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version 1.7, 2010/12/21 13:53:29 version 1.8, 2011/11/21 20:42:55
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   *> \brief \b DLAG2
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
   *
   *> \htmlonly
   *> Download DLAG2 + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlag2.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlag2.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlag2.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DLAG2( A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1,
   *                         WR2, WI )
   * 
   *       .. Scalar Arguments ..
   *       INTEGER            LDA, LDB
   *       DOUBLE PRECISION   SAFMIN, SCALE1, SCALE2, WI, WR1, WR2
   *       ..
   *       .. Array Arguments ..
   *       DOUBLE PRECISION   A( LDA, * ), B( LDB, * )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> DLAG2 computes the eigenvalues of a 2 x 2 generalized eigenvalue
   *> problem  A - w B, with scaling as necessary to avoid over-/underflow.
   *>
   *> The scaling factor "s" results in a modified eigenvalue equation
   *>
   *>     s A - w B
   *>
   *> where  s  is a non-negative scaling factor chosen so that  w,  w B,
   *> and  s A  do not overflow and, if possible, do not underflow, either.
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] A
   *> \verbatim
   *>          A is DOUBLE PRECISION array, dimension (LDA, 2)
   *>          On entry, the 2 x 2 matrix A.  It is assumed that its 1-norm
   *>          is less than 1/SAFMIN.  Entries less than
   *>          sqrt(SAFMIN)*norm(A) are subject to being treated as zero.
   *> \endverbatim
   *>
   *> \param[in] LDA
   *> \verbatim
   *>          LDA is INTEGER
   *>          The leading dimension of the array A.  LDA >= 2.
   *> \endverbatim
   *>
   *> \param[in] B
   *> \verbatim
   *>          B is DOUBLE PRECISION array, dimension (LDB, 2)
   *>          On entry, the 2 x 2 upper triangular matrix B.  It is
   *>          assumed that the one-norm of B is less than 1/SAFMIN.  The
   *>          diagonals should be at least sqrt(SAFMIN) times the largest
   *>          element of B (in absolute value); if a diagonal is smaller
   *>          than that, then  +/- sqrt(SAFMIN) will be used instead of
   *>          that diagonal.
   *> \endverbatim
   *>
   *> \param[in] LDB
   *> \verbatim
   *>          LDB is INTEGER
   *>          The leading dimension of the array B.  LDB >= 2.
   *> \endverbatim
   *>
   *> \param[in] SAFMIN
   *> \verbatim
   *>          SAFMIN is DOUBLE PRECISION
   *>          The smallest positive number s.t. 1/SAFMIN does not
   *>          overflow.  (This should always be DLAMCH('S') -- it is an
   *>          argument in order to avoid having to call DLAMCH frequently.)
   *> \endverbatim
   *>
   *> \param[out] SCALE1
   *> \verbatim
   *>          SCALE1 is DOUBLE PRECISION
   *>          A scaling factor used to avoid over-/underflow in the
   *>          eigenvalue equation which defines the first eigenvalue.  If
   *>          the eigenvalues are complex, then the eigenvalues are
   *>          ( WR1  +/-  WI i ) / SCALE1  (which may lie outside the
   *>          exponent range of the machine), SCALE1=SCALE2, and SCALE1
   *>          will always be positive.  If the eigenvalues are real, then
   *>          the first (real) eigenvalue is  WR1 / SCALE1 , but this may
   *>          overflow or underflow, and in fact, SCALE1 may be zero or
   *>          less than the underflow threshhold if the exact eigenvalue
   *>          is sufficiently large.
   *> \endverbatim
   *>
   *> \param[out] SCALE2
   *> \verbatim
   *>          SCALE2 is DOUBLE PRECISION
   *>          A scaling factor used to avoid over-/underflow in the
   *>          eigenvalue equation which defines the second eigenvalue.  If
   *>          the eigenvalues are complex, then SCALE2=SCALE1.  If the
   *>          eigenvalues are real, then the second (real) eigenvalue is
   *>          WR2 / SCALE2 , but this may overflow or underflow, and in
   *>          fact, SCALE2 may be zero or less than the underflow
   *>          threshhold if the exact eigenvalue is sufficiently large.
   *> \endverbatim
   *>
   *> \param[out] WR1
   *> \verbatim
   *>          WR1 is DOUBLE PRECISION
   *>          If the eigenvalue is real, then WR1 is SCALE1 times the
   *>          eigenvalue closest to the (2,2) element of A B**(-1).  If the
   *>          eigenvalue is complex, then WR1=WR2 is SCALE1 times the real
   *>          part of the eigenvalues.
   *> \endverbatim
   *>
   *> \param[out] WR2
   *> \verbatim
   *>          WR2 is DOUBLE PRECISION
   *>          If the eigenvalue is real, then WR2 is SCALE2 times the
   *>          other eigenvalue.  If the eigenvalue is complex, then
   *>          WR1=WR2 is SCALE1 times the real part of the eigenvalues.
   *> \endverbatim
   *>
   *> \param[out] WI
   *> \verbatim
   *>          WI is DOUBLE PRECISION
   *>          If the eigenvalue is real, then WI is zero.  If the
   *>          eigenvalue is complex, then WI is SCALE1 times the imaginary
   *>          part of the eigenvalues.  WI will always be non-negative.
