1DTGSEN(1) LAPACK routine (version 3.2) DTGSEN(1)
2
6 DTGSEN - reorders the generalized real Schur decomposition of a real
7 matrix pair (A, B) (in terms of an orthonormal equivalence trans- for‐
8 mation Q' * (A, B) * Z), so that a selected cluster of eigenvalues
9 appears in the leading diagonal blocks of the upper quasi-triangular
10 matrix A and the upper triangular B
11
13 SUBROUTINE DTGSEN( IJOB, WANTQ, WANTZ, SELECT, N, A, LDA, B, LDB,
14 ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, M, PL, PR,
15 DIF, WORK, LWORK, IWORK, LIWORK, INFO )
16 LOGICAL WANTQ, WANTZ
18 INTEGER IJOB, INFO, LDA, LDB, LDQ, LDZ, LIWORK, LWORK, M, N
20 DOUBLE PRECISION PL, PR
22 LOGICAL SELECT( * )
24 INTEGER IWORK( * )
26 DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ), B(
28 LDB, * ), BETA( * ), DIF( * ), Q( LDQ, * ), WORK( *
29 ), Z( LDZ, * )
30
32 DTGSEN reorders the generalized real Schur decomposition of a real
33 matrix pair (A, B) (in terms of an orthonormal equivalence trans- for‐
34 mation Q' * (A, B) * Z), so that a selected cluster of eigenvalues
35 appears in the leading diagonal blocks of the upper quasi-triangular
36 matrix A and the upper triangular B. The leading columns of Q and Z
37 form orthonormal bases of the corresponding left and right eigen- spa‐
38 ces (deflating subspaces). (A, B) must be in generalized real Schur
39 canonical form (as returned by DGGES), i.e. A is block upper triangular
40 with 1-by-1 and 2-by-2 diagonal blocks. B is upper triangular.
41 DTGSEN also computes the generalized eigenvalues
42 w(j) = (ALPHAR(j) + i*ALPHAI(j))/BETA(j)
43 of the reordered matrix pair (A, B).
44 Optionally, DTGSEN computes the estimates of reciprocal condition num‐
45 bers for eigenvalues and eigenspaces. These are Difu[(A11,B11),
46 (A22,B22)] and Difl[(A11,B11), (A22,B22)], i.e. the separation(s)
47 between the matrix pairs (A11, B11) and (A22,B22) that correspond to
48 the selected cluster and the eigenvalues outside the cluster, resp.,
49 and norms of "projections" onto left and right eigenspaces w.r.t. the
50 selected cluster in the (1,1)-block.
51
53 IJOB (input) INTEGER
54 Specifies whether condition numbers are required for the clus‐
55 ter of eigenvalues (PL and PR) or the deflating subspaces (Difu
56 and Difl):
57 =0: Only reorder w.r.t. SELECT. No extras.
58 =1: Reciprocal of norms of "projections" onto left and right
59 eigenspaces w.r.t. the selected cluster (PL and PR). =2: Upper
60 bounds on Difu and Difl. F-norm-based estimate
61 (DIF(1:2)).
62 =3: Estimate of Difu and Difl. 1-norm-based estimate
63 (DIF(1:2)). About 5 times as expensive as IJOB = 2. =4: Com‐
64 pute PL, PR and DIF (i.e. 0, 1 and 2 above): Economic version
65 to get it all. =5: Compute PL, PR and DIF (i.e. 0, 1 and 3
66 above)
67 WANTQ (input) LOGICAL
69 WANTZ (input) LOGICAL
71 SELECT (input) LOGICAL array, dimension (N)
73 SELECT specifies the eigenvalues in the selected cluster. To
74 select a real eigenvalue w(j), SELECT(j) must be set to .TRUE..
75 To select a complex conjugate pair of eigenvalues w(j) and
76 w(j+1), corresponding to a 2-by-2 diagonal block, either
77 SELECT(j) or SELECT(j+1) or both must be set to .TRUE.; a com‐
78 plex conjugate pair of eigenvalues must be either both included
79 in the cluster or both excluded.
80 N (input) INTEGER
82 The order of the matrices A and B. N >= 0.
83 A (input/output) DOUBLE PRECISION array, dimension(LDA,N)
85 On entry, the upper quasi-triangular matrix A, with (A, B) in
86 generalized real Schur canonical form. On exit, A is overwrit‐
87 ten by the reordered matrix A.
88 LDA (input) INTEGER
90 The leading dimension of the array A. LDA >= max(1,N).
91 B (input/output) DOUBLE PRECISION array, dimension(LDB,N)
93 On entry, the upper triangular matrix B, with (A, B) in gener‐
94 alized real Schur canonical form. On exit, B is overwritten by
95 the reordered matrix B.
