LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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cggbal.f
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1*> \brief \b CGGBAL
2*
3* =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6* http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
9*> Download CGGBAL + dependencies
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cggbal.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/cggbal.f">
13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cggbal.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE CGGBAL( JOB, N, A, LDA, B, LDB, ILO, IHI, LSCALE,
22* RSCALE, WORK, INFO )
23*
24* .. Scalar Arguments ..
25* CHARACTER JOB
26* INTEGER IHI, ILO, INFO, LDA, LDB, N
27* ..
28* .. Array Arguments ..
29* REAL LSCALE( * ), RSCALE( * ), WORK( * )
30* COMPLEX A( LDA, * ), B( LDB, * )
31* ..
32*
33*
34*> \par Purpose:
35* =============
36*>
37*> \verbatim
38*>
39*> CGGBAL balances a pair of general complex matrices (A,B). This
40*> involves, first, permuting A and B by similarity transformations to
41*> isolate eigenvalues in the first 1 to ILO$-$1 and last IHI+1 to N
42*> elements on the diagonal; and second, applying a diagonal similarity
43*> transformation to rows and columns ILO to IHI to make the rows
44*> and columns as close in norm as possible. Both steps are optional.
45*>
46*> Balancing may reduce the 1-norm of the matrices, and improve the
47*> accuracy of the computed eigenvalues and/or eigenvectors in the
48*> generalized eigenvalue problem A*x = lambda*B*x.
49*> \endverbatim
50*
51* Arguments:
52* ==========
53*
54*> \param[in] JOB
55*> \verbatim
56*> JOB is CHARACTER*1
57*> Specifies the operations to be performed on A and B:
58*> = 'N': none: simply set ILO = 1, IHI = N, LSCALE(I) = 1.0
59*> and RSCALE(I) = 1.0 for i=1,...,N;
60*> = 'P': permute only;
61*> = 'S': scale only;
62*> = 'B': both permute and scale.
63*> \endverbatim
64*>
65*> \param[in] N
66*> \verbatim
67*> N is INTEGER
68*> The order of the matrices A and B. N >= 0.
69*> \endverbatim
70*>
71*> \param[in,out] A
72*> \verbatim
73*> A is COMPLEX array, dimension (LDA,N)
74*> On entry, the input matrix A.
75*> On exit, A is overwritten by the balanced matrix.
76*> If JOB = 'N', A is not referenced.
77*> \endverbatim
78*>
79*> \param[in] LDA
80*> \verbatim
81*> LDA is INTEGER
82*> The leading dimension of the array A. LDA >= max(1,N).
83*> \endverbatim
84*>
85*> \param[in,out] B
86*> \verbatim
87*> B is COMPLEX array, dimension (LDB,N)
88*> On entry, the input matrix B.
89*> On exit, B is overwritten by the balanced matrix.
90*> If JOB = 'N', B is not referenced.
91*> \endverbatim
92*>
93*> \param[in] LDB
94*> \verbatim
95*> LDB is INTEGER
96*> The leading dimension of the array B. LDB >= max(1,N).
97*> \endverbatim
98*>
99*> \param[out] ILO
100*> \verbatim
101*> ILO is INTEGER
102*> \endverbatim
103*>
104*> \param[out] IHI
105*> \verbatim
106*> IHI is INTEGER
107*> ILO and IHI are set to integers such that on exit
108*> A(i,j) = 0 and B(i,j) = 0 if i > j and
109*> j = 1,...,ILO-1 or i = IHI+1,...,N.
110*> If JOB = 'N' or 'S', ILO = 1 and IHI = N.
111*> \endverbatim
112*>
113*> \param[out] LSCALE
114*> \verbatim
115*> LSCALE is REAL array, dimension (N)
116*> Details of the permutations and scaling factors applied
117*> to the left side of A and B. If P(j) is the index of the
118*> row interchanged with row j, and D(j) is the scaling factor
119*> applied to row j, then
120*> LSCALE(j) = P(j) for J = 1,...,ILO-1
121*> = D(j) for J = ILO,...,IHI
122*> = P(j) for J = IHI+1,...,N.
123*> The order in which the interchanges are made is N to IHI+1,
124*> then 1 to ILO-1.
