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
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
1099
1100
1101
1102
1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
1133
1134
1135
1136
1137
1138
1139
1140
1141
1142
1143
1144
1145
1146
1147
1148
1149
1150
1151
1152
1153
1154
1155
1156
1157
1158
1159
1160
1161
1162
1163
1164
1165
1166
1167
1168
1169
1170
1171
1172
1173
1174
1175
1176
1177
1178
1179
1180
1181
1182
1183
1184
1185
1186
1187
1188
1189
1190
1191
1192
1193
1194
1195
1196
1197
1198
1199
1200
1201
1202
1203
1204
1205
1206
1207
1208
1209
1210
1211
1212
1213
1214
1215
1216
1217
1218
1219
1220
1221
1222
1223
1224
1225
1226
1227
1228
1229
1230
1231
1232
1233
1234
1235
1236
1237
1238
1239
1240
1241
1242
1243
1244
1245
1246
1247
1248
1249
1250
1251
1252
1253
1254
1255
1256
1257
1258
1259
1260
1261
1262
1263
1264
1265
1266
1267
1268
1269
1270
1271
1272
1273
1274
1275
1276
1277
1278
1279
1280
1281
1282
1283
1284
1285
1286
1287
1288
1289
1290
1291
1292
1293
1294
1295
1296
1297
1298
1299
1300
1301
1302
1303
1304
1305
1306
1307
1308
1309
1310
1311
1312
1313
1314
1315
1316
1317
1318
1319
1320
1321
1322
1323
1324
1325
1326
1327
1328
1329
1330
1331
1332
1333
1334
1335
1336
1337
1338
1339
1340
1341
1342
1343
1344
1345
1346
1347
1348
1349
1350
1351
1352
1353
1354
1355
1356
1357
1358
1359
1360
1361
1362
1363
1364
1365
1366
1367
1368
1369
1370
1371
1372
1373
1374
1375
1376
1377
1378
1379
1380
1381
1382
1383
1384
1385
1386
1387
1388
1389
1390
1391
1392
1393
1394
1395
1396
1397
1398
1399
1400
1401
1402
1403
1404
1405
1406
1407
1408
1409
1410
1411
1412
1413
1414
1415
1416
1417
1418
1419
1420
1421
1422
1423
1424
1425
1426
1427
1428
1429
1430
1431
1432
1433
1434
1435
1436
1437
1438
1439
1440
1441
1442
1443
1444
1445
1446
1447
1448
1449
1450
1451
1452
1453
1454
1455
1456
1457
1458
1459
1460
1461
1462
1463
1464
1465
1466
1467
1468
1469
1470
1471
1472
1473
1474
1475
1476
1477
1478
1479
1480
1481
1482
1483
1484
1485
1486
1487
1488
1489
1490
1491
1492
1493
1494
1495
1496
1497
1498
1499
1500
1501
1502
1503
1504
1505
1506
1507
1508
1509
1510
1511
1512
1513
1514
1515
1516
1517
1518
1519
1520
1521
1522
1523
1524
1525
1526
1527
1528
1529
1530
1531
1532
1533
1534
1535
1536
1537
1538
1539
1540
1541
1542
1543
1544
1545
1546
1547
1548
1549
1550
1551
1552
1553
1554
1555
1556
1557
1558
1559
1560
1561
1562
1563
1564
1565
1566
1567
1568
1569
1570
1571
1572
1573
1574
1575
1576
1577
1578
1579
1580
1581
1582
1583
1584
1585
1586
1587
1588
1589
1590
1591
1592
1593
1594
1595
1596
1597
1598
1599
1600
1601
1602
1603
1604
1605
1606
1607
1608
1609
1610
1611
1612
1613
1614
1615
1616
1617
1618
1619
1620
1621
1622
1623
1624
1625
1626
1627
1628
1629
1630
1631
1632
1633
1634
1635
1636
1637
1638
1639
1640
1641
1642
1643
1644
1645
1646
1647
1648
1649
1650
1651
1652
1653
1654
1655
1656
1657
1658
1659
1660
1661
1662
1663
1664
1665
1666
1667
1668
1669
1670
1671
1672
1673
1674
1675
1676
1677
1678
1679
1680
1681
1682
1683
1684
1685
1686
1687
1688
1689
1690
1691
1692
1693
1694
1695
1696
1697
1698
1699
1700
1701
1702
1703
1704
1705
1706
1707
1708
1709
1710
1711
1712
1713
1714
1715
1716
1717
1718
1719
1720
1721
1722
1723
1724
1725
1726
1727
1728
1729
1730
1731
1732
1733
1734
1735
1736
1737
1738
1739
1740
1741
1742
1743
1744
1745
1746
1747
1748
1749
1750
1751
1752
1753
1754
1755
1756
1757
1758
1759
1760
1761
1762
1763
1764
1765
1766
1767
1768
1769
1770
1771
1772
1773
1774
1775
1776
1777
1778
1779
1780
1781
1782
1783
1784
1785
1786
1787
1788
1789
1790
1791
1792
1793
1794
1795
1796
1797
1798
1799
1800
1801
1802
1803
1804
1805
1806
1807
1808
1809
1810
1811
1812
1813
1814
1815
1816
1817
1818
1819
1820
1821
1822
1823
1824
1825
1826
1827
1828
1829
1830
1831
1832
1833
1834
1835
1836
1837
1838
1839
1840
1841
1842
1843
1844
1845
1846
1847
1848
1849
1850
1851
1852
1853
1854
1855
1856
1857
1858
1859
1860
1861
1862
1863
1864
1865
1866
1867
1868
1869
1870
1871
1872
1873
1874
1875
1876
1877
1878
1879
1880
1881
1882
1883
1884
1885
1886
1887
1888
1889
1890
1891
1892
1893
1894
1895
1896
1897
1898
1899
1900
1901
1902
1903
1904
1905
1906
1907
1908
1909
1910
1911
1912
1913
1914
1915
1916
1917
1918
1919
1920
1921
1922
1923
1924
1925
1926
1927
1928
1929
1930
1931
1932
1933
1934
1935
1936
1937
1938
1939
1940
1941
1942
1943
1944
1945
1946
1947
1948
1949
1950
1951
1952
1953
1954
1955
1956
1957
1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
2020
2021
2022
2023
2024
2025
2026
2027
2028
2029
2030
2031
2032
2033
2034
2035
2036
2037
2038
2039
2040
2041
2042
2043
2044
2045
2046
2047
2048
2049
2050
2051
2052
2053
2054
2055
2056
2057
2058
2059
2060
2061
2062
2063
2064
2065
2066
2067
2068
2069
2070
2071
2072
2073
2074
2075
2076
2077
2078
2079
2080
2081
2082
2083
2084
2085
2086
2087
2088
2089
2090
2091
2092
2093
2094
2095
2096
2097
2098
2099
2100
2101
2102
2103
2104
2105
2106
2107
2108
2109
2110
2111
2112
2113
2114
2115
2116
2117
2118
2119
2120
2121
2122
2123
2124
2125
2126
2127
2128
2129
2130
2131
2132
2133
2134
2135
2136
2137
2138
2139
2140
2141
2142
2143
2144
2145
2146
2147
2148
2149
2150
2151
2152
2153
2154
2155
2156
2157
2158
2159
2160
2161
2162
2163
2164
2165
2166
2167
2168
2169
2170
2171
2172
2173
2174
2175
2176
2177
2178
2179
2180
2181
2182
2183
2184
2185
2186
2187
2188
2189
2190
2191
2192
2193
2194
2195
2196
2197
2198
2199
2200
2201
2202
2203
2204
2205
2206
2207
2208
2209
2210
2211
2212
2213
2214
2215
2216
2217
2218
2219
2220
2221
2222
2223
2224
2225
2226
2227
2228
2229
2230
2231
2232
2233
2234
2235
2236
2237
2238
2239
2240
2241
2242
2243
2244
2245
2246
2247
2248
2249
2250
2251
2252
2253
2254
2255
2256
2257
2258
2259
2260
2261
2262
2263
2264
2265
2266
2267
2268
2269
2270
2271
2272
2273
2274
2275
2276
2277
2278
2279
2280
2281
2282
2283
2284
2285
2286
2287
2288
2289
2290
2291
2292
2293
2294
2295
2296
2297
2298
2299
2300
2301
2302
2303
2304
2305
2306
2307
2308
2309
2310
2311
2312
2313
2314
2315
2316
2317
2318
2319
2320
2321
2322
2323
2324
2325
2326
2327
2328
2329
2330
2331
2332
2333
2334
2335
2336
2337
2338
2339
2340
2341
2342
2343
2344
2345
2346
2347
2348
2349
2350
2351
2352
2353
2354
2355
2356
2357
2358
2359
2360
2361
2362
2363
2364
2365
2366
2367
2368
2369
2370
2371
2372
2373
2374
2375
2376
2377
2378
2379
2380
2381
2382
2383
2384
2385
2386
2387
2388
2389
2390
2391
2392
2393
2394
2395
2396
2397
2398
2399
2400
2401
2402
2403
2404
2405
2406
2407
2408
2409
2410
2411
2412
2413
2414
2415
2416
2417
2418
2419
2420
2421
2422
2423
2424
2425
2426
2427
2428
2429
2430
2431
2432
2433
2434
2435
2436
2437
2438
2439
2440
2441
2442
2443
2444
2445
2446
2447
2448
2449
2450
2451
2452
2453
2454
2455
2456
2457
2458
2459
2460
2461
2462
2463
2464
2465
2466
2467
2468
2469
2470
2471
2472
2473
2474
2475
2476
2477
2478
2479
2480
2481
2482
2483
2484
2485
2486
2487
2488
2489
2490
2491
2492
2493
2494
2495
2496
2497
2498
2499
2500
2501
2502
2503
2504
2505
2506
2507
2508
2509
2510
2511
2512
2513
2514
2515
2516
2517
2518
2519
2520
2521
2522
2523
2524
2525
2526
2527
2528
2529
2530
2531
2532
2533
2534
2535
2536
2537
2538
2539
2540
2541
2542
2543
2544
2545
2546
2547
2548
2549
2550
2551
2552
2553
2554
2555
2556
2557
2558
2559
2560
2561
2562
2563
2564
2565
2566
2567
2568
2569
2570
2571
2572
2573
2574
2575
2576
2577
2578
2579
2580
2581
2582
2583
2584
2585
2586
2587
2588
2589
2590
2591
2592
2593
2594
2595
2596
2597
2598
2599
2600
2601
2602
2603
2604
2605
2606
2607
2608
2609
2610
2611
2612
2613
2614
2615
2616
2617
2618
2619
2620
2621
2622
2623
2624
2625
2626
2627
2628
2629
2630
2631
2632
2633
2634
2635
2636
2637
2638
2639
2640
2641
2642
2643
2644
2645
2646
2647
2648
2649
2650
2651
2652
2653
2654
2655
2656
2657
2658
2659
2660
2661
2662
2663
2664
2665
2666
2667
2668
2669
2670
2671
2672
2673
2674
2675
2676
2677
2678
2679
2680
2681
2682
2683
2684
2685
2686
2687
2688
2689
2690
2691
2692
2693
2694
2695
2696
2697
2698
2699
2700
2701
2702
2703
2704
2705
2706
2707
2708
2709
2710
2711
2712
2713
2714
2715
2716
2717
2718
2719
2720
2721
2722
2723
2724
2725
2726
2727
2728
2729
2730
2731
2732
2733
2734
2735
2736
2737
2738
2739
2740
2741
2742
2743
2744
2745
2746
2747
2748
2749
2750
2751
2752
2753
2754
2755
2756
2757
2758
2759
2760
2761
2762
2763
2764
2765
2766
2767
2768
2769
2770
2771
2772
2773
2774
2775
2776
2777
2778
2779
2780
2781
2782
2783
2784
2785
2786
2787
2788
2789
2790
2791
2792
2793
2794
2795
2796
2797
2798
2799
2800
2801
2802
2803
2804
2805
2806
2807
2808
2809
2810
2811
2812
2813
2814
2815
2816
2817
2818
2819
2820
2821
2822
2823
2824
2825
2826
2827
2828
2829
2830
2831
2832
2833
2834
2835
2836
2837
2838
2839
2840
2841
2842
2843
2844
2845
2846
2847
2848
2849
2850
2851
2852
2853
2854
2855
2856
2857
2858
2859
2860
2861
2862
2863
2864
2865
2866
2867
2868
2869
2870
2871
2872
2873
2874
2875
2876
2877
2878
2879
2880
2881
2882
2883
2884
2885
2886
2887
2888
2889
2890
2891
2892
2893
2894
2895
2896
2897
2898
2899
2900
2901
2902
2903
2904
2905
2906
2907
2908
2909
2910
2911
2912
2913
2914
2915
2916
2917
2918
2919
2920
2921
2922
2923
2924
2925
2926
2927
2928
2929
2930
2931
2932
2933
2934
2935
2936
2937
2938
2939
2940
2941
2942
2943
2944
2945
2946
2947
2948
2949
2950
2951
2952
2953
2954
2955
2956
2957
2958
2959
2960
2961
2962
2963
2964
2965
2966
2967
2968
2969
2970
2971
2972
2973
2974
2975
2976
2977
2978
2979
2980
2981
2982
2983
2984
2985
2986
2987
2988
2989
2990
2991
2992
2993
2994
2995
2996
2997
2998
2999
3000
3001
3002
3003
3004
3005
3006
3007
3008
3009
3010
3011
3012
3013
3014
3015
3016
3017
3018
3019
3020
3021
3022
3023
3024
3025
3026
3027
3028
3029
3030
3031
3032
3033
3034
3035
3036
3037
3038
3039
3040
3041
3042
3043
3044
3045
3046
3047
3048
3049
3050
3051
3052
3053
3054
3055
3056
3057
3058
3059
3060
3061
3062
3063
3064
3065
3066
3067
3068
3069
3070
3071
3072
3073
3074
3075
3076
3077
3078
3079
3080
3081
3082
3083
3084
3085
3086
3087
3088
3089
3090
3091
3092
3093
3094
3095
3096
3097
3098
3099
3100
3101
3102
3103
3104
3105
3106
3107
3108
3109
3110
3111
3112
3113
3114
3115
3116
3117
3118
3119
3120
3121
3122
3123
3124
3125
3126
3127
3128
3129
3130
3131
3132
3133
3134
3135
3136
3137
3138
3139
3140
3141
3142
3143
3144
3145
3146
3147
3148
3149
3150
3151
3152
3153
3154
3155
3156
3157
3158
3159
3160
3161
3162
3163
3164
3165
3166
3167
3168
3169
3170
3171
3172
3173
3174
3175
3176
3177
3178
3179
3180
3181
3182
3183
3184
3185
3186
3187
3188
3189
3190
3191
3192
3193
3194
3195
3196
3197
3198
3199
3200
3201
3202
3203
3204
3205
3206
3207
3208
3209
3210
3211
3212
3213
3214
3215
3216
3217
3218
3219
3220
3221
3222
3223
3224
3225
3226
3227
3228
3229
3230
3231
3232
3233
3234
3235
3236
3237
3238
