*> \brief \b DLAEV2 computes the eigenvalues and eigenvectors of a 2-by-2 symmetric/Hermitian matrix. * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download DLAEV2 + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE DLAEV2( A, B, C, RT1, RT2, CS1, SN1 ) * * .. Scalar Arguments .. * DOUBLE PRECISION A, B, C, CS1, RT1, RT2, SN1 * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> DLAEV2 computes the eigendecomposition of a 2-by-2 symmetric matrix *> [ A B ] *> [ B C ]. *> On return, RT1 is the eigenvalue of larger absolute value, RT2 is the *> eigenvalue of smaller absolute value, and (CS1,SN1) is the unit right *> eigenvector for RT1, giving the decomposition *> *> [ CS1 SN1 ] [ A B ] [ CS1 -SN1 ] = [ RT1 0 ] *> [-SN1 CS1 ] [ B C ] [ SN1 CS1 ] [ 0 RT2 ]. *> \endverbatim * * Arguments: * ========== * *> \param[in] A *> \verbatim *> A is DOUBLE PRECISION *> The (1,1) element of the 2-by-2 matrix. *> \endverbatim *> *> \param[in] B *> \verbatim *> B is DOUBLE PRECISION *> The (1,2) element and the conjugate of the (2,1) element of *> the 2-by-2 matrix. *> \endverbatim *> *> \param[in] C *> \verbatim *> C is DOUBLE PRECISION *> The (2,2) element of the 2-by-2 matrix. *> \endverbatim *> *> \param[out] RT1 *> \verbatim *> RT1 is DOUBLE PRECISION *> The eigenvalue of larger absolute value. *> \endverbatim *> *> \param[out] RT2 *> \verbatim *> RT2 is DOUBLE PRECISION *> The eigenvalue of smaller absolute value. *> \endverbatim *> *> \param[out] CS1 *> \verbatim *> CS1 is DOUBLE PRECISION *> \endverbatim *> *> \param[out] SN1 *> \verbatim *> SN1 is DOUBLE PRECISION *> The vector (CS1, SN1) is a unit right eigenvector for RT1. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date December 2016 * *> \ingroup OTHERauxiliary * *> \par Further Details: * ===================== *> *> \verbatim *> *> RT1 is accurate to a few ulps barring over/underflow. *> *> RT2 may be inaccurate if there is massive cancellation in the *> determinant A*C-B*B; higher precision or correctly rounded or *> correctly truncated arithmetic would be needed to compute RT2 *> accurately in all cases. *> *> CS1 and SN1 are accurate to a few ulps barring over/underflow. *> *> Overflow is possible only if RT1 is within a factor of 5 of overflow. *> Underflow is harmless if the input data is 0 or exceeds *> underflow_threshold / macheps. *> \endverbatim *> * ===================================================================== SUBROUTINE DLAEV2( A, B, C, RT1, RT2, CS1, SN1 ) * * -- 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, CS1, RT1, RT2, SN1 * .. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D0 ) DOUBLE PRECISION TWO PARAMETER ( TWO = 2.0D0 ) DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D0 ) DOUBLE PRECISION HALF PARAMETER ( HALF = 0.5D0 ) * .. * .. Local Scalars .. INTEGER SGN1, SGN2 DOUBLE PRECISION AB, ACMN, ACMX, ACS, ADF, CS, CT, DF, RT, SM, \$ TB, TN * .. * .. Intrinsic Functions .. INTRINSIC ABS, SQRT * .. * .. Executable Statements .. * * Compute the eigenvalues * SM = A + C DF = A - C ADF = ABS( DF ) TB = B + B AB = ABS( TB ) IF( ABS( A ).GT.ABS( C ) ) THEN ACMX = A ACMN = C ELSE ACMX = C ACMN = A END IF IF( ADF.GT.AB ) THEN RT = ADF*SQRT( ONE+( AB / ADF )**2 ) ELSE IF( ADF.LT.AB ) THEN RT = AB*SQRT( ONE+( ADF / AB )**2 ) ELSE * * Includes case AB=ADF=0 * RT = AB*SQRT( TWO ) END IF IF( SM.LT.ZERO ) THEN RT1 = HALF*( SM-RT ) SGN1 = -1 * * Order of execution important. * To get fully accurate smaller eigenvalue, * next line needs to be executed in higher precision. * RT2 = ( ACMX / RT1 )*ACMN - ( B / RT1 )*B ELSE IF( SM.GT.ZERO ) THEN RT1 = HALF*( SM+RT ) SGN1 = 1 * * Order of execution important. * To get fully accurate smaller eigenvalue, * next line needs to be executed in higher precision. * RT2 = ( ACMX / RT1 )*ACMN - ( B / RT1 )*B ELSE * * Includes case RT1 = RT2 = 0 * RT1 = HALF*RT RT2 = -HALF*RT SGN1 = 1 END IF * * Compute the eigenvector * IF( DF.GE.ZERO ) THEN CS = DF + RT SGN2 = 1 ELSE CS = DF - RT SGN2 = -1 END IF ACS = ABS( CS ) IF( ACS.GT.AB ) THEN CT = -TB / CS SN1 = ONE / SQRT( ONE+CT*CT ) CS1 = CT*SN1 ELSE IF( AB.EQ.ZERO ) THEN CS1 = ONE SN1 = ZERO ELSE TN = -CS / TB CS1 = ONE / SQRT( ONE+TN*TN ) SN1 = TN*CS1 END IF END IF IF( SGN1.EQ.SGN2 ) THEN TN = CS1 CS1 = -SN1 SN1 = TN END IF RETURN * * End of DLAEV2 * END