1STGSEN(1) LAPACK routine (version 3.1.1) STGSEN(1)
2
6 STGSEN - the generalized real Schur decomposition of a real matrix pair
7 (A, B) (in terms of an orthonormal equivalence trans- formation Q' *
8 (A, B) * Z), so that a selected cluster of eigenvalues appears in the
9 leading diagonal blocks of the upper quasi-triangular matrix A and the
10 upper triangular B
11
13 SUBROUTINE STGSEN( 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 REAL PL, PR
22 LOGICAL SELECT( * )
24 INTEGER IWORK( * )
26 REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ), B( LDB, * ),
28 BETA( * ), DIF( * ), Q( LDQ, * ), WORK( * ), Z( LDZ,
29 * )
30
32 STGSEN 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 SGGES), i.e. A is block upper triangular
40 with 1-by-1 and 2-by-2 diagonal blocks. B is upper triangular.
41 STGSEN also computes the generalized eigenvalues
43 w(j) = (ALPHAR(j) + i*ALPHAI(j))/BETA(j)
45 of the reordered matrix pair (A, B).
47 Optionally, STGSEN computes the estimates of reciprocal condition num‐
49 bers for eigenvalues and eigenspaces. These are Difu[(A11,B11),
50 (A22,B22)] and Difl[(A11,B11), (A22,B22)], i.e. the separation(s)
51 between the matrix pairs (A11, B11) and (A22,B22) that correspond to
52 the selected cluster and the eigenvalues outside the cluster, resp.,
53 and norms of "projections" onto left and right eigenspaces w.r.t. the
54 selected cluster in the (1,1)-block.
55
58 IJOB (input) INTEGER
59 Specifies whether condition numbers are required for the clus‐
60 ter of eigenvalues (PL and PR) or the deflating subspaces (Difu
61 and Difl):
62 =0: Only reorder w.r.t. SELECT. No extras.
63 =1: Reciprocal of norms of "projections" onto left and right
64 eigenspaces w.r.t. the selected cluster (PL and PR). =2: Upper
65 bounds on Difu and Difl. F-norm-based estimate
66 (DIF(1:2)).
67 =3: Estimate of Difu and Difl. 1-norm-based estimate
68 (DIF(1:2)). About 5 times as expensive as IJOB = 2. =4: Com‐
69 pute PL, PR and DIF (i.e. 0, 1 and 2 above): Economic version
70 to get it all. =5: Compute PL, PR and DIF (i.e. 0, 1 and 3
71 above)
72 WANTQ (input) LOGICAL
74 WANTZ (input) LOGICAL
76 SELECT (input) LOGICAL array, dimension (N)
78 SELECT specifies the eigenvalues in the selected cluster. To
79 select a real eigenvalue w(j), SELECT(j) must be set to w(j)
80 and w(j+1), corresponding to a 2-by-2 diagonal block, either
81 SELECT(j) or SELECT(j+1) or both must be set to either both
82 included in the cluster or both excluded.
83 N (input) INTEGER
85 The order of the matrices A and B. N >= 0.
86 A (input/output) REAL array, dimension(LDA,N)
88 On entry, the upper quasi-triangular matrix A, with (A, B) in
89 generalized real Schur canonical form. On exit, A is overwrit‐
90 ten by the reordered matrix A.
91 LDA (input) INTEGER
93 The leading dimension of the array A. LDA >= max(1,N).
94 B (input/output) REAL array, dimension(LDB,N)
96 On entry, the upper triangular matrix B, with (A, B) in gener‐
97 alized real Schur canonical form. On exit, B is overwritten by
98 the reordered matrix B.
99 LDB (input) INTEGER
101 The leading dimension of the array B. LDB >= max(1,N).
102 ALPHAR (output) REAL array, dimension (N)
104 ALPHAI (output) REAL array, dimension (N) BETA (output)
105 REAL array, dimension (N) On exit, (ALPHAR(j) +
106 ALPHAI(j)*i)/BETA(j), j=1,...,N, will be the generalized eigen‐
107 values. ALPHAR(j) + ALPHAI(j)*i and BETA(j),j=1,...,N are the
108 diagonals of the complex Schur form (S,T) that would result if
109 the 2-by-2 diagonal blocks of the real generalized Schur form
110 of (A,B) were further reduced to triangular form using complex
111 unitary transformations. If ALPHAI(j) is zero, then the j-th
112 eigenvalue is real; if positive, then the j-th and (j+1)-st ei‐
113 genvalues are a complex conjugate pair, with ALPHAI(j+1) nega‐
114 tive.
