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SUBROUTINE CTGSEN( IJOB, WANTQ, WANTZ, SELECT, N, A, LDA, B, LDB,
$ ALPHA, BETA, Q, LDQ, Z, LDZ, M, PL, PR, DIF,
$ WORK, LWORK, IWORK, LIWORK, INFO )
*
* -- LAPACK routine (version 3.3.1) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* -- April 2011 --
*
* Modified to call CLACN2 in place of CLACON, 10 Feb 03, SJH.
*
* .. Scalar Arguments ..
LOGICAL WANTQ, WANTZ
INTEGER IJOB, INFO, LDA, LDB, LDQ, LDZ, LIWORK, LWORK,
$ M, N
REAL PL, PR
* ..
* .. Array Arguments ..
LOGICAL SELECT( * )
INTEGER IWORK( * )
REAL DIF( * )
COMPLEX A( LDA, * ), ALPHA( * ), B( LDB, * ),
$ BETA( * ), Q( LDQ, * ), WORK( * ), Z( LDZ, * )
* ..
*
* Purpose
* =======
*
* CTGSEN reorders the generalized Schur decomposition of a complex
* matrix pair (A, B) (in terms of an unitary equivalence trans-
* formation Q**H * (A, B) * Z), so that a selected cluster of eigenvalues
* appears in the leading diagonal blocks of the pair (A,B). The leading
* columns of Q and Z form unitary bases of the corresponding left and
* right eigenspaces (deflating subspaces). (A, B) must be in
* generalized Schur canonical form, that is, A and B are both upper
* triangular.
*
* CTGSEN also computes the generalized eigenvalues
*
* w(j)= ALPHA(j) / BETA(j)
*
* of the reordered matrix pair (A, B).
*
* Optionally, the routine computes estimates of reciprocal condition
* numbers for eigenvalues and eigenspaces. These are Difu[(A11,B11),
* (A22,B22)] and Difl[(A11,B11), (A22,B22)], i.e. the separation(s)
* between the matrix pairs (A11, B11) and (A22,B22) that correspond to
* the selected cluster and the eigenvalues outside the cluster, resp.,
* and norms of "projections" onto left and right eigenspaces w.r.t.
* the selected cluster in the (1,1)-block.
*
*
* Arguments
* =========
*
* IJOB (input) integer
* Specifies whether condition numbers are required for the
* cluster of eigenvalues (PL and PR) or the deflating subspaces
* (Difu and Difl):
* =0: Only reorder w.r.t. SELECT. No extras.
* =1: Reciprocal of norms of "projections" onto left and right
* eigenspaces w.r.t. the selected cluster (PL and PR).
* =2: Upper bounds on Difu and Difl. F-norm-based estimate
* (DIF(1:2)).
* =3: Estimate of Difu and Difl. 1-norm-based estimate
* (DIF(1:2)).
* About 5 times as expensive as IJOB = 2.
* =4: Compute PL, PR and DIF (i.e. 0, 1 and 2 above): Economic
* version to get it all.
* =5: Compute PL, PR and DIF (i.e. 0, 1 and 3 above)
*
* WANTQ (input) LOGICAL
* .TRUE. : update the left transformation matrix Q;
* .FALSE.: do not update Q.
*
* WANTZ (input) LOGICAL
* .TRUE. : update the right transformation matrix Z;
* .FALSE.: do not update Z.
*
* SELECT (input) LOGICAL array, dimension (N)
* SELECT specifies the eigenvalues in the selected cluster. To
* select an eigenvalue w(j), SELECT(j) must be set to
* .TRUE..
*
* N (input) INTEGER
* The order of the matrices A and B. N >= 0.
*
* A (input/output) COMPLEX array, dimension(LDA,N)
* On entry, the upper triangular matrix A, in generalized
* Schur canonical form.
* On exit, A is overwritten by the reordered matrix A.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* B (input/output) COMPLEX array, dimension(LDB,N)
* On entry, the upper triangular matrix B, in generalized
* Schur canonical form.
