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    1: *> \brief \b DLAEV2
    2: *
    3: *  =========== DOCUMENTATION ===========
    4: *
    5: * Online html documentation available at 
    6: *            http://www.netlib.org/lapack/explore-html/ 
    7: *
    8: *> \htmlonly
    9: *> Download DLAEV2 + dependencies 
   10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaev2.f"> 
   11: *> [TGZ]</a> 
   12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaev2.f"> 
   13: *> [ZIP]</a> 
   14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaev2.f"> 
   15: *> [TXT]</a>
   16: *> \endhtmlonly 
   17: *
   18: *  Definition:
   19: *  ===========
   20: *
   21: *       SUBROUTINE DLAEV2( A, B, C, RT1, RT2, CS1, SN1 )
   22: * 
   23: *       .. Scalar Arguments ..
   24: *       DOUBLE PRECISION   A, B, C, CS1, RT1, RT2, SN1
   25: *       ..
   26: *  
   27: *
   28: *> \par Purpose:
   29: *  =============
   30: *>
   31: *> \verbatim
   32: *>
   33: *> DLAEV2 computes the eigendecomposition of a 2-by-2 symmetric matrix
   34: *>    [  A   B  ]
   35: *>    [  B   C  ].
   36: *> On return, RT1 is the eigenvalue of larger absolute value, RT2 is the
   37: *> eigenvalue of smaller absolute value, and (CS1,SN1) is the unit right
   38: *> eigenvector for RT1, giving the decomposition
   39: *>
   40: *>    [ CS1  SN1 ] [  A   B  ] [ CS1 -SN1 ]  =  [ RT1  0  ]
   41: *>    [-SN1  CS1 ] [  B   C  ] [ SN1  CS1 ]     [  0  RT2 ].
   42: *> \endverbatim
   43: *
   44: *  Arguments:
   45: *  ==========
   46: *
   47: *> \param[in] A
   48: *> \verbatim
   49: *>          A is DOUBLE PRECISION
   50: *>          The (1,1) element of the 2-by-2 matrix.
   51: *> \endverbatim
   52: *>
   53: *> \param[in] B
   54: *> \verbatim
   55: *>          B is DOUBLE PRECISION
   56: *>          The (1,2) element and the conjugate of the (2,1) element of
   57: *>          the 2-by-2 matrix.
   58: *> \endverbatim
   59: *>
   60: *> \param[in] C
   61: *> \verbatim
   62: *>          C is DOUBLE PRECISION
   63: *>          The (2,2) element of the 2-by-2 matrix.
   64: *> \endverbatim
   65: *>
   66: *> \param[out] RT1
   67: *> \verbatim
   68: *>          RT1 is DOUBLE PRECISION
   69: *>          The eigenvalue of larger absolute value.
   70: *> \endverbatim
   71: *>
   72: *> \param[out] RT2
   73: *> \verbatim
   74: *>          RT2 is DOUBLE PRECISION
   75: *>          The eigenvalue of smaller absolute value.
   76: *> \endverbatim
   77: *>
   78: *> \param[out] CS1
   79: *> \verbatim
   80: *>          CS1 is DOUBLE PRECISION
   81: *> \endverbatim
   82: *>
   83: *> \param[out] SN1
   84: *> \verbatim
   85: *>          SN1 is DOUBLE PRECISION
   86: *>          The vector (CS1, SN1) is a unit right eigenvector for RT1.
   87: *> \endverbatim
   88: *
   89: *  Authors:
   90: *  ========
   91: *
   92: *> \author Univ. of Tennessee 
   93: *> \author Univ. of California Berkeley 
   94: *> \author Univ. of Colorado Denver 
   95: *> \author NAG Ltd. 
   96: *
   97: *> \date November 2011
   98: *
   99: *> \ingroup auxOTHERauxiliary
  100: *
  101: *> \par Further Details:
  102: *  =====================
  103: *>
  104: *> \verbatim
  105: *>
  106: *>  RT1 is accurate to a few ulps barring over/underflow.
  107: *>
  108: *>  RT2 may be inaccurate if there is massive cancellation in the
  109: *>  determinant A*C-B*B; higher precision or correctly rounded or
  110: *>  correctly truncated arithmetic would be needed to compute RT2
  111: *>  accurately in all cases.
  112: *>
  113: *>  CS1 and SN1 are accurate to a few ulps barring over/underflow.
  114: *>
  115: *>  Overflow is possible only if RT1 is within a factor of 5 of overflow.
  116: *>  Underflow is harmless if the input data is 0 or exceeds
  117: *>     underflow_threshold / macheps.
