1SLAQR5(1)           LAPACK auxiliary routine (version 3.2)           SLAQR5(1)
2
3
4

NAME

SYNOPSIS

7       SUBROUTINE SLAQR5( WANTT, WANTZ, KACC22, N, KTOP, KBOT, NSHFTS, SR, SI,
8                          H, LDH, ILOZ, IHIZ, Z, LDZ, V, LDV, U, LDU, NV,  WV,
9                          LDWV, NH, WH, LDWH )
10
11           INTEGER        IHIZ, ILOZ, KACC22, KBOT, KTOP, LDH, LDU, LDV, LDWH,
12                          LDWV, LDZ, N, NH, NSHFTS, NV
13
14           LOGICAL        WANTT, WANTZ
15
16           REAL           H( LDH, * ), SI( * ), SR( * ), U( LDU, * ), V(  LDV,
17                          * ), WH( LDWH, * ), WV( LDWV, * ), Z( LDZ, * )
18
19           REAL           ZERO, ONE
20
21           PARAMETER      ( ZERO = 0.0e0, ONE = 1.0e0 )
22
23           REAL           ALPHA,  BETA,  H11,  H12,  H21, H22, REFSUM, SAFMAX,
24                          SAFMIN, SCL, SMLNUM, SWAP, TST1, TST2, ULP
25
26           INTEGER        I, I2, I4, INCOL, J, J2, J4, JBOT, JCOL, JLEN, JROW,
27                          JTOP,  K,  K1,  KDU,  KMS,  KNZ, KRCOL, KZS, M, M22,
28                          MBOT, MEND, MSTART, MTOP, NBMPS, NDCOL, NS, NU
29
30           LOGICAL        ACCUM, BLK22, BMP22
31
32           REAL           SLAMCH
33
34           EXTERNAL       SLAMCH
35
36           INTRINSIC      ABS, MAX, MIN, MOD, REAL
37
38           REAL           VT( 3 )
39
40           EXTERNAL       SGEMM, SLABAD, SLACPY, SLAQR1, SLARFG, SLASET, STRMM
41
42           IF(            NSHFTS.LT.2 ) RETURN
43
44           IF(            KTOP.GE.KBOT ) RETURN
45
46           DO             10 I = 1, NSHFTS - 2, 2
47
48           IF(            SI( I ).NE.-SI( I+1 ) ) THEN
49
50           SWAP           = SR( I )
51
52           SR(            I ) = SR( I+1 )
53
54           SR(            I+1 ) = SR( I+2 )
55
56           SR(            I+2 ) = SWAP
57
58           SWAP           = SI( I )
59
60           SI(            I ) = SI( I+1 )
61
62           SI(            I+1 ) = SI( I+2 )
63
64           SI(            I+2 ) = SWAP
65
66           END            IF
67
68           10             CONTINUE
69
70           NS             = NSHFTS - MOD( NSHFTS, 2 )
71
72           SAFMIN         = SLAMCH( 'SAFE MINIMUM' )
73
74           SAFMAX         = ONE / SAFMIN
75
76           CALL           SLABAD( SAFMIN, SAFMAX )
77
78           ULP            = SLAMCH( 'PRECISION' )
79
80           SMLNUM         = SAFMIN*( REAL( N ) / ULP )
81
82           ACCUM          = ( KACC22.EQ.1 ) .OR. ( KACC22.EQ.2 )
83
84           BLK22          = ( NS.GT.2 ) .AND. ( KACC22.EQ.2 )
85
86           IF(            KTOP+2.LE.KBOT ) H( KTOP+2, KTOP ) = ZERO
87
88           NBMPS          = NS / 2
89
90           KDU            = 6*NBMPS - 3
91
92           DO             220 INCOL = 3*( 1-NBMPS ) + KTOP  -  1,  KBOT  -  2,
93                          3*NBMPS - 2
94
95           NDCOL          = INCOL + KDU
96
97           IF(            ACCUM  ) CALL SLASET( 'ALL', KDU, KDU, ZERO, ONE, U,
98                          LDU )
99
100           DO             150 KRCOL = INCOL, MIN( INCOL+3*NBMPS-3, KBOT-2 )
101
102           MTOP           = MAX( 1, ( ( KTOP-1 )-KRCOL+2 ) / 3+1 )
103
104           MBOT           = MIN( NBMPS, ( KBOT-KRCOL ) / 3 )
105
106           M22            = MBOT + 1
107
108           BMP22          = ( MBOT.LT.NBMPS ) .AND. ( KRCOL+3*( M22-1 )  ).EQ.
109                          ( KBOT-2 )
110
