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
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
|
*> \brief \b SLAHQR
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download SLAHQR + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/slahqr.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/slahqr.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/slahqr.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition
* ==========
*
* SUBROUTINE SLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
* ILOZ, IHIZ, Z, LDZ, INFO )
*
* .. Scalar Arguments ..
* INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, N
* LOGICAL WANTT, WANTZ
* ..
* .. Array Arguments ..
* REAL H( LDH, * ), WI( * ), WR( * ), Z( LDZ, * )
* ..
*
* Purpose
* =======
*
*>\details \b Purpose:
*>\verbatim
*> Purpose
*> =======
*>
*> SLAHQR is an auxiliary routine called by SHSEQR to update the
*> eigenvalues and Schur decomposition already computed by SHSEQR, by
*> dealing with the Hessenberg submatrix in rows and columns ILO to
*> IHI.
*>
*>\endverbatim
*
* Arguments
* =========
*
*> \param[in] WANTT
*> \verbatim
*> WANTT is LOGICAL
*> = .TRUE. : the full Schur form T is required;
*> = .FALSE.: only eigenvalues are required.
*> \endverbatim
*>
*> \param[in] WANTZ
*> \verbatim
*> WANTZ is LOGICAL
*> = .TRUE. : the matrix of Schur vectors Z is required;
*> = .FALSE.: Schur vectors are not required.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix H. N >= 0.
*> \endverbatim
*>
*> \param[in] ILO
*> \verbatim
*> ILO is INTEGER
*> \endverbatim
*>
*> \param[in] IHI
*> \verbatim
*> IHI is INTEGER
*> It is assumed that H is already upper quasi-triangular in
*> rows and columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless
*> ILO = 1). SLAHQR works primarily with the Hessenberg
*> submatrix in rows and columns ILO to IHI, but applies
*> transformations to all of H if WANTT is .TRUE..
*> 1 <= ILO <= max(1,IHI); IHI <= N.
*> \endverbatim
*>
*> \param[in,out] H
*> \verbatim
*> H is REAL array, dimension (LDH,N)
*> On entry, the upper Hessenberg matrix H.
*> On exit, if INFO is zero and if WANTT is .TRUE., H is upper
*> quasi-triangular in rows and columns ILO:IHI, with any
*> 2-by-2 diagonal blocks in standard form. If INFO is zero
*> and WANTT is .FALSE., the contents of H are unspecified on
*> exit. The output state of H if INFO is nonzero is given
*> below under the description of INFO.
*> \endverbatim
*>
*> \param[in] LDH
*> \verbatim
*> LDH is INTEGER
*> The leading dimension of the array H. LDH >= max(1,N).
*> \endverbatim
*>
*> \param[out] WR
*> \verbatim
*> WR is REAL array, dimension (N)
*> \endverbatim
*>
*> \param[out] WI
*> \verbatim
*> WI is REAL array, dimension (N)
*> The real and imaginary parts, respectively, of the computed
*> eigenvalues ILO to IHI are stored in the corresponding
*> elements of WR and WI. If two eigenvalues are computed as a
*> complex conjugate pair, they are stored in consecutive
*> elements of WR and WI, say the i-th and (i+1)th, with
*> WI(i) > 0 and WI(i+1) < 0. If WANTT is .TRUE., the
*> eigenvalues are stored in the same order as on the diagonal
*> of the Schur form returned in H, with WR(i) = H(i,i), and, if
*> H(i:i+1,i:i+1) is a 2-by-2 diagonal block,
*> WI(i) = sqrt(H(i+1,i)*H(i,i+1)) and WI(i+1) = -WI(i).
*> \endverbatim
*>
*> \param[in] ILOZ
*> \verbatim
*> ILOZ is INTEGER
*> \endverbatim
*>
*> \param[in] IHIZ
*> \verbatim
*> IHIZ is INTEGER
*> Specify the rows of Z to which transformations must be
*> applied if WANTZ is .TRUE..