125*> \endverbatim
126*>
127*> \param[out] RSCALE
128*> \verbatim
129*> RSCALE is REAL array, dimension (N)
130*> Details of the permutations and scaling factors applied
131*> to the right side of A and B. If P(j) is the index of the
132*> column interchanged with column j, and D(j) is the scaling
133*> factor applied to column j, then
134*> RSCALE(j) = P(j) for J = 1,...,ILO-1
135*> = D(j) for J = ILO,...,IHI
136*> = P(j) for J = IHI+1,...,N.
137*> The order in which the interchanges are made is N to IHI+1,
138*> then 1 to ILO-1.
139*> \endverbatim
140*>
141*> \param[out] WORK
142*> \verbatim
143*> WORK is REAL array, dimension (lwork)
144*> lwork must be at least max(1,6*N) when JOB = 'S' or 'B', and
145*> at least 1 when JOB = 'N' or 'P'.
146*> \endverbatim
147*>
148*> \param[out] INFO
149*> \verbatim
150*> INFO is INTEGER
151*> = 0: successful exit
152*> < 0: if INFO = -i, the i-th argument had an illegal value.
153*> \endverbatim
154*
155* Authors:
156* ========
157*
158*> \author Univ. of Tennessee
159*> \author Univ. of California Berkeley
160*> \author Univ. of Colorado Denver
161*> \author NAG Ltd.
162*
163*> \ingroup complexGBcomputational
164*
165*> \par Further Details:
166* =====================
167*>
168*> \verbatim
169*>
170*> See R.C. WARD, Balancing the generalized eigenvalue problem,
171*> SIAM J. Sci. Stat. Comp. 2 (1981), 141-152.
172*> \endverbatim
173*>
174* =====================================================================
175 SUBROUTINE cggbal( JOB, N, A, LDA, B, LDB, ILO, IHI, LSCALE,
176 $ RSCALE, WORK, INFO )
177*
178* -- LAPACK computational routine --
179* -- LAPACK is a software package provided by Univ. of Tennessee, --
180* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
181*
182* .. Scalar Arguments ..
183 CHARACTER JOB
184 INTEGER IHI, ILO, INFO, LDA, LDB, N
185* ..
186* .. Array Arguments ..
187 REAL LSCALE( * ), RSCALE( * ), WORK( * )
188 COMPLEX A( LDA, * ), B( LDB, * )
189* ..
190*
191* =====================================================================
192*
193* .. Parameters ..
194 REAL ZERO, HALF, ONE
195 parameter( zero = 0.0e+0, half = 0.5e+0, one = 1.0e+0 )
196 REAL THREE, SCLFAC
197 parameter( three = 3.0e+0, sclfac = 1.0e+1 )
198 COMPLEX CZERO
199 parameter( czero = ( 0.0e+0, 0.0e+0 ) )
200* ..
201* .. Local Scalars ..
202 INTEGER I, ICAB, IFLOW, IP1, IR, IRAB, IT, J, JC, JP1,
203 $ k, kount, l, lcab, lm1, lrab, lsfmax, lsfmin,
204 $ m, nr, nrp2
205 REAL ALPHA, BASL, BETA, CAB, CMAX, COEF, COEF2,
206 $ coef5, cor, ew, ewc, gamma, pgamma, rab, sfmax,
207 $ sfmin, sum, t, ta, tb, tc
208 COMPLEX CDUM
209* ..
210* .. External Functions ..
211 LOGICAL LSAME
212 INTEGER ICAMAX
213 REAL SDOT, SLAMCH
214 EXTERNAL lsame, icamax, sdot, slamch
215* ..
216* .. External Subroutines ..
217 EXTERNAL csscal, cswap, saxpy, sscal, xerbla
218* ..
219* .. Intrinsic Functions ..
220 INTRINSIC abs, aimag, int, log10, max, min, real, sign
221* ..
222* .. Statement Functions ..
223 REAL CABS1
224* ..
225* .. Statement Function definitions ..
226 cabs1( cdum ) = abs( real( cdum ) ) + abs( aimag( cdum ) )
227* ..
228* .. Executable Statements ..