3239
3240
3241
3242
3243
3244
3245
3246
3247
3248
3249
3250
3251
3252
3253
3254
3255
3256
3257
3258
3259
3260
3261
3262
3263
3264
3265
3266
3267
3268
3269
3270
3271
3272
3273
3274
3275
3276
3277
3278
3279
3280
3281
3282
3283
3284
3285
3286
3287
3288
3289
3290
3291
3292
3293
3294
3295
3296
3297
3298
3299
3300
3301
3302
3303
3304
3305
3306
3307
3308
3309
3310
3311
3312
3313
3314
3315
3316
3317
3318
3319
3320
3321
3322
3323
3324
3325
3326
3327
3328
3329
3330
3331
3332
3333
3334
3335
3336
3337
3338
3339
3340
3341
3342
3343
3344
3345
3346
3347
3348
3349
3350
3351
3352
3353
3354
3355
3356
3357
3358
3359
3360
3361
3362
3363
3364
3365
3366
3367
3368
3369
3370
3371
3372
3373
3374
3375
3376
3377
3378
3379
3380
3381
3382
3383
3384
3385
3386
3387
3388
3389
3390
3391
3392
3393
3394
3395
3396
3397
3398
3399
3400
3401
3402
3403
3404
3405
3406
3407
3408
3409
3410
3411
3412
3413
3414
3415
3416
3417
3418
3419
3420
3421
3422
3423
3424
3425
3426
3427
3428
3429
3430
3431
3432
3433
3434
3435
3436
3437
3438
3439
3440
3441
3442
3443
3444
3445
3446
3447
3448
3449
3450
3451
3452
3453
3454
3455
3456
3457
3458
3459
3460
3461
3462
3463
3464
3465
3466
3467
3468
3469
3470
3471
3472
3473
3474
3475
3476
3477
3478
3479
3480
3481
3482
3483
3484
3485
3486
3487
3488
3489
3490
3491
3492
3493
3494
3495
3496
3497
3498
3499
3500
3501
3502
3503
3504
3505
3506
3507
3508
3509
3510
3511
3512
3513
3514
3515
3516
3517
3518
3519
3520
3521
3522
3523
3524
3525
3526
3527
3528
3529
3530
3531
3532
3533
3534
3535
3536
3537
3538
3539
3540
3541
3542
3543
3544
3545
3546
3547
3548
3549
3550
3551
3552
3553
3554
3555
3556
3557
3558
3559
3560
3561
3562
3563
3564
3565
3566
3567
3568
3569
3570
3571
3572
3573
3574
3575
3576
3577
3578
3579
3580
3581
3582
3583
3584
3585
3586
3587
3588
3589
3590
3591
3592
3593
3594
3595
3596
3597
3598
3599
3600
3601
3602
3603
3604
3605
3606
3607
3608
3609
3610
3611
3612
3613
3614
3615
3616
3617
3618
3619
3620
3621
3622
3623
3624
3625
3626
3627
3628
3629
3630
3631
3632
3633
3634
3635
3636
3637
3638
3639
3640
3641
3642
3643
3644
3645
3646
3647
3648
3649
3650
3651
3652
3653
3654
3655
3656
3657
3658
3659
3660
3661
3662
3663
3664
3665
3666
3667
|
;;; calcalg2.el --- more algebraic functions for Calc
;; Copyright (C) 1990-1993, 2001-2017 Free Software Foundation, Inc.
;; Author: David Gillespie <daveg@synaptics.com>
;; This file is part of GNU Emacs.
;; GNU Emacs is free software: you can redistribute it and/or modify
;; it under the terms of the GNU General Public License as published by
;; the Free Software Foundation, either version 3 of the License, or
;; (at your option) any later version.
;; GNU Emacs is distributed in the hope that it will be useful,
;; but WITHOUT ANY WARRANTY; without even the implied warranty of
;; MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
;; GNU General Public License for more details.
;; You should have received a copy of the GNU General Public License
;; along with GNU Emacs. If not, see <https://www.gnu.org/licenses/>.
;;; Commentary:
;;; Code:
;; This file is autoloaded from calc-ext.el.
(require 'calc-ext)
(require 'calc-macs)
(defun calc-derivative (var num)
(interactive "sDifferentiate with respect to: \np")
(calc-slow-wrapper
(when (< num 0)
(error "Order of derivative must be positive"))
(let ((func (if (calc-is-hyperbolic) 'calcFunc-tderiv 'calcFunc-deriv))
n expr)
(if (or (equal var "") (equal var "$"))
(setq n 2
expr (calc-top-n 2)
var (calc-top-n 1))
(setq var (math-read-expr var))
(when (eq (car-safe var) 'error)
(error "Bad format in expression: %s" (nth 1 var)))
(setq n 1
expr (calc-top-n 1)))
(while (>= (setq num (1- num)) 0)
(setq expr (list func expr var)))
(calc-enter-result n "derv" expr))))
(defun calc-integral (var &optional arg)
(interactive "sIntegration variable: \nP")
(if arg
(calc-tabular-command 'calcFunc-integ "Integration" "intg" nil var nil nil)
(calc-slow-wrapper
(if (or (equal var "") (equal var "$"))
(calc-enter-result 2 "intg" (list 'calcFunc-integ
(calc-top-n 2)
(calc-top-n 1)))
(let ((var (math-read-expr var)))
(if (eq (car-safe var) 'error)
(error "Bad format in expression: %s" (nth 1 var)))
(calc-enter-result 1 "intg" (list 'calcFunc-integ
(calc-top-n 1)
var)))))))
(defun calc-num-integral (&optional varname lowname highname)
(interactive "sIntegration variable: ")
(calc-tabular-command 'calcFunc-ninteg "Integration" "nint"
nil varname lowname highname))
(defun calc-summation (arg &optional varname lowname highname)
(interactive "P\nsSummation variable: ")
(calc-tabular-command 'calcFunc-sum "Summation" "sum"
arg varname lowname highname))
(defun calc-alt-summation (arg &optional varname lowname highname)
(interactive "P\nsSummation variable: ")
(calc-tabular-command 'calcFunc-asum "Summation" "asum"
arg varname lowname highname))
(defun calc-product (arg &optional varname lowname highname)
(interactive "P\nsIndex variable: ")
(calc-tabular-command 'calcFunc-prod "Index" "prod"
arg varname lowname highname))
(defun calc-tabulate (arg &optional varname lowname highname)
(interactive "P\nsIndex variable: ")
(calc-tabular-command 'calcFunc-table "Index" "tabl"
arg varname lowname highname))
(defun calc-tabular-command (func prompt prefix arg varname lowname highname)
(calc-slow-wrapper
(let (var (low nil) (high nil) (step nil) stepname stepnum (num 1) expr)
(if (consp arg)
(setq stepnum 1)
(setq stepnum 0))
(if (or (equal varname "") (equal varname "$") (null varname))
(setq high (calc-top-n (+ stepnum 1))
low (calc-top-n (+ stepnum 2))
var (calc-top-n (+ stepnum 3))
num (+ stepnum 4))
(setq var (if (stringp varname) (math-read-expr varname) varname))
(if (eq (car-safe var) 'error)
(error "Bad format in expression: %s" (nth 1 var)))
(or lowname
(setq lowname (read-string (concat prompt " variable: " varname
", from: "))))
(if (or (equal lowname "") (equal lowname "$"))
(setq high (calc-top-n (+ stepnum 1))
low (calc-top-n (+ stepnum 2))
num (+ stepnum 3))
(setq low (if (stringp lowname) (math-read-expr lowname) lowname))
(if (eq (car-safe low) 'error)
(error "Bad format in expression: %s" (nth 1 low)))
(or highname
(setq highname (read-string (concat prompt " variable: " varname
", from: " lowname
", to: "))))
(if (or (equal highname "") (equal highname "$"))
(setq high (calc-top-n (+ stepnum 1))
num (+ stepnum 2))
(setq high (if (stringp highname) (math-read-expr highname)
highname))
(if (eq (car-safe high) 'error)
(error "Bad format in expression: %s" (nth 1 high)))
(if (consp arg)
(progn
(setq stepname (read-string (concat prompt " variable: "
varname
", from: " lowname
", to: " highname
", step: ")))
(if (or (equal stepname "") (equal stepname "$"))
(setq step (calc-top-n 1)
num 2)
(setq step (math-read-expr stepname))
(if (eq (car-safe step) 'error)
(error "Bad format in expression: %s"
(nth 1 step)))))))))
(or step
(if (consp arg)
(setq step (calc-top-n 1))
(if arg
(setq step (prefix-numeric-value arg)))))
(setq expr (calc-top-n num))
(calc-enter-result num prefix (append (list func expr var low high)
(and step (list step)))))))
(defun calc-solve-for (var)
(interactive "sVariable(s) to solve for: ")
(calc-slow-wrapper
(let ((func (if (calc-is-inverse)
(if (calc-is-hyperbolic) 'calcFunc-ffinv 'calcFunc-finv)
(if (calc-is-hyperbolic) 'calcFunc-fsolve 'calcFunc-solve))))
(if (or (equal var "") (equal var "$"))
(calc-enter-result 2 "solv" (list func
(calc-top-n 2)
(calc-top-n 1)))
(let ((var (if (and (string-match ",\\|[^ ] +[^ ]" var)
(not (string-match "\\[" var)))
(math-read-expr (concat "[" var "]"))
(math-read-expr var))))
(if (eq (car-safe var) 'error)
(error "Bad format in expression: %s" (nth 1 var)))
(calc-enter-result 1 "solv" (list func
(calc-top-n 1)
var)))))))
(defun calc-poly-roots (var)
(interactive "sVariable to solve for: ")
(calc-slow-wrapper
(if (or (equal var "") (equal var "$"))
(calc-enter-result 2 "prts" (list 'calcFunc-roots
(calc-top-n 2)
(calc-top-n 1)))
(let ((var (if (and (string-match ",\\|[^ ] +[^ ]" var)
(not (string-match "\\[" var)))
(math-read-expr (concat "[" var "]"))
(math-read-expr var))))
(if (eq (car-safe var) 'error)
(error "Bad format in expression: %s" (nth 1 var)))
(calc-enter-result 1 "prts" (list 'calcFunc-roots
(calc-top-n 1)
var))))))
(defun calc-taylor (var nterms)
(interactive "sTaylor expansion variable: \nNNumber of terms: ")
(calc-slow-wrapper
(let ((var (math-read-expr var)))
(if (eq (car-safe var) 'error)
(error "Bad format in expression: %s" (nth 1 var)))
(calc-enter-result 1 "tylr" (list 'calcFunc-taylor
(calc-top-n 1)
var
(prefix-numeric-value nterms))))))
;; The following are global variables used by math-derivative and some
;; related functions
(defvar math-deriv-var)
(defvar math-deriv-total)
(defvar math-deriv-symb)
(defvar math-decls-cache)
(defvar math-decls-all)
(defun math-derivative (expr)
(cond ((equal expr math-deriv-var)
1)
((or (Math-scalarp expr)
(eq (car expr) 'sdev)
(and (eq (car expr) 'var)
(or (not math-deriv-total)
(math-const-var expr)
(progn
(math-setup-declarations)
(memq 'const (nth 1 (or (assq (nth 2 expr)
math-decls-cache)
math-decls-all)))))))
0)
((eq (car expr) '+)
(math-add (math-derivative (nth 1 expr))
(math-derivative (nth 2 expr))))
((eq (car expr) '-)
(math-sub (math-derivative (nth 1 expr))
(math-derivative (nth 2 expr))))
((memq (car expr) '(calcFunc-eq calcFunc-neq calcFunc-lt
calcFunc-gt calcFunc-leq calcFunc-geq))
(list (car expr)
(math-derivative (nth 1 expr))
(math-derivative (nth 2 expr))))
((eq (car expr) 'neg)
(math-neg (math-derivative (nth 1 expr))))
((eq (car expr) '*)
(math-add (math-mul (nth 2 expr)
(math-derivative (nth 1 expr)))
(math-mul (nth 1 expr)
(math-derivative (nth 2 expr)))))
((eq (car expr) '/)
(math-sub (math-div (math-derivative (nth 1 expr))
(nth 2 expr))
(math-div (math-mul (nth 1 expr)
(math-derivative (nth 2 expr)))
(math-sqr (nth 2 expr)))))
((eq (car expr) '^)
(let ((du (math-derivative (nth 1 expr)))
(dv (math-derivative (nth 2 expr))))
(or (Math-zerop du)
(setq du (math-mul (nth 2 expr)
(math-mul (math-normalize
(list '^
(nth 1 expr)
(math-add (nth 2 expr) -1)))
du))))
(or (Math-zerop dv)
(setq dv (math-mul (math-normalize
(list 'calcFunc-ln (nth 1 expr)))
(math-mul expr dv))))
(math-add du dv)))
((eq (car expr) '%)
(math-derivative (nth 1 expr))) ; a reasonable definition
((eq (car expr) 'vec)
(math-map-vec 'math-derivative expr))
((and (memq (car expr) '(calcFunc-conj calcFunc-re calcFunc-im))
(= (length expr) 2))
(list (car expr) (math-derivative (nth 1 expr))))
