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    1: *> \brief \b DLAEV2 computes the eigenvalues and eigenvectors of a 2-by-2 symmetric/Hermitian matrix.
    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: *> \ingroup OTHERauxiliary
   98: *
   99: *> \par Further Details:
  100: *  =====================
  101: *>
  102: *> \verbatim
  103: *>
  104: *>  RT1 is accurate to a few ulps barring over/underflow.
  105: *>
  106: *>  RT2 may be inaccurate if there is massive cancellation in the
  107: *>  determinant A*C-B*B; higher precision or correctly rounded or
  108: *>  correctly truncated arithmetic would be needed to compute RT2
  109: *>  accurately in all cases.
  110: *>
  111: *>  CS1 and SN1 are accurate to a few ulps barring over/underflow.
  112: *>
  113: *>  Overflow is possible only if RT1 is within a factor of 5 of overflow.
  114: *>  Underflow is harmless if the input data is 0 or exceeds
  115: *>     underflow_threshold / macheps.
  116: *> \endverbatim
  117: *>
  118: *  =====================================================================
  119:       SUBROUTINE DLAEV2( A, B, C, RT1, RT2, CS1, SN1 )
  120: *
  121: *  -- LAPACK auxiliary routine --
  122: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
  123: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
  124: *
  125: *     .. Scalar Arguments ..
  126:       DOUBLE PRECISION   A, B, C, CS1, RT1, RT2, SN1
  127: *     ..
  128: *
  129: * =====================================================================
  130: *
  131: *     .. Parameters ..
  132:       DOUBLE PRECISION   ONE
  133:       PARAMETER          ( ONE = 1.0D0 )
  134:       DOUBLE PRECISION   TWO
  135:       PARAMETER          ( TWO = 2.0D0 )
  136:       DOUBLE PRECISION   ZERO
  137:       PARAMETER          ( ZERO = 0.0D0 )
  138:       DOUBLE PRECISION   HALF
  139:       PARAMETER          ( HALF = 0.5D0 )
  140: *     ..
  141: *     .. Local Scalars ..
  142:       INTEGER            SGN1, SGN2
  143:       DOUBLE PRECISION   AB, ACMN, ACMX, ACS, ADF, CS, CT, DF, RT, SM,
  144:      $                   TB, TN
  145: *     ..
  146: *     .. Intrinsic Functions ..
  147:       INTRINSIC          ABS, SQRT
  148: *     ..
  149: *     .. Executable Statements ..
  150: *
  151: *     Compute the eigenvalues
  152: *
  153:       SM = A + C
  154:       DF = A - C
  155:       ADF = ABS( DF )
  156:       TB = B + B
  157:       AB = ABS( TB )
  158:       IF( ABS( A ).GT.ABS( C ) ) THEN
  159:          ACMX = A
  160:          ACMN = C
  161:       ELSE
  162:          ACMX = C
  163:          ACMN = A
  164:       END IF
  165:       IF( ADF.GT.AB ) THEN
  166:          RT = ADF*SQRT( ONE+( AB / ADF )**2 )
  167:       ELSE IF( ADF.LT.AB ) THEN
  168:          RT = AB*SQRT( ONE+( ADF / AB )**2 )
  169:       ELSE
  170: *
  171: *        Includes case AB=ADF=0
  172: *
  173:          RT = AB*SQRT( TWO )
  174:       END IF
  175:       IF( SM.LT.ZERO ) THEN
  176:          RT1 = HALF*( SM-RT )
  177:          SGN1 = -1
  178: *
  179: *        Order of execution important.
  180: *        To get fully accurate smaller eigenvalue,
  181: *        next line needs to be executed in higher precision.
  182: *
  183:          RT2 = ( ACMX / RT1 )*ACMN - ( B / RT1 )*B
  184:       ELSE IF( SM.GT.ZERO ) THEN
  185:          RT1 = HALF*( SM+RT )
  186:          SGN1 = 1
  187: *
  188: *        Order of execution important.
  189: *        To get fully accurate smaller eigenvalue,
  190: *        next line needs to be executed in higher precision.
  191: *
  192:          RT2 = ( ACMX / RT1 )*ACMN - ( B / RT1 )*B
  193:       ELSE
  194: *
  195: *        Includes case RT1 = RT2 = 0
  196: *
  197:          RT1 = HALF*RT
  198:          RT2 = -HALF*RT
  199:          SGN1 = 1
  200:       END IF
  201: *
  202: *     Compute the eigenvector
  203: *
  204:       IF( DF.GE.ZERO ) THEN
  205:          CS = DF + RT
  206:          SGN2 = 1
  207:       ELSE
  208:          CS = DF - RT
  209:          SGN2 = -1
  210:       END IF
  211:       ACS = ABS( CS )
  212:       IF( ACS.GT.AB ) THEN
  213:          CT = -TB / CS
  214:          SN1 = ONE / SQRT( ONE+CT*CT )
  215:          CS1 = CT*SN1
  216:       ELSE
  217:          IF( AB.EQ.ZERO ) THEN
  218:             CS1 = ONE
  219:             SN1 = ZERO
  220:          ELSE
  221:             TN = -CS / TB
  222:             CS1 = ONE / SQRT( ONE+TN*TN )
  223:             SN1 = TN*CS1
  224:          END IF
  225:       END IF
  226:       IF( SGN1.EQ.SGN2 ) THEN
  227:          TN = CS1
  228:          CS1 = -SN1
  229:          SN1 = TN
  230:       END IF
  231:       RETURN
  232: *
  233: *     End of DLAEV2
  234: *
  235:       END