229*
230* Test the input parameters
231*
232 info = 0
233 IF( .NOT.lsame( job, 'N' ) .AND. .NOT.lsame( job, 'P' ) .AND.
234 $ .NOT.lsame( job, 'S' ) .AND. .NOT.lsame( job, 'B' ) ) THEN
235 info = -1
236 ELSE IF( n.LT.0 ) THEN
237 info = -2
238 ELSE IF( lda.LT.max( 1, n ) ) THEN
239 info = -4
240 ELSE IF( ldb.LT.max( 1, n ) ) THEN
241 info = -6
242 END IF
243 IF( info.NE.0 ) THEN
244 CALL xerbla( 'CGGBAL', -info )
245 RETURN
246 END IF
247*
248* Quick return if possible
249*
250 IF( n.EQ.0 ) THEN
251 ilo = 1
252 ihi = n
253 RETURN
254 END IF
255*
256 IF( n.EQ.1 ) THEN
257 ilo = 1
258 ihi = n
259 lscale( 1 ) = one
260 rscale( 1 ) = one
261 RETURN
262 END IF
263*
264 IF( lsame( job, 'N' ) ) THEN
265 ilo = 1
266 ihi = n
267 DO 10 i = 1, n
268 lscale( i ) = one
269 rscale( i ) = one
270 10 CONTINUE
271 RETURN
272 END IF
273*
274 k = 1
275 l = n
276 IF( lsame( job, 'S' ) )
277 $ GO TO 190
278*
279 GO TO 30
280*
281* Permute the matrices A and B to isolate the eigenvalues.
282*
283* Find row with one nonzero in columns 1 through L
284*
285 20 CONTINUE
286 l = lm1
287 IF( l.NE.1 )
288 $ GO TO 30
289*
290 rscale( 1 ) = one
291 lscale( 1 ) = one
292 GO TO 190
293*
294 30 CONTINUE
295 lm1 = l - 1
296 DO 80 i = l, 1, -1
297 DO 40 j = 1, lm1
298 jp1 = j + 1
299 IF( a( i, j ).NE.czero .OR. b( i, j ).NE.czero )
300 $ GO TO 50
301 40 CONTINUE
302 j = l
303 GO TO 70
304*
305 50 CONTINUE
306 DO 60 j = jp1, l
307 IF( a( i, j ).NE.czero .OR. b( i, j ).NE.czero )
308 $ GO TO 80
309 60 CONTINUE
310 j = jp1 - 1
311*
312 70 CONTINUE
313 m = l
314 iflow = 1
315 GO TO 160
316 80 CONTINUE
317 GO TO 100
318*
319* Find column with one nonzero in rows K through N
320*
321 90 CONTINUE
322 k = k + 1
323*
324 100 CONTINUE
325 DO 150 j = k, l
326 DO 110 i = k, lm1
327 ip1 = i + 1
328 IF( a( i, j ).NE.czero .OR. b( i, j ).NE.czero )
329 $ GO TO 120
330 110 CONTINUE
331 i = l
332 GO TO 140
333 120 CONTINUE
334 DO 130 i = ip1, l
335 IF( a( i, j ).NE.czero .OR. b( i, j ).NE.czero )
336 $ GO TO 150
337 130 CONTINUE
338 i = ip1 - 1
339 140 CONTINUE
340 m = k
341 iflow = 2
342 GO TO 160
343 150 CONTINUE
344 GO TO 190
345*
346* Permute rows M and I
347*
348 160 CONTINUE
349 lscale( m ) = i
350 IF( i.EQ.m )
351 $ GO TO 170
352 CALL cswap( n-k+1, a( i, k ), lda, a( m, k ), lda )
353 CALL cswap( n-k+1, b( i, k ), ldb, b( m, k ), ldb )
354*
355* Permute columns M and J
356*
357 170 CONTINUE
358 rscale( m ) = j
359 IF( j.EQ.m )
360 $ GO TO 180
361 CALL cswap( l, a( 1, j ), 1, a( 1, m ), 1 )
362 CALL cswap( l, b( 1, j ), 1, b( 1, m ), 1 )
363*
364 180 CONTINUE
365 GO TO ( 20, 90 )iflow
366*
367 190 CONTINUE
368 ilo = k
369 ihi = l
370*
371 IF( lsame( job, 'P' ) ) THEN
372 DO 195 i = ilo, ihi
373 lscale( i ) = one
374 rscale( i ) = one
375 195 CONTINUE
376 RETURN
377 END IF
378*
379 IF( ilo.EQ.ihi )
380 $ RETURN
381*
382* Balance the submatrix in rows ILO to IHI.
