dget52.f

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*> \brief \b DGET52
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at 
*            http://www.netlib.org/lapack/explore-html/ 
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGET52( LEFT, N, A, LDA, B, LDB, E, LDE, ALPHAR,
*                          ALPHAI, BETA, WORK, RESULT )
* 
*       .. Scalar Arguments ..
*       LOGICAL            LEFT
*       INTEGER            LDA, LDB, LDE, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
*      $                   B( LDB, * ), BETA( * ), E( LDE, * ),
*      $                   RESULT( 2 ), WORK( * )
*       ..
*  
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGET52  does an eigenvector check for the generalized eigenvalue
*> problem.
*>
*> The basic test for right eigenvectors is:
*>
*>                           | b(j) A E(j) -  a(j) B E(j) |
*>         RESULT(1) = max   -------------------------------
*>                      j    n ulp max( |b(j) A|, |a(j) B| )
*>
*> using the 1-norm.  Here, a(j)/b(j) = w is the j-th generalized
*> eigenvalue of A - w B, or, equivalently, b(j)/a(j) = m is the j-th
*> generalized eigenvalue of m A - B.
*>
*> For real eigenvalues, the test is straightforward.  For complex
*> eigenvalues, E(j) and a(j) are complex, represented by
*> Er(j) + i*Ei(j) and ar(j) + i*ai(j), resp., so the test for that
*> eigenvector becomes
*>
*>                 max( |Wr|, |Wi| )
*>     --------------------------------------------
*>     n ulp max( |b(j) A|, (|ar(j)|+|ai(j)|) |B| )
*>
*> where
*>
*>     Wr = b(j) A Er(j) - ar(j) B Er(j) + ai(j) B Ei(j)
*>
*>     Wi = b(j) A Ei(j) - ai(j) B Er(j) - ar(j) B Ei(j)
*>
*>                         T   T  _
*> For left eigenvectors, A , B , a, and b  are used.
*>
*> DGET52 also tests the normalization of E.  Each eigenvector is
*> supposed to be normalized so that the maximum "absolute value"
*> of its elements is 1, where in this case, "absolute value"
*> of a complex value x is  |Re(x)| + |Im(x)| ; let us call this
*> maximum "absolute value" norm of a vector v  M(v).
*> if a(j)=b(j)=0, then the eigenvector is set to be the jth coordinate
*> vector.  The normalization test is:
*>
*>         RESULT(2) =      max       | M(v(j)) - 1 | / ( n ulp )
*>                    eigenvectors v(j)
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] LEFT
*> \verbatim
*>          LEFT is LOGICAL
*>          =.TRUE.:  The eigenvectors in the columns of E are assumed
*>                    to be *left* eigenvectors.
*>          =.FALSE.: The eigenvectors in the columns of E are assumed
*>                    to be *right* eigenvectors.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The size of the matrices.  If it is zero, DGET52 does
*>          nothing.  It must be at least zero.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA, N)
*>          The matrix A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of A.  It must be at least 1
*>          and at least N.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB, N)
*>          The matrix B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of B.  It must be at least 1
*>          and at least N.
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*>          E is DOUBLE PRECISION array, dimension (LDE, N)
*>          The matrix of eigenvectors.  It must be O( 1 ).  Complex
*>          eigenvalues and eigenvectors always come in pairs, the
*>          eigenvalue and its conjugate being stored in adjacent
*>          elements of ALPHAR, ALPHAI, and BETA.  Thus, if a(j)/b(j)
*>          and a(j+1)/b(j+1) are a complex conjugate pair of
*>          generalized eigenvalues, then E(,j) contains the real part
*>          of the eigenvector and E(,j+1) contains the imaginary part.
*>          Note that whether E(,j) is a real eigenvector or part of a
*>          complex one is specified by whether ALPHAI(j) is zero or not.
*> \endverbatim
*>
*> \param[in] LDE
*> \verbatim
*>          LDE is INTEGER
*>          The leading dimension of E.  It must be at least 1 and at
*>          least N.
*> \endverbatim
*>
*> \param[in] ALPHAR
*> \verbatim
*>          ALPHAR is DOUBLE PRECISION array, dimension (N)
*>          The real parts of the values a(j) as described above, which,
*>          along with b(j), define the generalized eigenvalues.
