1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
|
<HTML>
<HEAD><TITLE>MB04BZ - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="MB04BZ">MB04BZ</A></H2>
<H3>
Eigenvalues of a complex skew-Hamiltonian/Hamiltonian pencil
</H3>
<A HREF ="#Specification"><B>[Specification]</B></A>
<A HREF ="#Arguments"><B>[Arguments]</B></A>
<A HREF ="#Method"><B>[Method]</B></A>
<A HREF ="#References"><B>[References]</B></A>
<A HREF ="#Comments"><B>[Comments]</B></A>
<A HREF ="#Example"><B>[Example]</B></A>
<P>
<B><FONT SIZE="+1">Purpose</FONT></B>
<PRE>
To compute the eigenvalues of a complex N-by-N skew-Hamiltonian/
Hamiltonian pencil aS - bH, with
( A D ) ( B F )
S = ( H ) and H = ( H ). (1)
( E A ) ( G -B )
This routine computes the eigenvalues using an embedding to a real
skew-Hamiltonian/skew-Hamiltonian pencil aB_S - bB_T, defined as
( Re(A) -Im(A) | Re(D) -Im(D) )
( | )
( Im(A) Re(A) | Im(D) Re(D) )
( | )
B_S = (-----------------+-----------------) , and
( | T T )
( Re(E) -Im(E) | Re(A ) Im(A ) )
( | T T )
( Im(E) Re(E) | -Im(A ) Re(A ) )
(2)
( -Im(B) -Re(B) | -Im(F) -Re(F) )
( | )
( Re(B) -Im(B) | Re(F) -Im(F) )
( | )
B_T = (-----------------+-----------------) , T = i*H.
( | T T )
( -Im(G) -Re(G) | -Im(B ) Re(B ) )
( | T T )
( Re(G) -Im(G) | -Re(B ) -Im(B ) )
Optionally, if JOB = 'T', the pencil aB_S - bB_H (B_H = -i*B_T) is
transformed by a unitary matrix Q to the structured Schur form
( BA BD ) ( BB BF )
B_Sout = ( H ) and B_Hout = ( H ), (3)
( 0 BA ) ( 0 -BB )
where BA and BB are upper triangular, BD is skew-Hermitian, and
BF is Hermitian. The embedding doubles the multiplicities of the
eigenvalues of the pencil aS - bH. Optionally, if COMPQ = 'C', the
unitary matrix Q is computed.
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
SUBROUTINE MB04BZ( JOB, COMPQ, N, A, LDA, DE, LDDE, B, LDB, FG,
$ LDFG, Q, LDQ, ALPHAR, ALPHAI, BETA, IWORK,
$ DWORK, LDWORK, ZWORK, LZWORK, BWORK, INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ, JOB
INTEGER INFO, LDA, LDB, LDDE, LDFG, LDQ, LDWORK,
$ LZWORK, N
C .. Array Arguments ..
LOGICAL BWORK( * )
INTEGER IWORK( * )
DOUBLE PRECISION ALPHAI( * ), ALPHAR( * ), BETA( * ), DWORK( * )
COMPLEX*16 A( LDA, * ), B( LDB, * ), DE( LDDE, * ),
$ FG( LDFG, * ), Q( LDQ, * ), ZWORK( * )
</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>
<B>Mode Parameters</B>
<PRE>
JOB CHARACTER*1
Specifies the computation to be performed, as follows:
= 'E': compute the eigenvalues only; S and H will not
necessarily be transformed as in (3).
= 'T': put S and H into the forms in (3) and return the
eigenvalues in ALPHAR, ALPHAI and BETA.
COMPQ CHARACTER*1
Specifies whether to compute the unitary transformation
matrix Q, as follows:
= 'N': Q is not computed;
= 'C': compute the unitary transformation matrix Q.
</PRE>
<B>Input/Output Parameters</B>
<PRE>
N (input) INTEGER
The order of the pencil aS - bH. N >= 0, even.
