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 "projections" 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 "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

          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 "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

          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.

June 2016

  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.