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|
*> \brief \b DGET53
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* SUBROUTINE DGET53( A, LDA, B, LDB, SCALE, WR, WI, RESULT, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, LDB
* DOUBLE PRECISION RESULT, SCALE, WI, WR
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), B( LDB, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGET53 checks the generalized eigenvalues computed by DLAG2.
*>
*> The basic test for an eigenvalue is:
*>
*> | det( s A - w B ) |
*> RESULT = ---------------------------------------------------
*> ulp max( s norm(A), |w| norm(B) )*norm( s A - w B )
*>
*> Two "safety checks" are performed:
*>
*> (1) ulp*max( s*norm(A), |w|*norm(B) ) must be at least
*> safe_minimum. This insures that the test performed is
*> not essentially det(0*A + 0*B)=0.
*>
*> (2) s*norm(A) + |w|*norm(B) must be less than 1/safe_minimum.
*> This insures that s*A - w*B will not overflow.
*>
*> If these tests are not passed, then s and w are scaled and
*> tested anyway, if this is possible.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA, 2)
*> The 2x2 matrix A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of A. It must be at least 2.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB, N)
*> The 2x2 upper-triangular matrix B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of B. It must be at least 2.
*> \endverbatim
*>
*> \param[in] SCALE
*> \verbatim
*> SCALE is DOUBLE PRECISION
*> The "scale factor" s in the formula s A - w B . It is
*> assumed to be non-negative.
*> \endverbatim
*>
*> \param[in] WR
*> \verbatim
*> WR is DOUBLE PRECISION
*> The real part of the eigenvalue w in the formula
*> s A - w B .
*> \endverbatim
*>
*> \param[in] WI
*> \verbatim
*> WI is DOUBLE PRECISION
*> The imaginary part of the eigenvalue w in the formula
*> s A - w B .
*> \endverbatim
*>
*> \param[out] RESULT
*> \verbatim
*> RESULT is DOUBLE PRECISION
*> If INFO is 2 or less, the value computed by the test
*> described above.
*> If INFO=3, this will just be 1/ulp.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> =0: The input data pass the "safety checks".
*> =1: s*norm(A) + |w|*norm(B) > 1/safe_minimum.
*> =2: ulp*max( s*norm(A), |w|*norm(B) ) < safe_minimum
*> =3: same as INFO=2, but s and w could not be scaled so
*> as to compute the test.
*> \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 DGET53( A, LDA, B, LDB, SCALE, WR, WI, RESULT, INFO )
*
* -- 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 ..
INTEGER INFO, LDA, LDB
DOUBLE PRECISION RESULT, SCALE, WI, WR
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION ABSW, ANORM, BNORM, CI11, CI12, CI22, CNORM,
$ CR11, CR12, CR21, CR22, CSCALE, DETI, DETR, S1,
$ SAFMIN, SCALES, SIGMIN, TEMP, ULP, WIS, WRS
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, SQRT
* ..
* .. Executable Statements ..
*
* Initialize
*
INFO = 0
RESULT = ZERO
SCALES = SCALE
WRS = WR
WIS = WI
*
* Machine constants and norms
*
SAFMIN = DLAMCH( 'Safe minimum' )
ULP = DLAMCH( 'Epsilon' )*DLAMCH( 'Base' )
ABSW = ABS( WRS ) + ABS( WIS )
ANORM = MAX( ABS( A( 1, 1 ) )+ABS( A( 2, 1 ) ),
$ ABS( A( 1, 2 ) )+ABS( A( 2, 2 ) ), SAFMIN )
BNORM = MAX( ABS( B( 1, 1 ) ), ABS( B( 1, 2 ) )+ABS( B( 2, 2 ) ),
$ SAFMIN )
*
* Check for possible overflow.
*
TEMP = ( SAFMIN*BNORM )*ABSW + ( SAFMIN*ANORM )*SCALES
IF( TEMP.GE.ONE ) THEN
*
* Scale down to avoid overflow
*
INFO = 1
TEMP = ONE / TEMP
SCALES = SCALES*TEMP
WRS = WRS*TEMP
WIS = WIS*TEMP
ABSW = ABS( WRS ) + ABS( WIS )
END IF
S1 = MAX( ULP*MAX( SCALES*ANORM, ABSW*BNORM ),
$ SAFMIN*MAX( SCALES, ABSW ) )
*
* Check for W and SCALE essentially zero.
*
IF( S1.LT.SAFMIN ) THEN
INFO = 2
IF( SCALES.LT.SAFMIN .AND. ABSW.LT.SAFMIN ) THEN
INFO = 3
RESULT = ONE / ULP
RETURN
END IF
*
* Scale up to avoid underflow
*
TEMP = ONE / MAX( SCALES*ANORM+ABSW*BNORM, SAFMIN )
SCALES = SCALES*TEMP
WRS = WRS*TEMP
WIS = WIS*TEMP
ABSW = ABS( WRS ) + ABS( WIS )
S1 = MAX( ULP*MAX( SCALES*ANORM, ABSW*BNORM ),
$ SAFMIN*MAX( SCALES, ABSW ) )
IF( S1.LT.SAFMIN ) THEN
INFO = 3
RESULT = ONE / ULP
RETURN
END IF
END IF
*
* Compute C = s A - w B
*
CR11 = SCALES*A( 1, 1 ) - WRS*B( 1, 1 )
CI11 = -WIS*B( 1, 1 )
CR21 = SCALES*A( 2, 1 )
CR12 = SCALES*A( 1, 2 ) - WRS*B( 1, 2 )
CI12 = -WIS*B( 1, 2 )
CR22 = SCALES*A( 2, 2 ) - WRS*B( 2, 2 )
CI22 = -WIS*B( 2, 2 )
*
* Compute the smallest singular value of s A - w B:
*
* |det( s A - w B )|
* sigma_min = ------------------
* norm( s A - w B )
*
CNORM = MAX( ABS( CR11 )+ABS( CI11 )+ABS( CR21 ),
$ ABS( CR12 )+ABS( CI12 )+ABS( CR22 )+ABS( CI22 ), SAFMIN )
CSCALE = ONE / SQRT( CNORM )
DETR = ( CSCALE*CR11 )*( CSCALE*CR22 ) -
$ ( CSCALE*CI11 )*( CSCALE*CI22 ) -
$ ( CSCALE*CR12 )*( CSCALE*CR21 )
DETI = ( CSCALE*CR11 )*( CSCALE*CI22 ) +
$ ( CSCALE*CI11 )*( CSCALE*CR22 ) -
$ ( CSCALE*CI12 )*( CSCALE*CR21 )
SIGMIN = ABS( DETR ) + ABS( DETI )
RESULT = SIGMIN / S1
RETURN
*
* End of DGET53
*
END
|
