LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
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ctgex2.f
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1*> \brief \b CTGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an unitary equivalence transformation.
2*
3* =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6* http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
9*> Download CTGEX2 + dependencies
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/ctgex2.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/ctgex2.f">
13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/ctgex2.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE CTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z,
22* LDZ, J1, INFO )
23*
24* .. Scalar Arguments ..
25* LOGICAL WANTQ, WANTZ
26* INTEGER INFO, J1, LDA, LDB, LDQ, LDZ, N
27* ..
28* .. Array Arguments ..
29* COMPLEX A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
30* $ Z( LDZ, * )
31* ..
32*
33*
34*> \par Purpose:
35* =============
36*>
37*> \verbatim
38*>
39*> CTGEX2 swaps adjacent diagonal 1 by 1 blocks (A11,B11) and (A22,B22)
40*> in an upper triangular matrix pair (A, B) by an unitary equivalence
41*> transformation.
42*>
43*> (A, B) must be in generalized Schur canonical form, that is, A and
44*> B are both upper triangular.
45*>
46*> Optionally, the matrices Q and Z of generalized Schur vectors are
47*> updated.
48*>
49*> Q(in) * A(in) * Z(in)**H = Q(out) * A(out) * Z(out)**H
50*> Q(in) * B(in) * Z(in)**H = Q(out) * B(out) * Z(out)**H
51*>
52*> \endverbatim
53*
54* Arguments:
55* ==========
56*
57*> \param[in] WANTQ
58*> \verbatim
59*> WANTQ is LOGICAL
60*> .TRUE. : update the left transformation matrix Q;
61*> .FALSE.: do not update Q.
62*> \endverbatim
63*>
64*> \param[in] WANTZ
65*> \verbatim
66*> WANTZ is LOGICAL
67*> .TRUE. : update the right transformation matrix Z;
68*> .FALSE.: do not update Z.
69*> \endverbatim
70*>
71*> \param[in] N
72*> \verbatim
73*> N is INTEGER
74*> The order of the matrices A and B. N >= 0.
75*> \endverbatim
76*>
77*> \param[in,out] A
78*> \verbatim
79*> A is COMPLEX array, dimension (LDA,N)
80*> On entry, the matrix A in the pair (A, B).
81*> On exit, the updated matrix A.
82*> \endverbatim
83*>
84*> \param[in] LDA
85*> \verbatim
86*> LDA is INTEGER
87*> The leading dimension of the array A. LDA >= max(1,N).
88*> \endverbatim
89*>
90*> \param[in,out] B
91*> \verbatim
92*> B is COMPLEX array, dimension (LDB,N)
93*> On entry, the matrix B in the pair (A, B).
94*> On exit, the updated matrix B.
95*> \endverbatim
96*>
97*> \param[in] LDB
98*> \verbatim
99*> LDB is INTEGER
100*> The leading dimension of the array B. LDB >= max(1,N).
101*> \endverbatim
102*>
103*> \param[in,out] Q
104*> \verbatim
105*> Q is COMPLEX array, dimension (LDQ,N)
106*> If WANTQ = .TRUE, on entry, the unitary matrix Q. On exit,
107*> the updated matrix Q.
108*> Not referenced if WANTQ = .FALSE..
109*> \endverbatim
110*>
111*> \param[in] LDQ
112*> \verbatim
113*> LDQ is INTEGER
114*> The leading dimension of the array Q. LDQ >= 1;
115*> If WANTQ = .TRUE., LDQ >= N.
116*> \endverbatim
117*>
118*> \param[in,out] Z
119*> \verbatim
120*> Z is COMPLEX array, dimension (LDZ,N)
121*> If WANTZ = .TRUE, on entry, the unitary matrix Z. On exit,
122*> the updated matrix Z.
123*> Not referenced if WANTZ = .FALSE..
124*> \endverbatim
125*>
126*> \param[in] LDZ
127*> \verbatim
128*> LDZ is INTEGER
129*> The leading dimension of the array Z. LDZ >= 1;
130*> If WANTZ = .TRUE., LDZ >= N.
131*> \endverbatim
132*>
133*> \param[in] J1
134*> \verbatim
135*> J1 is INTEGER
136*> The index to the first block (A11, B11).
137*> \endverbatim
138*>
139*> \param[out] INFO
140*> \verbatim
141*> INFO is INTEGER
142*> =0: Successful exit.
143*> =1: The transformed matrix pair (A, B) would be too far
144*> from generalized Schur form; the problem is ill-
145*> conditioned.
146*> \endverbatim
147*
148* Authors:
149* ========
150*
151*> \author Univ. of Tennessee
152*> \author Univ. of California Berkeley
153*> \author Univ. of Colorado Denver
154*> \author NAG Ltd.
155*
156*> \ingroup tgex2
157*
158*> \par Further Details:
159* =====================
160*>
161*> In the current code both weak and strong stability tests are
162*> performed. The user can omit the strong stability test by changing
163*> the internal logical parameter WANDS to .FALSE.. See ref. [2] for
164*> details.