383*
384 nr = ihi - ilo + 1
385 DO 200 i = ilo, ihi
386 rscale( i ) = zero
387 lscale( i ) = zero
388*
389 work( i ) = zero
390 work( i+n ) = zero
391 work( i+2*n ) = zero
392 work( i+3*n ) = zero
393 work( i+4*n ) = zero
394 work( i+5*n ) = zero
395 200 CONTINUE
396*
397* Compute right side vector in resulting linear equations
398*
399 basl = log10( sclfac )
400 DO 240 i = ilo, ihi
401 DO 230 j = ilo, ihi
402 IF( a( i, j ).EQ.czero ) THEN
403 ta = zero
404 GO TO 210
405 END IF
406 ta = log10( cabs1( a( i, j ) ) ) / basl
407*
408 210 CONTINUE
409 IF( b( i, j ).EQ.czero ) THEN
410 tb = zero
411 GO TO 220
412 END IF
413 tb = log10( cabs1( b( i, j ) ) ) / basl
414*
415 220 CONTINUE
416 work( i+4*n ) = work( i+4*n ) - ta - tb
417 work( j+5*n ) = work( j+5*n ) - ta - tb
418 230 CONTINUE
419 240 CONTINUE
420*
421 coef = one / real( 2*nr )
422 coef2 = coef*coef
423 coef5 = half*coef2
424 nrp2 = nr + 2
425 beta = zero
426 it = 1
427*
428* Start generalized conjugate gradient iteration
429*
430 250 CONTINUE
431*
432 gamma = sdot( nr, work( ilo+4*n ), 1, work( ilo+4*n ), 1 ) +
433 $ sdot( nr, work( ilo+5*n ), 1, work( ilo+5*n ), 1 )
434*
435 ew = zero
436 ewc = zero
437 DO 260 i = ilo, ihi
438 ew = ew + work( i+4*n )
439 ewc = ewc + work( i+5*n )
440 260 CONTINUE
441*
442 gamma = coef*gamma - coef2*( ew**2+ewc**2 ) - coef5*( ew-ewc )**2
443 IF( gamma.EQ.zero )
444 $ GO TO 350
445 IF( it.NE.1 )
446 $ beta = gamma / pgamma
447 t = coef5*( ewc-three*ew )
448 tc = coef5*( ew-three*ewc )
449*
450 CALL sscal( nr, beta, work( ilo ), 1 )
451 CALL sscal( nr, beta, work( ilo+n ), 1 )
452*
453 CALL saxpy( nr, coef, work( ilo+4*n ), 1, work( ilo+n ), 1 )
454 CALL saxpy( nr, coef, work( ilo+5*n ), 1, work( ilo ), 1 )
455*
456 DO 270 i = ilo, ihi
457 work( i ) = work( i ) + tc
458 work( i+n ) = work( i+n ) + t
459 270 CONTINUE
460*
461* Apply matrix to vector
462*
463 DO 300 i = ilo, ihi
464 kount = 0
465 sum = zero
466 DO 290 j = ilo, ihi
467 IF( a( i, j ).EQ.czero )
468 $ GO TO 280
469 kount = kount + 1
470 sum = sum + work( j )
471 280 CONTINUE
472 IF( b( i, j ).EQ.czero )
473 $ GO TO 290
474 kount = kount + 1
475 sum = sum + work( j )
476 290 CONTINUE
477 work( i+2*n ) = real( kount )*work( i+n ) + sum
478 300 CONTINUE
479*
480 DO 330 j = ilo, ihi
481 kount = 0
482 sum = zero
483 DO 320 i = ilo, ihi
484 IF( a( i, j ).EQ.czero )
485 $ GO TO 310
486 kount = kount + 1
487 sum = sum + work( i+n )
488 310 CONTINUE
489 IF( b( i, j ).EQ.czero )
490 $ GO TO 320
491 kount = kount + 1
492 sum = sum + work( i+n )
493 320 CONTINUE
494 work( j+3*n ) = real( kount )*work( j ) + sum