96 LDB (input) INTEGER
98 The leading dimension of the array B. LDB >= max(1,N).
99 ALPHAR (output) DOUBLE PRECISION array, dimension (N)
101 ALPHAI (output) DOUBLE PRECISION array, dimension (N) BETA
102 (output) DOUBLE PRECISION array, dimension (N) On exit,
103 (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will be the gen‐
104 eralized eigenvalues. ALPHAR(j) + ALPHAI(j)*i and
105 BETA(j),j=1,...,N are the diagonals of the complex Schur form
106 (S,T) that would result if the 2-by-2 diagonal blocks of the
107 real generalized Schur form of (A,B) were further reduced to
108 triangular form using complex unitary transformations. If
109 ALPHAI(j) is zero, then the j-th eigenvalue is real; if posi‐
110 tive, then the j-th and (j+1)-st eigenvalues are a complex con‐
111 jugate pair, with ALPHAI(j+1) negative.
112 Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
114 On entry, if WANTQ = .TRUE., Q is an N-by-N matrix. On exit, Q
115 has been postmultiplied by the left orthogonal transformation
116 matrix which reorder (A, B); The leading M columns of Q form
117 orthonormal bases for the specified pair of left eigenspaces
118 (deflating subspaces). If WANTQ = .FALSE., Q is not refer‐
119 enced.
120 LDQ (input) INTEGER
122 The leading dimension of the array Q. LDQ >= 1; and if WANTQ =
123 .TRUE., LDQ >= N.
124 Z (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
126 On entry, if WANTZ = .TRUE., Z is an N-by-N matrix. On exit, Z
127 has been postmultiplied by the left orthogonal transformation
128 matrix which reorder (A, B); The leading M columns of Z form
129 orthonormal bases for the specified pair of left eigenspaces
130 (deflating subspaces). If WANTZ = .FALSE., Z is not refer‐
131 enced.
132 LDZ (input) INTEGER
134 The leading dimension of the array Z. LDZ >= 1; If WANTZ =
135 .TRUE., LDZ >= N.
136 M (output) INTEGER
138 The dimension of the specified pair of left and right eigen-
139 spaces (deflating subspaces). 0 <= M <= N.
140 PL (output) DOUBLE PRECISION
142 PR (output) DOUBLE PRECISION If IJOB = 1, 4 or 5, PL, PR
143 are lower bounds on the reciprocal of the norm of "projections"
144 onto left and right eigenspaces with respect to the selected
145 cluster. 0 < PL, PR <= 1. If M = 0 or M = N, PL = PR = 1.
146 If IJOB = 0, 2 or 3, PL and PR are not referenced.
147 DIF (output) DOUBLE PRECISION array, dimension (2).
149 If IJOB >= 2, DIF(1:2) store the estimates of Difu and Difl.
150 If IJOB = 2 or 4, DIF(1:2) are F-norm-based upper bounds on
151 Difu and Difl. If IJOB = 3 or 5, DIF(1:2) are 1-norm-based
152 estimates of Difu and Difl. If M = 0 or N, DIF(1:2) = F-
153 norm([A, B]). If IJOB = 0 or 1, DIF is not referenced.
154 WORK (workspace/output) DOUBLE PRECISION array,
156 dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns
157 the optimal LWORK.
158 LWORK (input) INTEGER
160 The dimension of the array WORK. LWORK >= 4*N+16. If IJOB =
161 1, 2 or 4, LWORK >= MAX(4*N+16, 2*M*(N-M)). If IJOB = 3 or 5,
162 LWORK >= MAX(4*N+16, 4*M*(N-M)). If LWORK = -1, then a
163 workspace query is assumed; the routine only calculates the
164 optimal size of the WORK array, returns this value as the first
165 entry of the WORK array, and no error message related to LWORK
166 is issued by XERBLA.
167 IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
169 IF IJOB = 0, IWORK is not referenced. Otherwise, on exit, if
170 INFO = 0, IWORK(1) returns the optimal LIWORK.
171 LIWORK (input) INTEGER
173 The dimension of the array IWORK. LIWORK >= 1. If IJOB = 1, 2
174 or 4, LIWORK >= N+6. If IJOB = 3 or 5, LIWORK >= MAX(2*M*(N-
175 M), N+6). If LIWORK = -1, then a workspace query is assumed;
176 the routine only calculates the optimal size of the IWORK
177 array, returns this value as the first entry of the IWORK
178 array, and no error message related to LIWORK is issued by
179 XERBLA.
180 INFO (output) INTEGER
182 =0: Successful exit.
183 <0: If INFO = -i, the i-th argument had an illegal value.
184 =1: Reordering of (A, B) failed because the transformed matrix
185 pair (A, B) would be too far from generalized Schur form; the
186 problem is very ill-conditioned. (A, B) may have been par‐
187 tially reordered. If requested, 0 is returned in DIF(*), PL
188 and PR.
189
191 DTGSEN first collects the selected eigenvalues by computing orthogonal
192 U and W that move them to the top left corner of (A, B). In other
193 words, the selected eigenvalues are the eigenvalues of (A11, B11) in:
194 U'*(A, B)*W = (A11 A12) (B11 B12) n1
195 ( 0 A22),( 0 B22) n2
196 n1 n2 n1 n2
197 where N = n1+n2 and U' means the transpose of U. The first n1 columns
198 of U and W span the specified pair of left and right eigenspaces
199 (deflating subspaces) of (A, B).