*> 1 <= ILOZ <= ILO; IHI <= IHIZ <= N.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*> Z is REAL array, dimension (LDZ,N)
*> If WANTZ is .TRUE., on entry Z must contain the current
*> matrix Z of transformations accumulated by SHSEQR, and on
*> exit Z has been updated; transformations are applied only to
*> the submatrix Z(ILOZ:IHIZ,ILO:IHI).
*> If WANTZ is .FALSE., Z is not referenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> .GT. 0: If INFO = i, SLAHQR failed to compute all the
*> eigenvalues ILO to IHI in a total of 30 iterations
*> per eigenvalue; elements i+1:ihi of WR and WI
*> contain those eigenvalues which have been
*> successfully computed.
*>
*> If INFO .GT. 0 and WANTT is .FALSE., then on exit,
*> the remaining unconverged eigenvalues are the
*> eigenvalues of the upper Hessenberg matrix rows
*> and columns ILO thorugh INFO of the final, output
*> value of H.
*>
*> If INFO .GT. 0 and WANTT is .TRUE., then on exit
*> (*) (initial value of H)*U = U*(final value of H)
*> where U is an orthognal matrix. The final
*> value of H is upper Hessenberg and triangular in
*> rows and columns INFO+1 through IHI.
*>
*> If INFO .GT. 0 and WANTZ is .TRUE., then on exit
*> (final value of Z) = (initial value of Z)*U
*> where U is the orthogonal matrix in (*)
*> (regardless of the value of WANTT.)
*> \endverbatim
*>
*
* Authors
* =======
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup realOTHERauxiliary
*
*
* Further Details
* ===============
*>\details \b Further \b Details
*> \verbatim
*>
*> 02-96 Based on modifications by
*> David Day, Sandia National Laboratory, USA
*>
*> 12-04 Further modifications by
*> Ralph Byers, University of Kansas, USA
*> This is a modified version of SLAHQR from LAPACK version 3.0.
*> It is (1) more robust against overflow and underflow and
*> (2) adopts the more conservative Ahues & Tisseur stopping
*> criterion (LAWN 122, 1997).
*>
*> \endverbatim
*>
* =====================================================================
SUBROUTINE SLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
$ ILOZ, IHIZ, Z, LDZ, INFO )
*
* -- LAPACK auxiliary routine (version 3.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, N
LOGICAL WANTT, WANTZ
* ..
* .. Array Arguments ..
REAL H( LDH, * ), WI( * ), WR( * ), Z( LDZ, * )
* ..
*
* =========================================================
*
* .. Parameters ..
INTEGER ITMAX
PARAMETER ( ITMAX = 30 )
REAL ZERO, ONE, TWO
PARAMETER ( ZERO = 0.0e0, ONE = 1.0e0, TWO = 2.0e0 )
REAL DAT1, DAT2
PARAMETER ( DAT1 = 3.0e0 / 4.0e0, DAT2 = -0.4375e0 )
* ..
* .. Local Scalars ..
REAL AA, AB, BA, BB, CS, DET, H11, H12, H21, H21S,
$ H22, RT1I, RT1R, RT2I, RT2R, RTDISC, S, SAFMAX,
$ SAFMIN, SMLNUM, SN, SUM, T1, T2, T3, TR, TST,
$ ULP, V2, V3
INTEGER I, I1, I2, ITS, J, K, L, M, NH, NR, NZ
* ..
* .. Local Arrays ..
REAL V( 3 )
* ..
* .. External Functions ..
REAL SLAMCH
EXTERNAL SLAMCH
* ..
* .. External Subroutines ..
EXTERNAL SCOPY, SLABAD, SLANV2, SLARFG, SROT
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, REAL, SQRT
* ..
* .. Executable Statements ..
*
INFO = 0
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
IF( ILO.EQ.IHI ) THEN
WR( ILO ) = H( ILO, ILO )
WI( ILO ) = ZERO
RETURN
END IF
*
* ==== clear out the trash ====
DO 10 J = ILO, IHI - 3
H( J+2, J ) = ZERO
H( J+3, J ) = ZERO
10 CONTINUE
IF( ILO.LE.IHI-2 )
$ H( IHI, IHI-2 ) = ZERO
*
NH = IHI - ILO + 1
NZ = IHIZ - ILOZ + 1
*
* Set machine-dependent constants for the stopping criterion.