   *> \endverbatim
   *
   *  Authors:
   *  ========
   *
   *> \author Univ. of Tennessee 
   *> \author Univ. of California Berkeley 
   *> \author Univ. of Colorado Denver 
   *> \author NAG Ltd. 
   *
   *> \date November 2011
   *
   *> \ingroup doubleOTHERauxiliary
   *
   *  =====================================================================
       SUBROUTINE DLAG2( A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1,        SUBROUTINE DLAG2( A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1,
      $                  WR2, WI )       $                  WR2, WI )
 *  *
 *  -- LAPACK auxiliary routine (version 3.2) --  *  -- LAPACK auxiliary routine (version 3.4.0) --
 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --  *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--  *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
 *     November 2006  *     November 2011
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       INTEGER            LDA, LDB        INTEGER            LDA, LDB
Line 14 Line 169
       DOUBLE PRECISION   A( LDA, * ), B( LDB, * )        DOUBLE PRECISION   A( LDA, * ), B( LDB, * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  DLAG2 computes the eigenvalues of a 2 x 2 generalized eigenvalue  
 *  problem  A - w B, with scaling as necessary to avoid over-/underflow.  
 *  
 *  The scaling factor "s" results in a modified eigenvalue equation  
 *  
 *      s A - w B  
 *  
 *  where  s  is a non-negative scaling factor chosen so that  w,  w B,  
 *  and  s A  do not overflow and, if possible, do not underflow, either.  
 *  
 *  Arguments  
 *  =========  
 *  
 *  A       (input) DOUBLE PRECISION array, dimension (LDA, 2)  
 *          On entry, the 2 x 2 matrix A.  It is assumed that its 1-norm  
 *          is less than 1/SAFMIN.  Entries less than  
 *          sqrt(SAFMIN)*norm(A) are subject to being treated as zero.  
 *  
 *  LDA     (input) INTEGER  
 *          The leading dimension of the array A.  LDA >= 2.  
 *  
 *  B       (input) DOUBLE PRECISION array, dimension (LDB, 2)  
 *          On entry, the 2 x 2 upper triangular matrix B.  It is  
 *          assumed that the one-norm of B is less than 1/SAFMIN.  The  
 *          diagonals should be at least sqrt(SAFMIN) times the largest  
 *          element of B (in absolute value); if a diagonal is smaller  
 *          than that, then  +/- sqrt(SAFMIN) will be used instead of  
 *          that diagonal.  
 *  
 *  LDB     (input) INTEGER  
 *          The leading dimension of the array B.  LDB >= 2.  
 *  
 *  SAFMIN  (input) DOUBLE PRECISION  
 *          The smallest positive number s.t. 1/SAFMIN does not  
 *          overflow.  (This should always be DLAMCH('S') -- it is an  
 *          argument in order to avoid having to call DLAMCH frequently.)  
 *  
 *  SCALE1  (output) DOUBLE PRECISION  
 *          A scaling factor used to avoid over-/underflow in the  
 *          eigenvalue equation which defines the first eigenvalue.  If  
 *          the eigenvalues are complex, then the eigenvalues are  
 *          ( WR1  +/-  WI i ) / SCALE1  (which may lie outside the  
 *          exponent range of the machine), SCALE1=SCALE2, and SCALE1  
 *          will always be positive.  If the eigenvalues are real, then  
 *          the first (real) eigenvalue is  WR1 / SCALE1 , but this may  
 *          overflow or underflow, and in fact, SCALE1 may be zero or  
 *          less than the underflow threshhold if the exact eigenvalue  
 *          is sufficiently large.  
 *  
 *  SCALE2  (output) DOUBLE PRECISION  
 *          A scaling factor used to avoid over-/underflow in the  
 *          eigenvalue equation which defines the second eigenvalue.  If  
 *          the eigenvalues are complex, then SCALE2=SCALE1.  If the  
 *          eigenvalues are real, then the second (real) eigenvalue is  
 *          WR2 / SCALE2 , but this may overflow or underflow, and in  
 *          fact, SCALE2 may be zero or less than the underflow  
 *          threshhold if the exact eigenvalue is sufficiently large.  
 *  
 *  WR1     (output) DOUBLE PRECISION  
 *          If the eigenvalue is real, then WR1 is SCALE1 times the  
 *          eigenvalue closest to the (2,2) element of A B**(-1).  If the  
 *          eigenvalue is complex, then WR1=WR2 is SCALE1 times the real  
 *          part of the eigenvalues.  
 *  
 *  WR2     (output) DOUBLE PRECISION  
 *          If the eigenvalue is real, then WR2 is SCALE2 times the  
 *          other eigenvalue.  If the eigenvalue is complex, then  
 *          WR1=WR2 is SCALE1 times the real part of the eigenvalues.  
 *  
 *  WI      (output) DOUBLE PRECISION  
 *          If the eigenvalue is real, then WI is zero.  If the  
 *          eigenvalue is complex, then WI is SCALE1 times the imaginary  
 *          part of the eigenvalues.  WI will always be non-negative.  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
Removed from v.1.7  
changed lines
  Added in v.1.8

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