A (input/output) COMPLEX*16 array, dimension (LDA, K)
where K = N/2, if JOB = 'E', and K = N, if JOB = 'T'.
On entry, the leading N/2-by-N/2 part of this array must
contain the matrix A.
On exit, if JOB = 'T', the leading N-by-N part of this
array contains the upper triangular matrix BA in (3) (see
also METHOD). The strictly lower triangular part is not
zeroed, but it is preserved.
If JOB = 'E', this array is unchanged on exit.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1, K).
DE (input/output) COMPLEX*16 array, dimension
(LDDE, MIN(K+1,N))
On entry, the leading N/2-by-N/2 lower triangular part of
this array must contain the lower triangular part of the
skew-Hermitian matrix E, and the N/2-by-N/2 upper
triangular part of the submatrix in the columns 2 to N/2+1
of this array must contain the upper triangular part of
the skew-Hermitian matrix D.
On exit, if JOB = 'T', the leading N-by-N part of this
array contains the skew-Hermitian matrix BD in (3) (see
also METHOD). The strictly lower triangular part of the
input matrix is preserved.
If JOB = 'E', this array is unchanged on exit.
LDDE INTEGER
The leading dimension of the array DE. LDDE >= MAX(1, K).
B (input/output) COMPLEX*16 array, dimension (LDB, K)
On entry, the leading N/2-by-N/2 part of this array must
contain the matrix B.
On exit, if JOB = 'T', the leading N-by-N part of this
array contains the upper triangular matrix BB in (3) (see
also METHOD). The strictly lower triangular part is not
zeroed; the elements below the first subdiagonal of the
input matrix are preserved.
If JOB = 'E', this array is unchanged on exit.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1, K).
FG (input/output) COMPLEX*16 array, dimension
(LDFG, MIN(K+1,N))
On entry, the leading N/2-by-N/2 lower triangular part of
this array must contain the lower triangular part of the
Hermitian matrix G, and the N/2-by-N/2 upper triangular
part of the submatrix in the columns 2 to N/2+1 of this
array must contain the upper triangular part of the
Hermitian matrix F.
On exit, if JOB = 'T', the leading N-by-N part of this
array contains the Hermitian matrix BF in (3) (see also
METHOD). The strictly lower triangular part of the input
matrix is preserved. The diagonal elements might have tiny
imaginary parts.
If JOB = 'E', this array is unchanged on exit.
LDFG INTEGER
The leading dimension of the array FG. LDFG >= MAX(1, K).
Q (output) COMPLEX*16 array, dimension (LDQ, 2*N)
On exit, if COMPQ = 'C', the leading 2*N-by-2*N part of
this array contains the unitary transformation matrix Q
that reduced the matrices B_S and B_H to the form in (3).
However, if JOB = 'E', the reduction was possibly not
completed: the matrix B_H may have 2-by-2 diagonal blocks,
and the array Q returns the orthogonal matrix that
performed the partial reduction.
If COMPQ = 'N', this array is not referenced.
LDQ INTEGER
The leading dimension of the array Q.
LDQ >= 1, if COMPQ = 'N';
LDQ >= MAX(1, 2*N), if COMPQ = 'C'.
ALPHAR (output) DOUBLE PRECISION array, dimension (N)
The real parts of each scalar alpha defining an eigenvalue
of the pencil aS - bH.
ALPHAI (output) DOUBLE PRECISION array, dimension (N)
The imaginary parts of each scalar alpha defining an
eigenvalue of the pencil aS - bH.
If ALPHAI(j) is zero, then the j-th eigenvalue is real.
BETA (output) DOUBLE PRECISION array, dimension (N)
The scalars beta that define the eigenvalues of the pencil
aS - bH.
Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
beta = BETA(j) represent the j-th eigenvalue of the pencil
aS - bH, in the form lambda = alpha/beta. Since lambda may
overflow, the ratios should not, in general, be computed.
