207 lines
5.8 KiB
FortranFixed
207 lines
5.8 KiB
FortranFixed
SUBROUTINE DLANV2( A, B, C, D, RT1R, RT1I, RT2R, RT2I, CS, SN )
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*
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* -- LAPACK auxiliary routine (version 3.2.2) --
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* -- LAPACK is a software package provided by Univ. of Tennessee, --
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* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
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* June 2010
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*
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* .. Scalar Arguments ..
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DOUBLE PRECISION A, B, C, CS, D, RT1I, RT1R, RT2I, RT2R, SN
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* ..
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*
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* Purpose
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* =======
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*
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* DLANV2 computes the Schur factorization of a real 2-by-2 nonsymmetric
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* matrix in standard form:
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*
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* [ A B ] = [ CS -SN ] [ AA BB ] [ CS SN ]
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* [ C D ] [ SN CS ] [ CC DD ] [-SN CS ]
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*
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* where either
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* 1) CC = 0 so that AA and DD are real eigenvalues of the matrix, or
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* 2) AA = DD and BB*CC < 0, so that AA + or - sqrt(BB*CC) are complex
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* conjugate eigenvalues.
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*
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* Arguments
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* =========
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*
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* A (input/output) DOUBLE PRECISION
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* B (input/output) DOUBLE PRECISION
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* C (input/output) DOUBLE PRECISION
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* D (input/output) DOUBLE PRECISION
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* On entry, the elements of the input matrix.
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* On exit, they are overwritten by the elements of the
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* standardised Schur form.
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*
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* RT1R (output) DOUBLE PRECISION
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* RT1I (output) DOUBLE PRECISION
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* RT2R (output) DOUBLE PRECISION
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* RT2I (output) DOUBLE PRECISION
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* The real and imaginary parts of the eigenvalues. If the
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* eigenvalues are a complex conjugate pair, RT1I > 0.
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*
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* CS (output) DOUBLE PRECISION
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* SN (output) DOUBLE PRECISION
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* Parameters of the rotation matrix.
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*
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* Further Details
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* ===============
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*
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* Modified by V. Sima, Research Institute for Informatics, Bucharest,
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* Romania, to reduce the risk of cancellation errors,
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* when computing real eigenvalues, and to ensure, if possible, that
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* abs(RT1R) >= abs(RT2R).
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*
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* =====================================================================
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*
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* .. Parameters ..
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DOUBLE PRECISION ZERO, HALF, ONE
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PARAMETER ( ZERO = 0.0D+0, HALF = 0.5D+0, ONE = 1.0D+0 )
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DOUBLE PRECISION MULTPL
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PARAMETER ( MULTPL = 4.0D+0 )
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* ..
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* .. Local Scalars ..
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DOUBLE PRECISION AA, BB, BCMAX, BCMIS, CC, CS1, DD, EPS, P, SAB,
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$ SAC, SCALE, SIGMA, SN1, TAU, TEMP, Z
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* ..
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* .. External Functions ..
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DOUBLE PRECISION DLAMCH, DLAPY2
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EXTERNAL DLAMCH, DLAPY2
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC ABS, MAX, MIN, SIGN, SQRT
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* ..
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* .. Executable Statements ..
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*
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EPS = DLAMCH( 'P' )
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IF( C.EQ.ZERO ) THEN
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CS = ONE
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SN = ZERO
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GO TO 10
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*
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ELSE IF( B.EQ.ZERO ) THEN
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*
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* Swap rows and columns
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*
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CS = ZERO
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SN = ONE
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TEMP = D
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D = A
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A = TEMP
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B = -C
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C = ZERO
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GO TO 10
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ELSE IF( ( A-D ).EQ.ZERO .AND. SIGN( ONE, B ).NE.SIGN( ONE, C ) )
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$ THEN
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CS = ONE
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SN = ZERO
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GO TO 10
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ELSE
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*
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TEMP = A - D
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P = HALF*TEMP
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BCMAX = MAX( ABS( B ), ABS( C ) )
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BCMIS = MIN( ABS( B ), ABS( C ) )*SIGN( ONE, B )*SIGN( ONE, C )
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SCALE = MAX( ABS( P ), BCMAX )
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Z = ( P / SCALE )*P + ( BCMAX / SCALE )*BCMIS
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*
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* If Z is of the order of the machine accuracy, postpone the
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* decision on the nature of eigenvalues
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*
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IF( Z.GE.MULTPL*EPS ) THEN
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*
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* Real eigenvalues. Compute A and D.
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*
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Z = P + SIGN( SQRT( SCALE )*SQRT( Z ), P )
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A = D + Z
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D = D - ( BCMAX / Z )*BCMIS
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*
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* Compute B and the rotation matrix
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*
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TAU = DLAPY2( C, Z )
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CS = Z / TAU
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SN = C / TAU
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B = B - C
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C = ZERO
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ELSE
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*
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* Complex eigenvalues, or real (almost) equal eigenvalues.
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* Make diagonal elements equal.
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*
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SIGMA = B + C
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TAU = DLAPY2( SIGMA, TEMP )
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CS = SQRT( HALF*( ONE+ABS( SIGMA ) / TAU ) )
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SN = -( P / ( TAU*CS ) )*SIGN( ONE, SIGMA )
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*
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* Compute [ AA BB ] = [ A B ] [ CS -SN ]
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* [ CC DD ] [ C D ] [ SN CS ]
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*
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AA = A*CS + B*SN
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BB = -A*SN + B*CS
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CC = C*CS + D*SN
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DD = -C*SN + D*CS
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*
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* Compute [ A B ] = [ CS SN ] [ AA BB ]
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* [ C D ] [-SN CS ] [ CC DD ]
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*
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A = AA*CS + CC*SN
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B = BB*CS + DD*SN
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C = -AA*SN + CC*CS
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D = -BB*SN + DD*CS
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*
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TEMP = HALF*( A+D )
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A = TEMP
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D = TEMP
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*
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IF( C.NE.ZERO ) THEN
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IF( B.NE.ZERO ) THEN
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IF( SIGN( ONE, B ).EQ.SIGN( ONE, C ) ) THEN
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*
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* Real eigenvalues: reduce to upper triangular form
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*
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SAB = SQRT( ABS( B ) )
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SAC = SQRT( ABS( C ) )
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P = SIGN( SAB*SAC, C )
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TAU = ONE / SQRT( ABS( B+C ) )
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A = TEMP + P
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D = TEMP - P
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B = B - C
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C = ZERO
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CS1 = SAB*TAU
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SN1 = SAC*TAU
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TEMP = CS*CS1 - SN*SN1
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SN = CS*SN1 + SN*CS1
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CS = TEMP
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END IF
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ELSE
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B = -C
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C = ZERO
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TEMP = CS
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CS = -SN
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SN = TEMP
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END IF
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END IF
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END IF
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*
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END IF
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*
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10 CONTINUE
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*
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* Store eigenvalues in (RT1R,RT1I) and (RT2R,RT2I).
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*
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RT1R = A
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RT2R = D
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IF( C.EQ.ZERO ) THEN
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RT1I = ZERO
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RT2I = ZERO
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ELSE
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RT1I = SQRT( ABS( B ) )*SQRT( ABS( C ) )
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RT2I = -RT1I
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END IF
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RETURN
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*
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* End of DLANV2
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*
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END
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