((and (memq (car expr) '(calcFunc-subscr calcFunc-mrow calcFunc-mcol))
(= (length expr) 3))
(let ((d (math-derivative (nth 1 expr))))
(if (math-numberp d)
0 ; assume x and x_1 are independent vars
(list (car expr) d (nth 2 expr)))))
(t (or (and (symbolp (car expr))
(if (= (length expr) 2)
(let ((handler (get (car expr) 'math-derivative)))
(and handler
(let ((deriv (math-derivative (nth 1 expr))))
(if (Math-zerop deriv)
deriv
(math-mul (funcall handler (nth 1 expr))
deriv)))))
(let ((handler (get (car expr) 'math-derivative-n)))
(and handler
(funcall handler expr)))))
(and (not (eq math-deriv-symb 'pre-expand))
(let ((exp (math-expand-formula expr)))
(and exp
(or (let ((math-deriv-symb 'pre-expand))
(catch 'math-deriv (math-derivative expr)))
(math-derivative exp)))))
(if (or (Math-objvecp expr)
(eq (car expr) 'var)
(not (symbolp (car expr))))
(if math-deriv-symb
(throw 'math-deriv nil)
(list (if math-deriv-total 'calcFunc-tderiv 'calcFunc-deriv)
expr
math-deriv-var))
(let ((accum 0)
(arg expr)
(n 1)
derv)
(while (setq arg (cdr arg))
(or (Math-zerop (setq derv (math-derivative (car arg))))
(let ((func (intern (concat (symbol-name (car expr))
"'"
(if (> n 1)
(int-to-string n)
""))))
(prop (cond ((= (length expr) 2)
'math-derivative-1)
((= (length expr) 3)
'math-derivative-2)
((= (length expr) 4)
'math-derivative-3)
((= (length expr) 5)
'math-derivative-4)
((= (length expr) 6)
'math-derivative-5))))
(setq accum
(math-add
accum
(math-mul
derv
(let ((handler (get func prop)))
(or (and prop handler
(apply handler (cdr expr)))
(if (and math-deriv-symb
(not (get func
'calc-user-defn)))
(throw 'math-deriv nil)
(cons func (cdr expr))))))))))
(setq n (1+ n)))
accum))))))
(defun calcFunc-deriv (expr math-deriv-var &optional deriv-value math-deriv-symb)
(let* ((math-deriv-total nil)
(res (catch 'math-deriv (math-derivative expr))))
(or (eq (car-safe res) 'calcFunc-deriv)
(null res)
(setq res (math-normalize res)))
(and res
(if deriv-value
(math-expr-subst res math-deriv-var deriv-value)
res))))
(defun calcFunc-tderiv (expr math-deriv-var &optional deriv-value math-deriv-symb)
(math-setup-declarations)
(let* ((math-deriv-total t)
(res (catch 'math-deriv (math-derivative expr))))
(or (eq (car-safe res) 'calcFunc-tderiv)
(null res)
(setq res (math-normalize res)))
(and res
(if deriv-value
(math-expr-subst res math-deriv-var deriv-value)
res))))
(put 'calcFunc-inv\' 'math-derivative-1
(function (lambda (u) (math-neg (math-div 1 (math-sqr u))))))
(put 'calcFunc-sqrt\' 'math-derivative-1
(function (lambda (u) (math-div 1 (math-mul 2 (list 'calcFunc-sqrt u))))))
(put 'calcFunc-deg\' 'math-derivative-1
(function (lambda (u) (math-div-float '(float 18 1) (math-pi)))))
(put 'calcFunc-rad\' 'math-derivative-1
(function (lambda (u) (math-pi-over-180))))
(put 'calcFunc-ln\' 'math-derivative-1
(function (lambda (u) (math-div 1 u))))
(put 'calcFunc-log10\' 'math-derivative-1
(function (lambda (u)
(math-div (math-div 1 (math-normalize '(calcFunc-ln 10)))
u))))
(put 'calcFunc-lnp1\' 'math-derivative-1
(function (lambda (u) (math-div 1 (math-add u 1)))))
(put 'calcFunc-log\' 'math-derivative-2
(function (lambda (x b)
(and (not (Math-zerop b))
(let ((lnv (math-normalize
(list 'calcFunc-ln b))))
(math-div 1 (math-mul lnv x)))))))
(put 'calcFunc-log\'2 'math-derivative-2
(function (lambda (x b)
(let ((lnv (list 'calcFunc-ln b)))
(math-neg (math-div (list 'calcFunc-log x b)
(math-mul lnv b)))))))
(put 'calcFunc-exp\' 'math-derivative-1
(function (lambda (u) (math-normalize (list 'calcFunc-exp u)))))
(put 'calcFunc-expm1\' 'math-derivative-1
(function (lambda (u) (math-normalize (list 'calcFunc-expm1 u)))))
(put 'calcFunc-sin\' 'math-derivative-1
(function (lambda (u) (math-to-radians-2 (math-normalize
(list 'calcFunc-cos u)) t))))
(put 'calcFunc-cos\' 'math-derivative-1
(function (lambda (u) (math-neg (math-to-radians-2
(math-normalize
(list 'calcFunc-sin u)) t)))))
(put 'calcFunc-tan\' 'math-derivative-1
(function (lambda (u) (math-to-radians-2
(math-sqr
(math-normalize
(list 'calcFunc-sec u))) t))))
(put 'calcFunc-sec\' 'math-derivative-1
(function (lambda (u) (math-to-radians-2
(math-mul
(math-normalize
(list 'calcFunc-sec u))
(math-normalize
(list 'calcFunc-tan u))) t))))
(put 'calcFunc-csc\' 'math-derivative-1
(function (lambda (u) (math-neg
(math-to-radians-2
(math-mul
(math-normalize
(list 'calcFunc-csc u))
(math-normalize
(list 'calcFunc-cot u))) t)))))
(put 'calcFunc-cot\' 'math-derivative-1
(function (lambda (u) (math-neg
(math-to-radians-2
(math-sqr
(math-normalize
(list 'calcFunc-csc u))) t)))))
(put 'calcFunc-arcsin\' 'math-derivative-1
(function (lambda (u)
(math-from-radians-2
(math-div 1 (math-normalize
(list 'calcFunc-sqrt
(math-sub 1 (math-sqr u))))) t))))
(put 'calcFunc-arccos\' 'math-derivative-1
(function (lambda (u)
(math-from-radians-2
(math-div -1 (math-normalize
(list 'calcFunc-sqrt
(math-sub 1 (math-sqr u))))) t))))
(put 'calcFunc-arctan\' 'math-derivative-1
(function (lambda (u) (math-from-radians-2
(math-div 1 (math-add 1 (math-sqr u))) t))))
(put 'calcFunc-sinh\' 'math-derivative-1
(function (lambda (u) (math-normalize (list 'calcFunc-cosh u)))))
(put 'calcFunc-cosh\' 'math-derivative-1
(function (lambda (u) (math-normalize (list 'calcFunc-sinh u)))))
(put 'calcFunc-tanh\' 'math-derivative-1
(function (lambda (u) (math-sqr
(math-normalize
(list 'calcFunc-sech u))))))
(put 'calcFunc-sech\' 'math-derivative-1
(function (lambda (u) (math-neg
(math-mul
(math-normalize (list 'calcFunc-sech u))
(math-normalize (list 'calcFunc-tanh u)))))))
(put 'calcFunc-csch\' 'math-derivative-1
(function (lambda (u) (math-neg
(math-mul
(math-normalize (list 'calcFunc-csch u))
(math-normalize (list 'calcFunc-coth u)))))))
(put 'calcFunc-coth\' 'math-derivative-1
(function (lambda (u) (math-neg
(math-sqr
(math-normalize
(list 'calcFunc-csch u)))))))
(put 'calcFunc-arcsinh\' 'math-derivative-1
(function (lambda (u)
(math-div 1 (math-normalize
(list 'calcFunc-sqrt
(math-add (math-sqr u) 1)))))))
(put 'calcFunc-arccosh\' 'math-derivative-1
(function (lambda (u)
(math-div 1 (math-normalize
(list 'calcFunc-sqrt
(math-add (math-sqr u) -1)))))))
(put 'calcFunc-arctanh\' 'math-derivative-1
(function (lambda (u) (math-div 1 (math-sub 1 (math-sqr u))))))
(put 'calcFunc-bern\'2 'math-derivative-2
(function (lambda (n x)
(math-mul n (list 'calcFunc-bern (math-add n -1) x)))))
(put 'calcFunc-euler\'2 'math-derivative-2
(function (lambda (n x)
(math-mul n (list 'calcFunc-euler (math-add n -1) x)))))
(put 'calcFunc-gammag\'2 'math-derivative-2
(function (lambda (a x) (math-deriv-gamma a x 1))))
(put 'calcFunc-gammaG\'2 'math-derivative-2
(function (lambda (a x) (math-deriv-gamma a x -1))))
(put 'calcFunc-gammaP\'2 'math-derivative-2
(function (lambda (a x) (math-deriv-gamma a x
(math-div
1 (math-normalize
(list 'calcFunc-gamma
a)))))))
(put 'calcFunc-gammaQ\'2 'math-derivative-2
(function (lambda (a x) (math-deriv-gamma a x
(math-div
-1 (math-normalize
(list 'calcFunc-gamma
a)))))))
(defun math-deriv-gamma (a x scale)
(math-mul scale
(math-mul (math-pow x (math-add a -1))
(list 'calcFunc-exp (math-neg x)))))
(put 'calcFunc-betaB\' 'math-derivative-3
(function (lambda (x a b) (math-deriv-beta x a b 1))))
(put 'calcFunc-betaI\' 'math-derivative-3
(function (lambda (x a b) (math-deriv-beta x a b
(math-div
1 (list 'calcFunc-beta
a b))))))
(defun math-deriv-beta (x a b scale)
(math-mul (math-mul (math-pow x (math-add a -1))
(math-pow (math-sub 1 x) (math-add b -1)))
scale))
(put 'calcFunc-erf\' 'math-derivative-1
(function (lambda (x) (math-div 2
(math-mul (list 'calcFunc-exp
(math-sqr x))
(if calc-symbolic-mode
'(calcFunc-sqrt
(var pi var-pi))
(math-sqrt-pi)))))))
(put 'calcFunc-erfc\' 'math-derivative-1
(function (lambda (x) (math-div -2
(math-mul (list 'calcFunc-exp
(math-sqr x))
(if calc-symbolic-mode
'(calcFunc-sqrt
(var pi var-pi))
(math-sqrt-pi)))))))
(put 'calcFunc-besJ\'2 'math-derivative-2
(function (lambda (v z) (math-div (math-sub (list 'calcFunc-besJ
(math-add v -1)
z)
(list 'calcFunc-besJ
(math-add v 1)
z))
2))))
(put 'calcFunc-besY\'2 'math-derivative-2
(function (lambda (v z) (math-div (math-sub (list 'calcFunc-besY
(math-add v -1)
z)
(list 'calcFunc-besY
(math-add v 1)
z))
2))))
(put 'calcFunc-sum 'math-derivative-n
(function
(lambda (expr)
(if (math-expr-contains (cons 'vec (cdr (cdr expr))) math-deriv-var)
(throw 'math-deriv nil)
(cons 'calcFunc-sum
(cons (math-derivative (nth 1 expr))
(cdr (cdr expr))))))))
(put 'calcFunc-prod 'math-derivative-n
(function
(lambda (expr)
(if (math-expr-contains (cons 'vec (cdr (cdr expr))) math-deriv-var)
(throw 'math-deriv nil)
(math-mul expr
(cons 'calcFunc-sum
(cons (math-div (math-derivative (nth 1 expr))
(nth 1 expr))
(cdr (cdr expr)))))))))
(put 'calcFunc-integ 'math-derivative-n
(function
(lambda (expr)
(if (= (length expr) 3)
(if (equal (nth 2 expr) math-deriv-var)
(nth 1 expr)
(math-normalize
(list 'calcFunc-integ
(math-derivative (nth 1 expr))
(nth 2 expr))))
(if (= (length expr) 5)
(let ((lower (math-expr-subst (nth 1 expr) (nth 2 expr)
(nth 3 expr)))
(upper (math-expr-subst (nth 1 expr) (nth 2 expr)
(nth 4 expr))))
(math-add (math-sub (math-mul upper
(math-derivative (nth 4 expr)))
(math-mul lower
(math-derivative (nth 3 expr))))
(if (equal (nth 2 expr) math-deriv-var)
0
(math-normalize
(list 'calcFunc-integ
(math-derivative (nth 1 expr)) (nth 2 expr)
(nth 3 expr) (nth 4 expr)))))))))))
(put 'calcFunc-if 'math-derivative-n
(function
(lambda (expr)
(and (= (length expr) 4)
(list 'calcFunc-if (nth 1 expr)
(math-derivative (nth 2 expr))
(math-derivative (nth 3 expr)))))))
(put 'calcFunc-subscr 'math-derivative-n
(function
(lambda (expr)
(and (= (length expr) 3)
(list 'calcFunc-subscr (nth 1 expr)
(math-derivative (nth 2 expr)))))))
(defvar math-integ-var '(var X ---))
(defvar math-integ-var-2 '(var Y ---))
(defvar math-integ-vars (list 'f math-integ-var math-integ-var-2))
(defvar math-integ-var-list (list math-integ-var))
(defvar math-integ-var-list-list (list math-integ-var-list))
;; math-integ-depth is a local variable for math-try-integral, but is used
;; by math-integral and math-tracing-integral
;; which are called (directly or indirectly) by math-try-integral.
(defvar math-integ-depth)
;; math-integ-level is a local variable for math-try-integral, but is used
;; by math-integral, math-do-integral, math-tracing-integral,
;; math-sub-integration, math-integrate-by-parts and
;; math-integrate-by-substitution, which are called (directly or
;; indirectly) by math-try-integral.
