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*> \brief \b CGET52
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* SUBROUTINE CGET52( LEFT, N, A, LDA, B, LDB, E, LDE, ALPHA, BETA,
* WORK, RWORK, RESULT )
*
* .. Scalar Arguments ..
* LOGICAL LEFT
* INTEGER LDA, LDB, LDE, N
* ..
* .. Array Arguments ..
* REAL RESULT( 2 ), RWORK( * )
* COMPLEX A( LDA, * ), ALPHA( * ), B( LDB, * ),
* $ BETA( * ), E( LDE, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> CGET52 does an eigenvector check for the generalized eigenvalue
*> problem.
*>
*> The basic test for right eigenvectors is:
*>
*> | b(i) A E(i) - a(i) B E(i) |
*> RESULT(1) = max -------------------------------
*> i n ulp max( |b(i) A|, |a(i) B| )
*>
*> using the 1-norm. Here, a(i)/b(i) = w is the i-th generalized
*> eigenvalue of A - w B, or, equivalently, b(i)/a(i) = m is the i-th
*> generalized eigenvalue of m A - B.
*>
*> H H _ _
*> For left eigenvectors, A , B , a, and b are used.
*>
*> CGET52 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(i)=b(i)=0, then the eigenvector is set to be the jth coordinate
*> vector. The normalization test is:
*>
*> RESULT(2) = max | M(v(i)) - 1 | / ( n ulp )
*> eigenvectors v(i)
*> \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, CGET52 does
*> nothing. It must be at least zero.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is COMPLEX 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 COMPLEX 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 COMPLEX array, dimension (LDE, N)
*> The matrix of eigenvectors. It must be O( 1 ).
*> \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] ALPHA
*> \verbatim
*> ALPHA is COMPLEX array, dimension (N)
*> The values a(i) as described above, which, along with b(i),
*> define the generalized eigenvalues.
*> \endverbatim
*>
*> \param[in] BETA
*> \verbatim
*> BETA is COMPLEX array, dimension (N)
*> The values b(i) as described above, which, along with a(i),
*> define the generalized eigenvalues.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is COMPLEX array, dimension (N**2)
*> \endverbatim
*>
*> \param[out] RWORK
*> \verbatim
*> RWORK is REAL array, dimension (N)
*> \endverbatim
*>
*> \param[out] RESULT
*> \verbatim
*> RESULT is REAL 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 complex_eig
*
* =====================================================================
SUBROUTINE CGET52( LEFT, N, A, LDA, B, LDB, E, LDE, ALPHA, BETA,
$ WORK, RWORK, 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 ..
REAL RESULT( 2 ), RWORK( * )
COMPLEX A( LDA, * ), ALPHA( * ), B( LDB, * ),
$ BETA( * ), E( LDE, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
COMPLEX CZERO, CONE
PARAMETER ( CZERO = ( 0.0E+0, 0.0E+0 ),
$ CONE = ( 1.0E+0, 0.0E+0 ) )
* ..
* .. Local Scalars ..
CHARACTER NORMAB, TRANS
INTEGER J, JVEC
REAL ABMAX, ALFMAX, ANORM, BETMAX, BNORM, ENORM,
$ ENRMER, ERRNRM, SAFMAX, SAFMIN, SCALE, TEMP1,
$ ULP
COMPLEX ACOEFF, ALPHAI, BCOEFF, BETAI, X
* ..
* .. External Functions ..
REAL CLANGE, SLAMCH
EXTERNAL CLANGE, SLAMCH
* ..
* .. External Subroutines ..
EXTERNAL CGEMV
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, AIMAG, CONJG, MAX, REAL
* ..
* .. Statement Functions ..
REAL ABS1
* ..
* .. Statement Function definitions ..
ABS1( X ) = ABS( REAL( X ) ) + ABS( AIMAG( X ) )
* ..
* .. Executable Statements ..
*
RESULT( 1 ) = ZERO
RESULT( 2 ) = ZERO
IF( N.LE.0 )
$ RETURN
*
SAFMIN = SLAMCH( 'Safe minimum' )
SAFMAX = ONE / SAFMIN
ULP = SLAMCH( 'Epsilon' )*SLAMCH( 'Base' )
*
IF( LEFT ) THEN
TRANS = 'C'
NORMAB = 'I'
ELSE
TRANS = 'N'
NORMAB = 'O'
END IF
*
* Norm of A, B, and E:
*
ANORM = MAX( CLANGE( NORMAB, N, N, A, LDA, RWORK ), SAFMIN )
BNORM = MAX( CLANGE( NORMAB, N, N, B, LDB, RWORK ), SAFMIN )
ENORM = MAX( CLANGE( 'O', N, N, E, LDE, RWORK ), 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| )
*
DO 10 JVEC = 1, N
ALPHAI = ALPHA( JVEC )
BETAI = BETA( JVEC )
ABMAX = MAX( ABS1( ALPHAI ), ABS1( BETAI ) )
IF( ABS1( ALPHAI ).GT.ALFMAX .OR. ABS1( BETAI ).GT.BETMAX .OR.
$ ABMAX.LT.ONE ) THEN
SCALE = ONE / MAX( ABMAX, SAFMIN )
ALPHAI = SCALE*ALPHAI
BETAI = SCALE*BETAI
END IF
SCALE = ONE / MAX( ABS1( ALPHAI )*BNORM, ABS1( BETAI )*ANORM,
$ SAFMIN )
ACOEFF = SCALE*BETAI
BCOEFF = SCALE*ALPHAI
IF( LEFT ) THEN
ACOEFF = CONJG( ACOEFF )
BCOEFF = CONJG( BCOEFF )
END IF
CALL CGEMV( TRANS, N, N, ACOEFF, A, LDA, E( 1, JVEC ), 1,
$ CZERO, WORK( N*( JVEC-1 )+1 ), 1 )
CALL CGEMV( TRANS, N, N, -BCOEFF, B, LDA, E( 1, JVEC ), 1,
$ CONE, WORK( N*( JVEC-1 )+1 ), 1 )
10 CONTINUE
*
ERRNRM = CLANGE( 'One', N, N, WORK, N, RWORK ) / ENORM
*
* Compute RESULT(1)
*
RESULT( 1 ) = ERRNRM / ULP
*
* Normalization of E:
*
ENRMER = ZERO
DO 30 JVEC = 1, N
TEMP1 = ZERO
DO 20 J = 1, N
TEMP1 = MAX( TEMP1, ABS1( E( J, JVEC ) ) )
20 CONTINUE
ENRMER = MAX( ENRMER, TEMP1-ONE )
30 CONTINUE
*
* Compute RESULT(2) : the normalization error in E.
*
RESULT( 2 ) = ENRMER / ( REAL( N )*ULP )
*
RETURN
*
* End of CGET52
*
END
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