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



NAME
       CTGSEN

SYNOPSIS
       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)



PURPOSE
            CTGSEN reorders the generalized Schur decomposition of a complex
            matrix pair (A, B) (in terms of an unitary equivalence trans-
            formation Q**H * (A, B) * Z), so that a selected cluster of eigenvalues
            appears in the leading diagonal blocks of the pair (A,B). The leading
            columns of Q and Z form unitary bases of the corresponding left and
            right eigenspaces (deflating subspaces). (A, B) must be in
            generalized Schur canonical form, that is, A and B are both upper
            triangular.

            CTGSEN also computes the generalized eigenvalues

                     w(j)= ALPHA(j) / BETA(j)

            of the reordered matrix pair (A, B).

            Optionally, the routine computes estimates of reciprocal condition
            numbers for eigenvalues and eigenspaces. These are Difu[(A11,B11),
            (A22,B22)] and Difl[(A11,B11), (A22,B22)], i.e. the separation(s)
            between the matrix pairs (A11, B11) and (A22,B22) that correspond to
            the selected cluster and the eigenvalues outside the cluster, resp.,
            and norms of "projections" onto left and right eigenspaces w.r.t.
            the selected cluster in the (1,1)-block.




ARGUMENTS
           IJOB      (input)
                     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 an eigenvalue w(j), SELECT(j) must be set to
                     .TRUE..

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

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

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

           ALPHA     (output)
                     ALPHA is COMPLEX array, dimension (N)

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

           Z         (input/output)
                     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       (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
                     eigenspaces, (deflating subspaces) 0 <= M <= N.

           PL        (output)
                     PL is REAL

           PR        (output)
                     PR is REAL

                     If IJOB = 1, 4 or 5, PL, PR are lower bounds on the
                     reciprocal  of the norm of "projections" onto left and right
                     eigenspace with respect to the selected cluster.
                     0 < PL, PR <= 1.
                     If M = 0 or M = N, PL = PR  = 1.
                     If IJOB = 0, 2 or 3 PL, PR are not referenced.

           DIF       (output)
                     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      (output)
                     WORK is COMPLEX 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 >=  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     (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+2;
                     If IJOB = 3 or 5, LIWORK >= MAX(N+2, 2*M*(N-M));

                     If LIWORK = -1, then a workspace query is assumed; the
                     routine only calculates the optimal size of the IWORK array,
                     returns this value as the first entry of the IWORK array, and
                     no error message related to LIWORK is issued by XERBLA.

           INFO      (output)
                     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
             CTGSEN first collects the selected eigenvalues by computing unitary
             U and W that move them to the top left corner of (A, B). In other
             words, the selected eigenvalues are the eigenvalues of (A11, B11) in

                         U**H*(A, B)*W = (A11 A12) (B11 B12) n1
                                         ( 0  A22),( 0  B22) n2
                                           n1  n2    n1  n2

             where N = n1+n2 and U**H means the conjugate transpose of U. The first
             n1 columns of U and W span the specified pair of left and right
             eigenspaces (deflating subspaces) of (A, B).

             If (A, B) has been obtained from the generalized real Schur
             decomposition of a matrix pair (C, D) = Q*(A, B)*Z', then the
             reordered generalized Schur form of (C, D) is given by

                      (C, D) = (Q*U)*(U**H *(A, B)*W)*(Z*W)**H,

             and the first n1 columns of Q*U and Z*W span the corresponding
             deflating subspaces of (C, D) (Q and Z store Q*U and Z*W, resp.).

             Note that if the selected eigenvalue is sufficiently ill-conditioned,
             then its value may differ significantly from its value before
             reordering.

             The reciprocal condition numbers of the left and right eigenspaces
             spanned by the first n1 columns of U and W (or Q*U and Z*W) may
             be returned in DIF(1:2), corresponding to Difu and Difl, resp.

             The Difu and Difl are defined as:

                  Difu[(A11, B11), (A22, B22)] = sigma-min( Zu )
             and
                  Difl[(A11, B11), (A22, B22)] = Difu[(A22, B22), (A11, B11)],

             where sigma-min(Zu) is the smallest singular value of the
             (2*n1*n2)-by-(2*n1*n2) matrix

                  Zu = [ kron(In2, A11)  -kron(A22**H, In1) ]
                       [ kron(In2, B11)  -kron(B22**H, In1) ].

             Here, Inx is the identity matrix of size nx and A22**H is the
             conjuguate transpose of A22. kron(X, Y) is the Kronecker product between
             the matrices X and Y.

             When DIF(2) is small, small changes in (A, B) can cause large changes
             in the deflating subspace. An approximate (asymptotic) bound on the
             maximum angular error in the computed deflating subspaces is

                  EPS * norm((A, B)) / DIF(2),

             where EPS is the machine precision.

             The reciprocal norm of the projectors on the left and right
             eigenspaces associated with (A11, B11) may be returned in PL and PR.
             They are computed as follows. First we compute L and R so that
             P*(A, B)*Q is block diagonal, where

                  P = ( I -L ) n1           Q = ( I R ) n1
                      ( 0  I ) n2    and        ( 0 I ) n2
                        n1 n2                    n1 n2

             and (L, R) is the solution to the generalized Sylvester equation

                  A11*R - L*A22 = -A12
                  B11*R - L*B22 = -B12

             Then PL = (F-norm(L)**2+1)**(-1/2) and PR = (F-norm(R)**2+1)**(-1/2).
             An approximate (asymptotic) bound on the average absolute error of
             the selected eigenvalues is

                  EPS * norm((A, B)) / PL.

             There are also global error bounds which valid for perturbations up
             to a certain restriction:  A lower bound (x) on the smallest
             F-norm(E,F) for which an eigenvalue of (A11, B11) may move and
             coalesce with an eigenvalue of (A22, B22) under perturbation (E,F),
             (i.e. (A + E, B + F), is

              x = min(Difu,Difl)/((1/(PL*PL)+1/(PR*PR))**(1/2)+2*max(1/PL,1/PR)).

             An approximate bound on x can be computed from DIF(1:2), PL and PR.

             If y = ( F-norm(E,F) / x) <= 1, the angles between the perturbed
             (L', R') and unperturbed (L, R) left and right deflating subspaces
             associated with the selected cluster in the (1,1)-blocks can be
             bounded as

              max-angle(L, L') <= arctan( y * PL / (1 - y * (1 - PL * PL)**(1/2))
              max-angle(R, R') <= arctan( y * PR / (1 - y * (1 - PR * PR)**(1/2))

             See LAPACK User's Guide section 4.11 or the following references
             for more information.

             Note that if the default method for computing the Frobenius-norm-
             based estimate DIF is not wanted (see CLATDF), then the parameter
             IDIFJB (see below) should be changed from 3 to 4 (routine CLATDF
             (IJOB = 2 will be used)). See CTGSYL for more details.



LAPACK routine                  31 October 2017                      CTGSEN(3)