495 330 CONTINUE
496*
497 sum = sdot( nr, work( ilo+n ), 1, work( ilo+2*n ), 1 ) +
498 $ sdot( nr, work( ilo ), 1, work( ilo+3*n ), 1 )
499 alpha = gamma / sum
500*
501* Determine correction to current iteration
502*
503 cmax = zero
504 DO 340 i = ilo, ihi
505 cor = alpha*work( i+n )
506 IF( abs( cor ).GT.cmax )
507 $ cmax = abs( cor )
508 lscale( i ) = lscale( i ) + cor
509 cor = alpha*work( i )
510 IF( abs( cor ).GT.cmax )
511 $ cmax = abs( cor )
512 rscale( i ) = rscale( i ) + cor
513 340 CONTINUE
514 IF( cmax.LT.half )
515 $ GO TO 350
516*
517 CALL saxpy( nr, -alpha, work( ilo+2*n ), 1, work( ilo+4*n ), 1 )
518 CALL saxpy( nr, -alpha, work( ilo+3*n ), 1, work( ilo+5*n ), 1 )
519*
520 pgamma = gamma
521 it = it + 1
522 IF( it.LE.nrp2 )
523 $ GO TO 250
524*
525* End generalized conjugate gradient iteration
526*
527 350 CONTINUE
528 sfmin = slamch( 'S' )
529 sfmax = one / sfmin
530 lsfmin = int( log10( sfmin ) / basl+one )
531 lsfmax = int( log10( sfmax ) / basl )
532 DO 360 i = ilo, ihi
533 irab = icamax( n-ilo+1, a( i, ilo ), lda )
534 rab = abs( a( i, irab+ilo-1 ) )
535 irab = icamax( n-ilo+1, b( i, ilo ), ldb )
536 rab = max( rab, abs( b( i, irab+ilo-1 ) ) )
537 lrab = int( log10( rab+sfmin ) / basl+one )
538 ir = int( lscale( i ) + sign( half, lscale( i ) ) )
539 ir = min( max( ir, lsfmin ), lsfmax, lsfmax-lrab )
540 lscale( i ) = sclfac**ir
541 icab = icamax( ihi, a( 1, i ), 1 )
542 cab = abs( a( icab, i ) )
543 icab = icamax( ihi, b( 1, i ), 1 )
544 cab = max( cab, abs( b( icab, i ) ) )
545 lcab = int( log10( cab+sfmin ) / basl+one )
546 jc = int( rscale( i ) + sign( half, rscale( i ) ) )
547 jc = min( max( jc, lsfmin ), lsfmax, lsfmax-lcab )
548 rscale( i ) = sclfac**jc
549 360 CONTINUE
550*
551* Row scaling of matrices A and B
552*
553 DO 370 i = ilo, ihi
554 CALL csscal( n-ilo+1, lscale( i ), a( i, ilo ), lda )
555 CALL csscal( n-ilo+1, lscale( i ), b( i, ilo ), ldb )
556 370 CONTINUE
557*
558* Column scaling of matrices A and B
559*
560 DO 380 j = ilo, ihi
561 CALL csscal( ihi, rscale( j ), a( 1, j ), 1 )
562 CALL csscal( ihi, rscale( j ), b( 1, j ), 1 )
563 380 CONTINUE
564*
565 RETURN
566*
567* End of CGGBAL
568*
569 END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine csscal(N, SA, CX, INCX)
CSSCAL
Definition: csscal.f:78
subroutine cswap(N, CX, INCX, CY, INCY)
CSWAP
Definition: cswap.f:81
subroutine cggbal(JOB, N, A, LDA, B, LDB, ILO, IHI, LSCALE, RSCALE, WORK, INFO)
CGGBAL
Definition: cggbal.f:177
subroutine sscal(N, SA, SX, INCX)
SSCAL
Definition: sscal.f:79
subroutine saxpy(N, SA, SX, INCX, SY, INCY)
SAXPY
Definition: saxpy.f:89