</PRE>
<B>Workspace</B>
<PRE>
IWORK INTEGER array, dimension (2*N+4)
On exit, IWORK(1) contains the number, q, of unreliable,
possibly inaccurate (pairs of) eigenvalues, and the
absolute values in IWORK(2), ..., IWORK(q+1) are their
indices, as well as of the corresponding 1-by-1 and 2-by-2
diagonal blocks of the arrays B and A on exit, if
JOB = 'T'. Specifically, a positive value is an index of
a real or purely imaginary eigenvalue, corresponding to a
1-by-1 block, while the absolute value of a negative entry
in IWORK is an index to the first eigenvalue in a pair of
consecutively stored eigenvalues, corresponding to a
2-by-2 block. Moreover, IWORK(q+2),..., IWORK(2*q+1)
contain pointers to the starting elements in DWORK where
each block pair is stored. Specifically, if IWORK(i+1) > 0
then DWORK(r) and DWORK(r+1) store corresponding diagonal
elements of T11 and S11, respectively, and if
IWORK(i+1) < 0, then DWORK(r:r+3) and DWORK(r+4:r+7) store
the elements of the block in T11 and S11, respectively
(see Section METHOD), where r = IWORK(q+1+i). Moreover,
IWORK(2*q+2) contains the number of the 1-by-1 blocks, and
IWORK(2*q+3) contains the number of the 2-by-2 blocks,
corresponding to unreliable eigenvalues. IWORK(2*q+4)
contains the total number t of the 2-by-2 blocks.
If INFO = 0, then q = 0, therefore IWORK(1) = 0.
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0 or INFO = 3, DWORK(1) returns the
optimal LDWORK, and DWORK(2) and DWORK(3) contain the
Frobenius norms of the matrices B_S and B_T. These norms
are used in the tests to decide that some eigenvalues are
considered as numerically unreliable. Moreover, DWORK(4),
..., DWORK(3+2*s) contain the s pairs of values of the
1-by-1 diagonal elements of T11 and S11. The eigenvalue of
such a block pair is obtained from -i*T11(i,i)/S11(i,i).
Similarly, DWORK(4+2*s), ..., DWORK(3+2*s+8*t) contain the
t groups of pairs of 2-by-2 diagonal submatrices of T11
and S11, stored column-wise. The spectrum of such a block
pair is obtained from -i*ev, where ev are the eigenvalues
of (T11(i:i+1,i:i+1),S11(i:i+1,i:i+1)).
On exit, if INFO = -19, DWORK(1) returns the minimum value
of LDWORK.
LDWORK INTEGER
The dimension of the array DWORK. If COMPQ = 'N',
LDWORK >= MAX( 3, 4*N*N + 3*N ), if JOB = 'E';
LDWORK >= MAX( 3, 5*N*N + 3*N ), if JOB = 'T';
LDWORK >= MAX( 3, 11*N*N + 2*N ), if COMPQ = 'C'.
For good performance LDWORK should be generally larger.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
ZWORK COMPLEX*16 array, dimension (LZWORK)
On exit, if INFO = 0, ZWORK(1) returns the optimal LZWORK.
On exit, if INFO = -21, ZWORK(1) returns the minimum value
of LZWORK.
LZWORK INTEGER
The dimension of the array ZWORK.
LZWORK >= 1, if JOB = 'E'; otherwise,
LZWORK >= 6*N + 4, if COMPQ = 'N';
LZWORK >= 8*N + 4, if COMPQ = 'C'.
If LZWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
ZWORK array, returns this value as the first entry of
the ZWORK array, and no error message related to LZWORK
is issued by XERBLA.
BWORK LOGICAL array, dimension (LBWORK)
LBWORK >= 0, if JOB = 'E';
LBWORK >= N, if JOB = 'T'.
</PRE>
<B>Error Indicator</B>
<PRE>
INFO INTEGER
= 0: succesful exit;
< 0: if INFO = -i, the i-th argument had an illegal value;
= 1: QZ iteration failed in the SLICOT Library routine
MB04FD (QZ iteration did not converge or computation
of the shifts failed);
= 2: QZ iteration failed in the LAPACK routine ZHGEQZ when
trying to triangularize the 2-by-2 blocks;
= 3: warning: the pencil is numerically singular.