115 Q (input/output) REAL array, dimension (LDQ,N)
117 On entry, if WANTQ = .TRUE., Q is an N-by-N matrix. On exit, Q
118 has been postmultiplied by the left orthogonal transformation
119 matrix which reorder (A, B); The leading M columns of Q form
120 orthonormal bases for the specified pair of left eigenspaces
121 (deflating subspaces). If WANTQ = .FALSE., Q is not refer‐
122 enced.
123 LDQ (input) INTEGER
125 The leading dimension of the array Q. LDQ >= 1; and if WANTQ =
126 .TRUE., LDQ >= N.
127 Z (input/output) REAL array, dimension (LDZ,N)
129 On entry, if WANTZ = .TRUE., Z is an N-by-N matrix. On exit, Z
130 has been postmultiplied by the left orthogonal transformation
131 matrix which reorder (A, B); The leading M columns of Z form
132 orthonormal bases for the specified pair of left eigenspaces
133 (deflating subspaces). If WANTZ = .FALSE., Z is not refer‐
134 enced.
135 LDZ (input) INTEGER
137 The leading dimension of the array Z. LDZ >= 1; If WANTZ =
138 .TRUE., LDZ >= N.
139 M (output) INTEGER
141 The dimension of the specified pair of left and right eigen-
142 spaces (deflating subspaces). 0 <= M <= N.
143 PL (output) REAL
145 PR (output) REAL If IJOB = 1, 4 or 5, PL, PR are lower
146 bounds on the reciprocal of the norm of "projections" onto left
147 and right eigenspaces with respect to the selected cluster. 0
148 < PL, PR <= 1. If M = 0 or M = N, PL = PR = 1. If IJOB = 0,
149 2 or 3, PL and PR are not referenced.
150 DIF (output) REAL array, dimension (2).
152 If IJOB >= 2, DIF(1:2) store the estimates of Difu and Difl.
153 If IJOB = 2 or 4, DIF(1:2) are F-norm-based upper bounds on
154 Difu and Difl. If IJOB = 3 or 5, DIF(1:2) are 1-norm-based
155 estimates of Difu and Difl. If M = 0 or N, DIF(1:2) = F-
156 norm([A, B]). If IJOB = 0 or 1, DIF is not referenced.
157 WORK (workspace/output) REAL array, dimension (MAX(1,LWORK))
159 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
160 LWORK (input) INTEGER
162 The dimension of the array WORK. LWORK >= 4*N+16. If IJOB =
163 1, 2 or 4, LWORK >= MAX(4*N+16, 2*M*(N-M)). If IJOB = 3 or 5,
164 LWORK >= MAX(4*N+16, 4*M*(N-M)).
165 If LWORK = -1, then a workspace query is assumed; the routine
167 only calculates the optimal size of the WORK array, returns
168 this value as the first entry of the WORK array, and no error
169 message related to LWORK is issued by XERBLA.
170 IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
172 IF IJOB = 0, IWORK is not referenced. Otherwise, on exit, if
173 INFO = 0, IWORK(1) returns the optimal LIWORK.
174 LIWORK (input) INTEGER
176 The dimension of the array IWORK. LIWORK >= 1. If IJOB = 1, 2
177 or 4, LIWORK >= N+6. If IJOB = 3 or 5, LIWORK >= MAX(2*M*(N-
178 M), N+6).
179 If LIWORK = -1, then a workspace query is assumed; the routine
181 only calculates the optimal size of the IWORK array, returns
182 this value as the first entry of the IWORK array, and no error
183 message related to LIWORK is issued by XERBLA.
184 INFO (output) INTEGER
186 =0: Successful exit.
187 <0: If INFO = -i, the i-th argument had an illegal value.
188 =1: Reordering of (A, B) failed because the transformed matrix
189 pair (A, B) would be too far from generalized Schur form; the
190 problem is very ill-conditioned. (A, B) may have been par‐
191 tially reordered. If requested, 0 is returned in DIF(*), PL
192 and PR.
193
195 STGSEN first collects the selected eigenvalues by computing orthogonal
196 U and W that move them to the top left corner of (A, B). In other
197 words, the selected eigenvalues are the eigenvalues of (A11, B11) in:
198 U'*(A, B)*W = (A11 A12) (B11 B12) n1
200 ( 0 A22),( 0 B22) n2
201 n1 n2 n1 n2
202 where N = n1+n2 and U' means the transpose of U. The first n1 columns
204 of U and W span the specified pair of left and right eigenspaces
205 (deflating subspaces) of (A, B).
206 If (A, B) has been obtained from the generalized real Schur decomposi‐
208 tion of a matrix pair (C, D) = Q*(A, B)*Z', then the reordered general‐
209 ized real Schur form of (C, D) is given by
210 (C, D) = (Q*U)*(U'*(A, B)*W)*(Z*W)',
212 and the first n1 columns of Q*U and Z*W span the corresponding deflat‐
214 ing subspaces of (C, D) (Q and Z store Q*U and Z*W, resp.).
