!>
!> 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  onto left and right eigenspaces w.r.t.
!> the selected cluster in the (1,1)-block.
!>
!> 

IJOB

!>          IJOB is 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  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

!>          WANTQ is LOGICAL
!>          .TRUE. : update the left transformation matrix Q;
!>          .FALSE.: do not update Q.
!> 

WANTZ

!>          WANTZ is LOGICAL
!>          .TRUE. : update the right transformation matrix Z;
!>          .FALSE.: do not update Z.
!> 

SELECT

!>          SELECT is 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

!>          N is INTEGER
!>          The order of the matrices A and B. N >= 0.
!> 

A

!>          A is 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

!>          LDA is INTEGER
!>          The leading dimension of the array A. LDA >= max(1,N).
!> 

B

!>          B is 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

!>          LDB is INTEGER
!>          The leading dimension of the array B. LDB >= max(1,N).
!> 

ALPHA

!>          ALPHA is COMPLEX array, dimension (N)
!> 

BETA

!>          BETA is 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

!>          Q is 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

!>          LDQ is INTEGER
!>          The leading dimension of the array Q. LDQ >= 1.
!>          If WANTQ = .TRUE., LDQ >= N.
!> 

Z

!>          Z is 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

!>          LDZ is INTEGER
!>          The leading dimension of the array Z. LDZ >= 1.
!>          If WANTZ = .TRUE., LDZ >= N.
!> 

M

!>          M is INTEGER
!>          The dimension of the specified pair of left and right
!>          eigenspaces, (deflating subspaces) 0 <= M <= N.
!> 

PL

!>          PL is REAL
!> 

PR

!>          PR is REAL
!>
!>          If IJOB = 1, 4 or 5, PL, PR are lower bounds on the
!>          reciprocal  of the norm of  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

!>          DIF is 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

!>          WORK is COMPLEX array, dimension (MAX(1,LWORK))
!>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
!> 

LWORK

!>          LWORK is 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

!>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
!>          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
!> 

LIWORK

!>          LIWORK is 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

!>          INFO is 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.
!> 

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

!>
!>  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
!>  conjugate 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.
!> 

Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

[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.

!>
!> DTGSEN reorders the generalized real Schur decomposition of a real
!> matrix pair (A, B) (in terms of an orthonormal equivalence trans-
!> formation Q**T * (A, B) * Z), so that a selected cluster of eigenvalues
!> appears in the leading diagonal blocks of the upper quasi-triangular
!> matrix A and the upper triangular B. The leading columns of Q and
!> Z form orthonormal bases of the corresponding left and right eigen-
!> spaces (deflating subspaces). (A, B) must be in generalized real
!> Schur canonical 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.
!>
!> DTGSEN also computes the generalized eigenvalues
!>
!>             w(j) = (ALPHAR(j) + i*ALPHAI(j))/BETA(j)
!>
!> of the reordered matrix pair (A, B).
!>
!> Optionally, DTGSEN computes the 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  onto left and right eigenspaces w.r.t.
!> the selected cluster in the (1,1)-block.
!> 

IJOB

!>          IJOB is 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  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

!>          WANTQ is LOGICAL
!>          .TRUE. : update the left transformation matrix Q;
!>          .FALSE.: do not update Q.
!> 

WANTZ

!>          WANTZ is LOGICAL
!>          .TRUE. : update the right transformation matrix Z;
!>          .FALSE.: do not update Z.
!> 

SELECT

!>          SELECT is LOGICAL array, dimension (N)
!>          SELECT specifies the eigenvalues in the selected cluster.
!>          To select a real eigenvalue w(j), SELECT(j) must be set to
!>          .TRUE.. To select a complex conjugate pair of eigenvalues
!>          w(j) and w(j+1), corresponding to a 2-by-2 diagonal block,
!>          either SELECT(j) or SELECT(j+1) or both must be set to
!>          .TRUE.; a complex conjugate pair of eigenvalues must be
!>          either both included in the cluster or both excluded.
!> 

N

!>          N is INTEGER
!>          The order of the matrices A and B. N >= 0.
!> 

A

!>          A is DOUBLE PRECISION array, dimension(LDA,N)
!>          On entry, the upper quasi-triangular matrix A, with (A, B) in
!>          generalized real Schur canonical form.
!>          On exit, A is overwritten by the reordered matrix A.
!> 

LDA

!>          LDA is INTEGER
!>          The leading dimension of the array A. LDA >= max(1,N).
!> 