* On exit, B is overwritten by the reordered matrix B.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,N).
*
* ALPHA (output) COMPLEX array, dimension (N)
* BETA (output) COMPLEX array, dimension (N)
* The diagonal elements of A and B, respectively,
* when the pair (A,B) has been reduced to generalized Schur
* form. ALPHA(i)/BETA(i) i=1,...,N are the generalized
* eigenvalues.
*
* Q (input/output) COMPLEX array, dimension (LDQ,N)
* On entry, if WANTQ = .TRUE., Q is an N-by-N matrix.
* On exit, Q has been postmultiplied by the left unitary
* transformation matrix which reorder (A, B); The leading M
* columns of Q form orthonormal bases for the specified pair of
* left eigenspaces (deflating subspaces).
* If WANTQ = .FALSE., Q is not referenced.
*
* LDQ (input) INTEGER
* The leading dimension of the array Q. LDQ >= 1.
* If WANTQ = .TRUE., LDQ >= N.
*
* Z (input/output) COMPLEX array, dimension (LDZ,N)
* On entry, if WANTZ = .TRUE., Z is an N-by-N matrix.
* On exit, Z has been postmultiplied by the left unitary
* transformation matrix which reorder (A, B); The leading M
* columns of Z form orthonormal bases for the specified pair of
* left eigenspaces (deflating subspaces).
* If WANTZ = .FALSE., Z is not referenced.
*
* LDZ (input) INTEGER
* The leading dimension of the array Z. LDZ >= 1.
* If WANTZ = .TRUE., LDZ >= N.
*
* M (output) INTEGER
* The dimension of the specified pair of left and right
* eigenspaces, (deflating subspaces) 0 <= M <= N.
*
* PL (output) REAL
* PR (output) REAL
* If IJOB = 1, 4 or 5, PL, PR are lower bounds on the
* reciprocal of the norm of "projections" onto left and right
* eigenspace with respect to the selected cluster.
* 0 < PL, PR <= 1.
* If M = 0 or M = N, PL = PR = 1.
* If IJOB = 0, 2 or 3 PL, PR are not referenced.
*
* DIF (output) REAL array, dimension (2).
* If IJOB >= 2, DIF(1:2) store the estimates of Difu and Difl.
* If IJOB = 2 or 4, DIF(1:2) are F-norm-based upper bounds on
* Difu and Difl. If IJOB = 3 or 5, DIF(1:2) are 1-norm-based
* estimates of Difu and Difl, computed using reversed
* communication with CLACN2.
* If M = 0 or N, DIF(1:2) = F-norm([A, B]).
* If IJOB = 0 or 1, DIF is not referenced.
*
* WORK (workspace/output) COMPLEX array, dimension (MAX(1,LWORK))
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) INTEGER
* The dimension of the array WORK. LWORK >= 1
* If IJOB = 1, 2 or 4, LWORK >= 2*M*(N-M)
* If IJOB = 3 or 5, LWORK >= 4*M*(N-M)
*
* 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.
*
* IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
* On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
*
* LIWORK (input) INTEGER
* The dimension of the array IWORK. LIWORK >= 1.
* If IJOB = 1, 2 or 4, LIWORK >= N+2;
* If IJOB = 3 or 5, LIWORK >= MAX(N+2, 2*M*(N-M));
*
* If LIWORK = -1, then a workspace query is assumed; the
* routine only calculates the optimal size of the IWORK array,
* returns this value as the first entry of the IWORK array, and
* no error message related to LIWORK is issued by XERBLA.
*
* INFO (output) INTEGER
* =0: Successful exit.
* <0: If INFO = -i, the i-th argument had an illegal value.
* =1: Reordering of (A, B) failed because the transformed
* matrix pair (A, B) would be too far from generalized
* Schur form; the problem is very ill-conditioned.
* (A, B) may have been partially reordered.