111           DO             20 M = MTOP, MBOT
112
113           K              = KRCOL + 3*( M-1 )
114
115           IF(            K.EQ.KTOP-1 ) THEN
116
117           CALL           SLAQR1(  3,  H(  KTOP, KTOP ), LDH, SR( 2*M-1 ), SI(
118                          2*M-1 ), SR( 2*M ), SI( 2*M ), V( 1, M ) )
119
120           ALPHA          = V( 1, M )
121
122           CALL           SLARFG( 3, ALPHA, V( 2, M ), 1, V( 1, M ) )
123
124           ELSE
125
126           BETA           = H( K+1, K )
127
128           V(             2, M ) = H( K+2, K )
129
130           V(             3, M ) = H( K+3, K )
131
132           CALL           SLARFG( 3, BETA, V( 2, M ), 1, V( 1, M ) )
133
134           IF(            H( K+3, K ).NE.ZERO .OR. H(  K+3,  K+1  ).NE.   ZERO
135                          .OR. H( K+3, K+2 ).EQ.ZERO ) THEN
136
137           H(             K+1, K ) = BETA
138
139           H(             K+2, K ) = ZERO
140
141           H(             K+3, K ) = ZERO
142
143           ELSE
144
145           CALL           SLAQR1(  3,  H(  K+1,  K+1  ), LDH, SR( 2*M-1 ), SI(
146                          2*M-1 ), SR( 2*M ), SI( 2*M ), VT )
147
148           ALPHA          = VT( 1 )
149
150           CALL           SLARFG( 3, ALPHA, VT( 2 ), 1, VT( 1 ) )
151
152           REFSUM         = VT( 1 )*( H( K+1, K )+VT( 2 )* H( K+2, K ) )
153
154           IF(            ABS( H( K+2, K )-REFSUM*VT( 2 ) )+ ABS( REFSUM*VT( 3
155                          )  ).GT.ULP*  (  ABS( H( K, K ) )+ABS( H( K+1, K+1 )
156                          )+ABS( H( K+2, K+2 ) ) ) ) THEN
157
158           H(             K+1, K ) = BETA
159
160           H(             K+2, K ) = ZERO
161
162           H(             K+3, K ) = ZERO
163
164           ELSE
165
166           H(             K+1, K ) = H( K+1, K ) - REFSUM
167
168           H(             K+2, K ) = ZERO
169
170           H(             K+3, K ) = ZERO
171
172           V(             1, M ) = VT( 1 )
173
174           V(             2, M ) = VT( 2 )
175
176           V(             3, M ) = VT( 3 )
177
178           END            IF
179
180           END            IF
181
182           END            IF
183
184           20             CONTINUE
185
186           K              = KRCOL + 3*( M22-1 )
187
188           IF(            BMP22 ) THEN
189
190           IF(            K.EQ.KTOP-1 ) THEN
191
192           CALL           SLAQR1( 2, H( K+1, K+1 ), LDH, SR(  2*M22-1  ),  SI(
193                          2*M22-1 ), SR( 2*M22 ), SI( 2*M22 ), V( 1, M22 ) )
194
195           BETA           = V( 1, M22 )
196
197           CALL           SLARFG( 2, BETA, V( 2, M22 ), 1, V( 1, M22 ) )
198
199           ELSE
200
201           BETA           = H( K+1, K )
202
203           V(             2, M22 ) = H( K+2, K )
204
205           CALL           SLARFG( 2, BETA, V( 2, M22 ), 1, V( 1, M22 ) )
206
207           H(             K+1, K ) = BETA
208
209           H(             K+2, K ) = ZERO
210
211           END            IF
212
213           END            IF
214
215           IF(            ACCUM ) THEN
216
217           JBOT           = MIN( NDCOL, KBOT )
218
219           ELSE           IF( WANTT ) THEN
220
221           JBOT           = N
222
223           ELSE
224
225           JBOT           = KBOT
226
227           END            IF
228
229           DO             40 J = MAX( KTOP, KRCOL ), JBOT
230
231           MEND           = MIN( MBOT, ( J-KRCOL+2 ) / 3 )
