DTGSEN(3)      LAPACK routine of NEC Numeric Library Collection      DTGSEN(3)



NAME
       DTGSEN

SYNOPSIS
       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)



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




ARGUMENTS
           IJOB      (input)
                     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 "projections" onto left and right
                          eigenspaces w.r.t. the selected cluster (PL and PR).
                      =2: Upper bounds on Difu and Difl. F-norm-based estimate
                          (DIF(1:2)).
                      =3: Estimate of Difu and Difl. 1-norm-based estimate
                          (DIF(1:2)).
                          About 5 times as expensive as IJOB = 2.
                      =4: Compute PL, PR and DIF (i.e. 0, 1 and 2 above): Economic
                          version to get it all.
                      =5: Compute PL, PR and DIF (i.e. 0, 1 and 3 above)

           WANTQ     (input)
                     WANTQ is LOGICAL
                     .TRUE. : update the left transformation matrix Q;
                     .FALSE.: do not update Q.

           WANTZ     (input)
                     WANTZ is LOGICAL
                     .TRUE. : update the right transformation matrix Z;
                     .FALSE.: do not update Z.

           SELECT    (input)
                     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         (input)
                     N is INTEGER
                     The order of the matrices A and B. N >= 0.

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

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

           ALPHAR    (output)
                     ALPHAR is DOUBLE PRECISION array, dimension (N)

           ALPHAI    (output)
                     ALPHAI is DOUBLE PRECISION array, dimension (N)

           BETA      (output)
                     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         (input/output)
                     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       (input)
                     LDQ is INTEGER
                     The leading dimension of the array Q.  LDQ >= 1;
                     and if WANTQ = .TRUE., LDQ >= N.

           Z         (input/output)
                     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       (input)
                     LDZ is INTEGER
                     The leading dimension of the array Z. LDZ >= 1;
                     If WANTZ = .TRUE., LDZ >= N.

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

           PL        (output)
                     PL is DOUBLE PRECISION


           PR        (output)
                     PR is DOUBLE PRECISION

                     If IJOB = 1, 4 or 5, PL, PR are lower bounds on the
                     reciprocal of the norm of "projections" 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       (output)
                     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      (output)
                     WORK is DOUBLE PRECISION array,
                     dimension (MAX(1,LWORK))
                     On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

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

           LIWORK    (input)
                     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      (output)
                     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.






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.












LAPACK routine                  31 October 2017                      DTGSEN(3)