*
SAFMIN = SLAMCH( 'SAFE MINIMUM' )
SAFMAX = ONE / SAFMIN
CALL SLABAD( SAFMIN, SAFMAX )
ULP = SLAMCH( 'PRECISION' )
SMLNUM = SAFMIN*( REAL( NH ) / ULP )
*
* I1 and I2 are the indices of the first row and last column of H
* to which transformations must be applied. If eigenvalues only are
* being computed, I1 and I2 are set inside the main loop.
*
IF( WANTT ) THEN
I1 = 1
I2 = N
END IF
*
* The main loop begins here. I is the loop index and decreases from
* IHI to ILO in steps of 1 or 2. Each iteration of the loop works
* with the active submatrix in rows and columns L to I.
* Eigenvalues I+1 to IHI have already converged. Either L = ILO or
* H(L,L-1) is negligible so that the matrix splits.
*
I = IHI
20 CONTINUE
L = ILO
IF( I.LT.ILO )
$ GO TO 160
*
* Perform QR iterations on rows and columns ILO to I until a
* submatrix of order 1 or 2 splits off at the bottom because a
* subdiagonal element has become negligible.
*
DO 140 ITS = 0, ITMAX
*
* Look for a single small subdiagonal element.
*
DO 30 K = I, L + 1, -1
IF( ABS( H( K, K-1 ) ).LE.SMLNUM )
$ GO TO 40
TST = ABS( H( K-1, K-1 ) ) + ABS( H( K, K ) )
IF( TST.EQ.ZERO ) THEN
IF( K-2.GE.ILO )
$ TST = TST + ABS( H( K-1, K-2 ) )
IF( K+1.LE.IHI )
$ TST = TST + ABS( H( K+1, K ) )
END IF
* ==== The following is a conservative small subdiagonal
* . deflation criterion due to Ahues & Tisseur (LAWN 122,
* . 1997). It has better mathematical foundation and
* . improves accuracy in some cases. ====
IF( ABS( H( K, K-1 ) ).LE.ULP*TST ) THEN
AB = MAX( ABS( H( K, K-1 ) ), ABS( H( K-1, K ) ) )
BA = MIN( ABS( H( K, K-1 ) ), ABS( H( K-1, K ) ) )
AA = MAX( ABS( H( K, K ) ),
$ ABS( H( K-1, K-1 )-H( K, K ) ) )
BB = MIN( ABS( H( K, K ) ),
$ ABS( H( K-1, K-1 )-H( K, K ) ) )
S = AA + AB
IF( BA*( AB / S ).LE.MAX( SMLNUM,
$ ULP*( BB*( AA / S ) ) ) )GO TO 40
END IF
30 CONTINUE
40 CONTINUE
L = K
IF( L.GT.ILO ) THEN
*
* H(L,L-1) is negligible
*
H( L, L-1 ) = ZERO
END IF
*
* Exit from loop if a submatrix of order 1 or 2 has split off.
*
IF( L.GE.I-1 )
$ GO TO 150
*
* Now the active submatrix is in rows and columns L to I. If
* eigenvalues only are being computed, only the active submatrix
* need be transformed.
*
IF( .NOT.WANTT ) THEN
I1 = L
I2 = I
END IF
*
IF( ITS.EQ.10 ) THEN
*
* Exceptional shift.
*
S = ABS( H( L+1, L ) ) + ABS( H( L+2, L+1 ) )
H11 = DAT1*S + H( L, L )
H12 = DAT2*S
H21 = S
H22 = H11
ELSE IF( ITS.EQ.20 ) THEN
*
* Exceptional shift.
