*> \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 *> Download DLANV2 + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \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. * *> \date December 2016 * *> \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 (version 3.7.0) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * December 2016 * * .. Scalar Arguments .. DOUBLE PRECISION A, B, C, CS, D, RT1I, RT1R, RT2I, RT2R, SN * .. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ZERO, HALF, ONE PARAMETER ( ZERO = 0.0D+0, HALF = 0.5D+0, ONE = 1.0D+0 ) 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 * .. * .. External Functions .. DOUBLE PRECISION DLAMCH, DLAPY2 EXTERNAL DLAMCH, DLAPY2 * .. * .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN, SIGN, SQRT * .. * .. Executable Statements .. * EPS = DLAMCH( '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 = 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. * SIGMA = B + C 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 * 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 DLANV2 * END