</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
First, T = i*H is set. Then, the embeddings, B_S and B_T, of the
matrices S and T, are determined and, subsequently, the SLICOT
Library routine MB04FD is applied to compute the structured Schur
form, i.e., the factorizations
~ T T ( S11 S12 )
B_S = J Q J B_S Q = ( T ) and
( 0 S11 )
~ T T ( T11 T12 )
B_T = J Q J B_T Q = ( T ),
( 0 T11 )
where Q is real orthogonal, S11 is upper triangular, and T11 is
upper quasi-triangular. If JOB = 'T', then the matrices above are
~
further transformed so that the 2-by-2 blocks in i*B_T are split
into 1-by-1 blocks. If COMPQ = 'C', the transformations are
accumulated in the unitary matrix Q.
See also page 22 in [1] for more details.
</PRE>
<A name="References"><B><FONT SIZE="+1">References</FONT></B></A>
<PRE>
[1] Benner, P., Byers, R., Mehrmann, V. and Xu, H.
Numerical Computation of Deflating Subspaces of Embedded
Hamiltonian Pencils.
Tech. Rep. SFB393/99-15, Technical University Chemnitz,
Germany, June 1999.
</PRE>
<A name="Numerical Aspects"><B><FONT SIZE="+1">Numerical Aspects</FONT></B></A>
<PRE> 3
The algorithm is numerically backward stable and needs O(N )
complex floating point operations.
</PRE>
<A name="Comments"><B><FONT SIZE="+1">Further Comments</FONT></B></A>
<PRE>
The returned eigenvalues are those of the pencil (-i*T11,S11),
where i is the purely imaginary unit.
If JOB = 'E', the returned matrix T11 is quasi-triangular. Note
that the off-diagonal elements of the 2-by-2 blocks of S11 are
zero by construction.
If JOB = 'T', the returned eigenvalues correspond to the diagonal
elements of BB and BA.
This routine does not perform any scaling of the matrices. Scaling
might sometimes be useful, and it should be done externally.
</PRE>
<A name="Example"><B><FONT SIZE="+1">Example</FONT></B></A>
<P>
<B>Program Text</B>
<PRE>
* MB04BZ EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 50 )
INTEGER LDA, LDB, LDDE, LDFG, LDQ, LDWORK, LZWORK
PARAMETER ( LDA = NMAX, LDB = NMAX, LDDE = NMAX,
$ LDFG = NMAX, LDQ = 2*NMAX,
$ LDWORK = 11*NMAX*NMAX + 2*NMAX,
$ LZWORK = 8*NMAX + 4 )
*
* .. Local Scalars ..
CHARACTER COMPQ, JOB
INTEGER I, INFO, J, M, N
*
* .. Local Arrays ..
COMPLEX*16 A( LDA, NMAX ), B( LDB, NMAX ),
$ DE( LDDE, NMAX ), FG( LDFG, NMAX ),
$ Q( LDQ, 2*NMAX ), ZWORK( LZWORK )
DOUBLE PRECISION ALPHAI( NMAX ), ALPHAR( NMAX ),
$ BETA( NMAX ), DWORK( LDWORK )
INTEGER IWORK( 2*NMAX+3 )
LOGICAL BWORK( NMAX )
*
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
*
* .. External Subroutines ..
EXTERNAL MB04BZ
*
* .. Intrinsic Functions ..
INTRINSIC MOD
*
* .. Executable Statements ..
*
WRITE( NOUT, FMT = 99999 )
* Skip the heading in the data file and read in the data.