*>          Complex eigenvalues always come in complex conjugate pairs
*>          a(j)/b(j) and a(j+1)/b(j+1), which are stored in adjacent
*>          elements in ALPHAR, ALPHAI, and BETA.  Thus, if the j-th
*>          and (j+1)-st eigenvalues form a pair, ALPHAR(j+1)/BETA(j+1)
*>          is assumed to be equal to ALPHAR(j)/BETA(j).
*> \endverbatim
*>
*> \param[in] ALPHAI
*> \verbatim
*>          ALPHAI is DOUBLE PRECISION array, dimension (N)
*>          The imaginary parts of the values a(j) as described above,
*>          which, along with b(j), define the generalized eigenvalues.
*>          If ALPHAI(j)=0, then the eigenvalue is real, otherwise it
*>          is part of a complex conjugate pair.  Complex eigenvalues
*>          always come in complex conjugate pairs a(j)/b(j) and
*>          a(j+1)/b(j+1), which are stored in adjacent elements in
*>          ALPHAR, ALPHAI, and BETA.  Thus, if the j-th and (j+1)-st
*>          eigenvalues form a pair, ALPHAI(j+1)/BETA(j+1) is assumed to
*>          be equal to  -ALPHAI(j)/BETA(j).  Also, nonzero values in
*>          ALPHAI are assumed to always come in adjacent pairs.
*> \endverbatim
*>
*> \param[in] BETA
*> \verbatim
*>          BETA is DOUBLE PRECISION array, dimension (N)
*>          The values b(j) as described above, which, along with a(j),
*>          define the generalized eigenvalues.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (N**2+N)
*> \endverbatim
*>
*> \param[out] RESULT
*> \verbatim
*>          RESULT is DOUBLE PRECISION array, dimension (2)
*>          The values computed by the test described above.  If A E or
*>          B E is likely to overflow, then RESULT(1:2) is set to
*>          10 / ulp.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee 
*> \author Univ. of California Berkeley 
*> \author Univ. of Colorado Denver 
*> \author NAG Ltd. 
*
*> \date November 2011
*
*> \ingroup double_eig
*
*  =====================================================================
      SUBROUTINE dget52( LEFT, N, A, LDA, B, LDB, E, LDE, ALPHAR,
     $                   alphai, beta, work, result )
*
*  -- LAPACK test routine (version 3.4.0) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     November 2011
*
*     .. Scalar Arguments ..
      LOGICAL            LEFT
      INTEGER            LDA, LDB, LDE, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( lda, * ), ALPHAI( * ), ALPHAR( * ),
     $                   b( ldb, * ), beta( * ), e( lde, * ),
     $                   result( 2 ), work( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE, TEN
      parameter                ( zero = 0.0d0, one = 1.0d0, ten = 10.0d0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            ILCPLX
      CHARACTER          NORMAB, TRANS
      INTEGER            J, JVEC
      DOUBLE PRECISION   ABMAX, ACOEF, ALFMAX, ANORM, BCOEFI, BCOEFR,
     $                   betmax, bnorm, enorm, enrmer, errnrm, safmax,
     $                   safmin, salfi, salfr, sbeta, scale, temp1, ulp
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH, DLANGE
      EXTERNAL           dlamch, dlange
*     ..
*     .. External Subroutines ..
      EXTERNAL           dgemv
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, dble, max
*     ..
*     .. Executable Statements ..
*
      result( 1 ) = zero
      result( 2 ) = zero
      IF( n.LE.0 )
     $   RETURN
*
      safmin = dlamch( 'Safe minimum' )
      safmax = one / safmin
      ulp = dlamch( 'Epsilon' )*dlamch( 'Base' )
*
      IF( left ) THEN
         trans = 'T'
         normab = 'I'
      ELSE
         trans = 'N'
         normab = 'O'
      END IF
*
*     Norm of A, B, and E:
*
      anorm = max( dlange( normab, n, n, a, lda, work ), safmin )
      bnorm = max( dlange( normab, n, n, b, ldb, work ), safmin )
      enorm = max( dlange( 'O', n, n, e, lde, work ), ulp )
      alfmax = safmax / max( one, bnorm )
      betmax = safmax / max( one, anorm )
*
*     Compute error matrix.
*     Column i = ( b(i) A - a(i) B ) E(i) / max( |a(i) B| |b(i) A| )
*
      ilcplx = .false.
      DO 10 jvec = 1, n
         IF( ilcplx ) THEN
*
*           2nd Eigenvalue/-vector of pair -- do nothing
*
            ilcplx = .false.
