*> \brief \b DTGSNA
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTGSNA + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtgsna.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtgsna.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtgsna.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DTGSNA( JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL,
* LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK,
* IWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER HOWMNY, JOB
* INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, M, MM, N
* ..
* .. Array Arguments ..
* LOGICAL SELECT( * )
* INTEGER IWORK( * )
* DOUBLE PRECISION A( LDA, * ), B( LDB, * ), DIF( * ), S( * ),
* $ VL( LDVL, * ), VR( LDVR, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DTGSNA estimates reciprocal condition numbers for specified
*> eigenvalues and/or eigenvectors of a matrix pair (A, B) in
*> generalized real Schur canonical form (or of any matrix pair
*> (Q*A*Z**T, Q*B*Z**T) with orthogonal matrices Q and Z, where
*> Z**T denotes the transpose of Z.
*>
*> (A, B) must be in generalized real Schur form (as returned by DGGES),
*> i.e. A is block upper triangular with 1-by-1 and 2-by-2 diagonal
*> blocks. B is upper triangular.
*>
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOB
*> \verbatim
*> JOB is CHARACTER*1
*> Specifies whether condition numbers are required for
*> eigenvalues (S) or eigenvectors (DIF):
*> = 'E': for eigenvalues only (S);
*> = 'V': for eigenvectors only (DIF);
*> = 'B': for both eigenvalues and eigenvectors (S and DIF).
*> \endverbatim
*>
*> \param[in] HOWMNY
*> \verbatim
*> HOWMNY is CHARACTER*1
*> = 'A': compute condition numbers for all eigenpairs;
*> = 'S': compute condition numbers for selected eigenpairs
*> specified by the array SELECT.
*> \endverbatim
*>
*> \param[in] SELECT
*> \verbatim
*> SELECT is LOGICAL array, dimension (N)
*> If HOWMNY = 'S', SELECT specifies the eigenpairs for which
*> condition numbers are required. To select condition numbers
*> for the eigenpair corresponding to a real eigenvalue w(j),
*> SELECT(j) must be set to .TRUE.. To select condition numbers
*> corresponding to a complex conjugate pair of eigenvalues w(j)
*> and w(j+1), either SELECT(j) or SELECT(j+1) or both, must be
*> set to .TRUE..
*> If HOWMNY = 'A', SELECT is not referenced.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the square matrix pair (A, B). N >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> The upper quasi-triangular matrix A in the pair (A,B).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,N)
*> The upper triangular matrix B in the pair (A,B).
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[in] VL
*> \verbatim
*> VL is DOUBLE PRECISION array, dimension (LDVL,M)
*> If JOB = 'E' or 'B', VL must contain left eigenvectors of
*> (A, B), corresponding to the eigenpairs specified by HOWMNY
*> and SELECT. The eigenvectors must be stored in consecutive
*> columns of VL, as returned by DTGEVC.
*> If JOB = 'V', VL is not referenced.
*> \endverbatim
*>
*> \param[in] LDVL
*> \verbatim
*> LDVL is INTEGER
*> The leading dimension of the array VL. LDVL >= 1.
*> If JOB = 'E' or 'B', LDVL >= N.
*> \endverbatim
*>
*> \param[in] VR
*> \verbatim
*> VR is DOUBLE PRECISION array, dimension (LDVR,M)
*> If JOB = 'E' or 'B', VR must contain right eigenvectors of
*> (A, B), corresponding to the eigenpairs specified by HOWMNY
*> and SELECT. The eigenvectors must be stored in consecutive
*> columns ov VR, as returned by DTGEVC.
*> If JOB = 'V', VR is not referenced.
*> \endverbatim
*>
*> \param[in] LDVR
*> \verbatim
*> LDVR is INTEGER
*> The leading dimension of the array VR. LDVR >= 1.
*> If JOB = 'E' or 'B', LDVR >= N.
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*> S is DOUBLE PRECISION array, dimension (MM)
*> If JOB = 'E' or 'B', the reciprocal condition numbers of the
*> selected eigenvalues, stored in consecutive elements of the
*> array. For a complex conjugate pair of eigenvalues two
*> consecutive elements of S are set to the same value. Thus
*> S(j), DIF(j), and the j-th columns of VL and VR all
*> correspond to the same eigenpair (but not in general the
*> j-th eigenpair, unless all eigenpairs are selected).
*> If JOB = 'V', S is not referenced.