READ( NIN, FMT = * )
READ( NIN, FMT = * ) JOB, COMPQ, N
IF( N.LT.0 .OR. N.GT.NMAX .OR. MOD( N, 2 ).NE.0 ) THEN
WRITE( NOUT, FMT = 99998 ) N
ELSE
M = N/2
READ( NIN, FMT = * ) ( ( A( I, J ), J = 1, M ), I = 1, M )
READ( NIN, FMT = * ) ( ( DE( I, J ), J = 1, M+1 ), I = 1, M )
READ( NIN, FMT = * ) ( ( B( I, J ), J = 1, M ), I = 1, M )
READ( NIN, FMT = * ) ( ( FG( I, J ), J = 1, M+1 ), I = 1, M )
* Compute the eigenvalues of a complex skew-Hamiltonian/
* Hamiltonian pencil.
CALL MB04BZ( JOB, COMPQ, N, A, LDA, DE, LDDE, B, LDB, FG, LDFG,
$ Q, LDQ, ALPHAR, ALPHAI, BETA, IWORK, DWORK,
$ LDWORK, ZWORK, LZWORK, BWORK, INFO )
*
IF( INFO.NE.0 ) THEN
WRITE( NOUT, FMT = 99997 ) INFO
ELSE
IF( LSAME( JOB, 'T' ) ) THEN
WRITE( NOUT, FMT = 99996 )
DO 10 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( A( I, J ), J = 1, N )
10 CONTINUE
WRITE( NOUT, FMT = 99994 )
DO 20 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( DE( I, J ), J = 1, N )
20 CONTINUE
WRITE( NOUT, FMT = 99993 )
DO 30 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( B( I, J ), J = 1, N )
30 CONTINUE
WRITE( NOUT, FMT = 99992 )
DO 40 I = 1, N
WRITE( NOUT, FMT = 99995 ) ( FG( I, J ), J = 1, N )
40 CONTINUE
END IF
IF( LSAME( COMPQ, 'C' ) ) THEN
WRITE( NOUT, FMT = 99991 )
DO 50 I = 1, 2*N
WRITE( NOUT, FMT = 99995 ) ( Q( I, J ), J = 1, 2*N )
50 CONTINUE
END IF
WRITE( NOUT, FMT = 99990 )
WRITE( NOUT, FMT = 99989 ) ( ALPHAR( I ), I = 1, N )
WRITE( NOUT, FMT = 99988 )
WRITE( NOUT, FMT = 99989 ) ( ALPHAI( I ), I = 1, N )
WRITE( NOUT, FMT = 99987 )
WRITE( NOUT, FMT = 99989 ) ( BETA( I ), I = 1, N )
END IF
END IF
STOP
*
99999 FORMAT ( 'MB04BZ EXAMPLE PROGRAM RESULTS', 1X )
99998 FORMAT ( 'N is out of range.', /, 'N = ', I5 )
99997 FORMAT ( 'INFO on exit from MB04BZ = ', I2 )
99996 FORMAT (/'The matrix A on exit is ' )
99995 FORMAT (20( 1X, F9.4, SP, F9.4, S, 'i ') )
99994 FORMAT (/'The matrix D on exit is ' )
99993 FORMAT (/'The matrix B on exit is ' )
99992 FORMAT (/'The matrix F on exit is ' )
99991 FORMAT (/'The matrix Q is ' )
99990 FORMAT (/'The vector ALPHAR is ' )
99989 FORMAT ( 50( 1X, F8.4 ) )
99988 FORMAT (/'The vector ALPHAI is ' )
99987 FORMAT (/'The vector BETA is ' )
END
</PRE>
<B>Program Data</B>
<PRE>
MB04BZ EXAMPLE PROGRAM DATA
T C 4
(0.0604,0.6568) (0.5268,0.2919)
(0.3992,0.6279) (0.4167,0.4316)
(0,0.4896) (0,0.9516) (0.3724,0.0526)
(0.9840,0.3394) (0,0.9203) (0,0.7378)
(0.2691,0.4177) (0.5478,0.3014)
(0.4228,0.9830) (0.9427,0.7010)
0.6663 0.6981 (0.1781,0.8818)
(0.5391,0.1711) 0.6665 0.1280
</PRE>