*
S = ABS( H( I, I-1 ) ) + ABS( H( I-1, I-2 ) )
H11 = DAT1*S + H( I, I )
H12 = DAT2*S
H21 = S
H22 = H11
ELSE
*
* Prepare to use Francis' double shift
* (i.e. 2nd degree generalized Rayleigh quotient)
*
H11 = H( I-1, I-1 )
H21 = H( I, I-1 )
H12 = H( I-1, I )
H22 = H( I, I )
END IF
S = ABS( H11 ) + ABS( H12 ) + ABS( H21 ) + ABS( H22 )
IF( S.EQ.ZERO ) THEN
RT1R = ZERO
RT1I = ZERO
RT2R = ZERO
RT2I = ZERO
ELSE
H11 = H11 / S
H21 = H21 / S
H12 = H12 / S
H22 = H22 / S
TR = ( H11+H22 ) / TWO
DET = ( H11-TR )*( H22-TR ) - H12*H21
RTDISC = SQRT( ABS( DET ) )
IF( DET.GE.ZERO ) THEN
*
* ==== complex conjugate shifts ====
*
RT1R = TR*S
RT2R = RT1R
RT1I = RTDISC*S
RT2I = -RT1I
ELSE
*
* ==== real shifts (use only one of them) ====
*
RT1R = TR + RTDISC
RT2R = TR - RTDISC
IF( ABS( RT1R-H22 ).LE.ABS( RT2R-H22 ) ) THEN
RT1R = RT1R*S
RT2R = RT1R
ELSE
RT2R = RT2R*S
RT1R = RT2R
END IF
RT1I = ZERO
RT2I = ZERO
END IF
END IF
*
* Look for two consecutive small subdiagonal elements.
*
DO 50 M = I - 2, L, -1
* Determine the effect of starting the double-shift QR
* iteration at row M, and see if this would make H(M,M-1)
* negligible. (The following uses scaling to avoid
* overflows and most underflows.)
*
H21S = H( M+1, M )
S = ABS( H( M, M )-RT2R ) + ABS( RT2I ) + ABS( H21S )
H21S = H( M+1, M ) / S
V( 1 ) = H21S*H( M, M+1 ) + ( H( M, M )-RT1R )*
$ ( ( H( M, M )-RT2R ) / S ) - RT1I*( RT2I / S )
V( 2 ) = H21S*( H( M, M )+H( M+1, M+1 )-RT1R-RT2R )
V( 3 ) = H21S*H( M+2, M+1 )
S = ABS( V( 1 ) ) + ABS( V( 2 ) ) + ABS( V( 3 ) )
V( 1 ) = V( 1 ) / S
V( 2 ) = V( 2 ) / S
V( 3 ) = V( 3 ) / S
IF( M.EQ.L )
$ GO TO 60
IF( ABS( H( M, M-1 ) )*( ABS( V( 2 ) )+ABS( V( 3 ) ) ).LE.
$ ULP*ABS( V( 1 ) )*( ABS( H( M-1, M-1 ) )+ABS( H( M,
$ M ) )+ABS( H( M+1, M+1 ) ) ) )GO TO 60
50 CONTINUE
60 CONTINUE
*
* Double-shift QR step
*
DO 130 K = M, I - 1
*
* The first iteration of this loop determines a reflection G
* from the vector V and applies it from left and right to H,
* thus creating a nonzero bulge below the subdiagonal.
*
* Each subsequent iteration determines a reflection G to
* restore the Hessenberg form in the (K-1)th column, and thus
* chases the bulge one step toward the bottom of the active
* submatrix. NR is the order of G.
*
NR = MIN( 3, I-K+1 )
IF( K.GT.M )
$ CALL SCOPY( NR, H( K, K-1 ), 1, V, 1 )
CALL SLARFG( NR, V( 1 ), V( 2 ), 1, T1 )
IF( K.GT.M ) THEN
H( K, K-1 ) = V( 1 )
H( K+1, K-1 ) = ZERO
IF( K.LT.I-1 )
$ H( K+2, K-1 ) = ZERO
ELSE IF( M.GT.L ) THEN
* ==== Use the following instead of
* . H( K, K-1 ) = -H( K, K-1 ) to
* . avoid a bug when v(2) and v(3)
* . underflow. ====
H( K, K-1 ) = H( K, K-1 )*( ONE-T1 )
END IF
V2 = V( 2 )
T2 = T1*V2
IF( NR.EQ.3 ) THEN
V3 = V( 3 )
T3 = T1*V3
*
* Apply G from the left to transform the rows of the matrix
* in columns K to I2.
