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SUBROUTINE DGET53( A, LDA, B, LDB, SCALE, WR, WI, RESULT, INFO )
*
* -- LAPACK test routine (version 3.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* March 31, 1993
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LDB
DOUBLE PRECISION RESULT, SCALE, WI, WR
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * )
* ..
*
* Purpose
* =======
*
* 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.
*
* Arguments
* =========
*
* A (input) DOUBLE PRECISION array, dimension (LDA, 2)
* The 2x2 matrix A.
*
* LDA (input) INTEGER
* The leading dimension of A. It must be at least 2.
*
* B (input) DOUBLE PRECISION array, dimension (LDB, N)
* The 2x2 upper-triangular matrix B.
*
* LDB (input) INTEGER
* The leading dimension of B. It must be at least 2.
*
* SCALE (input) DOUBLE PRECISION
* The "scale factor" s in the formula s A - w B . It is
* assumed to be non-negative.
*
* WR (input) DOUBLE PRECISION
* The real part of the eigenvalue w in the formula
* s A - w B .
*
* WI (input) DOUBLE PRECISION
* The imaginary part of the eigenvalue w in the formula
* s A - w B .
*
* RESULT (output) 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.
*
* INFO (output) 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.
*
* =====================================================================
*
* .. 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
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