         ELSE
            salfr = alphar( jvec )
            salfi = alphai( jvec )
            sbeta = beta( jvec )
            IF( salfi.EQ.zero ) THEN
*
*              Real eigenvalue and -vector
*
               abmax = max( abs( salfr ), abs( sbeta ) )
               IF( abs( salfr ).GT.alfmax .OR. abs( sbeta ).GT.
     $             betmax .OR. abmax.LT.one ) THEN
                  scale = one / max( abmax, safmin )
                  salfr = scale*salfr
                  sbeta = scale*sbeta
               END IF
               scale = one / max( abs( salfr )*bnorm,
     $                 abs( sbeta )*anorm, safmin )
               acoef = scale*sbeta
               bcoefr = scale*salfr
               CALL dgemv( trans, n, n, acoef, a, lda, e( 1, jvec ), 1,
     $                     zero, work( n*( jvec-1 )+1 ), 1 )
               CALL dgemv( trans, n, n, -bcoefr, b, lda, e( 1, jvec ),
     $                     1, one, work( n*( jvec-1 )+1 ), 1 )
            ELSE
*
*              Complex conjugate pair
*
               ilcplx = .true.
               IF( jvec.EQ.n ) THEN
                  result( 1 ) = ten / ulp
                  RETURN
               END IF
               abmax = max( abs( salfr )+abs( salfi ), abs( sbeta ) )
               IF( abs( salfr )+abs( salfi ).GT.alfmax .OR.
     $             abs( sbeta ).GT.betmax .OR. abmax.LT.one ) THEN
                  scale = one / max( abmax, safmin )
                  salfr = scale*salfr
                  salfi = scale*salfi
                  sbeta = scale*sbeta
               END IF
               scale = one / max( ( abs( salfr )+abs( salfi ) )*bnorm,
     $                 abs( sbeta )*anorm, safmin )
               acoef = scale*sbeta
               bcoefr = scale*salfr
               bcoefi = scale*salfi
               IF( left ) THEN
                  bcoefi = -bcoefi
               END IF
*
               CALL dgemv( trans, n, n, acoef, a, lda, e( 1, jvec ), 1,
     $                     zero, work( n*( jvec-1 )+1 ), 1 )
               CALL dgemv( trans, n, n, -bcoefr, b, lda, e( 1, jvec ),
     $                     1, one, work( n*( jvec-1 )+1 ), 1 )
               CALL dgemv( trans, n, n, bcoefi, b, lda, e( 1, jvec+1 ),
     $                     1, one, work( n*( jvec-1 )+1 ), 1 )
*
               CALL dgemv( trans, n, n, acoef, a, lda, e( 1, jvec+1 ),
     $                     1, zero, work( n*jvec+1 ), 1 )
               CALL dgemv( trans, n, n, -bcoefi, b, lda, e( 1, jvec ),
     $                     1, one, work( n*jvec+1 ), 1 )
               CALL dgemv( trans, n, n, -bcoefr, b, lda, e( 1, jvec+1 ),
     $                     1, one, work( n*jvec+1 ), 1 )
            END IF
         END IF
   10 CONTINUE
*
      errnrm = dlange( 'One', n, n, work, n, work( n**2+1 ) ) / enorm
*
*     Compute RESULT(1)
*
      result( 1 ) = errnrm / ulp
*
*     Normalization of E:
*
      enrmer = zero
      ilcplx = .false.
      DO 40 jvec = 1, n
         IF( ilcplx ) THEN
            ilcplx = .false.
         ELSE
            temp1 = zero
            IF( alphai( jvec ).EQ.zero ) THEN
               DO 20 j = 1, n
                  temp1 = max( temp1, abs( e( j, jvec ) ) )
   20          CONTINUE
               enrmer = max( enrmer, temp1-one )
            ELSE
               ilcplx = .true.
               DO 30 j = 1, n
                  temp1 = max( temp1, abs( e( j, jvec ) )+
     $                    abs( e( j, jvec+1 ) ) )
   30          CONTINUE
               enrmer = max( enrmer, temp1-one )
            END IF
         END IF
   40 CONTINUE
*
*     Compute RESULT(2) : the normalization error in E.
*
      result( 2 ) = enrmer / ( dble( n )*ulp )
*
      RETURN
*
*     End of DGET52
*
      END