* If requested, 0 is returned in DIF(*), PL and PR.
*
*
* Further Details
* ===============
*
* CTGSEN first collects the selected eigenvalues by computing unitary
* U and W that move them to the top left corner of (A, B). In other
* words, the selected eigenvalues are the eigenvalues of (A11, B11) in
*
* U**H*(A, B)*W = (A11 A12) (B11 B12) n1
* ( 0 A22),( 0 B22) n2
* n1 n2 n1 n2
*
* where N = n1+n2 and U**H means the conjugate transpose of U. The first
* n1 columns of U and W span the specified pair of left and right
* eigenspaces (deflating subspaces) of (A, B).
*
* If (A, B) has been obtained from the generalized real Schur
* decomposition of a matrix pair (C, D) = Q*(A, B)*Z', then the
* reordered generalized Schur form of (C, D) is given by
*
* (C, D) = (Q*U)*(U**H *(A, B)*W)*(Z*W)**H,
*
* and the first n1 columns of Q*U and Z*W span the corresponding
* deflating subspaces of (C, D) (Q and Z store Q*U and Z*W, resp.).
*
* Note that if the selected eigenvalue is sufficiently ill-conditioned,
* then its value may differ significantly from its value before
* reordering.
*
* The reciprocal condition numbers of the left and right eigenspaces
* spanned by the first n1 columns of U and W (or Q*U and Z*W) may
* be returned in DIF(1:2), corresponding to Difu and Difl, resp.
*
* The Difu and Difl are defined as:
*
* Difu[(A11, B11), (A22, B22)] = sigma-min( Zu )
* and
* Difl[(A11, B11), (A22, B22)] = Difu[(A22, B22), (A11, B11)],
*
* where sigma-min(Zu) is the smallest singular value of the
* (2*n1*n2)-by-(2*n1*n2) matrix
*
* Zu = [ kron(In2, A11) -kron(A22**H, In1) ]
* [ kron(In2, B11) -kron(B22**H, In1) ].
*
* Here, Inx is the identity matrix of size nx and A22**H is the
* conjuguate transpose of A22. kron(X, Y) is the Kronecker product between
* the matrices X and Y.
*
* When DIF(2) is small, small changes in (A, B) can cause large changes
* in the deflating subspace. An approximate (asymptotic) bound on the
* maximum angular error in the computed deflating subspaces is
*
* EPS * norm((A, B)) / DIF(2),
*
* where EPS is the machine precision.
*
* The reciprocal norm of the projectors on the left and right
* eigenspaces associated with (A11, B11) may be returned in PL and PR.
* They are computed as follows. First we compute L and R so that
* P*(A, B)*Q is block diagonal, where
*
* P = ( I -L ) n1 Q = ( I R ) n1
* ( 0 I ) n2 and ( 0 I ) n2
* n1 n2 n1 n2
*
* and (L, R) is the solution to the generalized Sylvester equation
*
* A11*R - L*A22 = -A12
* B11*R - L*B22 = -B12
*
* Then PL = (F-norm(L)**2+1)**(-1/2) and PR = (F-norm(R)**2+1)**(-1/2).
* An approximate (asymptotic) bound on the average absolute error of
* the selected eigenvalues is
*
* EPS * norm((A, B)) / PL.
*
* There are also global error bounds which valid for perturbations up
* to a certain restriction: A lower bound (x) on the smallest
* F-norm(E,F) for which an eigenvalue of (A11, B11) may move and
* coalesce with an eigenvalue of (A22, B22) under perturbation (E,F),
* (i.e. (A + E, B + F), is
*
* x = min(Difu,Difl)/((1/(PL*PL)+1/(PR*PR))**(1/2)+2*max(1/PL,1/PR)).
*
* An approximate bound on x can be computed from DIF(1:2), PL and PR.