165*
166*> \par Contributors:
167* ==================
168*>
169*> Bo Kagstrom and Peter Poromaa, Department of Computing Science,
170*> Umea University, S-901 87 Umea, Sweden.
171*
172*> \par References:
173* ================
174*>
175*> [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
176*> Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
177*> M.S. Moonen et al (eds), Linear Algebra for Large Scale and
178*> Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
179*> \n
180*> [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified
181*> Eigenvalues of a Regular Matrix Pair (A, B) and Condition
182*> Estimation: Theory, Algorithms and Software, Report UMINF-94.04,
183*> Department of Computing Science, Umea University, S-901 87 Umea,
184*> Sweden, 1994. Also as LAPACK Working Note 87. To appear in
185*> Numerical Algorithms, 1996.
186*>
187* =====================================================================
188 SUBROUTINE ctgex2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z,
189 $ LDZ, J1, INFO )
190*
191* -- LAPACK auxiliary routine --
192* -- LAPACK is a software package provided by Univ. of Tennessee, --
193* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
194*
195* .. Scalar Arguments ..
196 LOGICAL WANTQ, WANTZ
197 INTEGER INFO, J1, LDA, LDB, LDQ, LDZ, N
198* ..
199* .. Array Arguments ..
200 COMPLEX A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
201 $ z( ldz, * )
202* ..
203*
204* =====================================================================
205*
206* .. Parameters ..
207 COMPLEX CZERO, CONE
208 parameter( czero = ( 0.0e+0, 0.0e+0 ),
209 $ cone = ( 1.0e+0, 0.0e+0 ) )
210 REAL TWENTY
211 parameter( twenty = 2.0e+1 )
212 INTEGER LDST
213 parameter( ldst = 2 )
214 LOGICAL WANDS
215 parameter( wands = .true. )
216* ..
217* .. Local Scalars ..
218 LOGICAL STRONG, WEAK
219 INTEGER I, M
220 REAL CQ, CZ, EPS, SA, SB, SCALE, SMLNUM, SUM,
221 $ thresha, threshb
222 COMPLEX CDUM, F, G, SQ, SZ
223* ..
224* .. Local Arrays ..
225 COMPLEX S( LDST, LDST ), T( LDST, LDST ), WORK( 8 )
226* ..
227* .. External Functions ..
228 REAL SLAMCH
229 EXTERNAL slamch
230* ..
231* .. External Subroutines ..
232 EXTERNAL clacpy, clartg, classq, crot
233* ..
234* .. Intrinsic Functions ..
235 INTRINSIC abs, conjg, max, real, sqrt
236* ..
237* .. Executable Statements ..
238*
239 info = 0
240*
241* Quick return if possible
242*
243 IF( n.LE.1 )
244 $ RETURN
245*
246 m = ldst
247 weak = .false.
248 strong = .false.
249*
250* Make a local copy of selected block in (A, B)
251*
252 CALL clacpy( 'Full', m, m, a( j1, j1 ), lda, s, ldst )
253 CALL clacpy( 'Full', m, m, b( j1, j1 ), ldb, t, ldst )
254*
255* Compute the threshold for testing the acceptance of swapping.
256*
257 eps = slamch( 'P' )
258 smlnum = slamch( 'S' ) / eps
259 scale = real( czero )
260 sum = real( cone )
261 CALL clacpy( 'Full', m, m, s, ldst, work, m )
262 CALL clacpy( 'Full', m, m, t, ldst, work( m*m+1 ), m )
263 CALL classq( m*m, work, 1, scale, sum )
264 sa = scale*sqrt( sum )
265 scale = dble( czero )
266 sum = dble( cone )
267 CALL classq( m*m, work(m*m+1), 1, scale, sum )
268 sb = scale*sqrt( sum )
269*
270* THRES has been changed from
271* THRESH = MAX( TEN*EPS*SA, SMLNUM )
272* to
273* THRESH = MAX( TWENTY*EPS*SA, SMLNUM )
274* on 04/01/10.
275* "Bug" reported by Ondra Kamenik, confirmed by Julie Langou, fixed by
276* Jim Demmel and Guillaume Revy. See forum post 1783.
277*
278 thresha = max( twenty*eps*sa, smlnum )
279 threshb = max( twenty*eps*sb, smlnum )
280*
281* Compute unitary QL and RQ that swap 1-by-1 and 1-by-1 blocks
282* using Givens rotations and perform the swap tentatively.