B

!>          B is DOUBLE PRECISION array, dimension(LDB,N)
!>          On entry, the upper triangular matrix B, with (A, B) in
!>          generalized real Schur canonical form.
!>          On exit, B is overwritten by the reordered matrix B.
!> 

LDB

!>          LDB is INTEGER
!>          The leading dimension of the array B. LDB >= max(1,N).
!> 

ALPHAR

!>          ALPHAR is DOUBLE PRECISION array, dimension (N)
!> 

ALPHAI

!>          ALPHAI is DOUBLE PRECISION array, dimension (N)
!> 

BETA

!>          BETA is DOUBLE PRECISION array, dimension (N)
!>
!>          On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
!>          be the generalized eigenvalues.  ALPHAR(j) + ALPHAI(j)*i
!>          and BETA(j),j=1,...,N  are the diagonals of the complex Schur
!>          form (S,T) that would result if the 2-by-2 diagonal blocks of
!>          the real generalized Schur form of (A,B) were further reduced
!>          to triangular form using complex unitary transformations.
!>          If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
!>          positive, then the j-th and (j+1)-st eigenvalues are a
!>          complex conjugate pair, with ALPHAI(j+1) negative.
!> 

Q

!>          Q is DOUBLE PRECISION 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 orthogonal
!>          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

!>          LDQ is INTEGER
!>          The leading dimension of the array Q.  LDQ >= 1;
!>          and if WANTQ = .TRUE., LDQ >= N.
!> 

Z

!>          Z is DOUBLE PRECISION 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 orthogonal
!>          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

!>          LDZ is INTEGER
!>          The leading dimension of the array Z. LDZ >= 1;
!>          If WANTZ = .TRUE., LDZ >= N.
!> 

M

!>          M is INTEGER
!>          The dimension of the specified pair of left and right eigen-
!>          spaces (deflating subspaces). 0 <= M <= N.
!> 

PL

!>          PL is DOUBLE PRECISION
!> 

PR

!>          PR is DOUBLE PRECISION
!>
!>          If IJOB = 1, 4 or 5, PL, PR are lower bounds on the
!>          reciprocal of the norm of  onto left and right
!>          eigenspaces 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 and PR are not referenced.
!> 

DIF

!>          DIF is DOUBLE PRECISION 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.
!>          If M = 0 or N, DIF(1:2) = F-norm([A, B]).
!>          If IJOB = 0 or 1, DIF is not referenced.
!> 

WORK

!>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
!>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
!> 

LWORK

!>          LWORK is INTEGER
!>          The dimension of the array WORK. LWORK >=  4*N+16.
!>          If IJOB = 1, 2 or 4, LWORK >= MAX(4*N+16, 2*M*(N-M)).
!>          If IJOB = 3 or 5, LWORK >= MAX(4*N+16, 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

!>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
!>          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
!> 

LIWORK

!>          LIWORK is INTEGER
!>          The dimension of the array IWORK. LIWORK >= 1.
!>          If IJOB = 1, 2 or 4, LIWORK >=  N+6.
!>          If IJOB = 3 or 5, LIWORK >= MAX(2*M*(N-M), N+6).
!>
!>          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

!>          INFO is 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.
!> 

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

!>
!>  DTGSEN first collects the selected eigenvalues by computing
!>  orthogonal 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**T*(A, B)*W = (A11 A12) (B11 B12) n1
!>                              ( 0  A22),( 0  B22) n2
!>                                n1  n2    n1  n2
!>
!>  where N = n1+n2 and U**T means the 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**T, then the
!>  reordered generalized real Schur form of (C, D) is given by
!>
!>           (C, D) = (Q*U)*(U**T*(A, B)*W)*(Z*W)**T,
!>
!>  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**T, In1) ]
!>            [ kron(In2, B11)  -kron(B22**T, In1) ].
!>
!>  Here, Inx is the identity matrix of size nx and A22**T is the
!>  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 DLATDF), then the parameter
!>  IDIFJB (see below) should be changed from 3 to 4 (routine DLATDF
!>  (IJOB = 2 will be used)). See DTGSYL for more details.
!> 

Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

!>
!>  [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.
!> 

!>
!> STGSEN reorders the generalized real Schur decomposition of a real
!> matrix pair (A, B) (in terms of an orthonormal equivalence trans-
!> formation Q**T * (A, B) * Z), so that a selected cluster of eigenvalues
!> appears in the leading diagonal blocks of the upper quasi-triangular
!> matrix A and the upper triangular B. The leading columns of Q and
!> Z form orthonormal bases of the corresponding left and right eigen-
!> spaces (deflating subspaces). (A, B) must be in generalized real
!> Schur canonical form (as returned by SGGES), i.e. A is block upper
!> triangular with 1-by-1 and 2-by-2 diagonal blocks. B is upper
!> triangular.
!>
!> STGSEN also computes the generalized eigenvalues
!>
!>             w(j) = (ALPHAR(j) + i*ALPHAI(j))/BETA(j)
!>
!> of the reordered matrix pair (A, B).
!>
!> Optionally, STGSEN computes the 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  onto left and right eigenspaces w.r.t.
!> the selected cluster in the (1,1)-block.
!> 

IJOB

!>          IJOB is 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  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

!>          WANTQ is LOGICAL
!>          .TRUE. : update the left transformation matrix Q;
!>          .FALSE.: do not update Q.
!> 

WANTZ

!>          WANTZ is LOGICAL
!>          .TRUE. : update the right transformation matrix Z;
!>          .FALSE.: do not update Z.
!> 

SELECT

!>          SELECT is LOGICAL array, dimension (N)
!>          SELECT specifies the eigenvalues in the selected cluster.
!>          To select a real eigenvalue w(j), SELECT(j) must be set to
!>          .TRUE.. To select a complex conjugate pair of eigenvalues
!>          w(j) and w(j+1), corresponding to a 2-by-2 diagonal block,
!>          either SELECT(j) or SELECT(j+1) or both must be set to
!>          .TRUE.; a complex conjugate pair of eigenvalues must be
!>          either both included in the cluster or both excluded.
!> 

N

!>          N is INTEGER
!>          The order of the matrices A and B. N >= 0.
!> 

A

!>          A is REAL array, dimension(LDA,N)
!>          On entry, the upper quasi-triangular matrix A, with (A, B) in
!>          generalized real Schur canonical form.
!>          On exit, A is overwritten by the reordered matrix A.
!> 

LDA

!>          LDA is INTEGER
!>          The leading dimension of the array A. LDA >= max(1,N).
!> 

B

!>          B is REAL array, dimension(LDB,N)
!>          On entry, the upper triangular matrix B, with (A, B) in
!>          generalized real Schur canonical form.
!>          On exit, B is overwritten by the reordered matrix B.
!> 

LDB

!>          LDB is INTEGER
!>          The leading dimension of the array B. LDB >= max(1,N).
!> 

ALPHAR

!>          ALPHAR is REAL array, dimension (N)
!> 

ALPHAI

!>          ALPHAI is REAL array, dimension (N)
!> 

BETA

!>          BETA is REAL array, dimension (N)
!>
!>          On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
!>          be the generalized eigenvalues.  ALPHAR(j) + ALPHAI(j)*i
!>          and BETA(j),j=1,...,N  are the diagonals of the complex Schur
!>          form (S,T) that would result if the 2-by-2 diagonal blocks of
!>          the real generalized Schur form of (A,B) were further reduced
!>          to triangular form using complex unitary transformations.
!>          If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
!>          positive, then the j-th and (j+1)-st eigenvalues are a
!>          complex conjugate pair, with ALPHAI(j+1) negative.
!> 

Q

!>          Q is REAL 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 orthogonal
!>          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

!>          LDQ is INTEGER
!>          The leading dimension of the array Q.  LDQ >= 1;
!>          and if WANTQ = .TRUE., LDQ >= N.
!> 

Z

!>          Z is REAL 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 orthogonal
!>          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

!>          LDZ is INTEGER
!>          The leading dimension of the array Z. LDZ >= 1;
!>          If WANTZ = .TRUE., LDZ >= N.
!> 

M

!>          M is INTEGER
!>          The dimension of the specified pair of left and right eigen-
!>          spaces (deflating subspaces). 0 <= M <= N.
!> 

PL

!>          PL is REAL
!> 

PR

!>          PR is REAL
!>
!>          If IJOB = 1, 4 or 5, PL, PR are lower bounds on the
!>          reciprocal of the norm of  onto left and right
!>          eigenspaces 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 and PR are not referenced.
!> 

DIF

!>          DIF is 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.
!>          If M = 0 or N, DIF(1:2) = F-norm([A, B]).
!>          If IJOB = 0 or 1, DIF is not referenced.
!> 