232
233           DO             30 M = MTOP, MEND
234
235           K              = KRCOL + 3*( M-1 )
236
237           REFSUM         =  V( 1, M )*( H( K+1, J )+V( 2, M )* H( K+2, J )+V(
238                          3, M )*H( K+3, J ) )
239
240           H(             K+1, J ) = H( K+1, J ) - REFSUM
241
242           H(             K+2, J ) = H( K+2, J ) - REFSUM*V( 2, M )
243
244           H(             K+3, J ) = H( K+3, J ) - REFSUM*V( 3, M )
245
246           30             CONTINUE
247
248           40             CONTINUE
249
250           IF(            BMP22 ) THEN
251
252           K              = KRCOL + 3*( M22-1 )
253
254           DO             50 J = MAX( K+1, KTOP ), JBOT
255
256           REFSUM         = V( 1, M22 )*( H( K+1, J )+V( 2, M22 )* H( K+2, J )
257                          )
258
259           H(             K+1, J ) = H( K+1, J ) - REFSUM
260
261           H(             K+2, J ) = H( K+2, J ) - REFSUM*V( 2, M22 )
262
263           50             CONTINUE
264
265           END            IF
266
267           IF(            ACCUM ) THEN
268
269           JTOP           = MAX( KTOP, INCOL )
270
271           ELSE           IF( WANTT ) THEN
272
273           JTOP           = 1
274
275           ELSE
276
277           JTOP           = KTOP
278
279           END            IF
280
281           DO             90 M = MTOP, MBOT
282
283           IF(            V( 1, M ).NE.ZERO ) THEN
284
285           K              = KRCOL + 3*( M-1 )
286
287           DO             60 J = JTOP, MIN( KBOT, K+3 )
288
289           REFSUM         =  V( 1, M )*( H( J, K+1 )+V( 2, M )* H( J, K+2 )+V(
290                          3, M )*H( J, K+3 ) )
291
292           H(             J, K+1 ) = H( J, K+1 ) - REFSUM
293
294           H(             J, K+2 ) = H( J, K+2 ) - REFSUM*V( 2, M )
295
296           H(             J, K+3 ) = H( J, K+3 ) - REFSUM*V( 3, M )
297
298           60             CONTINUE
299
300           IF(            ACCUM ) THEN
301
302           KMS            = K - INCOL
303
304           DO             70 J = MAX( 1, KTOP-INCOL ), KDU
305
306           REFSUM         = V( 1, M )*( U( J, KMS+1 )+V( 2, M )* U(  J,  KMS+2
307                          )+V( 3, M )*U( J, KMS+3 ) )
308
309           U(             J, KMS+1 ) = U( J, KMS+1 ) - REFSUM
310
311           U(             J, KMS+2 ) = U( J, KMS+2 ) - REFSUM*V( 2, M )
312
313           U(             J, KMS+3 ) = U( J, KMS+3 ) - REFSUM*V( 3, M )
314
315           70             CONTINUE
316
317           ELSE           IF( WANTZ ) THEN
318
319           DO             80 J = ILOZ, IHIZ
320
321           REFSUM         =  V( 1, M )*( Z( J, K+1 )+V( 2, M )* Z( J, K+2 )+V(
322                          3, M )*Z( J, K+3 ) )
323
324           Z(             J, K+1 ) = Z( J, K+1 ) - REFSUM
325
326           Z(             J, K+2 ) = Z( J, K+2 ) - REFSUM*V( 2, M )
327
328           Z(             J, K+3 ) = Z( J, K+3 ) - REFSUM*V( 3, M )
329
330           80             CONTINUE
331
332           END            IF
333
334           END            IF
335
336           90             CONTINUE
337
338           K              = KRCOL + 3*( M22-1 )
339
340           IF(            BMP22 .AND. ( V( 1, M22 ).NE.ZERO ) ) THEN
341
342           DO             100 J = JTOP, MIN( KBOT, K+3 )
343
344           REFSUM         = V( 1, M22 )*( H( J, K+1 )+V( 2, M22 )* H( J, K+2 )