*> \endverbatim
*>
*> \param[out] DIF
*> \verbatim
*> DIF is DOUBLE PRECISION array, dimension (MM)
*> If JOB = 'V' or 'B', the estimated reciprocal condition
*> numbers of the selected eigenvectors, stored in consecutive
*> elements of the array. For a complex eigenvector two
*> consecutive elements of DIF are set to the same value. If
*> the eigenvalues cannot be reordered to compute DIF(j), DIF(j)
*> is set to 0; this can only occur when the true value would be
*> very small anyway.
*> If JOB = 'E', DIF is not referenced.
*> \endverbatim
*>
*> \param[in] MM
*> \verbatim
*> MM is INTEGER
*> The number of elements in the arrays S and DIF. MM >= M.
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*> M is INTEGER
*> The number of elements of the arrays S and DIF used to store
*> the specified condition numbers; for each selected real
*> eigenvalue one element is used, and for each selected complex
*> conjugate pair of eigenvalues, two elements are used.
*> If HOWMNY = 'A', M is set to N.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= max(1,N).
*> If JOB = 'V' or 'B' LWORK >= 2*N*(N+2)+16.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (N + 6)
*> If JOB = 'E', IWORK is not referenced.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> =0: Successful exit
*> <0: If INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup doubleOTHERcomputational
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The reciprocal of the condition number of a generalized eigenvalue
*> w = (a, b) is defined as
*>
*> S(w) = (|u**TAv|**2 + |u**TBv|**2)**(1/2) / (norm(u)*norm(v))
*>
*> where u and v are the left and right eigenvectors of (A, B)
*> corresponding to w; |z| denotes the absolute value of the complex
*> number, and norm(u) denotes the 2-norm of the vector u.
*> The pair (a, b) corresponds to an eigenvalue w = a/b (= u**TAv/u**TBv)
*> of the matrix pair (A, B). If both a and b equal zero, then (A B) is
*> singular and S(I) = -1 is returned.
*>
*> An approximate error bound on the chordal distance between the i-th
*> computed generalized eigenvalue w and the corresponding exact
*> eigenvalue lambda is
*>
*> chord(w, lambda) <= EPS * norm(A, B) / S(I)
*>
*> where EPS is the machine precision.
*>
*> The reciprocal of the condition number DIF(i) of right eigenvector u
*> and left eigenvector v corresponding to the generalized eigenvalue w
*> is defined as follows:
*>
*> a) If the i-th eigenvalue w = (a,b) is real
*>
*> Suppose U and V are orthogonal transformations such that
*>
*> U**T*(A, B)*V = (S, T) = ( a * ) ( b * ) 1
*> ( 0 S22 ),( 0 T22 ) n-1
*> 1 n-1 1 n-1
*>
*> Then the reciprocal condition number DIF(i) is
*>
*> Difl((a, b), (S22, T22)) = sigma-min( Zl ),
*>
*> where sigma-min(Zl) denotes the smallest singular value of the
*> 2(n-1)-by-2(n-1) matrix
*>
*> Zl = [ kron(a, In-1) -kron(1, S22) ]
*> [ kron(b, In-1) -kron(1, T22) ] .
*>
*> Here In-1 is the identity matrix of size n-1. kron(X, Y) is the
*> Kronecker product between the matrices X and Y.
*>
*> Note that if the default method for computing DIF(i) is wanted
*> (see DLATDF), then the parameter DIFDRI (see below) should be
*> changed from 3 to 4 (routine DLATDF(IJOB = 2 will be used)).
*> See DTGSYL for more details.
*>
*> b) If the i-th and (i+1)-th eigenvalues are complex conjugate pair,
*>
*> Suppose U and V are orthogonal transformations such that
*>
*> U**T*(A, B)*V = (S, T) = ( S11 * ) ( T11 * ) 2
*> ( 0 S22 ),( 0 T22) n-2
*> 2 n-2 2 n-2
*>
*> and (S11, T11) corresponds to the complex conjugate eigenvalue
*> pair (w, conjg(w)). There exist unitary matrices U1 and V1 such
*> that
*>
*> U1**T*S11*V1 = ( s11 s12 ) and U1**T*T11*V1 = ( t11 t12 )
*> ( 0 s22 ) ( 0 t22 )
*>
*> where the generalized eigenvalues w = s11/t11 and
*> conjg(w) = s22/t22.