(defvar math-integ-level)
;; math-integral-limit is a local variable for calcFunc-integ, but is
;; used by math-tracing-integral, math-sub-integration and
;; math-try-integration.
(defvar math-integral-limit)
(defmacro math-tracing-integral (&rest parts)
`(and trace-buffer
(with-current-buffer trace-buffer
(goto-char (point-max))
(and (bolp)
(insert (make-string (- math-integral-limit
math-integ-level) 32)
(format "%2d " math-integ-depth)
(make-string math-integ-level 32)))
;;(condition-case err
(insert ,@parts)
;; (error (insert (prin1-to-string err))))
(sit-for 0))))
;;; The following wrapper caches results and avoids infinite recursion.
;;; Each cache entry is: ( A B ) Integral of A is B;
;;; ( A N ) Integral of A failed at level N;
;;; ( A busy ) Currently working on integral of A;
;;; ( A parts ) Currently working, integ-by-parts;
;;; ( A parts2 ) Currently working, integ-by-parts;
;;; ( A cancelled ) Ignore this cache entry;
;;; ( A [B] ) Same result as for math-cur-record = B.
;; math-cur-record is a local variable for math-try-integral, but is used
;; by math-integral, math-replace-integral-parts and math-integrate-by-parts
;; which are called (directly or indirectly) by math-try-integral, as well as
;; by calc-dump-integral-cache
(defvar math-cur-record)
;; math-enable-subst and math-any-substs are local variables for
;; calcFunc-integ, but are used by math-integral and math-try-integral.
(defvar math-enable-subst)
(defvar math-any-substs)
;; math-integ-msg is a local variable for math-try-integral, but is
;; used (both locally and non-locally) by math-integral.
(defvar math-integ-msg)
(defvar math-integral-cache nil)
(defvar math-integral-cache-state nil)
(defun math-integral (expr &optional simplify same-as-above)
(let* ((simp math-cur-record)
(math-cur-record (assoc expr math-integral-cache))
(math-integ-depth (1+ math-integ-depth))
(val 'cancelled))
(math-tracing-integral "Integrating "
(math-format-value expr 1000)
"...\n")
(and math-cur-record
(progn
(math-tracing-integral "Found "
(math-format-value (nth 1 math-cur-record) 1000))
(and (consp (nth 1 math-cur-record))
(math-replace-integral-parts math-cur-record))
(math-tracing-integral " => "
(math-format-value (nth 1 math-cur-record) 1000)
"\n")))
(or (and math-cur-record
(not (eq (nth 1 math-cur-record) 'cancelled))
(or (not (integerp (nth 1 math-cur-record)))
(>= (nth 1 math-cur-record) math-integ-level)))
(and (math-integral-contains-parts expr)
(progn
(setq val nil)
t))
(unwind-protect
(progn
(let (math-integ-msg)
(if (eq calc-display-working-message 'lots)
(progn
(calc-set-command-flag 'clear-message)
(setq math-integ-msg (format
"Working... Integrating %s"
(math-format-flat-expr expr 0)))
(message "%s" math-integ-msg)))
(if math-cur-record
(setcar (cdr math-cur-record)
(if same-as-above (vector simp) 'busy))
(setq math-cur-record
(list expr (if same-as-above (vector simp) 'busy))
math-integral-cache (cons math-cur-record
math-integral-cache)))
(if (eq simplify 'yes)
(progn
(math-tracing-integral "Simplifying...")
(setq simp (math-simplify expr))
(setq val (if (equal simp expr)
(progn
(math-tracing-integral " no change\n")
(math-do-integral expr))
(math-tracing-integral " simplified\n")
(math-integral simp 'no t))))
(or (setq val (math-do-integral expr))
(eq simplify 'no)
(let ((simp (math-simplify expr)))
(or (equal simp expr)
(progn
(math-tracing-integral "Trying again after "
"simplification...\n")
(setq val (math-integral simp 'no t))))))))
(if (eq calc-display-working-message 'lots)
(message "%s" math-integ-msg)))
(setcar (cdr math-cur-record) (or val
(if (or math-enable-subst
(not math-any-substs))
math-integ-level
'cancelled)))))
(setq val math-cur-record)
(while (vectorp (nth 1 val))
(setq val (aref (nth 1 val) 0)))
(setq val (if (memq (nth 1 val) '(parts parts2))
(progn
(setcar (cdr val) 'parts2)
(list 'var 'PARTS val))
(and (consp (nth 1 val))
(nth 1 val))))
(math-tracing-integral "Integral of "
(math-format-value expr 1000)
" is "
(math-format-value val 1000)
"\n")
val))
(defun math-integral-contains-parts (expr)
(if (Math-primp expr)
(and (eq (car-safe expr) 'var)
(eq (nth 1 expr) 'PARTS)
(listp (nth 2 expr)))
(while (and (setq expr (cdr expr))
(not (math-integral-contains-parts (car expr)))))
expr))
(defun math-replace-integral-parts (expr)
(or (Math-primp expr)
(while (setq expr (cdr expr))
(and (consp (car expr))
(if (eq (car (car expr)) 'var)
(and (eq (nth 1 (car expr)) 'PARTS)
(consp (nth 2 (car expr)))
(if (listp (nth 1 (nth 2 (car expr))))
(progn
(setcar expr (nth 1 (nth 2 (car expr))))
(math-replace-integral-parts (cons 'foo expr)))
(setcar (cdr math-cur-record) 'cancelled)))
(math-replace-integral-parts (car expr)))))))
(defvar math-linear-subst-tried t
"Non-nil means that a linear substitution has been tried.")
;; The variable math-has-rules is a local variable for math-try-integral,
;; but is used by math-do-integral, which is called (non-directly) by
;; math-try-integral.
(defvar math-has-rules)
;; math-old-integ is a local variable for math-do-integral, but is
;; used by math-sub-integration.
(defvar math-old-integ)
;; The variables math-t1, math-t2 and math-t3 are local to
;; math-do-integral, math-try-solve-for and math-decompose-poly, but
;; are used by functions they call (directly or indirectly);
;; math-do-integral calls math-do-integral-methods;
;; math-try-solve-for calls math-try-solve-prod,
;; math-solve-find-root-term and math-solve-find-root-in-prod;
;; math-decompose-poly calls math-solve-poly-funny-powers and
;; math-solve-crunch-poly.
(defvar math-t1)
(defvar math-t2)
(defvar math-t3)
(defun math-do-integral (expr)
(let ((math-linear-subst-tried nil)
math-t1 math-t2)
(or (cond ((not (math-expr-contains expr math-integ-var))
(math-mul expr math-integ-var))
((equal expr math-integ-var)
(math-div (math-sqr expr) 2))
((eq (car expr) '+)
(and (setq math-t1 (math-integral (nth 1 expr)))
(setq math-t2 (math-integral (nth 2 expr)))
(math-add math-t1 math-t2)))
((eq (car expr) '-)
(and (setq math-t1 (math-integral (nth 1 expr)))
(setq math-t2 (math-integral (nth 2 expr)))
(math-sub math-t1 math-t2)))
((eq (car expr) 'neg)
(and (setq math-t1 (math-integral (nth 1 expr)))
(math-neg math-t1)))
((eq (car expr) '*)
(cond ((not (math-expr-contains (nth 1 expr) math-integ-var))
(and (setq math-t1 (math-integral (nth 2 expr)))
(math-mul (nth 1 expr) math-t1)))
((not (math-expr-contains (nth 2 expr) math-integ-var))
(and (setq math-t1 (math-integral (nth 1 expr)))
(math-mul math-t1 (nth 2 expr))))
((memq (car-safe (nth 1 expr)) '(+ -))
(math-integral (list (car (nth 1 expr))
(math-mul (nth 1 (nth 1 expr))
(nth 2 expr))
(math-mul (nth 2 (nth 1 expr))
(nth 2 expr)))
'yes t))
((memq (car-safe (nth 2 expr)) '(+ -))
(math-integral (list (car (nth 2 expr))
(math-mul (nth 1 (nth 2 expr))
(nth 1 expr))
(math-mul (nth 2 (nth 2 expr))
(nth 1 expr)))
'yes t))))
((eq (car expr) '/)
(cond ((and (not (math-expr-contains (nth 1 expr)
math-integ-var))
(not (math-equal-int (nth 1 expr) 1)))
(and (setq math-t1 (math-integral (math-div 1 (nth 2 expr))))
(math-mul (nth 1 expr) math-t1)))
((not (math-expr-contains (nth 2 expr) math-integ-var))
(and (setq math-t1 (math-integral (nth 1 expr)))
(math-div math-t1 (nth 2 expr))))
((and (eq (car-safe (nth 1 expr)) '*)
(not (math-expr-contains (nth 1 (nth 1 expr))
math-integ-var)))
(and (setq math-t1 (math-integral
(math-div (nth 2 (nth 1 expr))
(nth 2 expr))))
(math-mul math-t1 (nth 1 (nth 1 expr)))))
((and (eq (car-safe (nth 1 expr)) '*)
(not (math-expr-contains (nth 2 (nth 1 expr))
math-integ-var)))
(and (setq math-t1 (math-integral
(math-div (nth 1 (nth 1 expr))
(nth 2 expr))))
(math-mul math-t1 (nth 2 (nth 1 expr)))))
((and (eq (car-safe (nth 2 expr)) '*)
(not (math-expr-contains (nth 1 (nth 2 expr))
math-integ-var)))
(and (setq math-t1 (math-integral
(math-div (nth 1 expr)
(nth 2 (nth 2 expr)))))
(math-div math-t1 (nth 1 (nth 2 expr)))))
((and (eq (car-safe (nth 2 expr)) '*)
(not (math-expr-contains (nth 2 (nth 2 expr))
math-integ-var)))
(and (setq math-t1 (math-integral
(math-div (nth 1 expr)
(nth 1 (nth 2 expr)))))
(math-div math-t1 (nth 2 (nth 2 expr)))))
((eq (car-safe (nth 2 expr)) 'calcFunc-exp)
(math-integral
(math-mul (nth 1 expr)
(list 'calcFunc-exp
(math-neg (nth 1 (nth 2 expr)))))))))
((eq (car expr) '^)
(cond ((not (math-expr-contains (nth 1 expr) math-integ-var))
(or (and (setq math-t1 (math-is-polynomial (nth 2 expr)
math-integ-var 1))
(math-div expr
(math-mul (nth 1 math-t1)
(math-normalize
(list 'calcFunc-ln
(nth 1 expr))))))
(math-integral
(list 'calcFunc-exp
(math-mul (nth 2 expr)
(math-normalize
(list 'calcFunc-ln
(nth 1 expr)))))
'yes t)))
((not (math-expr-contains (nth 2 expr) math-integ-var))
(if (and (integerp (nth 2 expr)) (< (nth 2 expr) 0))
(math-integral
(list '/ 1 (math-pow (nth 1 expr) (- (nth 2 expr))))
nil t)
(or (and (setq math-t1 (math-is-polynomial (nth 1 expr)
math-integ-var
1))
(setq math-t2 (math-add (nth 2 expr) 1))
(math-div (math-pow (nth 1 expr) math-t2)
(math-mul math-t2 (nth 1 math-t1))))
(and (Math-negp (nth 2 expr))
(math-integral
(math-div 1
(math-pow (nth 1 expr)
(math-neg
(nth 2 expr))))
nil t))
nil))))))
;; Integral of a polynomial.
(and (setq math-t1 (math-is-polynomial expr math-integ-var 20))
(let ((accum 0)
(n 1))
(while math-t1
(if (setq accum (math-add accum
(math-div (math-mul (car math-t1)
(math-pow
math-integ-var
n))
n))
math-t1 (cdr math-t1))
(setq n (1+ n))))
accum))
;; Try looking it up!
(cond ((= (length expr) 2)
(and (symbolp (car expr))
(setq math-t1 (get (car expr) 'math-integral))
(progn
(while (and math-t1
(not (setq math-t2 (funcall (car math-t1)
(nth 1 expr)))))
(setq math-t1 (cdr math-t1)))
(and math-t2 (math-normalize math-t2)))))
((= (length expr) 3)
(and (symbolp (car expr))
(setq math-t1 (get (car expr) 'math-integral-2))
(progn
(while (and math-t1
(not (setq math-t2 (funcall (car math-t1)
(nth 1 expr)
(nth 2 expr)))))
(setq math-t1 (cdr math-t1)))
(and math-t2 (math-normalize math-t2))))))
;; Integral of a rational function.
(and (math-ratpoly-p expr math-integ-var)
(setq math-t1 (calcFunc-apart expr math-integ-var))
(not (equal math-t1 expr))
(math-integral math-t1))
;; Try user-defined integration rules.
(and math-has-rules
(let ((math-old-integ (symbol-function 'calcFunc-integ))
(input (list 'calcFunc-integtry expr math-integ-var))
res part)
(unwind-protect
(progn
(fset 'calcFunc-integ 'math-sub-integration)
(setq res (math-rewrite input
'(var IntegRules var-IntegRules)
1))
(fset 'calcFunc-integ math-old-integ)
(and (not (equal res input))
(if (setq part (math-expr-calls
res '(calcFunc-integsubst)))
(and (memq (length part) '(3 4 5))
(let ((parts (mapcar
(function
(lambda (x)
(math-expr-subst
x (nth 2 part)
math-integ-var)))
(cdr part))))
(math-integrate-by-substitution
expr (car parts) t
(or (nth 2 parts)
(list 'calcFunc-integfailed
math-integ-var))
(nth 3 parts))))
(if (not (math-expr-calls res
'(calcFunc-integtry
calcFunc-integfailed)))
res))))
(fset 'calcFunc-integ math-old-integ))))
;; See if the function is a symbolic derivative.