  118: *> \endverbatim
  119: *>
  120: *  =====================================================================
  121:       SUBROUTINE DLAEV2( A, B, C, RT1, RT2, CS1, SN1 )
  122: *
  123: *  -- LAPACK auxiliary routine (version 3.4.0) --
  124: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
  125: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
  126: *     November 2011
  127: *
  128: *     .. Scalar Arguments ..
  129:       DOUBLE PRECISION   A, B, C, CS1, RT1, RT2, SN1
  130: *     ..
  131: *
  132: * =====================================================================
  133: *
  134: *     .. Parameters ..
  135:       DOUBLE PRECISION   ONE
  136:       PARAMETER          ( ONE = 1.0D0 )
  137:       DOUBLE PRECISION   TWO
  138:       PARAMETER          ( TWO = 2.0D0 )
  139:       DOUBLE PRECISION   ZERO
  140:       PARAMETER          ( ZERO = 0.0D0 )
  141:       DOUBLE PRECISION   HALF
  142:       PARAMETER          ( HALF = 0.5D0 )
  143: *     ..
  144: *     .. Local Scalars ..
  145:       INTEGER            SGN1, SGN2
  146:       DOUBLE PRECISION   AB, ACMN, ACMX, ACS, ADF, CS, CT, DF, RT, SM,
  147:      $                   TB, TN
  148: *     ..
  149: *     .. Intrinsic Functions ..
  150:       INTRINSIC          ABS, SQRT
  151: *     ..
  152: *     .. Executable Statements ..
  153: *
  154: *     Compute the eigenvalues
  155: *
  156:       SM = A + C
  157:       DF = A - C
  158:       ADF = ABS( DF )
  159:       TB = B + B
  160:       AB = ABS( TB )
  161:       IF( ABS( A ).GT.ABS( C ) ) THEN
  162:          ACMX = A
  163:          ACMN = C
  164:       ELSE
  165:          ACMX = C
  166:          ACMN = A
  167:       END IF
  168:       IF( ADF.GT.AB ) THEN
  169:          RT = ADF*SQRT( ONE+( AB / ADF )**2 )
  170:       ELSE IF( ADF.LT.AB ) THEN
  171:          RT = AB*SQRT( ONE+( ADF / AB )**2 )
  172:       ELSE
  173: *
  174: *        Includes case AB=ADF=0
  175: *
  176:          RT = AB*SQRT( TWO )
  177:       END IF
  178:       IF( SM.LT.ZERO ) THEN
  179:          RT1 = HALF*( SM-RT )
  180:          SGN1 = -1
  181: *
  182: *        Order of execution important.
  183: *        To get fully accurate smaller eigenvalue,
  184: *        next line needs to be executed in higher precision.
  185: *
  186:          RT2 = ( ACMX / RT1 )*ACMN - ( B / RT1 )*B
  187:       ELSE IF( SM.GT.ZERO ) THEN
  188:          RT1 = HALF*( SM+RT )
  189:          SGN1 = 1
  190: *
  191: *        Order of execution important.
  192: *        To get fully accurate smaller eigenvalue,
  193: *        next line needs to be executed in higher precision.
  194: *
  195:          RT2 = ( ACMX / RT1 )*ACMN - ( B / RT1 )*B
  196:       ELSE
  197: *
  198: *        Includes case RT1 = RT2 = 0
  199: *
  200:          RT1 = HALF*RT
  201:          RT2 = -HALF*RT
  202:          SGN1 = 1
  203:       END IF
  204: *
  205: *     Compute the eigenvector
  206: *
  207:       IF( DF.GE.ZERO ) THEN
  208:          CS = DF + RT
  209:          SGN2 = 1
  210:       ELSE
  211:          CS = DF - RT
  212:          SGN2 = -1
  213:       END IF
  214:       ACS = ABS( CS )
  215:       IF( ACS.GT.AB ) THEN
  216:          CT = -TB / CS
  217:          SN1 = ONE / SQRT( ONE+CT*CT )
  218:          CS1 = CT*SN1
  219:       ELSE
  220:          IF( AB.EQ.ZERO ) THEN
  221:             CS1 = ONE
  222:             SN1 = ZERO
  223:          ELSE
  224:             TN = -CS / TB
  225:             CS1 = ONE / SQRT( ONE+TN*TN )
  226:             SN1 = TN*CS1
  227:          END IF
  228:       END IF
  229:       IF( SGN1.EQ.SGN2 ) THEN
  230:          TN = CS1
  231:          CS1 = -SN1
  232:          SN1 = TN
  233:       END IF
  234:       RETURN
  235: *
  236: *     End of DLAEV2
  237: *
  238:       END