345                          )
346
347           H(             J, K+1 ) = H( J, K+1 ) - REFSUM
348
349           H(             J, K+2 ) = H( J, K+2 ) - REFSUM*V( 2, M22 )
350
351           100            CONTINUE
352
353           IF(            ACCUM ) THEN
354
355           KMS            = K - INCOL
356
357           DO             110 J = MAX( 1, KTOP-INCOL ), KDU
358
359           REFSUM         =  V(  1,  M22  )*( U( J, KMS+1 )+V( 2, M22 )* U( J,
360                          KMS+2 ) )
361
362           U(             J, KMS+1 ) = U( J, KMS+1 ) - REFSUM
363
364           U(             J, KMS+2 ) = U( J, KMS+2 ) - REFSUM*V( 2, M22 )
365
366           110            CONTINUE
367
368           ELSE           IF( WANTZ ) THEN
369
370           DO             120 J = ILOZ, IHIZ
371
372           REFSUM         = V( 1, M22 )*( Z( J, K+1 )+V( 2, M22 )* Z( J, K+2 )
373                          )
374
375           Z(             J, K+1 ) = Z( J, K+1 ) - REFSUM
376
377           Z(             J, K+2 ) = Z( J, K+2 ) - REFSUM*V( 2, M22 )
378
379           120            CONTINUE
380
381           END            IF
382
383           END            IF
384
385           MSTART         = MTOP
386
387           IF(            KRCOL+3*( MSTART-1 ).LT.KTOP ) MSTART = MSTART + 1
388
389           MEND           = MBOT
390
391           IF(            BMP22 ) MEND = MEND + 1
392
393           IF(            KRCOL.EQ.KBOT-2 ) MEND = MEND + 1
394
395           DO             130 M = MSTART, MEND
396
397           K              = MIN( KBOT-1, KRCOL+3*( M-1 ) )
398
399           IF(            H( K+1, K ).NE.ZERO ) THEN
400
401           TST1           = ABS( H( K, K ) ) + ABS( H( K+1, K+1 ) )
402
403           IF(            TST1.EQ.ZERO ) THEN
404
405           IF(            K.GE.KTOP+1 ) TST1 = TST1 + ABS( H( K, K-1 ) )
406
407           IF(            K.GE.KTOP+2 ) TST1 = TST1 + ABS( H( K, K-2 ) )
408
409           IF(            K.GE.KTOP+3 ) TST1 = TST1 + ABS( H( K, K-3 ) )
410
411           IF(            K.LE.KBOT-2 ) TST1 = TST1 + ABS( H( K+2, K+1 ) )
412
413           IF(            K.LE.KBOT-3 ) TST1 = TST1 + ABS( H( K+3, K+1 ) )
414
415           IF(            K.LE.KBOT-4 ) TST1 = TST1 + ABS( H( K+4, K+1 ) )
416
417           END            IF
418
419           IF(            ABS( H( K+1, K ) ).LE.MAX( SMLNUM, ULP*TST1 ) ) THEN
420
421           H12            = MAX( ABS( H( K+1, K ) ), ABS( H( K, K+1 ) ) )
422
423           H21            = MIN( ABS( H( K+1, K ) ), ABS( H( K, K+1 ) ) )
424
425           H11            =  MAX( ABS( H( K+1, K+1 ) ), ABS( H( K, K )-H( K+1,
426                          K+1 ) ) )
427
428           H22            = MIN( ABS( H( K+1, K+1 ) ), ABS( H( K, K )-H(  K+1,
429                          K+1 ) ) )
430
431           SCL            = H11 + H12
432
433           TST2           = H22*( H11 / SCL )
434
435           IF(            TST2.EQ.ZERO  .OR.  H21*( H12 / SCL ).LE.  MAX( SML‐
436                          NUM, ULP*TST2 ) )H( K+1, K ) = ZERO
437
438           END            IF
439
440           END            IF
441
442           130            CONTINUE
443
444           MEND           = MIN( NBMPS, ( KBOT-KRCOL-1 ) / 3 )
445
446           DO             140 M = MTOP, MEND
447
448           K              = KRCOL + 3*( M-1 )
449
450           REFSUM         = V( 1, M )*V( 3, M )*H( K+4, K+3 )
451
452           H(             K+4, K+1 ) = -REFSUM
453