(and (string-match "'" (symbol-name (car expr)))
(let ((name (symbol-name (car expr)))
(p expr) (n 0) (which nil) (bad nil))
(while (setq n (1+ n) p (cdr p))
(if (equal (car p) math-integ-var)
(if which (setq bad t) (setq which n))
(if (math-expr-contains (car p) math-integ-var)
(setq bad t))))
(and which (not bad)
(let ((prime (if (= which 1) "'" (format "'%d" which))))
(and (string-match (concat prime "\\('['0-9]*\\|$\\)")
name)
(cons (intern
(concat
(substring name 0 (match-beginning 0))
(substring name (+ (match-beginning 0)
(length prime)))))
(cdr expr)))))))
;; Try transformation methods (parts, substitutions).
(and (> math-integ-level 0)
(math-do-integral-methods expr))
;; Try expanding the function's definition.
(let ((res (math-expand-formula expr)))
(and res
(math-integral res))))))
(defun math-sub-integration (expr &rest rest)
(or (if (or (not rest)
(and (< math-integ-level math-integral-limit)
(eq (car rest) math-integ-var)))
(math-integral expr)
(let ((res (apply math-old-integ expr rest)))
(and (or (= math-integ-level math-integral-limit)
(not (math-expr-calls res 'calcFunc-integ)))
res)))
(list 'calcFunc-integfailed expr)))
;; math-so-far is a local variable for math-do-integral-methods, but
;; is used by math-integ-try-linear-substitutions and
;; math-integ-try-substitutions.
(defvar math-so-far)
;; math-integ-expr is a local variable for math-do-integral-methods,
;; but is used by math-integ-try-linear-substitutions and
;; math-integ-try-substitutions.
(defvar math-integ-expr)
(defun math-do-integral-methods (math-integ-expr)
(let ((math-so-far math-integ-var-list-list)
rat-in)
;; Integration by substitution, for various likely sub-expressions.
;; (In first pass, we look only for sub-exprs that are linear in X.)
(or (math-integ-try-linear-substitutions math-integ-expr)
(math-integ-try-substitutions math-integ-expr)
;; If function has sines and cosines, try tan(x/2) substitution.
(and (let ((p (setq rat-in (math-expr-rational-in math-integ-expr))))
(while (and p
(memq (car (car p)) '(calcFunc-sin
calcFunc-cos
calcFunc-tan
calcFunc-sec
calcFunc-csc
calcFunc-cot))
(equal (nth 1 (car p)) math-integ-var))
(setq p (cdr p)))
(null p))
(or (and (math-integ-parts-easy math-integ-expr)
(math-integ-try-parts math-integ-expr t))
(math-integrate-by-good-substitution
math-integ-expr (list 'calcFunc-tan (math-div math-integ-var 2)))))
;; If function has sinh and cosh, try tanh(x/2) substitution.
(and (let ((p rat-in))
(while (and p
(memq (car (car p)) '(calcFunc-sinh
calcFunc-cosh
calcFunc-tanh
calcFunc-sech
calcFunc-csch
calcFunc-coth
calcFunc-exp))
(equal (nth 1 (car p)) math-integ-var))
(setq p (cdr p)))
(null p))
(or (and (math-integ-parts-easy math-integ-expr)
(math-integ-try-parts math-integ-expr t))
(math-integrate-by-good-substitution
math-integ-expr (list 'calcFunc-tanh (math-div math-integ-var 2)))))
;; If function has square roots, try sin, tan, or sec substitution.
(and (let ((p rat-in))
(setq math-t1 nil)
(while (and p
(or (equal (car p) math-integ-var)
(and (eq (car (car p)) 'calcFunc-sqrt)
(setq math-t1 (math-is-polynomial
(nth 1 (setq math-t2 (car p)))
math-integ-var 2)))))
(setq p (cdr p)))
(and (null p) math-t1))
(if (cdr (cdr math-t1))
(if (math-guess-if-neg (nth 2 math-t1))
(let* ((c (math-sqrt (math-neg (nth 2 math-t1))))
(d (math-div (nth 1 math-t1) (math-mul -2 c)))
(a (math-sqrt (math-add (car math-t1) (math-sqr d)))))
(math-integrate-by-good-substitution
math-integ-expr (list 'calcFunc-arcsin
(math-div-thru
(math-add (math-mul c math-integ-var) d)
a))))
(let* ((c (math-sqrt (nth 2 math-t1)))
(d (math-div (nth 1 math-t1) (math-mul 2 c)))
(aa (math-sub (car math-t1) (math-sqr d))))
(if (and nil (not (and (eq d 0) (eq c 1))))
(math-integrate-by-good-substitution
math-integ-expr (math-add (math-mul c math-integ-var) d))
(if (math-guess-if-neg aa)
(math-integrate-by-good-substitution
math-integ-expr (list 'calcFunc-arccosh
(math-div-thru
(math-add (math-mul c math-integ-var)
d)
(math-sqrt (math-neg aa)))))
(math-integrate-by-good-substitution
math-integ-expr (list 'calcFunc-arcsinh
(math-div-thru
(math-add (math-mul c math-integ-var)
d)
(math-sqrt aa))))))))
(math-integrate-by-good-substitution math-integ-expr math-t2)) )
;; Try integration by parts.
(math-integ-try-parts math-integ-expr)
;; Give up.
nil)))
(defun math-integ-parts-easy (expr)
(cond ((Math-primp expr) t)
((memq (car expr) '(+ - *))
(and (math-integ-parts-easy (nth 1 expr))
(math-integ-parts-easy (nth 2 expr))))
((eq (car expr) '/)
(and (math-integ-parts-easy (nth 1 expr))
(math-atomic-factorp (nth 2 expr))))
((eq (car expr) '^)
(and (natnump (nth 2 expr))
(math-integ-parts-easy (nth 1 expr))))
((eq (car expr) 'neg)
(math-integ-parts-easy (nth 1 expr)))
(t t)))
;; math-prev-parts-v is local to calcFunc-integ (as well as
;; math-integrate-by-parts), but is used by math-integ-try-parts.
(defvar math-prev-parts-v)
;; math-good-parts is local to calcFunc-integ (as well as
;; math-integ-try-parts), but is used by math-integrate-by-parts.
(defvar math-good-parts)
(defun math-integ-try-parts (expr &optional math-good-parts)
;; Integration by parts:
;; integ(f(x) g(x),x) = f(x) h(x) - integ(h(x) f'(x),x)
;; where h(x) = integ(g(x),x).
(or (let ((exp (calcFunc-expand expr)))
(and (not (equal exp expr))
(math-integral exp)))
(and (eq (car expr) '*)
(let ((first-bad (or (math-polynomial-p (nth 1 expr)
math-integ-var)
(equal (nth 2 expr) math-prev-parts-v))))
(or (and first-bad ; so try this one first
(math-integrate-by-parts (nth 1 expr) (nth 2 expr)))
(math-integrate-by-parts (nth 2 expr) (nth 1 expr))
(and (not first-bad)
(math-integrate-by-parts (nth 1 expr) (nth 2 expr))))))
(and (eq (car expr) '/)
(math-expr-contains (nth 1 expr) math-integ-var)
(let ((recip (math-div 1 (nth 2 expr))))
(or (math-integrate-by-parts (nth 1 expr) recip)
(math-integrate-by-parts recip (nth 1 expr)))))
(and (eq (car expr) '^)
(math-integrate-by-parts (math-pow (nth 1 expr)
(math-sub (nth 2 expr) 1))
(nth 1 expr)))))
(defun math-integrate-by-parts (u vprime)
(let ((math-integ-level (if (or math-good-parts
(math-polynomial-p u math-integ-var))
math-integ-level
(1- math-integ-level)))
(math-doing-parts t)
v temp)
(and (>= math-integ-level 0)
(unwind-protect
(progn
(setcar (cdr math-cur-record) 'parts)
(math-tracing-integral "Integrating by parts, u = "
(math-format-value u 1000)
", v' = "
(math-format-value vprime 1000)
"\n")
(and (setq v (math-integral vprime))
(setq temp (calcFunc-deriv u math-integ-var nil t))
(setq temp (let ((math-prev-parts-v v))
(math-integral (math-mul v temp) 'yes)))
(setq temp (math-sub (math-mul u v) temp))
(if (eq (nth 1 math-cur-record) 'parts)
(calcFunc-expand temp)
(setq v (list 'var 'PARTS math-cur-record)
temp (let (calc-next-why)
(math-simplify-extended
(math-solve-for (math-sub v temp) 0 v nil)))
temp (if (and (eq (car-safe temp) '/)
(math-zerop (nth 2 temp)))
nil temp)))))
(setcar (cdr math-cur-record) 'busy)))))
;;; This tries two different formulations, hoping the algebraic simplifier
;;; will be strong enough to handle at least one.
(defun math-integrate-by-substitution (expr u &optional user uinv uinvprime)
(and (> math-integ-level 0)
(let ((math-integ-level (max (- math-integ-level 2) 0)))
(math-integrate-by-good-substitution expr u user uinv uinvprime))))
(defun math-integrate-by-good-substitution (expr u &optional user
uinv uinvprime)
(let ((math-living-dangerously t)
deriv temp)
(and (setq uinv (if uinv
(math-expr-subst uinv math-integ-var
math-integ-var-2)
(let (calc-next-why)
(math-solve-for u
math-integ-var-2
math-integ-var nil))))
(progn
(math-tracing-integral "Integrating by substitution, u = "
(math-format-value u 1000)
"\n")
(or (and (setq deriv (calcFunc-deriv u
math-integ-var nil
(not user)))
(setq temp (math-integral (math-expr-subst
(math-expr-subst
(math-expr-subst
(math-div expr deriv)
u
math-integ-var-2)
math-integ-var
uinv)
math-integ-var-2
math-integ-var)
'yes)))
(and (setq deriv (or uinvprime
(calcFunc-deriv uinv
math-integ-var-2
math-integ-var
(not user))))
(setq temp (math-integral (math-mul
(math-expr-subst
(math-expr-subst
(math-expr-subst
expr
u
math-integ-var-2)
math-integ-var
uinv)
math-integ-var-2
math-integ-var)
deriv)
'yes)))))
(math-simplify-extended
(math-expr-subst temp math-integ-var u)))))
;;; Look for substitutions of the form u = a x + b.
(defun math-integ-try-linear-substitutions (sub-expr)
(setq math-linear-subst-tried t)
(and (not (Math-primp sub-expr))
(or (and (not (memq (car sub-expr) '(+ - * / neg)))
(not (and (eq (car sub-expr) '^)
(integerp (nth 2 sub-expr))))
(math-expr-contains sub-expr math-integ-var)
(let ((res nil))
(while (and (setq sub-expr (cdr sub-expr))
(or (not (math-linear-in (car sub-expr)
math-integ-var))
(assoc (car sub-expr) math-so-far)
(progn
(setq math-so-far (cons (list (car sub-expr))
math-so-far))
(not (setq res
(math-integrate-by-substitution
math-integ-expr (car sub-expr))))))))
res))
(let ((res nil))
(while (and (setq sub-expr (cdr sub-expr))
(not (setq res (math-integ-try-linear-substitutions
(car sub-expr))))))
res))))
;;; Recursively try different substitutions based on various sub-expressions.
(defun math-integ-try-substitutions (sub-expr &optional allow-rat)
(and (not (Math-primp sub-expr))
(not (assoc sub-expr math-so-far))
(math-expr-contains sub-expr math-integ-var)
(or (and (if (and (not (memq (car sub-expr) '(+ - * / neg)))
(not (and (eq (car sub-expr) '^)
(integerp (nth 2 sub-expr)))))
(setq allow-rat t)
(prog1 allow-rat (setq allow-rat nil)))
(not (eq sub-expr math-integ-expr))
(or (math-integrate-by-substitution math-integ-expr sub-expr)
(and (eq (car sub-expr) '^)
(integerp (nth 2 sub-expr))
(< (nth 2 sub-expr) 0)
(math-integ-try-substitutions
(math-pow (nth 1 sub-expr) (- (nth 2 sub-expr)))
t))))
(let ((res nil))
(setq math-so-far (cons (list sub-expr) math-so-far))
(while (and (setq sub-expr (cdr sub-expr))
(not (setq res (math-integ-try-substitutions
(car sub-expr) allow-rat)))))
res))))
;; The variable math-expr-parts is local to math-expr-rational-in,
;; but is used by math-expr-rational-in-rec
(defvar math-expr-parts)
(defun math-expr-rational-in (expr)
(let ((math-expr-parts nil))
(math-expr-rational-in-rec expr)
(mapcar 'car math-expr-parts)))
(defun math-expr-rational-in-rec (expr)
(cond ((Math-primp expr)
(and (equal expr math-integ-var)
(not (assoc expr math-expr-parts))
(setq math-expr-parts (cons (list expr) math-expr-parts))))
((or (memq (car expr) '(+ - * / neg))
(and (eq (car expr) '^) (integerp (nth 2 expr))))
(math-expr-rational-in-rec (nth 1 expr))
(and (nth 2 expr) (math-expr-rational-in-rec (nth 2 expr))))
((and (eq (car expr) '^)
(eq (math-quarter-integer (nth 2 expr)) 2))
(math-expr-rational-in-rec (list 'calcFunc-sqrt (nth 1 expr))))
(t
(and (not (assoc expr math-expr-parts))
(math-expr-contains expr math-integ-var)
(setq math-expr-parts (cons (list expr) math-expr-parts))))))
(defun math-expr-calls (expr funcs &optional arg-contains)
(if (consp expr)
(if (or (memq (car expr) funcs)
(and (eq (car expr) '^) (eq (car funcs) 'calcFunc-sqrt)
(eq (math-quarter-integer (nth 2 expr)) 2)))
(and (or (not arg-contains)
(math-expr-contains expr arg-contains))
expr)
(and (not (Math-primp expr))
(let ((res nil))
(while (and (setq expr (cdr expr))
(not (setq res (math-expr-calls
(car expr) funcs arg-contains)))))
res)))))
(defun math-fix-const-terms (expr except-vars)
(cond ((not (math-expr-depends expr except-vars)) 0)
((Math-primp expr) expr)
((eq (car expr) '+)
(math-add (math-fix-const-terms (nth 1 expr) except-vars)
(math-fix-const-terms (nth 2 expr) except-vars)))
((eq (car expr) '-)
(math-sub (math-fix-const-terms (nth 1 expr) except-vars)
(math-fix-const-terms (nth 2 expr) except-vars)))
(t expr)))
;; Command for debugging the Calculator's symbolic integrator.
