NAME

stgsen - reorder the generalized real Schur decomposition of a real matrix pair (A, B) (in terms of an orthonormal equivalence trans- formation Q' * (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


SYNOPSIS

  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)
  INTEGER IJOB, N, LDA, LDB, LDQ, LDZ, M, LWORK, LIWORK, INFO
  INTEGER IWORK(*)
  LOGICAL WANTQ, WANTZ
  LOGICAL SELECT(*)
  REAL PL, PR
  REAL A(LDA,*), B(LDB,*), ALPHAR(*), ALPHAI(*), BETA(*), Q(LDQ,*), Z(LDZ,*), DIF(*), WORK(*)
  SUBROUTINE STGSEN_64( 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)
  INTEGER*8 IJOB, N, LDA, LDB, LDQ, LDZ, M, LWORK, LIWORK, INFO
  INTEGER*8 IWORK(*)
  LOGICAL*8 WANTQ, WANTZ
  LOGICAL*8 SELECT(*)
  REAL PL, PR
  REAL A(LDA,*), B(LDB,*), ALPHAR(*), ALPHAI(*), BETA(*), Q(LDQ,*), Z(LDZ,*), DIF(*), WORK(*)

F95 INTERFACE

  SUBROUTINE TGSEN( 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])
  INTEGER :: IJOB, N, LDA, LDB, LDQ, LDZ, M, LWORK, LIWORK, INFO
  INTEGER, DIMENSION(:) :: IWORK
  LOGICAL :: WANTQ, WANTZ
  LOGICAL, DIMENSION(:) :: SELECT
  REAL :: PL, PR
  REAL, DIMENSION(:) :: ALPHAR, ALPHAI, BETA, DIF, WORK
  REAL, DIMENSION(:,:) :: A, B, Q, Z
  SUBROUTINE TGSEN_64( 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])
  INTEGER(8) :: IJOB, N, LDA, LDB, LDQ, LDZ, M, LWORK, LIWORK, INFO
  INTEGER(8), DIMENSION(:) :: IWORK
  LOGICAL(8) :: WANTQ, WANTZ
  LOGICAL(8), DIMENSION(:) :: SELECT
  REAL :: PL, PR
  REAL, DIMENSION(:) :: ALPHAR, ALPHAI, BETA, DIF, WORK
  REAL, DIMENSION(:,:) :: A, B, Q, Z

C INTERFACE

#include <sunperf.h>

void stgsen(int ijob, logical wantq, logical wantz, logical *select, int n, float *a, int lda, float *b, int ldb, float *alphar, float *alphai, float *beta, float *q, int ldq, float *z, int ldz, int *m, float *pl, float *pr, float *dif, int *info);

void stgsen_64(long ijob, logical wantq, logical wantz, logical *select, long n, float *a, long lda, float *b, long ldb, float *alphar, float *alphai, float *beta, float *q, long ldq, float *z, long ldz, long *m, float *pl, float *pr, float *dif, long *info);


PURPOSE

stgsen reorders the generalized real Schur decomposition of a real matrix pair (A, B) (in terms of an orthonormal equivalence trans- formation Q' * (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.


ARGUMENTS


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

where N = n1+n2 and U' 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', then the reordered generalized real Schur form of (C, D) is given by

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

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:

ifu[(A11, B11), (A22, B22)] = sigma-min( Zu )

and ifl[(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

u = [ kron(In2, A11) -kron(A22', In1) ]

          [ kron(In2, B11)  -kron(B22', In1) ].

Here, Inx is the identity matrix of size nx and A22' 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 PS * 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

  = ( 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 11*R - L*A22 = -A12 11*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

PS * 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.

Based on contributions by

   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.