN ]
    *       [ C  D ]   [ SN  CS ] [ CC  DD ] [-SN  CS ]
    *
    *  where either
    *  1) CC = 0 so that AA and DD are real eigenvalues of the matrix, or
    *  2) AA = DD and BB*CC < 0, so that AA + or - sqrt(BB*CC) are complex
    *  conjugate eigenvalues.
    *
    *  Arguments
    *  =========
    *
    *  A       (input/output) REAL
    *  B       (input/output) REAL
    *  C       (input/output) REAL
    *  D       (input/output) REAL
    *          On entry, the elements of the input matrix.
    *          On exit, they are overwritten by the elements of the
    *          standardised Schur form.
    *
    *  RT1R    (output) REAL
    *  RT1I    (output) REAL
    *  RT2R    (output) REAL
    *  RT2I    (output) REAL
    *          The real and imaginary parts of the eigenvalues. If the
    *          eigenvalues are a complex conjugate pair, RT1I > 0.
    *
    *  CS      (output) REAL
    *  SN      (output) REAL
    *          Parameters of the rotation matrix.
    *
    *  Further Details
    *  ===============
    *
    *  Modified by V. Sima, Research Institute for Informatics, Bucharest,
    *  Romania, to reduce the risk of cancellation errors,
    *  when computing real eigenvalues, and to ensure, if possible, that
    *  abs(RT1R) >= abs(RT2R).
    *
    *  =====================================================================
    *
    *     .. Parameters ..
          REAL               ZERO, HALF, ONE
          PARAMETER          ( ZERO = 0.0E+0, HALF = 0.5E+0, ONE = 1.0E+0 )
          REAL               MULTPL
          PARAMETER          ( MULTPL = 4.0E+0 )
    *     ..
    *     .. Local Scalars ..
          REAL               AA, BB, BCMAX, BCMIS, CC, CS1, DD, EPS, P, SAB,
         $                   SAC, SCALE, SIGMA, SN1, TAU, TEMP, Z
    *     ..
    *     .. External Functions ..
          REAL               SLAMCH, SLAPY2
          EXTERNAL           SLAMCH, SLAPY2
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          ABS, MAX, MIN, SIGN, SQRT
    *     ..
    *     .. Executable Statements ..
    *
          EPS = SLAMCH( 'P' )
          IF( C.EQ.ZERO ) THEN
             CS = ONE
             SN = ZERO
             GO TO 10
    *
          ELSE IF( B.EQ.ZERO ) THEN
    *
    *        Swap rows and columns
    *
             CS = ZERO
             SN = ONE
             TEMP = D
             D = A
             A = TEMP
             B = -C
             C = ZERO
             GO TO 10
          ELSE IF( (A-D).EQ.ZERO .AND. SIGN( ONE, B ).NE.
         $   SIGN( ONE, C ) ) THEN
             CS = ONE
             SN = ZERO
             GO TO 10
          ELSE
    *
             TEMP = A - D
             P = HALF*TEMP
             BCMAX = MAX( ABS( B ), ABS( C ) )
             BCMIS = MIN( ABS( B ), ABS( C ) )*SIGN( ONE, B )*SIGN( ONE, C )
             SCALE = MAX( ABS( P ), BCMAX )
             Z = ( P / SCALE )*P + ( BCMAX / SCALE )*BCMIS
    *
    *        If Z is of the order of the machine accuracy, postpone the
    *        decision on the nature of eigenvalues
    *
             IF( Z.GE.MULTPL*EPS ) THEN
    *
    *           Real eigenvalues. Compute A and D.
    *
                Z = P + SIGN( SQRT( SCALE )*SQRT( Z ), P )
                A = D + Z
                D = D - ( BCMAX / Z )*BCMIS
    *
    *           Compute B and the rotation matrix
    *
                TAU = SLAPY2( C, Z )
                CS = Z / TAU
                SN = C / TAU
                B = B - C
                C = ZERO
             ELSE
    *
    *           Complex eigenvalues, or real (almost) equal eigenvalues.
    *           Make diagonal elements equal.
    *
                SIGMA = B + C
                TAU = SLAPY2( SIGMA, TEMP )
                CS = SQRT( HALF*( ONE+ABS( SIGMA ) / TAU ) )
                SN = -( P / ( TAU*CS ) )*SIGN( ONE, SIGMA )
    *
    *           Compute [ AA  BB ] = [ A  B ] [ CS -SN ]
    *                   [ CC  DD ]   [ C  D ] [ SN  CS ]
    *
                AA = A*CS + B*SN
                BB = -A*SN + B*CS
                CC = C*CS + D*SN
                DD = -C*SN + D*CS
    *
    *           Compute [ A  B ] = [ CS  SN ] [ AA  BB ]
    *                   [ C  D ]   [-SN  CS ] [ CC  DD ]
    *
                A = AA*CS + CC*SN
                B = BB*CS + DD*SN
                C = -AA*SN + CC*CS
                D = -BB*SN + DD*CS
    *
                TEMP = HALF*( A+D )
                A = TEMP
                D = TEMP
    *
                IF( C.NE.ZERO ) THEN
                   IF( B.NE.ZERO ) THEN
                      IF( SIGN( ONE, B ).EQ.SIGN( ONE, C ) ) THEN
    *
    *                    Real eigenvalues: reduce to upper triangular form
    *
                         SAB = SQRT( ABS( B ) )
                         SAC = SQRT( ABS( C ) )
                         P = SIGN( SAB*SAC, C )
                         TAU = ONE / SQRT( ABS( B+C ) )
                         A = TEMP + P
                         D = TEMP - P
                         B = B - C
                         C = ZERO
                         CS1 = SAB*TAU
                         SN1 = SAC*TAU
                         TEMP = CS*CS1 - SN*SN1
                         SN = CS*SN1 + SN*CS1
                         CS = TEMP
                      END IF
                   ELSE
                      B = -C
                      C = ZERO
                      TEMP = CS
                      CS = -SN
                      SN = TEMP
                   END IF
                END IF
             END IF
    *
          END IF
    *
       10 CONTINUE
    *
    *     Store eigenvalues in (RT1R,RT1I) and (RT2R,RT2I).
    *
          RT1R = A
          RT2R = D
          IF( C.EQ.ZERO ) THEN
             RT1I = ZERO
             RT2I = ZERO
          ELSE
             RT1I = SQRT( ABS( B ) )*SQRT( ABS( C ) )
             RT2I = -RT1I
          END IF
          RETURN
    *
    *     End of SLANV2
    *
          END