*
* If y = ( F-norm(E,F) / x) <= 1, the angles between the perturbed
* (L', R') and unperturbed (L, R) left and right deflating subspaces
* associated with the selected cluster in the (1,1)-blocks can be
* bounded as
*
* max-angle(L, L') <= arctan( y * PL / (1 - y * (1 - PL * PL)**(1/2))
* max-angle(R, R') <= arctan( y * PR / (1 - y * (1 - PR * PR)**(1/2))
*
* See LAPACK User's Guide section 4.11 or the following references
* for more information.
*
* Note that if the default method for computing the Frobenius-norm-
* based estimate DIF is not wanted (see CLATDF), then the parameter
* IDIFJB (see below) should be changed from 3 to 4 (routine CLATDF
* (IJOB = 2 will be used)). See CTGSYL for more details.
*
* Based on contributions by
* Bo Kagstrom and Peter Poromaa, Department of Computing Science,
* Umea University, S-901 87 Umea, Sweden.
*
* References
* ==========
*
* [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.
*
* =====================================================================
*
* .. Parameters ..
INTEGER IDIFJB
PARAMETER ( IDIFJB = 3 )
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY, SWAP, WANTD, WANTD1, WANTD2, WANTP
INTEGER I, IERR, IJB, K, KASE, KS, LIWMIN, LWMIN, MN2,
$ N1, N2
REAL DSCALE, DSUM, RDSCAL, SAFMIN
COMPLEX TEMP1, TEMP2
* ..
* .. Local Arrays ..
INTEGER ISAVE( 3 )
* ..
* .. External Subroutines ..
REAL SLAMCH
EXTERNAL CLACN2, CLACPY, CLASSQ, CSCAL, CTGEXC, CTGSYL,
$ SLAMCH, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, CMPLX, CONJG, MAX, SQRT
* ..
* .. Executable Statements ..
*
* Decode and test the input parameters
*
INFO = 0
LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
*
IF( IJOB.LT.0 .OR. IJOB.GT.5 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -5
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -9
ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
INFO = -13
ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
INFO = -15
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'CTGSEN', -INFO )
RETURN
END IF
*
IERR = 0
*
WANTP = IJOB.EQ.1 .OR. IJOB.GE.4
WANTD1 = IJOB.EQ.2 .OR. IJOB.EQ.4
WANTD2 = IJOB.EQ.3 .OR. IJOB.EQ.5
WANTD = WANTD1 .OR. WANTD2
*
* Set M to the dimension of the specified pair of deflating
* subspaces.
*
M = 0
DO 10 K = 1, N
ALPHA( K ) = A( K, K )
BETA( K ) = B( K, K )
IF( K.LT.N ) THEN
IF( SELECT( K ) )
$ M = M + 1
ELSE
IF( SELECT( N ) )
$ M = M + 1
END IF
10 CONTINUE
*
IF( IJOB.EQ.1 .OR. IJOB.EQ.2 .OR. IJOB.EQ.4 ) THEN
LWMIN = MAX( 1, 2*M*(N-M) )
LIWMIN = MAX( 1, N+2 )
ELSE IF( IJOB.EQ.3 .OR. IJOB.EQ.5 ) THEN
LWMIN = MAX( 1, 4*M*(N-M) )
LIWMIN = MAX( 1, 2*M*(N-M), N+2 )
ELSE
LWMIN = 1
LIWMIN = 1
END IF
*
WORK( 1 ) = LWMIN
IWORK( 1 ) = LIWMIN
*
IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
INFO = -21
ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
INFO = -23
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'CTGSEN', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible.
*
IF( M.EQ.N .OR. M.EQ.0 ) THEN
IF( WANTP ) THEN
PL = ONE
PR = ONE
END IF
IF( WANTD ) THEN
DSCALE = ZERO
DSUM = ONE
DO 20 I = 1, N
CALL CLASSQ( N, A( 1, I ), 1, DSCALE, DSUM )
CALL CLASSQ( N, B( 1, I ), 1, DSCALE, DSUM )
20 CONTINUE
DIF( 1 ) = DSCALE*SQRT( DSUM )
DIF( 2 ) = DIF( 1 )
END IF
GO TO 70
END IF
*
* Get machine constant
*
SAFMIN = SLAMCH( 'S' )
*
* Collect the selected blocks at the top-left corner of (A, B).