<B>Program Results</B>
<PRE>
MB04BZ EXAMPLE PROGRAM RESULTS
The matrix A on exit is
0.7430 +0.0000i 0.0389 -0.4330i -0.1155 -0.1366i -0.6586 -0.3210i
0.3992 +0.6279i 0.7548 +0.0000i 0.6099 -0.2308i 0.2140 +0.1260i
0.0000 +0.0000i 0.0000 +0.0000i 1.4085 +0.0000i 0.0848 +0.4972i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 1.4725 +0.0000i
The matrix D on exit is
0.0000 -0.6858i 0.1839 -0.0474i -0.4428 -0.1290i 0.4759 +0.0380i
0.9840 +0.3394i 0.0000 +0.6858i -0.6339 +0.1358i 0.4204 -0.2140i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 -0.2110i -0.0159 -0.0338i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.2110i
The matrix B on exit is
-1.5832 +0.5069i -0.0097 +0.0866i 0.1032 -0.1431i -0.0426 +0.7942i
0.0000 +0.0000i 1.6085 +0.5150i -0.1342 -0.8180i 0.5143 +0.0178i
0.0000 +0.0000i 0.0000 +0.0000i -0.0842 -0.1642i 0.0246 -0.0264i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0880 -0.1716i
The matrix F on exit is
0.3382 0.0000i 0.0234 +0.0907i -0.1619 +0.9033i -0.8227 +0.0204i
0.5391 +0.1711i -0.3382 +0.0000i -0.6525 +0.2455i -0.3532 -0.6409i
0.0000 +0.0000i 0.0000 +0.0000i 0.0120 0.0000i 0.0019 -0.0009i
0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i -0.0120 +0.0000i
The matrix Q is
0.1422 +0.5446i -0.3877 -0.1273i -0.4363 +0.1705i 0.0348 -0.5440i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i 0.0000 +0.0000i
0.1594 -0.2382i 0.1967 -0.2467i -0.1376 -0.0961i -0.1070 -0.2058i -0.1273 +0.0585i -0.0852 +0.1020i 0.6125 -0.1059i -0.0172 +0.5589i
-0.3659 -0.0211i -0.0291 +0.4967i -0.0729 +0.4236i 0.3169 -0.0008i 0.2947 -0.1080i 0.1614 -0.2342i 0.2867 -0.0578i -0.0170 +0.2603i
0.1846 +0.4089i -0.2815 -0.2018i 0.3220 -0.1600i -0.0526 +0.3937i 0.2747 -0.0655i 0.1045 -0.2159i 0.2085 -0.3104i -0.3052 +0.1463i
-0.0201 -0.2898i 0.2131 -0.0081i -0.2165 -0.1055i -0.1324 -0.3133i 0.1660 -0.1635i 0.2250 -0.1390i -0.1590 -0.4634i -0.5310 -0.2239i
0.1342 -0.1295i 0.1128 -0.1990i -0.0712 -0.1686i -0.1490 -0.1336i 0.6198 +0.0113i 0.0281 -0.4762i -0.0462 +0.3244i 0.3464 +0.0086i
0.2305 -0.1358i 0.1292 -0.3311i -0.0106 +0.4992i 0.3906 +0.0997i 0.1429 +0.3376i -0.4310 -0.0866i -0.0894 -0.1336i -0.1601 -0.1055i
-0.2601 +0.0835i -0.0940 +0.3652i -0.0213 -0.3116i -0.2502 -0.0995i 0.1361 +0.4589i -0.5898 -0.0730i 0.0294 -0.1192i -0.1253 +0.0085i
The vector ALPHAR is
-1.5832 1.5832 -0.0842 0.0842
The vector ALPHAI is
0.5069 0.5069 -0.1642 -0.1642
The vector BETA is
0.7430 0.7430 1.4085 1.4085
</PRE>
<HR>
<p>
<A HREF=..\libindex.html><B>Return to index</B></A></BODY>
</HTML>
|