454           H(             K+4, K+2 ) = -REFSUM*V( 2, M )
455
456           H(             K+4, K+3 ) = H( K+4, K+3 ) - REFSUM*V( 3, M )
457
458           140            CONTINUE
459
460           150            CONTINUE
461
462           IF(            ACCUM ) THEN
463
464           IF(            WANTT ) THEN
465
466           JTOP           = 1
467
468           JBOT           = N
469
470           ELSE
471
472           JTOP           = KTOP
473
474           JBOT           = KBOT
475
476           END            IF
477
478           IF(            ( .NOT.BLK22  )  .OR.  (  INCOL.LT.KTOP  )  .OR.   (
479                          NDCOL.GT.KBOT ) .OR. ( NS.LE.2 ) ) THEN
480
481           K1             = MAX( 1, KTOP-INCOL )
482
483           NU             = ( KDU-MAX( 0, NDCOL-KBOT ) ) - K1 + 1
484
485           DO             160 JCOL = MIN( NDCOL, KBOT ) + 1, JBOT, NH
486
487           JLEN           = MIN( NH, JBOT-JCOL+1 )
488
489           CALL           SGEMM(  'C',  'N',  NU,  JLEN, NU, ONE, U( K1, K1 ),
490                          LDU, H( INCOL+K1, JCOL ), LDH, ZERO, WH, LDWH )
491
492           CALL           SLACPY( 'ALL', NU, JLEN, WH, LDWH, H( INCOL+K1, JCOL
493                          ), LDH )
494
495           160            CONTINUE
496
497           DO             170 JROW = JTOP, MAX( KTOP, INCOL ) - 1, NV
498
499           JLEN           = MIN( NV, MAX( KTOP, INCOL )-JROW )
500
501           CALL           SGEMM(  'N',  'N',  JLEN,  NU,  NU,  ONE,  H(  JROW,
502                          INCOL+K1 ), LDH, U( K1, K1 ), LDU, ZERO, WV, LDWV )
503
504           CALL           SLACPY( 'ALL', JLEN, NU, WV, LDWV, H( JROW, INCOL+K1
505                          ), LDH )
506
507           170            CONTINUE
508
509           IF(            WANTZ ) THEN
510
511           DO             180 JROW = ILOZ, IHIZ, NV
512
513           JLEN           = MIN( NV, IHIZ-JROW+1 )
514
515           CALL           SGEMM(  'N',  'N',  JLEN,  NU,  NU,  ONE,  Z(  JROW,
516                          INCOL+K1 ), LDZ, U( K1, K1 ), LDU, ZERO, WV, LDWV )
517
518           CALL           SLACPY( 'ALL', JLEN, NU, WV, LDWV, Z( JROW, INCOL+K1
519                          ), LDZ )
520
521           180            CONTINUE
522
523           END            IF
524
525           ELSE
526
527           I2             = ( KDU+1 ) / 2
528
529           I4             = KDU
530
531           J2             = I4 - I2
532
533           J4             = KDU
534
535           KZS            = ( J4-J2 ) - ( NS+1 )
536
537           KNZ            = NS + 1
538
539           DO             190 JCOL = MIN( NDCOL, KBOT ) + 1, JBOT, NH
540
541           JLEN           = MIN( NH, JBOT-JCOL+1 )
542
543           CALL           SLACPY(  'ALL',  KNZ,  JLEN,  H( INCOL+1+J2, JCOL ),
544                          LDH, WH( KZS+1, 1 ), LDWH )
545
546           CALL           SLASET( 'ALL', KZS, JLEN, ZERO, ZERO, WH, LDWH )
547
548           CALL           STRMM( 'L', 'U', 'C', 'N', KNZ, JLEN, ONE, U(  J2+1,
549                          1+KZS ), LDU, WH( KZS+1, 1 ), LDWH )
550
551           CALL           SGEMM(  'C',  'N',  I2,  JLEN,  J2,  ONE, U, LDU, H(
552                          INCOL+1, JCOL ), LDH, ONE, WH, LDWH )
553
554           CALL           SLACPY( 'ALL', J2, JLEN, H( INCOL+1,  JCOL  ),  LDH,
555                          WH( I2+1, 1 ), LDWH )
556
557           CALL           STRMM( 'L', 'L', 'C', 'N', J2, JLEN, ONE, U( 1, I2+1