*
KS = 0
DO 30 K = 1, N
SWAP = SELECT( K )
IF( SWAP ) THEN
KS = KS + 1
*
* Swap the K-th block to position KS. Compute unitary Q
* and Z that will swap adjacent diagonal blocks in (A, B).
*
IF( K.NE.KS )
$ CALL CTGEXC( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z,
$ LDZ, K, KS, IERR )
*
IF( IERR.GT.0 ) THEN
*
* Swap is rejected: exit.
*
INFO = 1
IF( WANTP ) THEN
PL = ZERO
PR = ZERO
END IF
IF( WANTD ) THEN
DIF( 1 ) = ZERO
DIF( 2 ) = ZERO
END IF
GO TO 70
END IF
END IF
30 CONTINUE
IF( WANTP ) THEN
*
* Solve generalized Sylvester equation for R and L:
* A11 * R - L * A22 = A12
* B11 * R - L * B22 = B12
*
N1 = M
N2 = N - M
I = N1 + 1
CALL CLACPY( 'Full', N1, N2, A( 1, I ), LDA, WORK, N1 )
CALL CLACPY( 'Full', N1, N2, B( 1, I ), LDB, WORK( N1*N2+1 ),
$ N1 )
IJB = 0
CALL CTGSYL( 'N', IJB, N1, N2, A, LDA, A( I, I ), LDA, WORK,
$ N1, B, LDB, B( I, I ), LDB, WORK( N1*N2+1 ), N1,
$ DSCALE, DIF( 1 ), WORK( N1*N2*2+1 ),
$ LWORK-2*N1*N2, IWORK, IERR )
*
* Estimate the reciprocal of norms of "projections" onto
* left and right eigenspaces
*
RDSCAL = ZERO
DSUM = ONE
CALL CLASSQ( N1*N2, WORK, 1, RDSCAL, DSUM )
PL = RDSCAL*SQRT( DSUM )
IF( PL.EQ.ZERO ) THEN
PL = ONE
ELSE
PL = DSCALE / ( SQRT( DSCALE*DSCALE / PL+PL )*SQRT( PL ) )
END IF
RDSCAL = ZERO
DSUM = ONE
CALL CLASSQ( N1*N2, WORK( N1*N2+1 ), 1, RDSCAL, DSUM )
PR = RDSCAL*SQRT( DSUM )
IF( PR.EQ.ZERO ) THEN
PR = ONE
ELSE
PR = DSCALE / ( SQRT( DSCALE*DSCALE / PR+PR )*SQRT( PR ) )
END IF
END IF
IF( WANTD ) THEN
*
* Compute estimates Difu and Difl.
*
IF( WANTD1 ) THEN
N1 = M
N2 = N - M
I = N1 + 1
IJB = IDIFJB
*
* Frobenius norm-based Difu estimate.
*
CALL CTGSYL( 'N', IJB, N1, N2, A, LDA, A( I, I ), LDA, WORK,
$ N1, B, LDB, B( I, I ), LDB, WORK( N1*N2+1 ),
$ N1, DSCALE, DIF( 1 ), WORK( N1*N2*2+1 ),
$ LWORK-2*N1*N2, IWORK, IERR )
*
* Frobenius norm-based Difl estimate.
*
CALL CTGSYL( 'N', IJB, N2, N1, A( I, I ), LDA, A, LDA, WORK,
$ N2, B( I, I ), LDB, B, LDB, WORK( N1*N2+1 ),
$ N2, DSCALE, DIF( 2 ), WORK( N1*N2*2+1 ),
$ LWORK-2*N1*N2, IWORK, IERR )
ELSE
*
* Compute 1-norm-based estimates of Difu and Difl using
* reversed communication with CLACN2. In each step a
* generalized Sylvester equation or a transposed variant
* is solved.