*>
*> Then the reciprocal condition number DIF(i) is bounded by
*>
*> min( d1, max( 1, |real(s11)/real(s22)| )*d2 )
*>
*> where, d1 = Difl((s11, t11), (s22, t22)) = sigma-min(Z1), where
*> Z1 is the complex 2-by-2 matrix
*>
*> Z1 = [ s11 -s22 ]
*> [ t11 -t22 ],
*>
*> This is done by computing (using real arithmetic) the
*> roots of the characteristical polynomial det(Z1**T * Z1 - lambda I),
*> where Z1**T denotes the transpose of Z1 and det(X) denotes
*> the determinant of X.
*>
*> and d2 is an upper bound on Difl((S11, T11), (S22, T22)), i.e. an
*> upper bound on sigma-min(Z2), where Z2 is (2n-2)-by-(2n-2)
*>
*> Z2 = [ kron(S11**T, In-2) -kron(I2, S22) ]
*> [ kron(T11**T, In-2) -kron(I2, T22) ]
*>
*> Note that if the default method for computing DIF is wanted (see
*> DLATDF), then the parameter DIFDRI (see below) should be changed
*> from 3 to 4 (routine DLATDF(IJOB = 2 will be used)). See DTGSYL
*> for more details.
*>
*> For each eigenvalue/vector specified by SELECT, DIF stores a
*> Frobenius norm-based estimate of Difl.
*>
*> An approximate error bound for the i-th computed eigenvector VL(i) or
*> VR(i) is given by
*>
*> EPS * norm(A, B) / DIF(i).
*>
*> See ref. [2-3] for more details and further references.
*> \endverbatim
*
*> \par Contributors:
* ==================
*>
*> Bo Kagstrom and Peter Poromaa, Department of Computing Science,
*> Umea University, S-901 87 Umea, Sweden.
*
*> \par References:
* ================
*>
*> \verbatim
*>
*> [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
*> Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
*> M.S. Moonen et al (eds), Linear Algebra for Large Scale and
*> Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
*>
*> [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified
*> Eigenvalues of a Regular Matrix Pair (A, B) and Condition
*> Estimation: Theory, Algorithms and Software,
*> Report UMINF - 94.04, Department of Computing Science, Umea
*> University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working
*> Note 87. To appear in Numerical Algorithms, 1996.
*>
*> [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
*> for Solving the Generalized Sylvester Equation and Estimating the
*> Separation between Regular Matrix Pairs, Report UMINF - 93.23,
*> Department of Computing Science, Umea University, S-901 87 Umea,
*> Sweden, December 1993, Revised April 1994, Also as LAPACK Working
*> Note 75. To appear in ACM Trans. on Math. Software, Vol 22,
*> No 1, 1996.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DTGSNA( JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL,
$ LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK,
$ IWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER HOWMNY, JOB
INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, M, MM, N
* ..
* .. Array Arguments ..
LOGICAL SELECT( * )
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), DIF( * ), S( * ),
$ VL( LDVL, * ), VR( LDVR, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
INTEGER DIFDRI
PARAMETER ( DIFDRI = 3 )
DOUBLE PRECISION ZERO, ONE, TWO, FOUR
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0,
$ FOUR = 4.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY, PAIR, SOMCON, WANTBH, WANTDF, WANTS
INTEGER I, IERR, IFST, ILST, IZ, K, KS, LWMIN, N1, N2
DOUBLE PRECISION ALPHAI, ALPHAR, ALPRQT, BETA, C1, C2, COND,
$ EPS, LNRM, RNRM, ROOT1, ROOT2, SCALE, SMLNUM,
$ TMPII, TMPIR, TMPRI, TMPRR, UHAV, UHAVI, UHBV,
$ UHBVI
* ..
* .. Local Arrays ..
DOUBLE PRECISION DUMMY( 1 ), DUMMY1( 1 )
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DDOT, DLAMCH, DLAPY2, DNRM2
EXTERNAL LSAME, DDOT, DLAMCH, DLAPY2, DNRM2
* ..
* .. External Subroutines ..
EXTERNAL DGEMV, DLACPY, DLAG2, DTGEXC, DTGSYL, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
* Decode and test the input parameters
*
WANTBH = LSAME( JOB, 'B' )
WANTS = LSAME( JOB, 'E' ) .OR. WANTBH
WANTDF = LSAME( JOB, 'V' ) .OR. WANTBH
*
SOMCON = LSAME( HOWMNY, 'S' )
*
INFO = 0
LQUERY = ( LWORK.EQ.-1 )
*
IF( .NOT.WANTS .AND. .NOT.WANTDF ) THEN
INFO = -1
ELSE IF( .NOT.LSAME( HOWMNY, 'A' ) .AND. .NOT.SOMCON ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -6
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -8
ELSE IF( WANTS .AND. LDVL.LT.N ) THEN
INFO = -10
ELSE IF( WANTS .AND. LDVR.LT.N ) THEN
INFO = -12
ELSE
*
* Set M to the number of eigenpairs for which condition numbers
* are required, and test MM.