558                          ), LDU, WH( I2+1, 1 ), LDWH )
559
560           CALL           SGEMM( 'C', 'N', I4-I2, JLEN, J4-J2, ONE,  U(  J2+1,
561                          I2+1  ),  LDU,  H( INCOL+1+J2, JCOL ), LDH, ONE, WH(
562                          I2+1, 1 ), LDWH )
563
564           CALL           SLACPY( 'ALL', KDU, JLEN, WH, LDWH, H( INCOL+1, JCOL
565                          ), LDH )
566
567           190            CONTINUE
568
569           DO             200 JROW = JTOP, MAX( INCOL, KTOP ) - 1, NV
570
571           JLEN           = MIN( NV, MAX( INCOL, KTOP )-JROW )
572
573           CALL           SLACPY(  'ALL',  JLEN,  KNZ,  H( JROW, INCOL+1+J2 ),
574                          LDH, WV( 1, 1+KZS ), LDWV )
575
576           CALL           SLASET( 'ALL', JLEN, KZS, ZERO, ZERO, WV, LDWV )
577
578           CALL           STRMM( 'R', 'U', 'N', 'N', JLEN, KNZ, ONE, U(  J2+1,
579                          1+KZS ), LDU, WV( 1, 1+KZS ), LDWV )
580
581           CALL           SGEMM( 'N', 'N', JLEN, I2, J2, ONE, H( JROW, INCOL+1
582                          ), LDH, U, LDU, ONE, WV, LDWV )
583
584           CALL           SLACPY( 'ALL', JLEN, J2, H( JROW,  INCOL+1  ),  LDH,
585                          WV( 1, 1+I2 ), LDWV )
586
587           CALL           STRMM(  'R',  'L', 'N', 'N', JLEN, I4-I2, ONE, U( 1,
588                          I2+1 ), LDU, WV( 1, 1+I2 ), LDWV )
589
590           CALL           SGEMM( 'N', 'N', JLEN, I4-I2, J4-J2, ONE,  H(  JROW,
591                          INCOL+1+J2 ), LDH, U( J2+1, I2+1 ), LDU, ONE, WV( 1,
592                          1+I2 ), LDWV )
593
594           CALL           SLACPY( 'ALL', JLEN, KDU, WV, LDWV, H( JROW, INCOL+1
595                          ), LDH )
596
597           200            CONTINUE
598
599           IF(            WANTZ ) THEN
600
601           DO             210 JROW = ILOZ, IHIZ, NV
602
603           JLEN           = MIN( NV, IHIZ-JROW+1 )
604
605           CALL           SLACPY(  'ALL',  JLEN,  KNZ,  Z( JROW, INCOL+1+J2 ),
606                          LDZ, WV( 1, 1+KZS ), LDWV )
607
608           CALL           SLASET( 'ALL', JLEN, KZS, ZERO, ZERO, WV, LDWV )
609
610           CALL           STRMM( 'R', 'U', 'N', 'N', JLEN, KNZ, ONE, U(  J2+1,
611                          1+KZS ), LDU, WV( 1, 1+KZS ), LDWV )
612
613           CALL           SGEMM( 'N', 'N', JLEN, I2, J2, ONE, Z( JROW, INCOL+1
614                          ), LDZ, U, LDU, ONE, WV, LDWV )
615
616           CALL           SLACPY( 'ALL', JLEN, J2, Z( JROW,  INCOL+1  ),  LDZ,
617                          WV( 1, 1+I2 ), LDWV )
618
619           CALL           STRMM(  'R',  'L', 'N', 'N', JLEN, I4-I2, ONE, U( 1,
620                          I2+1 ), LDU, WV( 1, 1+I2 ), LDWV )
621
622           CALL           SGEMM( 'N', 'N', JLEN, I4-I2, J4-J2, ONE,  Z(  JROW,
623                          INCOL+1+J2 ), LDZ, U( J2+1, I2+1 ), LDU, ONE, WV( 1,
624                          1+I2 ), LDWV )
625
626           CALL           SLACPY( 'ALL', JLEN, KDU, WV, LDWV, Z( JROW, INCOL+1
627                          ), LDZ )
628
629           210            CONTINUE
630
631           END            IF
632
633           END            IF
634
635           END            IF
636
637           220            CONTINUE
638
639           END
640

PURPOSE

642 LAPACK auxiliary routine (versionNo3v.e2m)ber 2008                       SLAQR5(1)