(defun calc-dump-integral-cache (&optional arg)
(interactive "P")
(let ((buf (current-buffer)))
(unwind-protect
(let ((p math-integral-cache)
math-cur-record)
(display-buffer (get-buffer-create "*Integral Cache*"))
(set-buffer (get-buffer "*Integral Cache*"))
(erase-buffer)
(while p
(setq math-cur-record (car p))
(or arg (math-replace-integral-parts math-cur-record))
(insert (math-format-flat-expr (car math-cur-record) 0)
" --> "
(if (symbolp (nth 1 math-cur-record))
(concat "(" (symbol-name (nth 1 math-cur-record)) ")")
(math-format-flat-expr (nth 1 math-cur-record) 0))
"\n")
(setq p (cdr p)))
(goto-char (point-min)))
(set-buffer buf))))
;; The variable math-max-integral-limit is local to calcFunc-integ,
;; but is used by math-try-integral.
(defvar math-max-integral-limit)
(defun math-try-integral (expr)
(let ((math-integ-level math-integral-limit)
(math-integ-depth 0)
(math-integ-msg "Working...done")
(math-cur-record nil) ; a technicality
(math-integrating t)
(calc-prefer-frac t)
(calc-symbolic-mode t)
(math-has-rules (calc-has-rules 'var-IntegRules)))
(or (math-integral expr 'yes)
(and math-any-substs
(setq math-enable-subst t)
(math-integral expr 'yes))
(and (> math-max-integral-limit math-integral-limit)
(setq math-integral-limit math-max-integral-limit
math-integ-level math-integral-limit)
(math-integral expr 'yes)))))
(defvar var-IntegLimit nil)
(defun calcFunc-integ (expr var &optional low high)
(cond
;; Do these even if the parts turn out not to be integrable.
((eq (car-safe expr) '+)
(math-add (calcFunc-integ (nth 1 expr) var low high)
(calcFunc-integ (nth 2 expr) var low high)))
((eq (car-safe expr) '-)
(math-sub (calcFunc-integ (nth 1 expr) var low high)
(calcFunc-integ (nth 2 expr) var low high)))
((eq (car-safe expr) 'neg)
(math-neg (calcFunc-integ (nth 1 expr) var low high)))
((and (eq (car-safe expr) '*)
(not (math-expr-contains (nth 1 expr) var)))
(math-mul (nth 1 expr) (calcFunc-integ (nth 2 expr) var low high)))
((and (eq (car-safe expr) '*)
(not (math-expr-contains (nth 2 expr) var)))
(math-mul (calcFunc-integ (nth 1 expr) var low high) (nth 2 expr)))
((and (eq (car-safe expr) '/)
(not (math-expr-contains (nth 1 expr) var))
(not (math-equal-int (nth 1 expr) 1)))
(math-mul (nth 1 expr)
(calcFunc-integ (math-div 1 (nth 2 expr)) var low high)))
((and (eq (car-safe expr) '/)
(not (math-expr-contains (nth 2 expr) var)))
(math-div (calcFunc-integ (nth 1 expr) var low high) (nth 2 expr)))
((and (eq (car-safe expr) '/)
(eq (car-safe (nth 1 expr)) '*)
(not (math-expr-contains (nth 1 (nth 1 expr)) var)))
(math-mul (nth 1 (nth 1 expr))
(calcFunc-integ (math-div (nth 2 (nth 1 expr)) (nth 2 expr))
var low high)))
((and (eq (car-safe expr) '/)
(eq (car-safe (nth 1 expr)) '*)
(not (math-expr-contains (nth 2 (nth 1 expr)) var)))
(math-mul (nth 2 (nth 1 expr))
(calcFunc-integ (math-div (nth 1 (nth 1 expr)) (nth 2 expr))
var low high)))
((and (eq (car-safe expr) '/)
(eq (car-safe (nth 2 expr)) '*)
(not (math-expr-contains (nth 1 (nth 2 expr)) var)))
(math-div (calcFunc-integ (math-div (nth 1 expr) (nth 2 (nth 2 expr)))
var low high)
(nth 1 (nth 2 expr))))
((and (eq (car-safe expr) '/)
(eq (car-safe (nth 2 expr)) '*)
(not (math-expr-contains (nth 2 (nth 2 expr)) var)))
(math-div (calcFunc-integ (math-div (nth 1 expr) (nth 1 (nth 2 expr)))
var low high)
(nth 2 (nth 2 expr))))
((eq (car-safe expr) 'vec)
(cons 'vec (mapcar (function (lambda (x) (calcFunc-integ x var low high)))
(cdr expr))))
(t
(let ((state (list calc-angle-mode
;;calc-symbolic-mode
;;calc-prefer-frac
calc-internal-prec
(calc-var-value 'var-IntegRules)
(calc-var-value 'var-IntegSimpRules))))
(or (equal state math-integral-cache-state)
(setq math-integral-cache-state state
math-integral-cache nil)))
(let* ((math-max-integral-limit (or (and (natnump var-IntegLimit)
var-IntegLimit)
3))
(math-integral-limit 1)
(sexpr (math-expr-subst expr var math-integ-var))
(trace-buffer (get-buffer "*Trace*"))
(calc-language (if (eq calc-language 'big) nil calc-language))
(math-any-substs t)
(math-enable-subst nil)
(math-prev-parts-v nil)
(math-doing-parts nil)
(math-good-parts nil)
(res
(if trace-buffer
(let ((calcbuf (current-buffer))
(calcwin (selected-window)))
(unwind-protect
(progn
(if (get-buffer-window trace-buffer)
(select-window (get-buffer-window trace-buffer)))
(set-buffer trace-buffer)
(goto-char (point-max))
(or (assq 'scroll-stop (buffer-local-variables))
(progn
(make-local-variable 'scroll-step)
(setq scroll-step 3)))
(insert "\n\n\n")
(set-buffer calcbuf)
(math-try-integral sexpr))
(select-window calcwin)
(set-buffer calcbuf)))
(math-try-integral sexpr))))
(if res
(progn
(if (calc-has-rules 'var-IntegAfterRules)
(setq res (math-rewrite res '(var IntegAfterRules
var-IntegAfterRules))))
(math-simplify
(if (and low high)
(math-sub (math-expr-subst res math-integ-var high)
(math-expr-subst res math-integ-var low))
(setq res (math-fix-const-terms res math-integ-vars))
(if low
(math-expr-subst res math-integ-var low)
(math-expr-subst res math-integ-var var)))))
(append (list 'calcFunc-integ expr var)
(and low (list low))
(and high (list high))))))))
(math-defintegral calcFunc-inv
(math-integral (math-div 1 u)))
(math-defintegral calcFunc-conj
(let ((int (math-integral u)))
(and int
(list 'calcFunc-conj int))))
(math-defintegral calcFunc-deg
(let ((int (math-integral u)))
(and int
(list 'calcFunc-deg int))))
(math-defintegral calcFunc-rad
(let ((int (math-integral u)))
(and int
(list 'calcFunc-rad int))))
(math-defintegral calcFunc-re
(let ((int (math-integral u)))
(and int
(list 'calcFunc-re int))))
(math-defintegral calcFunc-im
(let ((int (math-integral u)))
(and int
(list 'calcFunc-im int))))
(math-defintegral calcFunc-sqrt
(and (equal u math-integ-var)
(math-mul '(frac 2 3)
(list 'calcFunc-sqrt (math-pow u 3)))))
(math-defintegral calcFunc-exp
(or (and (equal u math-integ-var)
(list 'calcFunc-exp u))
(let ((p (math-is-polynomial u math-integ-var 2)))
(and (nth 2 p)
(let ((sqa (math-sqrt (math-neg (nth 2 p)))))
(math-div
(math-mul
(math-mul (math-div (list 'calcFunc-sqrt '(var pi var-pi))
sqa)
(math-normalize
(list 'calcFunc-exp
(math-div (math-sub (math-mul (car p)
(nth 2 p))
(math-div
(math-sqr (nth 1 p))
4))
(nth 2 p)))))
(list 'calcFunc-erf
(math-sub (math-mul sqa math-integ-var)
(math-div (nth 1 p) (math-mul 2 sqa)))))
2))))))
(math-defintegral calcFunc-ln
(or (and (equal u math-integ-var)
(math-sub (math-mul u (list 'calcFunc-ln u)) u))
(and (eq (car u) '*)
(math-integral (math-add (list 'calcFunc-ln (nth 1 u))
(list 'calcFunc-ln (nth 2 u)))))
(and (eq (car u) '/)
(math-integral (math-sub (list 'calcFunc-ln (nth 1 u))
(list 'calcFunc-ln (nth 2 u)))))
(and (eq (car u) '^)
(math-integral (math-mul (nth 2 u)
(list 'calcFunc-ln (nth 1 u)))))))
(math-defintegral calcFunc-log10
(and (equal u math-integ-var)
(math-sub (math-mul u (list 'calcFunc-ln u))
(math-div u (list 'calcFunc-ln 10)))))
(math-defintegral-2 calcFunc-log
(math-integral (math-div (list 'calcFunc-ln u)
(list 'calcFunc-ln v))))
(math-defintegral calcFunc-sin
(or (and (equal u math-integ-var)
(math-neg (math-from-radians-2 (list 'calcFunc-cos u))))
(and (nth 2 (math-is-polynomial u math-integ-var 2))
(math-integral (math-to-exponentials (list 'calcFunc-sin u))))))
(math-defintegral calcFunc-cos
(or (and (equal u math-integ-var)
(math-from-radians-2 (list 'calcFunc-sin u)))
(and (nth 2 (math-is-polynomial u math-integ-var 2))
(math-integral (math-to-exponentials (list 'calcFunc-cos u))))))
(math-defintegral calcFunc-tan
(and (equal u math-integ-var)
(math-from-radians-2
(list 'calcFunc-ln (list 'calcFunc-sec u)))))
(math-defintegral calcFunc-sec
(and (equal u math-integ-var)
(math-from-radians-2
(list 'calcFunc-ln
(math-add
(list 'calcFunc-sec u)
(list 'calcFunc-tan u))))))
(math-defintegral calcFunc-csc
(and (equal u math-integ-var)
(math-from-radians-2
(list 'calcFunc-ln
(math-sub
(list 'calcFunc-csc u)
(list 'calcFunc-cot u))))))
(math-defintegral calcFunc-cot
(and (equal u math-integ-var)
(math-from-radians-2
(list 'calcFunc-ln (list 'calcFunc-sin u)))))
(math-defintegral calcFunc-arcsin
(and (equal u math-integ-var)
(math-add (math-mul u (list 'calcFunc-arcsin u))
(math-from-radians-2
(list 'calcFunc-sqrt (math-sub 1 (math-sqr u)))))))
(math-defintegral calcFunc-arccos
(and (equal u math-integ-var)
(math-sub (math-mul u (list 'calcFunc-arccos u))
(math-from-radians-2
(list 'calcFunc-sqrt (math-sub 1 (math-sqr u)))))))
(math-defintegral calcFunc-arctan
(and (equal u math-integ-var)
(math-sub (math-mul u (list 'calcFunc-arctan u))
(math-from-radians-2
(math-div (list 'calcFunc-ln (math-add 1 (math-sqr u)))
2)))))
(math-defintegral calcFunc-sinh
(and (equal u math-integ-var)
(list 'calcFunc-cosh u)))
(math-defintegral calcFunc-cosh
(and (equal u math-integ-var)
(list 'calcFunc-sinh u)))
(math-defintegral calcFunc-tanh
(and (equal u math-integ-var)
(list 'calcFunc-ln (list 'calcFunc-cosh u))))
(math-defintegral calcFunc-sech
(and (equal u math-integ-var)
(list 'calcFunc-arctan (list 'calcFunc-sinh u))))
(math-defintegral calcFunc-csch
(and (equal u math-integ-var)
(list 'calcFunc-ln (list 'calcFunc-tanh (math-div u 2)))))
(math-defintegral calcFunc-coth
(and (equal u math-integ-var)
(list 'calcFunc-ln (list 'calcFunc-sinh u))))
(math-defintegral calcFunc-arcsinh
(and (equal u math-integ-var)
(math-sub (math-mul u (list 'calcFunc-arcsinh u))
(list 'calcFunc-sqrt (math-add (math-sqr u) 1)))))
(math-defintegral calcFunc-arccosh
(and (equal u math-integ-var)
(math-sub (math-mul u (list 'calcFunc-arccosh u))
(list 'calcFunc-sqrt (math-sub 1 (math-sqr u))))))
(math-defintegral calcFunc-arctanh
(and (equal u math-integ-var)
(math-sub (math-mul u (list 'calcFunc-arctan u))
(math-div (list 'calcFunc-ln
(math-add 1 (math-sqr u)))
2))))
;;; (Ax + B) / (ax^2 + bx + c)^n forms.