*
IF( SOMCON ) THEN
M = 0
PAIR = .FALSE.
DO 10 K = 1, N
IF( PAIR ) THEN
PAIR = .FALSE.
ELSE
IF( K.LT.N ) THEN
IF( A( K+1, K ).EQ.ZERO ) THEN
IF( SELECT( K ) )
$ M = M + 1
ELSE
PAIR = .TRUE.
IF( SELECT( K ) .OR. SELECT( K+1 ) )
$ M = M + 2
END IF
ELSE
IF( SELECT( N ) )
$ M = M + 1
END IF
END IF
10 CONTINUE
ELSE
M = N
END IF
*
IF( N.EQ.0 ) THEN
LWMIN = 1
ELSE IF( LSAME( JOB, 'V' ) .OR. LSAME( JOB, 'B' ) ) THEN
LWMIN = 2*N*( N + 2 ) + 16
ELSE
LWMIN = N
END IF
WORK( 1 ) = LWMIN
*
IF( MM.LT.M ) THEN
INFO = -15
ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
INFO = -18
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DTGSNA', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Get machine constants
*
EPS = DLAMCH( 'P' )
SMLNUM = DLAMCH( 'S' ) / EPS
KS = 0
PAIR = .FALSE.
*
DO 20 K = 1, N
*
* Determine whether A(k,k) begins a 1-by-1 or 2-by-2 block.
*
IF( PAIR ) THEN
PAIR = .FALSE.
GO TO 20
ELSE
IF( K.LT.N )
$ PAIR = A( K+1, K ).NE.ZERO
END IF
*
* Determine whether condition numbers are required for the k-th
* eigenpair.
*
IF( SOMCON ) THEN
IF( PAIR ) THEN
IF( .NOT.SELECT( K ) .AND. .NOT.SELECT( K+1 ) )
$ GO TO 20
ELSE
IF( .NOT.SELECT( K ) )
$ GO TO 20
END IF
END IF
*
KS = KS + 1
*
IF( WANTS ) THEN
*
* Compute the reciprocal condition number of the k-th
* eigenvalue.
*
IF( PAIR ) THEN
*
* Complex eigenvalue pair.
*
RNRM = DLAPY2( DNRM2( N, VR( 1, KS ), 1 ),
$ DNRM2( N, VR( 1, KS+1 ), 1 ) )
LNRM = DLAPY2( DNRM2( N, VL( 1, KS ), 1 ),
$ DNRM2( N, VL( 1, KS+1 ), 1 ) )
CALL DGEMV( 'N', N, N, ONE, A, LDA, VR( 1, KS ), 1, ZERO,
$ WORK, 1 )
TMPRR = DDOT( N, WORK, 1, VL( 1, KS ), 1 )
TMPRI = DDOT( N, WORK, 1, VL( 1, KS+1 ), 1 )
CALL DGEMV( 'N', N, N, ONE, A, LDA, VR( 1, KS+1 ), 1,
$ ZERO, WORK, 1 )
TMPII = DDOT( N, WORK, 1, VL( 1, KS+1 ), 1 )
TMPIR = DDOT( N, WORK, 1, VL( 1, KS ), 1 )
UHAV = TMPRR + TMPII
UHAVI = TMPIR - TMPRI
CALL DGEMV( 'N', N, N, ONE, B, LDB, VR( 1, KS ), 1, ZERO,
$ WORK, 1 )
TMPRR = DDOT( N, WORK, 1, VL( 1, KS ), 1 )
TMPRI = DDOT( N, WORK, 1, VL( 1, KS+1 ), 1 )
CALL DGEMV( 'N', N, N, ONE, B, LDB, VR( 1, KS+1 ), 1,
$ ZERO, WORK, 1 )
TMPII = DDOT( N, WORK, 1, VL( 1, KS+1 ), 1 )
TMPIR = DDOT( N, WORK, 1, VL( 1, KS ), 1 )
UHBV = TMPRR + TMPII
UHBVI = TMPIR - TMPRI
UHAV = DLAPY2( UHAV, UHAVI )
UHBV = DLAPY2( UHBV, UHBVI )
COND = DLAPY2( UHAV, UHBV )
S( KS ) = COND / ( RNRM*LNRM )
S( KS+1 ) = S( KS )
*
ELSE
*
* Real eigenvalue.