215 Note that if the selected eigenvalue is sufficiently ill-conditioned,
217 then its value may differ significantly from its value before reorder‐
218 ing.
219 The reciprocal condition numbers of the left and right eigenspaces
221 spanned by the first n1 columns of U and W (or Q*U and Z*W) may be
222 returned in DIF(1:2), corresponding to Difu and Difl, resp.
223 The Difu and Difl are defined as:
225 Difu[(A11, B11), (A22, B22)] = sigma-min( Zu )
227 and
228 Difl[(A11, B11), (A22, B22)] = Difu[(A22, B22), (A11, B11)],
229 where sigma-min(Zu) is the smallest singular value of the
231 (2*n1*n2)-by-(2*n1*n2) matrix
232 Zu = [ kron(In2, A11) -kron(A22', In1) ]
234 [ kron(In2, B11) -kron(B22', In1) ].
235 Here, Inx is the identity matrix of size nx and A22' is the transpose
237 of A22. kron(X, Y) is the Kronecker product between the matrices X and
238 Y.
239 When DIF(2) is small, small changes in (A, B) can cause large changes
241 in the deflating subspace. An approximate (asymptotic) bound on the
242 maximum angular error in the computed deflating subspaces is
243 EPS * norm((A, B)) / DIF(2),
245 where EPS is the machine precision.
247 The reciprocal norm of the projectors on the left and right eigenspaces
249 associated with (A11, B11) may be returned in PL and PR. They are com‐
250 puted as follows. First we compute L and R so that P*(A, B)*Q is block
251 diagonal, where
252 P = ( I -L ) n1 Q = ( I R ) n1
254 ( 0 I ) n2 and ( 0 I ) n2
255 n1 n2 n1 n2
256 and (L, R) is the solution to the generalized Sylvester equation
258 A11*R - L*A22 = -A12
260 B11*R - L*B22 = -B12
261 Then PL = (F-norm(L)**2+1)**(-1/2) and PR = (F-norm(R)**2+1)**(-1/2).
263 An approximate (asymptotic) bound on the average absolute error of the
264 selected eigenvalues is
265 EPS * norm((A, B)) / PL.
267 There are also global error bounds which valid for perturbations up to
269 a certain restriction: A lower bound (x) on the smallest F-norm(E,F)
270 for which an eigenvalue of (A11, B11) may move and coalesce with an ei‐
271 genvalue of (A22, B22) under perturbation (E,F), (i.e. (A + E, B + F),
272 is
273 x = min(Difu,Difl)/((1/(PL*PL)+1/(PR*PR))**(1/2)+2*max(1/PL,1/PR)).
275 An approximate bound on x can be computed from DIF(1:2), PL and PR.
277 If y = ( F-norm(E,F) / x) <= 1, the angles between the perturbed (L',
279 R') and unperturbed (L, R) left and right deflating subspaces associ‐
280 ated with the selected cluster in the (1,1)-blocks can be bounded as
281 max-angle(L, L') <= arctan( y * PL / (1 - y * (1 - PL * PL)**(1/2))
283 max-angle(R, R') <= arctan( y * PR / (1 - y * (1 - PR * PR)**(1/2))
284 See LAPACK User's Guide section 4.11 or the following references for
286 more information.
287 Note that if the default method for computing the Frobenius-norm- based
289 estimate DIF is not wanted (see SLATDF), then the parameter IDIFJB (see
290 below) should be changed from 3 to 4 (routine SLATDF (IJOB = 2 will be
291 used)). See STGSYL for more details.
292 Based on contributions by
294 Bo Kagstrom and Peter Poromaa, Department of Computing Science,
295 Umea University, S-901 87 Umea, Sweden.
296 References
298 ==========
299 [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
301 Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
302 M.S. Moonen et al (eds), Linear Algebra for Large Scale and
303 Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
304 [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified
306 Eigenvalues of a Regular Matrix Pair (A, B) and Condition
307 Estimation: Theory, Algorithms and Software,
308 Report UMINF - 94.04, Department of Computing Science, Umea
309 University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working
310 Note 87. To appear in Numerical Algorithms, 1996.
311 [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
313 for Solving the Generalized Sylvester Equation and Estimating the
314 Separation between Regular Matrix Pairs, Report UMINF - 93.23,
315 Department of Computing Science, Umea University, S-901 87 Umea,
316 Sweden, December 1993, Revised April 1994, Also as LAPACK Working
317 Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1,
318 1996.
319 LAPACK routine (version 3.1.1) February 2007 STGSEN(1)