*> \brief \b DLANV2 computes the Schur factorization of a real 2-by-2 nonsymmetric matrix in standard form.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlanv2.f">
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlanv2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLANV2( A, B, C, D, RT1R, RT1I, RT2R, RT2I, CS, SN )
*
* .. Scalar Arguments ..
* DOUBLE PRECISION A, B, C, CS, D, RT1I, RT1R, RT2I, RT2R, SN
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLANV2 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.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> C is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION
*> On entry, the elements of the input matrix.
*> On exit, they are overwritten by the elements of the
*> standardised Schur form.
*> \endverbatim
*>
*> \param[out] RT1R
*> \verbatim
*> RT1R is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[out] RT1I
*> \verbatim
*> RT1I is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[out] RT2R
*> \verbatim
*> RT2R is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[out] RT2I
*> \verbatim
*> RT2I is DOUBLE PRECISION
*> The real and imaginary parts of the eigenvalues. If the
*> eigenvalues are a complex conjugate pair, RT1I > 0.
*> \endverbatim
*>
*> \param[out] CS
*> \verbatim
*> CS is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[out] SN
*> \verbatim
*> SN is DOUBLE PRECISION
*> Parameters of the rotation matrix.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup doubleOTHERauxiliary
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> 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).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DLANV2( A, B, C, D, RT1R, RT1I, RT2R, RT2I, CS, SN )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
DOUBLE PRECISION A, B, C, CS, D, RT1I, RT1R, RT2I, RT2R, SN
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, HALF, ONE, TWO
PARAMETER ( ZERO = 0.0D+0, HALF = 0.5D+0, ONE = 1.0D+0,
$ TWO = 2.0D0 )
DOUBLE PRECISION MULTPL
PARAMETER ( MULTPL = 4.0D+0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION AA, BB, BCMAX, BCMIS, CC, CS1, DD, EPS, P, SAB,
$ SAC, SCALE, SIGMA, SN1, TAU, TEMP, Z, SAFMIN,
$ SAFMN2, SAFMX2
INTEGER COUNT
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH, DLAPY2
EXTERNAL DLAMCH, DLAPY2
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, SIGN, SQRT
* ..
* .. Executable Statements ..
*
SAFMIN = DLAMCH( 'S' )
EPS = DLAMCH( 'P' )
SAFMN2 = DLAMCH( 'B' )**INT( LOG( SAFMIN / EPS ) /
$ LOG( DLAMCH( 'B' ) ) / TWO )
SAFMX2 = ONE / SAFMN2
IF( C.EQ.ZERO ) THEN
CS = ONE
SN = ZERO
*
ELSE IF( B.EQ.ZERO ) THEN
*
* Swap rows and columns
*
CS = ZERO
SN = ONE
TEMP = D
D = A
A = TEMP
B = -C
C = ZERO
*
ELSE IF( ( A-D ).EQ.ZERO .AND. SIGN( ONE, B ).NE.SIGN( ONE, C ) )
$ THEN
CS = ONE
SN = ZERO
*
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 = DLAPY2( 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.
*
COUNT = 0
SIGMA = B + C
10 CONTINUE
COUNT = COUNT + 1
SCALE = MAX( ABS(TEMP), ABS(SIGMA) )
IF( SCALE.GE.SAFMX2 ) THEN
SIGMA = SIGMA * SAFMN2
TEMP = TEMP * SAFMN2
IF (COUNT .LE. 20)
$ GOTO 10
END IF
IF( SCALE.LE.SAFMN2 ) THEN
SIGMA = SIGMA * SAFMX2
TEMP = TEMP * SAFMX2
IF (COUNT .LE. 20)
$ GOTO 10
END IF
P = HALF*TEMP
TAU = DLAPY2( 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
*
* 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 DLANV2
*
END