tgsen(3)

NAME
tgsen - tgsen: reorder generalized Schur form

SYNOPSIS

  Functions

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)
CTGSEN
subroutine dtgsen (ijob, wantq, wantz, select, n, a, lda, b, ldb, alphar, alphai, beta, q, ldq, z, ldz, m, pl, pr, dif, work, lwork, iwork, liwork, info)
DTGSEN
subroutine stgsen (ijob, wantq, wantz, select, n, a, lda, b, ldb, alphar, alphai, beta, q, ldq, z, ldz, m, pl, pr, dif, work, lwork, iwork, liwork, info)
STGSEN
subroutine ztgsen (ijob, wantq, wantz, select, n, a, lda, b, ldb, alpha, beta, q, ldq, z, ldz, m, pl, pr, dif, work, lwork, iwork, liwork, info)
ZTGSEN

Detailed Description

Function Documentation

  subroutine ctgsen (integer ijob, logical wantq, logical wantz, logical, dimension( * ) select, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( * ) alpha, complex, dimension( * ) beta, complex, dimension( ldq, * ) q, integer ldq, complex, dimension( ldz, * ) z, integer ldz, integer m, real pl, real pr, real, dimension( * ) dif, complex, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, integer info)

CTGSEN

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

Parameters
    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.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

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
    !>  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.
    !>

Contributors:
    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.

  subroutine dtgsen (integer ijob, logical wantq, logical wantz, logical, dimension( * ) select, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( * ) alphar, double precision, dimension( * ) alphai, double precision, dimension( * ) beta, double precision, dimension( ldq, * ) q, integer ldq, double precision, dimension( ldz, * ) z, integer ldz, integer m, double precision pl, double precision pr, double precision, dimension( * ) dif, double precision, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, integer info)

DTGSEN

Purpose:

    !>
    !> 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.
    !>

Parameters
    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.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>  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.
    !>

Contributors:
    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.
    !>

  subroutine stgsen (integer ijob, logical wantq, logical wantz, logical, dimension( * ) select, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real, dimension( * ) alphar, real, dimension( * ) alphai, real, dimension( * ) beta, real, dimension( ldq, * ) q, integer ldq, real, dimension( ldz, * ) z, integer ldz, integer m, real pl, real pr, real, dimension( * ) dif, real, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, integer info)

STGSEN

Purpose:

    !>
    !> 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.
    !>

Parameters
    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.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>  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.
    !>

Contributors:
    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.
    !>

  subroutine ztgsen (integer ijob, logical wantq, logical wantz, logical, dimension( * ) select, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( * ) alpha, complex*16, dimension( * ) beta, complex*16, dimension( ldq, * ) q, integer ldq, complex*16, dimension( ldz, * ) z, integer ldz, integer m, double precision pl, double precision pr, double precision, dimension( * ) dif, complex*16, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, integer info)

ZTGSEN

Purpose:

    !>
    !> 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.
    !>
    !>

Parameters
    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.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>  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.
    !>

Contributors:
    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.

Author

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