(math-defintegral-2 /
(math-integral-rational-funcs u v))
(defun math-integral-rational-funcs (u v)
(let ((pu (math-is-polynomial u math-integ-var 1))
(vpow 1) pv)
(and pu
(catch 'int-rat
(if (and (eq (car-safe v) '^) (natnump (nth 2 v)))
(setq vpow (nth 2 v)
v (nth 1 v)))
(and (setq pv (math-is-polynomial v math-integ-var 2))
(let ((int (math-mul-thru
(car pu)
(math-integral-q02 (car pv) (nth 1 pv)
(nth 2 pv) v vpow))))
(if (cdr pu)
(setq int (math-add int
(math-mul-thru
(nth 1 pu)
(math-integral-q12
(car pv) (nth 1 pv)
(nth 2 pv) v vpow)))))
int))))))
(defun math-integral-q12 (a b c v vpow)
(let (q)
(cond ((not c)
(cond ((= vpow 1)
(math-sub (math-div math-integ-var b)
(math-mul (math-div a (math-sqr b))
(list 'calcFunc-ln v))))
((= vpow 2)
(math-div (math-add (list 'calcFunc-ln v)
(math-div a v))
(math-sqr b)))
(t
(let ((nm1 (math-sub vpow 1))
(nm2 (math-sub vpow 2)))
(math-div (math-sub
(math-div a (math-mul nm1 (math-pow v nm1)))
(math-div 1 (math-mul nm2 (math-pow v nm2))))
(math-sqr b))))))
((math-zerop
(setq q (math-sub (math-mul 4 (math-mul a c)) (math-sqr b))))
(let ((part (math-div b (math-mul 2 c))))
(math-mul-thru (math-pow c vpow)
(math-integral-q12 part 1 nil
(math-add math-integ-var part)
(* vpow 2)))))
((= vpow 1)
(and (math-ratp q) (math-negp q)
(let ((calc-symbolic-mode t))
(math-ratp (math-sqrt (math-neg q))))
(throw 'int-rat nil)) ; should have used calcFunc-apart first
(math-sub (math-div (list 'calcFunc-ln v) (math-mul 2 c))
(math-mul-thru (math-div b (math-mul 2 c))
(math-integral-q02 a b c v 1))))
(t
(let ((n (1- vpow)))
(math-sub (math-neg (math-div
(math-add (math-mul b math-integ-var)
(math-mul 2 a))
(math-mul n (math-mul q (math-pow v n)))))
(math-mul-thru (math-div (math-mul b (1- (* 2 n)))
(math-mul n q))
(math-integral-q02 a b c v n))))))))
(defun math-integral-q02 (a b c v vpow)
(let (q rq part)
(cond ((not c)
(cond ((= vpow 1)
(math-div (list 'calcFunc-ln v) b))
(t
(math-div (math-pow v (- 1 vpow))
(math-mul (- 1 vpow) b)))))
((math-zerop
(setq q (math-sub (math-mul 4 (math-mul a c)) (math-sqr b))))
(let ((part (math-div b (math-mul 2 c))))
(math-mul-thru (math-pow c vpow)
(math-integral-q02 part 1 nil
(math-add math-integ-var part)
(* vpow 2)))))
((progn
(setq part (math-add (math-mul 2 (math-mul c math-integ-var)) b))
(> vpow 1))
(let ((n (1- vpow)))
(math-add (math-div part (math-mul n (math-mul q (math-pow v n))))
(math-mul-thru (math-div (math-mul (- (* 4 n) 2) c)
(math-mul n q))
(math-integral-q02 a b c v n)))))
((math-guess-if-neg q)
(setq rq (list 'calcFunc-sqrt (math-neg q)))
;;(math-div-thru (list 'calcFunc-ln
;; (math-div (math-sub part rq)
;; (math-add part rq)))
;; rq)
(math-div (math-mul -2 (list 'calcFunc-arctanh
(math-div part rq)))
rq))
(t
(setq rq (list 'calcFunc-sqrt q))
(math-div (math-mul 2 (math-to-radians-2
(list 'calcFunc-arctan
(math-div part rq))))
rq)))))
(math-defintegral calcFunc-erf
(and (equal u math-integ-var)
(math-add (math-mul u (list 'calcFunc-erf u))
(math-div 1 (math-mul (list 'calcFunc-exp (math-sqr u))
(list 'calcFunc-sqrt
'(var pi var-pi)))))))
(math-defintegral calcFunc-erfc
(and (equal u math-integ-var)
(math-sub (math-mul u (list 'calcFunc-erfc u))
(math-div 1 (math-mul (list 'calcFunc-exp (math-sqr u))
(list 'calcFunc-sqrt
'(var pi var-pi)))))))
(defvar math-tabulate-initial nil)
(defvar math-tabulate-function nil)
;; These variables are local to calcFunc-table, but are used by
;; math-scan-for-limits.
(defvar calc-low)
(defvar calc-high)
(defvar math-var)
(defun calcFunc-table (expr math-var &optional calc-low calc-high step)
(or calc-low
(setq calc-low '(neg (var inf var-inf)) calc-high '(var inf var-inf)))
(or calc-high (setq calc-high calc-low calc-low 1))
(and (or (math-infinitep calc-low) (math-infinitep calc-high))
(not step)
(math-scan-for-limits expr))
(and step (math-zerop step) (math-reject-arg step 'nonzerop))
(let ((known (+ (if (Math-objectp calc-low) 1 0)
(if (Math-objectp calc-high) 1 0)
(if (or (null step) (Math-objectp step)) 1 0)))
(count '(var inf var-inf))
vec)
(or (= known 2) ; handy optimization
(equal calc-high '(var inf var-inf))
(progn
(setq count (math-div (math-sub calc-high calc-low) (or step 1)))
(or (Math-objectp count)
(setq count (math-simplify count)))
(if (Math-messy-integerp count)
(setq count (math-trunc count)))))
(if (Math-negp count)
(setq count -1))
(if (integerp count)
(let ((var-DUMMY nil)
(vec math-tabulate-initial)
(math-working-step-2 (1+ count))
(math-working-step 0))
(setq expr (math-evaluate-expr
(math-expr-subst expr math-var '(var DUMMY var-DUMMY))))
(while (>= count 0)
(setq math-working-step (1+ math-working-step)
var-DUMMY calc-low
vec (cond ((eq math-tabulate-function 'calcFunc-sum)
(math-add vec (math-evaluate-expr expr)))
((eq math-tabulate-function 'calcFunc-prod)
(math-mul vec (math-evaluate-expr expr)))
(t
(cons (math-evaluate-expr expr) vec)))
calc-low (math-add calc-low (or step 1))
count (1- count)))
(if math-tabulate-function
vec
(cons 'vec (nreverse vec))))
(if (Math-integerp count)
(calc-record-why 'fixnump calc-high)
(if (Math-num-integerp calc-low)
(if (Math-num-integerp calc-high)
(calc-record-why 'integerp step)
(calc-record-why 'integerp calc-high))
(calc-record-why 'integerp calc-low)))
(append (list (or math-tabulate-function 'calcFunc-table)
expr math-var)
(and (not (and (equal calc-low '(neg (var inf var-inf)))
(equal calc-high '(var inf var-inf))))
(list calc-low calc-high))
(and step (list step))))))
(defun math-scan-for-limits (x)
(cond ((Math-primp x))
((and (eq (car x) 'calcFunc-subscr)
(Math-vectorp (nth 1 x))
(math-expr-contains (nth 2 x) math-var))
(let* ((calc-next-why nil)
(low-val (math-solve-for (nth 2 x) 1 math-var nil))
(high-val (math-solve-for (nth 2 x) (1- (length (nth 1 x)))
math-var nil))
temp)
(and low-val (math-realp low-val)
high-val (math-realp high-val))
(and (Math-lessp high-val low-val)
(setq temp low-val low-val high-val high-val temp))
(setq calc-low (math-max calc-low (math-ceiling low-val))
calc-high (math-min calc-high (math-floor high-val)))))
(t
(while (setq x (cdr x))
(math-scan-for-limits (car x))))))
(defvar math-disable-sums nil)
(defun calcFunc-sum (expr var &optional low high step)
(if math-disable-sums (math-reject-arg))
(let* ((res (let* ((calc-internal-prec (+ calc-internal-prec 2)))
(math-sum-rec expr var low high step)))
(math-disable-sums t))
(math-normalize res)))
(defun math-sum-rec (expr var &optional low high step)
(or low (setq low '(neg (var inf var-inf)) high '(var inf var-inf)))
(and low (not high) (setq high low low 1))
(let (t1 t2 val)
(setq val
(cond
((not (math-expr-contains expr var))
(math-mul expr (math-add (math-div (math-sub high low) (or step 1))
1)))
((and step (not (math-equal-int step 1)))
(if (math-negp step)
(math-sum-rec expr var high low (math-neg step))
(let ((lo (math-simplify (math-div low step))))
(if (math-known-num-integerp lo)
(math-sum-rec (math-normalize
(math-expr-subst expr var
(math-mul step var)))
var lo (math-simplify (math-div high step)))
(math-sum-rec (math-normalize
(math-expr-subst expr var
(math-add (math-mul step var)
low)))
var 0
(math-simplify (math-div (math-sub high low)
step)))))))
((memq (setq t1 (math-compare low high)) '(0 1))
(if (eq t1 0)
(math-expr-subst expr var low)
0))
((setq t1 (math-is-polynomial expr var 20))
(let ((poly nil)
(n 0))
(while t1
(setq poly (math-poly-mix poly 1
(math-sum-integer-power n) (car t1))
n (1+ n)
t1 (cdr t1)))
(setq n (math-build-polynomial-expr poly high))
(if (= low 1)
n
(math-sub n (math-build-polynomial-expr poly
(math-sub low 1))))))
((and (memq (car expr) '(+ -))
(setq t1 (math-sum-rec (nth 1 expr) var low high)
t2 (math-sum-rec (nth 2 expr) var low high))
(not (and (math-expr-calls t1 '(calcFunc-sum))
(math-expr-calls t2 '(calcFunc-sum)))))
(list (car expr) t1 t2))
((and (eq (car expr) '*)
(setq t1 (math-sum-const-factors expr var)))
(math-mul (car t1) (math-sum-rec (cdr t1) var low high)))
((and (eq (car expr) '*) (memq (car-safe (nth 1 expr)) '(+ -)))
(math-sum-rec (math-add-or-sub (math-mul (nth 1 (nth 1 expr))
(nth 2 expr))
(math-mul (nth 2 (nth 1 expr))
(nth 2 expr))
nil (eq (car (nth 1 expr)) '-))
var low high))
((and (eq (car expr) '*) (memq (car-safe (nth 2 expr)) '(+ -)))
(math-sum-rec (math-add-or-sub (math-mul (nth 1 expr)
(nth 1 (nth 2 expr)))
(math-mul (nth 1 expr)
(nth 2 (nth 2 expr)))
nil (eq (car (nth 2 expr)) '-))
var low high))
((and (eq (car expr) '/)
(not (math-primp (nth 1 expr)))
(setq t1 (math-sum-const-factors (nth 1 expr) var)))
(math-mul (car t1)
(math-sum-rec (math-div (cdr t1) (nth 2 expr))
var low high)))
((and (eq (car expr) '/)
(setq t1 (math-sum-const-factors (nth 2 expr) var)))
(math-div (math-sum-rec (math-div (nth 1 expr) (cdr t1))
var low high)
(car t1)))
((eq (car expr) 'neg)
(math-neg (math-sum-rec (nth 1 expr) var low high)))
((and (eq (car expr) '^)
(not (math-expr-contains (nth 1 expr) var))
(setq t1 (math-is-polynomial (nth 2 expr) var 1)))
(let ((x (math-pow (nth 1 expr) (nth 1 t1))))
(math-div (math-mul (math-sub (math-pow x (math-add 1 high))
(math-pow x low))
(math-pow (nth 1 expr) (car t1)))
(math-sub x 1))))
((and (setq t1 (math-to-exponentials expr))
(setq t1 (math-sum-rec t1 var low high))
(not (math-expr-calls t1 '(calcFunc-sum))))
(math-to-exps t1))
((memq (car expr) '(calcFunc-ln calcFunc-log10))
(list (car expr) (calcFunc-prod (nth 1 expr) var low high)))
((and (eq (car expr) 'calcFunc-log)
(= (length expr) 3)
(not (math-expr-contains (nth 2 expr) var)))
(list 'calcFunc-log
(calcFunc-prod (nth 1 expr) var low high)
(nth 2 expr)))))
(if (equal val '(var nan var-nan)) (setq val nil))
(or val
(let* ((math-tabulate-initial 0)
(math-tabulate-function 'calcFunc-sum))
(calcFunc-table expr var low high)))))
(defun calcFunc-asum (expr var low &optional high step no-mul-flag)
(or high (setq high low low 1))
(if (and step (not (math-equal-int step 1)))
(if (math-negp step)
(math-mul (math-pow -1 low)
(calcFunc-asum expr var high low (math-neg step) t))
(let ((lo (math-simplify (math-div low step))))
(if (math-num-integerp lo)
(calcFunc-asum (math-normalize
(math-expr-subst expr var
(math-mul step var)))
var lo (math-simplify (math-div high step)))
(calcFunc-asum (math-normalize
(math-expr-subst expr var
(math-add (math-mul step var)
low)))
var 0
(math-simplify (math-div (math-sub high low)
step))))))
(math-mul (if no-mul-flag 1 (math-pow -1 low))
(calcFunc-sum (math-mul (math-pow -1 var) expr) var low high))))
(defun math-sum-const-factors (expr var)
(let ((const nil)
(not-const nil)
(p expr))
(while (eq (car-safe p) '*)
(if (math-expr-contains (nth 1 p) var)
(setq not-const (cons (nth 1 p) not-const))
(setq const (cons (nth 1 p) const)))
(setq p (nth 2 p)))
(if (math-expr-contains p var)
(setq not-const (cons p not-const))
(setq const (cons p const)))
(and const
(cons (let ((temp (car const)))
(while (setq const (cdr const))
(setq temp (list '* (car const) temp)))
temp)
(let ((temp (or (car not-const) 1)))
(while (setq not-const (cdr not-const))
(setq temp (list '* (car not-const) temp)))
temp)))))
(defvar math-sum-int-pow-cache (list '(0 1)))
;; Following is from CRC Math Tables, 27th ed, pp. 52-53.