283*
284 f = s( 2, 2 )*t( 1, 1 ) - t( 2, 2 )*s( 1, 1 )
285 g = s( 2, 2 )*t( 1, 2 ) - t( 2, 2 )*s( 1, 2 )
286 sa = abs( s( 2, 2 ) ) * abs( t( 1, 1 ) )
287 sb = abs( s( 1, 1 ) ) * abs( t( 2, 2 ) )
288 CALL clartg( g, f, cz, sz, cdum )
289 sz = -sz
290 CALL crot( 2, s( 1, 1 ), 1, s( 1, 2 ), 1, cz, conjg( sz ) )
291 CALL crot( 2, t( 1, 1 ), 1, t( 1, 2 ), 1, cz, conjg( sz ) )
292 IF( sa.GE.sb ) THEN
293 CALL clartg( s( 1, 1 ), s( 2, 1 ), cq, sq, cdum )
294 ELSE
295 CALL clartg( t( 1, 1 ), t( 2, 1 ), cq, sq, cdum )
296 END IF
297 CALL crot( 2, s( 1, 1 ), ldst, s( 2, 1 ), ldst, cq, sq )
298 CALL crot( 2, t( 1, 1 ), ldst, t( 2, 1 ), ldst, cq, sq )
299*
300* Weak stability test: |S21| <= O(EPS F-norm((A)))
301* and |T21| <= O(EPS F-norm((B)))
302*
303 weak = abs( s( 2, 1 ) ).LE.thresha .AND.
304 $ abs( t( 2, 1 ) ).LE.threshb
305 IF( .NOT.weak )
306 $ GO TO 20
307*
308 IF( wands ) THEN
309*
310* Strong stability test:
311* F-norm((A-QL**H*S*QR, B-QL**H*T*QR)) <= O(EPS*F-norm((A, B)))
312*
313 CALL clacpy( 'Full', m, m, s, ldst, work, m )
314 CALL clacpy( 'Full', m, m, t, ldst, work( m*m+1 ), m )
315 CALL crot( 2, work, 1, work( 3 ), 1, cz, -conjg( sz ) )
316 CALL crot( 2, work( 5 ), 1, work( 7 ), 1, cz, -conjg( sz ) )
317 CALL crot( 2, work, 2, work( 2 ), 2, cq, -sq )
318 CALL crot( 2, work( 5 ), 2, work( 6 ), 2, cq, -sq )
319 DO 10 i = 1, 2
320 work( i ) = work( i ) - a( j1+i-1, j1 )
321 work( i+2 ) = work( i+2 ) - a( j1+i-1, j1+1 )
322 work( i+4 ) = work( i+4 ) - b( j1+i-1, j1 )
323 work( i+6 ) = work( i+6 ) - b( j1+i-1, j1+1 )
324 10 CONTINUE
325 scale = dble( czero )
326 sum = dble( cone )
327 CALL classq( m*m, work, 1, scale, sum )
328 sa = scale*sqrt( sum )
329 scale = dble( czero )
330 sum = dble( cone )
331 CALL classq( m*m, work(m*m+1), 1, scale, sum )
332 sb = scale*sqrt( sum )
333 strong = sa.LE.thresha .AND. sb.LE.threshb
334 IF( .NOT.strong )
335 $ GO TO 20
336 END IF
337*
338* If the swap is accepted ("weakly" and "strongly"), apply the
339* equivalence transformations to the original matrix pair (A,B)
340*
341 CALL crot( j1+1, a( 1, j1 ), 1, a( 1, j1+1 ), 1, cz, conjg( sz ) )
342 CALL crot( j1+1, b( 1, j1 ), 1, b( 1, j1+1 ), 1, cz, conjg( sz ) )
343 CALL crot( n-j1+1, a( j1, j1 ), lda, a( j1+1, j1 ), lda, cq, sq )
344 CALL crot( n-j1+1, b( j1, j1 ), ldb, b( j1+1, j1 ), ldb, cq, sq )
345*
346* Set N1 by N2 (2,1) blocks to 0
347*
348 a( j1+1, j1 ) = czero
349 b( j1+1, j1 ) = czero
350*
351* Accumulate transformations into Q and Z if requested.
352*
353 IF( wantz )
354 $ CALL crot( n, z( 1, j1 ), 1, z( 1, j1+1 ), 1, cz, conjg( sz ) )
355 IF( wantq )
356 $ CALL crot( n, q( 1, j1 ), 1, q( 1, j1+1 ), 1, cq, conjg( sq ) )
357*
358* Exit with INFO = 0 if swap was successfully performed.
359*
360 RETURN
361*
362* Exit with INFO = 1 if swap was rejected.
363*
364 20 CONTINUE
365 info = 1
366 RETURN
367*
368* End of CTGEX2
369*
370 END
subroutine clacpy(uplo, m, n, a, lda, b, ldb)
CLACPY copies all or part of one two-dimensional array to another.
Definition clacpy.f:103
subroutine clartg(f, g, c, s, r)
CLARTG generates a plane rotation with real cosine and complex sine.
Definition clartg.f90:116
subroutine classq(n, x, incx, scale, sumsq)
CLASSQ updates a sum of squares represented in scaled form.
Definition classq.f90:124
subroutine crot(n, cx, incx, cy, incy, c, s)
CROT applies a plane rotation with real cosine and complex sine to a pair of complex vectors.
Definition crot.f:103
subroutine ctgex2(wantq, wantz, n, a, lda, b, ldb, q, ldq, z, ldz, j1, info)
CTGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an unitary equiva...
Definition ctgex2.f:190