HP 48gII hp 48gII_user's manual_English_E_HDPMSG48E67_V2.pdf - Page 360
X +16Y- Z = 41., X + 2Y + 3Z = 2
UPC - 829160013183
View all HP 48gII manuals
Add to My Manuals
Save this manual to your list of manuals |
Page 360 highlights
pivoting operations. When row and column exchanges are allowed in pivoting, the procedure is known as full pivoting. When exchanging rows and columns in partial or full pivoting, it is necessary to keep track of the exchanges because the order of the unknowns in the solution is altered by those exchanges. One way to keep track of column exchanges in partial or full pivoting mode, is to create a permutation matrix P = In×n, at the beginning of the procedure. Any row or column exchange required in the augmented matrix Aaug is also registered as a row or column exchange, respectively, in the permutation matrix. When the solution is achieved, then, we multiply the permutation matrix by the unknown vector x to obtain the order of the unknowns in the solution. In other words, the final solution is given by P⋅x = b', where b' is the last column of the augmented matrix after the solution has been found. Example of Gauss-Jordan elimination with full pivoting Let's illustrate full pivoting with an example. Solve the following system of equations using full pivoting and the Gauss-Jordan elimination procedure: X + 2Y + 3Z = 2, 2X + 3Z = -1, 8X +16Y- Z = 41. The augmented matrix and the permutation matrix are as follows: 1 2 3 2 1 0 0 A aug = 2 0 3 −1, P = 0 1 0. 8 16 −1 41 0 0 1 Store the augmented matrix in variable AAUG, then press , @AAUG to get a copy in the stack. We want to keep the CSWP (Column Swap) command readily available, for which we use: ,N~~cs~ (find CSWP), @@OK@@. You'll get an error message, press $, and ignore the message. Next, get the ROW menu available by pressing: „Ø @)CREAT @)@ROW@. Now we are ready to start the Gauss-Jordan elimination with full pivoting. We will need to keep track of the permutation matrix by hand, so take your notebook and write the P matrix shown above. Page 11-34
-
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
-
618
-
619
-
620
-
621
-
622
-
623
-
624
-
625
-
626
-
627
-
628
-
629
-
630
-
631
-
632
-
633
-
634
-
635
-
636
-
637
-
638
-
639
-
640
-
641
-
642
-
643
-
644
-
645
-
646
-
647
-
648
-
649
-
650
-
651
-
652
-
653
-
654
-
655
-
656
-
657
-
658
-
659
-
660
-
661
-
662
-
663
-
664
-
665
-
666
-
667
-
668
-
669
-
670
-
671
-
672
-
673
-
674
-
675
-
676
-
677
-
678
-
679
-
680
-
681
-
682
-
683
-
684
-
685
-
686
-
687
-
688
-
689
-
690
-
691
-
692
-
693
-
694
-
695
-
696
-
697
-
698
-
699
-
700
-
701
-
702
-
703
-
704
-
705
-
706
-
707
-
708
-
709
-
710
-
711
-
712
-
713
-
714
-
715
-
716
-
717
-
718
-
719
-
720
-
721
-
722
-
723
-
724
-
725
-
726
-
727
-
728
-
729
-
730
-
731
-
732
-
733
-
734
-
735
-
736
-
737
-
738
-
739
-
740
-
741
-
742
-
743
-
744
-
745
-
746
-
747
-
748
-
749
-
750
-
751
-
752
-
753
-
754
-
755
-
756
-
757
-
758
-
759
-
760
-
761
-
762
-
763
-
764
-
765
-
766
-
767
-
768
-
769
-
770
-
771
-
772
-
773
-
774
-
775
-
776
-
777
-
778
-
779
-
780
-
781
-
782
-
783
-
784
-
785
-
786
-
787
-
788
-
789
-
790
-
791
-
792
-
793
-
794
-
795
-
796
-
797
-
798
-
799
-
800
-
801
-
802
-
803
-
804
-
805
-
806
-
807
-
808
-
809
-
810
-
811
-
812
-
813
-
814
-
815
-
816
-
817
-
818
-
819
-
820
-
821
-
822
-
823
-
824
-
825
-
826
-
827
-
828
-
829
-
830
-
831
-
832
-
833
-
834
-
835
-
836
-
837
-
838
-
839
-
840
-
841
-
842
-
843
-
844
-
845
-
846
-
847
-
848
-
849
-
850
-
851
-
852
-
853
-
854
-
855
-
856
-
857
-
858
-
859
-
860
-
861
-
862
-
863
-
864