dtgsen (l)  Linux Manuals
dtgsen: reorders the generalized real Schur decomposition of a real matrix pair (A, B) (in terms of an orthonormal equivalence trans formation Qaq * (A, B) * Z), so that a selected cluster of eigenvalues appears in the leading diagonal blocks of the upper quasitriangular matrix A and the upper triangular B
NAME
DTGSEN  reorders the generalized real Schur decomposition of a real matrix pair (A, B) (in terms of an orthonormal equivalence trans formation Qaq * (A, B) * Z), so that a selected cluster of eigenvalues appears in the leading diagonal blocks of the upper quasitriangular matrix A and the upper triangular BSYNOPSIS
 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 )
 LOGICAL WANTQ, WANTZ
 INTEGER IJOB, INFO, LDA, LDB, LDQ, LDZ, LIWORK, LWORK, M, N
 DOUBLE PRECISION PL, PR
 LOGICAL SELECT( * )
 INTEGER IWORK( * )
 DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ), B( LDB, * ), BETA( * ), DIF( * ), Q( LDQ, * ), WORK( * ), Z( LDZ, * )
PURPOSE
DTGSEN reorders the generalized real Schur decomposition of a real matrix pair (A, B) (in terms of an orthonormal equivalence trans formation Qaq * (A, B) * Z), so that a selected cluster of eigenvalues appears in the leading diagonal blocks of the upper quasitriangular 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 1by1 and 2by2 diagonal blocks. B is upper triangular.DTGSEN also computes the generalized eigenvalues
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) 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. Fnormbased estimate
(DIF(1:2)).
=3: Estimate of Difu and Difl. 1normbased 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) LOGICAL

 WANTZ (input) LOGICAL

 SELECT (input) 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 2by2 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) INTEGER
 The order of the matrices A and B. N >= 0.
 A (input/output) DOUBLE PRECISION array, dimension(LDA,N)
 On entry, the upper quasitriangular matrix A, with (A, B) in generalized real Schur canonical form. On exit, A is overwritten by the reordered matrix A.
 LDA (input) INTEGER
 The leading dimension of the array A. LDA >= max(1,N).
 B (input/output) 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) INTEGER
 The leading dimension of the array B. LDB >= max(1,N).
 ALPHAR (output) DOUBLE PRECISION array, dimension (N)
 ALPHAI (output) DOUBLE PRECISION array, dimension (N) BETA (output) 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 2by2 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 jth eigenvalue is real; if positive, then the jth and (j+1)st eigenvalues are a complex conjugate pair, with ALPHAI(j+1) negative.
 Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
 On entry, if WANTQ = .TRUE., Q is an NbyN 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) INTEGER
 The leading dimension of the array Q. LDQ >= 1; and if WANTQ = .TRUE., LDQ >= N.
 Z (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
 On entry, if WANTZ = .TRUE., Z is an NbyN 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) INTEGER
 The leading dimension of the array Z. LDZ >= 1; If WANTZ = .TRUE., LDZ >= N.
 M (output) INTEGER
 The dimension of the specified pair of left and right eigen spaces (deflating subspaces). 0 <= M <= N.
 PL (output) DOUBLE PRECISION
 PR (output) 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) 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 Fnormbased upper bounds on
Difu and Difl. If IJOB = 3 or 5, DIF(1:2) are 1normbased estimates of Difu and Difl. If M = 0 or N, DIF(1:2) = Fnorm([A, B]). If IJOB = 0 or 1, DIF is not referenced.  WORK (workspace/output) DOUBLE PRECISION array,
 dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
 LWORK (input) INTEGER
 The dimension of the array WORK. LWORK >= 4*N+16. If IJOB = 1, 2 or 4, LWORK >= MAX(4*N+16, 2*M*(NM)). If IJOB = 3 or 5, LWORK >= MAX(4*N+16, 4*M*(NM)). 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 (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
 IF IJOB = 0, IWORK is not referenced. Otherwise, on exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
 LIWORK (input) 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*(NM), 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) INTEGER

=0: Successful exit.
<0: If INFO = i, the ith 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 illconditioned. (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:where N = n1+n2 and Uaq 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)*Zaq, then the reordered generalized real Schur form of (C, D) is given by
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 illconditioned, 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:
and
Here, Inx is the identity matrix of size nx and A22aq 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
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
and (L, R) is the solution to the generalized Sylvester equation
Then PL = (Fnorm(L)**2+1)**(1/2) and PR = (Fnorm(R)**2+1)**(1/2). An approximate (asymptotic) bound on the average absolute error of the selected eigenvalues is
There are also global error bounds which valid for perturbations up to a certain restriction: A lower bound (x) on the smallest Fnorm(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
Note that if the default method for computing the Frobeniusnorm 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.
Based on contributions by
Bo Kagstrom and Peter Poromaa, Department of Computing Science,
Umea University, S901 87 Umea, Sweden.
References
==========
[1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
[3] B. Kagstrom and P. Poromaa, LAPACKStyle Algorithms and Software