WORK

!>          WORK is REAL array, dimension (MAX(1,LWORK))
!>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
!> 

LWORK

!>          LWORK is INTEGER
!>          The dimension of the array WORK. LWORK >=  4*N+16.
!>          If IJOB = 1, 2 or 4, LWORK >= MAX(4*N+16, 2*M*(N-M)).
!>          If IJOB = 3 or 5, LWORK >= MAX(4*N+16, 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

!>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
!>          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
!> 

LIWORK

!>          LIWORK is INTEGER
!>          The dimension of the array IWORK. LIWORK >= 1.
!>          If IJOB = 1, 2 or 4, LIWORK >=  N+6.
!>          If IJOB = 3 or 5, LIWORK >= MAX(2*M*(N-M), N+6).
!>
!>          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

!>          INFO is 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.
!> 

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

!>
!>  STGSEN first collects the selected eigenvalues by computing
!>  orthogonal 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**T*(A, B)*W = (A11 A12) (B11 B12) n1
!>                              ( 0  A22),( 0  B22) n2
!>                                n1  n2    n1  n2
!>
!>  where N = n1+n2 and U**T means the 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**T, then the
!>  reordered generalized real Schur form of (C, D) is given by
!>
!>           (C, D) = (Q*U)*(U**T*(A, B)*W)*(Z*W)**T,
!>
!>  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**T, In1) ]
!>            [ kron(In2, B11)  -kron(B22**T, In1) ].
!>
!>  Here, Inx is the identity matrix of size nx and A22**T is the
!>  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 SLATDF), then the parameter
!>  IDIFJB (see below) should be changed from 3 to 4 (routine SLATDF
!>  (IJOB = 2 will be used)). See STGSYL for more details.
!> 

Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

!>
!>  [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.
!> 

!>
!> ZTGSEN 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.
!>
!> ZTGSEN 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  onto left and right eigenspaces w.r.t.
!> the selected cluster in the (1,1)-block.
!>
!> 

IJOB

!>          IJOB is 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  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

!>          WANTQ is LOGICAL
!>          .TRUE. : update the left transformation matrix Q;
!>          .FALSE.: do not update Q.
!> 

WANTZ

!>          WANTZ is LOGICAL
!>          .TRUE. : update the right transformation matrix Z;
!>          .FALSE.: do not update Z.
!> 

SELECT

!>          SELECT is 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

!>          N is INTEGER
!>          The order of the matrices A and B. N >= 0.
!> 

A

!>          A is COMPLEX*16 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

!>          LDA is INTEGER
!>          The leading dimension of the array A. LDA >= max(1,N).
!> 

B

!>          B is COMPLEX*16 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

!>          LDB is INTEGER
!>          The leading dimension of the array B. LDB >= max(1,N).
!> 

ALPHA

!>          ALPHA is COMPLEX*16 array, dimension (N)
!> 

BETA

!>          BETA is COMPLEX*16 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

!>          Q is COMPLEX*16 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

!>          LDQ is INTEGER
!>          The leading dimension of the array Q. LDQ >= 1.
!>          If WANTQ = .TRUE., LDQ >= N.
!> 

Z

!>          Z is COMPLEX*16 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

!>          LDZ is INTEGER
!>          The leading dimension of the array Z. LDZ >= 1.
!>          If WANTZ = .TRUE., LDZ >= N.
!> 

M

!>          M is INTEGER
!>          The dimension of the specified pair of left and right
!>          eigenspaces, (deflating subspaces) 0 <= M <= N.
!> 

PL

!>          PL is DOUBLE PRECISION
!> 

PR

!>          PR is DOUBLE PRECISION
!>
!>          If IJOB = 1, 4 or 5, PL, PR are lower bounds on the
!>          reciprocal  of the norm of  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

!>          DIF is DOUBLE PRECISION 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 ZLACN2.
!>          If M = 0 or N, DIF(1:2) = F-norm([A, B]).
!>          If IJOB = 0 or 1, DIF is not referenced.
!> 

WORK

!>          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
!>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
!> 

LWORK

!>          LWORK is 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

!>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
!>          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
!> 

LIWORK

!>          LIWORK is 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

!>          INFO is 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.
!> 

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

!>
!>  ZTGSEN 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**H, 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
!>  conjugate 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 ZLATDF), then the parameter
!>  IDIFJB (see below) should be changed from 3 to 4 (routine ZLATDF
!>  (IJOB = 2 will be used)). See ZTGSYL for more details.
!> 

Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

[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.