*
KASE = 0
N1 = M
N2 = N - M
I = N1 + 1
IJB = 0
MN2 = 2*N1*N2
*
* 1-norm-based estimate of Difu.
*
40 CONTINUE
CALL CLACN2( MN2, WORK( MN2+1 ), WORK, DIF( 1 ), KASE,
$ ISAVE )
IF( KASE.NE.0 ) THEN
IF( KASE.EQ.1 ) THEN
*
* Solve generalized Sylvester equation
*
CALL CTGSYL( 'N', IJB, N1, N2, A, LDA, A( I, I ), LDA,
$ WORK, N1, B, LDB, B( I, I ), LDB,
$ WORK( N1*N2+1 ), N1, DSCALE, DIF( 1 ),
$ WORK( N1*N2*2+1 ), LWORK-2*N1*N2, IWORK,
$ IERR )
ELSE
*
* Solve the transposed variant.
*
CALL CTGSYL( 'C', IJB, N1, N2, A, LDA, A( I, I ), LDA,
$ WORK, N1, B, LDB, B( I, I ), LDB,
$ WORK( N1*N2+1 ), N1, DSCALE, DIF( 1 ),
$ WORK( N1*N2*2+1 ), LWORK-2*N1*N2, IWORK,
$ IERR )
END IF
GO TO 40
END IF
DIF( 1 ) = DSCALE / DIF( 1 )
*
* 1-norm-based estimate of Difl.
*
50 CONTINUE
CALL CLACN2( MN2, WORK( MN2+1 ), WORK, DIF( 2 ), KASE,
$ ISAVE )
IF( KASE.NE.0 ) THEN
IF( KASE.EQ.1 ) THEN
*
* Solve generalized Sylvester equation
*
CALL CTGSYL( 'N', IJB, N2, N1, A( I, I ), LDA, A, LDA,
$ WORK, N2, B( I, I ), LDB, B, LDB,
$ WORK( N1*N2+1 ), N2, DSCALE, DIF( 2 ),
$ WORK( N1*N2*2+1 ), LWORK-2*N1*N2, IWORK,
$ IERR )
ELSE
*
* Solve the transposed variant.
*
CALL CTGSYL( 'C', IJB, N2, N1, A( I, I ), LDA, A, LDA,
$ WORK, N2, B, LDB, B( I, I ), LDB,
$ WORK( N1*N2+1 ), N2, DSCALE, DIF( 2 ),
$ WORK( N1*N2*2+1 ), LWORK-2*N1*N2, IWORK,
$ IERR )
END IF
GO TO 50
END IF
DIF( 2 ) = DSCALE / DIF( 2 )
END IF
END IF
*
* If B(K,K) is complex, make it real and positive (normalization
* of the generalized Schur form) and Store the generalized
* eigenvalues of reordered pair (A, B)
*
DO 60 K = 1, N
DSCALE = ABS( B( K, K ) )
IF( DSCALE.GT.SAFMIN ) THEN
TEMP1 = CONJG( B( K, K ) / DSCALE )
TEMP2 = B( K, K ) / DSCALE
B( K, K ) = DSCALE
CALL CSCAL( N-K, TEMP1, B( K, K+1 ), LDB )
CALL CSCAL( N-K+1, TEMP1, A( K, K ), LDA )
IF( WANTQ )
$ CALL CSCAL( N, TEMP2, Q( 1, K ), 1 )
ELSE
B( K, K ) = CMPLX( ZERO, ZERO )
END IF
*
ALPHA( K ) = A( K, K )
BETA( K ) = B( K, K )
*
60 CONTINUE
*
70 CONTINUE
*
WORK( 1 ) = LWMIN
IWORK( 1 ) = LIWMIN
*
RETURN
*
* End of CTGSEN
*
END
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