*
RNRM = DNRM2( N, VR( 1, KS ), 1 )
LNRM = DNRM2( N, VL( 1, KS ), 1 )
CALL DGEMV( 'N', N, N, ONE, A, LDA, VR( 1, KS ), 1, ZERO,
$ WORK, 1 )
UHAV = DDOT( N, WORK, 1, VL( 1, KS ), 1 )
CALL DGEMV( 'N', N, N, ONE, B, LDB, VR( 1, KS ), 1, ZERO,
$ WORK, 1 )
UHBV = DDOT( N, WORK, 1, VL( 1, KS ), 1 )
COND = DLAPY2( UHAV, UHBV )
IF( COND.EQ.ZERO ) THEN
S( KS ) = -ONE
ELSE
S( KS ) = COND / ( RNRM*LNRM )
END IF
END IF
END IF
*
IF( WANTDF ) THEN
IF( N.EQ.1 ) THEN
DIF( KS ) = DLAPY2( A( 1, 1 ), B( 1, 1 ) )
GO TO 20
END IF
*
* Estimate the reciprocal condition number of the k-th
* eigenvectors.
IF( PAIR ) THEN
*
* Copy the 2-by 2 pencil beginning at (A(k,k), B(k, k)).
* Compute the eigenvalue(s) at position K.
*
WORK( 1 ) = A( K, K )
WORK( 2 ) = A( K+1, K )
WORK( 3 ) = A( K, K+1 )
WORK( 4 ) = A( K+1, K+1 )
WORK( 5 ) = B( K, K )
WORK( 6 ) = B( K+1, K )
WORK( 7 ) = B( K, K+1 )
WORK( 8 ) = B( K+1, K+1 )
CALL DLAG2( WORK, 2, WORK( 5 ), 2, SMLNUM*EPS, BETA,
$ DUMMY1( 1 ), ALPHAR, DUMMY( 1 ), ALPHAI )
ALPRQT = ONE
C1 = TWO*( ALPHAR*ALPHAR+ALPHAI*ALPHAI+BETA*BETA )
C2 = FOUR*BETA*BETA*ALPHAI*ALPHAI
ROOT1 = C1 + SQRT( C1*C1-4.0D0*C2 )
ROOT1 = ROOT1 / TWO
ROOT2 = C2 / ROOT1
COND = MIN( SQRT( ROOT1 ), SQRT( ROOT2 ) )
END IF
*
* Copy the matrix (A, B) to the array WORK and swap the
* diagonal block beginning at A(k,k) to the (1,1) position.
*
CALL DLACPY( 'Full', N, N, A, LDA, WORK, N )
CALL DLACPY( 'Full', N, N, B, LDB, WORK( N*N+1 ), N )
IFST = K
ILST = 1
*
CALL DTGEXC( .FALSE., .FALSE., N, WORK, N, WORK( N*N+1 ), N,
$ DUMMY, 1, DUMMY1, 1, IFST, ILST,
$ WORK( N*N*2+1 ), LWORK-2*N*N, IERR )
*
IF( IERR.GT.0 ) THEN
*
* Ill-conditioned problem - swap rejected.
*
DIF( KS ) = ZERO
ELSE
*
* Reordering successful, solve generalized Sylvester
* equation for R and L,
* A22 * R - L * A11 = A12
* B22 * R - L * B11 = B12,
* and compute estimate of Difl((A11,B11), (A22, B22)).
*
N1 = 1
IF( WORK( 2 ).NE.ZERO )
$ N1 = 2
N2 = N - N1
IF( N2.EQ.0 ) THEN
DIF( KS ) = COND
ELSE
I = N*N + 1
IZ = 2*N*N + 1
CALL DTGSYL( 'N', DIFDRI, N2, N1, WORK( N*N1+N1+1 ),
$ N, WORK, N, WORK( N1+1 ), N,
$ WORK( N*N1+N1+I ), N, WORK( I ), N,
$ WORK( N1+I ), N, SCALE, DIF( KS ),
$ WORK( IZ+1 ), LWORK-2*N*N, IWORK, IERR )
*
IF( PAIR )
$ DIF( KS ) = MIN( MAX( ONE, ALPRQT )*DIF( KS ),
$ COND )
END IF
END IF
IF( PAIR )
$ DIF( KS+1 ) = DIF( KS )
END IF
IF( PAIR )
$ KS = KS + 1
*
20 CONTINUE
WORK( 1 ) = LWMIN
RETURN
*
* End of DTGSNA
*
END