*
DO 70 J = K, I2
SUM = H( K, J ) + V2*H( K+1, J ) + V3*H( K+2, J )
H( K, J ) = H( K, J ) - SUM*T1
H( K+1, J ) = H( K+1, J ) - SUM*T2
H( K+2, J ) = H( K+2, J ) - SUM*T3
70 CONTINUE
*
* Apply G from the right to transform the columns of the
* matrix in rows I1 to min(K+3,I).
*
DO 80 J = I1, MIN( K+3, I )
SUM = H( J, K ) + V2*H( J, K+1 ) + V3*H( J, K+2 )
H( J, K ) = H( J, K ) - SUM*T1
H( J, K+1 ) = H( J, K+1 ) - SUM*T2
H( J, K+2 ) = H( J, K+2 ) - SUM*T3
80 CONTINUE
*
IF( WANTZ ) THEN
*
* Accumulate transformations in the matrix Z
*
DO 90 J = ILOZ, IHIZ
SUM = Z( J, K ) + V2*Z( J, K+1 ) + V3*Z( J, K+2 )
Z( J, K ) = Z( J, K ) - SUM*T1
Z( J, K+1 ) = Z( J, K+1 ) - SUM*T2
Z( J, K+2 ) = Z( J, K+2 ) - SUM*T3
90 CONTINUE
END IF
ELSE IF( NR.EQ.2 ) THEN
*
* Apply G from the left to transform the rows of the matrix
* in columns K to I2.
*
DO 100 J = K, I2
SUM = H( K, J ) + V2*H( K+1, J )
H( K, J ) = H( K, J ) - SUM*T1
H( K+1, J ) = H( K+1, J ) - SUM*T2
100 CONTINUE
*
* Apply G from the right to transform the columns of the
* matrix in rows I1 to min(K+3,I).
*
DO 110 J = I1, I
SUM = H( J, K ) + V2*H( J, K+1 )
H( J, K ) = H( J, K ) - SUM*T1
H( J, K+1 ) = H( J, K+1 ) - SUM*T2
110 CONTINUE
*
IF( WANTZ ) THEN
*
* Accumulate transformations in the matrix Z
*
DO 120 J = ILOZ, IHIZ
SUM = Z( J, K ) + V2*Z( J, K+1 )
Z( J, K ) = Z( J, K ) - SUM*T1
Z( J, K+1 ) = Z( J, K+1 ) - SUM*T2
120 CONTINUE
END IF
END IF
130 CONTINUE
*
140 CONTINUE
*
* Failure to converge in remaining number of iterations
*
INFO = I
RETURN
*
150 CONTINUE
*
IF( L.EQ.I ) THEN
*
* H(I,I-1) is negligible: one eigenvalue has converged.
*
WR( I ) = H( I, I )
WI( I ) = ZERO
ELSE IF( L.EQ.I-1 ) THEN
*
* H(I-1,I-2) is negligible: a pair of eigenvalues have converged.
*
* Transform the 2-by-2 submatrix to standard Schur form,
* and compute and store the eigenvalues.
*
CALL SLANV2( H( I-1, I-1 ), H( I-1, I ), H( I, I-1 ),
$ H( I, I ), WR( I-1 ), WI( I-1 ), WR( I ), WI( I ),
$ CS, SN )
*
IF( WANTT ) THEN
*
* Apply the transformation to the rest of H.
*
IF( I2.GT.I )
$ CALL SROT( I2-I, H( I-1, I+1 ), LDH, H( I, I+1 ), LDH,
$ CS, SN )
CALL SROT( I-I1-1, H( I1, I-1 ), 1, H( I1, I ), 1, CS, SN )
END IF
IF( WANTZ ) THEN
*
* Apply the transformation to Z.
*
CALL SROT( NZ, Z( ILOZ, I-1 ), 1, Z( ILOZ, I ), 1, CS, SN )
END IF
END IF
*
* return to start of the main loop with new value of I.
*
I = L - 1
GO TO 20
*
160 CONTINUE
RETURN
*
* End of SLAHQR
*
END
|