200 If (A, B) has been obtained from the generalized real Schur decomposi‐
201 tion of a matrix pair (C, D) = Q*(A, B)*Z', then the reordered general‐
202 ized real Schur form of (C, D) is given by
203 (C, D) = (Q*U)*(U'*(A, B)*W)*(Z*W)',
204 and the first n1 columns of Q*U and Z*W span the corresponding deflat‐
205 ing subspaces of (C, D) (Q and Z store Q*U and Z*W, resp.). Note that
206 if the selected eigenvalue is sufficiently ill-conditioned, then its
207 value may differ significantly from its value before reordering.
208 The reciprocal condition numbers of the left and right eigenspaces
209 spanned by the first n1 columns of U and W (or Q*U and Z*W) may be
210 returned in DIF(1:2), corresponding to Difu and Difl, resp. The Difu
211 and Difl are defined as:
212 Difu[(A11, B11), (A22, B22)] = sigma-min( Zu )
213 and
214 Difl[(A11, B11), (A22, B22)] = Difu[(A22, B22), (A11, B11)], where
215 sigma-min(Zu) is the smallest singular value of the
216 (2*n1*n2)-by-(2*n1*n2) matrix
217 Zu = [ kron(In2, A11) -kron(A22', In1) ]
218 [ kron(In2, B11) -kron(B22', In1) ].
219 Here, Inx is the identity matrix of size nx and A22' is the transpose
220 of A22. kron(X, Y) is the Kronecker product between the matrices X and
221 Y.
222 When DIF(2) is small, small changes in (A, B) can cause large changes
223 in the deflating subspace. An approximate (asymptotic) bound on the
224 maximum angular error in the computed deflating subspaces is
225 EPS * norm((A, B)) / DIF(2),
226 where EPS is the machine precision.
227 The reciprocal norm of the projectors on the left and right eigenspaces
228 associated with (A11, B11) may be returned in PL and PR. They are com‐
229 puted as follows. First we compute L and R so that P*(A, B)*Q is block
230 diagonal, where
231 P = ( I -L ) n1 Q = ( I R ) n1
232 ( 0 I ) n2 and ( 0 I ) n2
233 n1 n2 n1 n2
234 and (L, R) is the solution to the generalized Sylvester equation
235 A11*R - L*A22 = -A12
236 B11*R - L*B22 = -B12
237 Then PL = (F-norm(L)**2+1)**(-1/2) and PR = (F-norm(R)**2+1)**(-1/2).
238 An approximate (asymptotic) bound on the average absolute error of the
239 selected eigenvalues is
240 EPS * norm((A, B)) / PL.
241 There are also global error bounds which valid for perturbations up to
242 a certain restriction: A lower bound (x) on the smallest F-norm(E,F)
243 for which an eigenvalue of (A11, B11) may move and coalesce with an ei‐
244 genvalue of (A22, B22) under perturbation (E,F), (i.e. (A + E, B + F),
245 is
246 x = min(Difu,Difl)/((1/(PL*PL)+1/(PR*PR))**(1/2)+2*max(1/PL,1/PR)).
247 An approximate bound on x can be computed from DIF(1:2), PL and PR. If
248 y = ( F-norm(E,F) / x) <= 1, the angles between the perturbed (L', R')
249 and unperturbed (L, R) left and right deflating subspaces associated
250 with the selected cluster in the (1,1)-blocks can be bounded as
251 max-angle(L, L') <= arctan( y * PL / (1 - y * (1 - PL * PL)**(1/2))
252 max-angle(R, R') <= arctan( y * PR / (1 - y * (1 - PR * PR)**(1/2))
253 See LAPACK User's Guide section 4.11 or the following references for
254 more information.
255 Note that if the default method for computing the Frobenius-norm- based
256 estimate DIF is not wanted (see DLATDF), then the parameter IDIFJB (see
257 below) should be changed from 3 to 4 (routine DLATDF (IJOB = 2 will be
258 used)). See DTGSYL for more details.
259 Based on contributions by
260 Bo Kagstrom and Peter Poromaa, Department of Computing Science,
261 Umea University, S-901 87 Umea, Sweden.
262 References
263 ==========
264 [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
265 Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
266 M.S. Moonen et al (eds), Linear Algebra for Large Scale and
267 Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
268 [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified
269 Eigenvalues of a Regular Matrix Pair (A, B) and Condition
270 Estimation: Theory, Algorithms and Software,
271 Report UMINF - 94.04, Department of Computing Science, Umea
272 University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working
273 Note 87. To appear in Numerical Algorithms, 1996.
274 [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
275 for Solving the Generalized Sylvester Equation and Estimating the
276 Separation between Regular Matrix Pairs, Report UMINF - 93.23,
277 Department of Computing Science, Umea University, S-901 87 Umea,
278 Sweden, December 1993, Revised April 1994, Also as LAPACK Working
279 Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1,
280 1996.
281 LAPACK routine (version 3.2) November 2008 DTGSEN(1)