(defun math-sum-integer-power (pow)
(let ((calc-prefer-frac t)
(n (length math-sum-int-pow-cache)))
(while (<= n pow)
(let* ((new (list 0 0))
(lin new)
(pp (cdr (nth (1- n) math-sum-int-pow-cache)))
(p 2)
(sum 0)
q)
(while pp
(setq q (math-div (car pp) p)
new (cons (math-mul q n) new)
sum (math-add sum q)
p (1+ p)
pp (cdr pp)))
(setcar lin (math-sub 1 (math-mul n sum)))
(setq math-sum-int-pow-cache
(nconc math-sum-int-pow-cache (list (nreverse new)))
n (1+ n))))
(nth pow math-sum-int-pow-cache)))
(defun math-to-exponentials (expr)
(and (consp expr)
(= (length expr) 2)
(let ((x (nth 1 expr))
(pi (if calc-symbolic-mode '(var pi var-pi) (math-pi)))
(i (if calc-symbolic-mode '(var i var-i) '(cplx 0 1))))
(cond ((eq (car expr) 'calcFunc-exp)
(list '^ '(var e var-e) x))
((eq (car expr) 'calcFunc-sin)
(or (eq calc-angle-mode 'rad)
(setq x (list '/ (list '* x pi) 180)))
(list '/ (list '-
(list '^ '(var e var-e) (list '* x i))
(list '^ '(var e var-e)
(list 'neg (list '* x i))))
(list '* 2 i)))
((eq (car expr) 'calcFunc-cos)
(or (eq calc-angle-mode 'rad)
(setq x (list '/ (list '* x pi) 180)))
(list '/ (list '+
(list '^ '(var e var-e)
(list '* x i))
(list '^ '(var e var-e)
(list 'neg (list '* x i))))
2))
((eq (car expr) 'calcFunc-sinh)
(list '/ (list '-
(list '^ '(var e var-e) x)
(list '^ '(var e var-e) (list 'neg x)))
2))
((eq (car expr) 'calcFunc-cosh)
(list '/ (list '+
(list '^ '(var e var-e) x)
(list '^ '(var e var-e) (list 'neg x)))
2))
(t nil)))))
(defun math-to-exps (expr)
(cond (calc-symbolic-mode expr)
((Math-primp expr)
(if (equal expr '(var e var-e)) (math-e) expr))
((and (eq (car expr) '^)
(equal (nth 1 expr) '(var e var-e)))
(list 'calcFunc-exp (nth 2 expr)))
(t
(cons (car expr) (mapcar 'math-to-exps (cdr expr))))))
(defvar math-disable-prods nil)
(defun calcFunc-prod (expr var &optional low high step)
(if math-disable-prods (math-reject-arg))
(let* ((res (let* ((calc-internal-prec (+ calc-internal-prec 2)))
(math-prod-rec expr var low high step)))
(math-disable-prods t))
(math-normalize res)))
(defun math-prod-rec (expr var &optional low high step)
(or low (setq low '(neg (var inf var-inf)) high '(var inf var-inf)))
(and low (not high) (setq high '(var inf var-inf)))
(let (t1 t2 t3 val)
(setq val
(cond
((not (math-expr-contains expr var))
(math-pow expr (math-add (math-div (math-sub high low) (or step 1))
1)))
((and step (not (math-equal-int step 1)))
(if (math-negp step)
(math-prod-rec expr var high low (math-neg step))
(let ((lo (math-simplify (math-div low step))))
(if (math-known-num-integerp lo)
(math-prod-rec (math-normalize
(math-expr-subst expr var
(math-mul step var)))
var lo (math-simplify (math-div high step)))
(math-prod-rec (math-normalize
(math-expr-subst expr var
(math-add (math-mul step
var)
low)))
var 0
(math-simplify (math-div (math-sub high low)
step)))))))
((and (memq (car expr) '(* /))
(setq t1 (math-prod-rec (nth 1 expr) var low high)
t2 (math-prod-rec (nth 2 expr) var low high))
(not (and (math-expr-calls t1 '(calcFunc-prod))
(math-expr-calls t2 '(calcFunc-prod)))))
(list (car expr) t1 t2))
((and (eq (car expr) '^)
(not (math-expr-contains (nth 2 expr) var)))
(math-pow (math-prod-rec (nth 1 expr) var low high)
(nth 2 expr)))
((and (eq (car expr) '^)
(not (math-expr-contains (nth 1 expr) var)))
(math-pow (nth 1 expr)
(calcFunc-sum (nth 2 expr) var low high)))
((eq (car expr) 'sqrt)
(math-normalize (list 'calcFunc-sqrt
(list 'calcFunc-prod (nth 1 expr)
var low high))))
((eq (car expr) 'neg)
(math-mul (math-pow -1 (math-add (math-sub high low) 1))
(math-prod-rec (nth 1 expr) var low high)))
((eq (car expr) 'calcFunc-exp)
(list 'calcFunc-exp (calcFunc-sum (nth 1 expr) var low high)))
((and (setq t1 (math-is-polynomial expr var 1))
(setq t2
(cond
((or (and (math-equal-int (nth 1 t1) 1)
(setq low (math-simplify
(math-add low (car t1)))
high (math-simplify
(math-add high (car t1)))))
(and (math-equal-int (nth 1 t1) -1)
(setq t2 low
low (math-simplify
(math-sub (car t1) high))
high (math-simplify
(math-sub (car t1) t2)))))
(if (or (math-zerop low) (math-zerop high))
0
(if (and (or (math-negp low) (math-negp high))
(or (math-num-integerp low)
(math-num-integerp high)))
(if (math-posp high)
0
(math-mul (math-pow -1
(math-add
(math-add low high) 1))
(list '/
(list 'calcFunc-fact
(math-neg low))
(list 'calcFunc-fact
(math-sub -1 high)))))
(list '/
(list 'calcFunc-fact high)
(list 'calcFunc-fact (math-sub low 1))))))
((and (or (and (math-equal-int (nth 1 t1) 2)
(setq t2 (math-simplify
(math-add (math-mul low 2)
(car t1)))
t3 (math-simplify
(math-add (math-mul high 2)
(car t1)))))
(and (math-equal-int (nth 1 t1) -2)
(setq t2 (math-simplify
(math-sub (car t1)
(math-mul high 2)))
t3 (math-simplify
(math-sub (car t1)
(math-mul low
2))))))
(or (math-integerp t2)
(and (math-messy-integerp t2)
(setq t2 (math-trunc t2)))
(math-integerp t3)
(and (math-messy-integerp t3)
(setq t3 (math-trunc t3)))))
(if (or (math-zerop t2) (math-zerop t3))
0
(if (or (math-evenp t2) (math-evenp t3))
(if (or (math-negp t2) (math-negp t3))
(if (math-posp high)
0
(list '/
(list 'calcFunc-dfact
(math-neg t2))
(list 'calcFunc-dfact
(math-sub -2 t3))))
(list '/
(list 'calcFunc-dfact t3)
(list 'calcFunc-dfact
(math-sub t2 2))))
(if (math-negp t3)
(list '*
(list '^ -1
(list '/ (list '- (list '- t2 t3)
2)
2))
(list '/
(list 'calcFunc-dfact
(math-neg t2))
(list 'calcFunc-dfact
(math-sub -2 t3))))
(if (math-posp t2)
(list '/
(list 'calcFunc-dfact t3)
(list 'calcFunc-dfact
(math-sub t2 2)))
nil))))))))
t2)))
(if (equal val '(var nan var-nan)) (setq val nil))
(or val
(let* ((math-tabulate-initial 1)
(math-tabulate-function 'calcFunc-prod))
(calcFunc-table expr var low high)))))
(defvar math-solve-ranges nil)
(defvar math-solve-sign)
;;; Attempt to reduce math-solve-lhs = math-solve-rhs to
;;; math-solve-var = math-solve-rhs', where math-solve-var appears
;;; in math-solve-lhs but not in math-solve-rhs or math-solve-rhs';
;;; return math-solve-rhs'.
;;; Uses global values: math-solve-var, math-solve-full.
(defvar math-solve-var)
(defvar math-solve-full)
;; The variables math-solve-lhs, math-solve-rhs and math-try-solve-sign
;; are local to math-try-solve-for, but are used by math-try-solve-prod.
;; (math-solve-lhs and math-solve-rhs are also local to
;; math-decompose-poly, but used by math-solve-poly-funny-powers.)
(defvar math-solve-lhs)
(defvar math-solve-rhs)
(defvar math-try-solve-sign)
(defun math-try-solve-for
(math-solve-lhs math-solve-rhs &optional math-try-solve-sign no-poly)
(let (math-t1 math-t2 math-t3)
(cond ((equal math-solve-lhs math-solve-var)
(setq math-solve-sign math-try-solve-sign)
(if (eq math-solve-full 'all)
(let ((vec (list 'vec (math-evaluate-expr math-solve-rhs)))
newvec var p)
(while math-solve-ranges
(setq p (car math-solve-ranges)
var (car p)
newvec (list 'vec))
(while (setq p (cdr p))
(setq newvec (nconc newvec
(cdr (math-expr-subst
vec var (car p))))))
(setq vec newvec
math-solve-ranges (cdr math-solve-ranges)))
(math-normalize vec))
math-solve-rhs))
((Math-primp math-solve-lhs)
nil)
((and (eq (car math-solve-lhs) '-)
(eq (car-safe (nth 1 math-solve-lhs)) (car-safe (nth 2 math-solve-lhs)))
(Math-zerop math-solve-rhs)
(= (length (nth 1 math-solve-lhs)) 2)
(= (length (nth 2 math-solve-lhs)) 2)
(setq math-t1 (get (car (nth 1 math-solve-lhs)) 'math-inverse))
(setq math-t2 (funcall math-t1 '(var SOLVEDUM SOLVEDUM)))
(eq (math-expr-contains-count math-t2 '(var SOLVEDUM SOLVEDUM)) 1)
(setq math-t3 (math-solve-above-dummy math-t2))
(setq math-t1 (math-try-solve-for
(math-sub (nth 1 (nth 1 math-solve-lhs))
(math-expr-subst
math-t2 math-t3
(nth 1 (nth 2 math-solve-lhs))))
0)))
math-t1)
((eq (car math-solve-lhs) 'neg)
(math-try-solve-for (nth 1 math-solve-lhs) (math-neg math-solve-rhs)
(and math-try-solve-sign (- math-try-solve-sign))))
((and (not (eq math-solve-full 't)) (math-try-solve-prod)))
((and (not no-poly)
(setq math-t2
(math-decompose-poly math-solve-lhs
math-solve-var 15 math-solve-rhs)))
(setq math-t1 (cdr (nth 1 math-t2))
math-t1 (let ((math-solve-ranges math-solve-ranges))
(cond ((= (length math-t1) 5)
(apply 'math-solve-quartic (car math-t2) math-t1))
((= (length math-t1) 4)
(apply 'math-solve-cubic (car math-t2) math-t1))
((= (length math-t1) 3)
(apply 'math-solve-quadratic (car math-t2) math-t1))
((= (length math-t1) 2)
(apply 'math-solve-linear
(car math-t2) math-try-solve-sign math-t1))
(math-solve-full
(math-poly-all-roots (car math-t2) math-t1))
(calc-symbolic-mode nil)
(t
(math-try-solve-for
(car math-t2)
(math-poly-any-root (reverse math-t1) 0 t)
nil t)))))
(if math-t1
(if (eq (nth 2 math-t2) 1)
math-t1
(math-solve-prod math-t1 (math-try-solve-for (nth 2 math-t2) 0 nil t)))
(calc-record-why "*Unable to find a symbolic solution")
nil))
((and (math-solve-find-root-term math-solve-lhs nil)
(eq (math-expr-contains-count math-solve-lhs math-t1) 1)) ; just in case
(math-try-solve-for (math-simplify
(math-sub (if (or math-t3 (math-evenp math-t2))
(math-pow math-t1 math-t2)
(math-neg (math-pow math-t1 math-t2)))
(math-expand-power
(math-sub (math-normalize
(math-expr-subst
math-solve-lhs math-t1 0))
math-solve-rhs)
math-t2 math-solve-var)))
0))
((eq (car math-solve-lhs) '+)
(cond ((not (math-expr-contains (nth 1 math-solve-lhs) math-solve-var))
(math-try-solve-for (nth 2 math-solve-lhs)
(math-sub math-solve-rhs (nth 1 math-solve-lhs))
math-try-solve-sign))
((not (math-expr-contains (nth 2 math-solve-lhs) math-solve-var))
(math-try-solve-for (nth 1 math-solve-lhs)
(math-sub math-solve-rhs (nth 2 math-solve-lhs))
math-try-solve-sign))))
((eq (car math-solve-lhs) 'calcFunc-eq)
(math-try-solve-for (math-sub (nth 1 math-solve-lhs) (nth 2 math-solve-lhs))
math-solve-rhs math-try-solve-sign no-poly))
((eq (car math-solve-lhs) '-)
(cond ((or (and (eq (car-safe (nth 1 math-solve-lhs)) 'calcFunc-sin)
(eq (car-safe (nth 2 math-solve-lhs)) 'calcFunc-cos))
(and (eq (car-safe (nth 1 math-solve-lhs)) 'calcFunc-cos)
(eq (car-safe (nth 2 math-solve-lhs)) 'calcFunc-sin)))
(math-try-solve-for (math-sub (nth 1 math-solve-lhs)
(list (car (nth 1 math-solve-lhs))
(math-sub
(math-quarter-circle t)
(nth 1 (nth 2 math-solve-lhs)))))
math-solve-rhs))
((not (math-expr-contains (nth 1 math-solve-lhs) math-solve-var))
(math-try-solve-for (nth 2 math-solve-lhs)
(math-sub (nth 1 math-solve-lhs) math-solve-rhs)
(and math-try-solve-sign
(- math-try-solve-sign))))
((not (math-expr-contains (nth 2 math-solve-lhs) math-solve-var))
(math-try-solve-for (nth 1 math-solve-lhs)
(math-add math-solve-rhs (nth 2 math-solve-lhs))
math-try-solve-sign))))
((and (eq math-solve-full 't) (math-try-solve-prod)))
((and (eq (car math-solve-lhs) '%)
(not (math-expr-contains (nth 2 math-solve-lhs) math-solve-var)))
(math-try-solve-for (nth 1 math-solve-lhs) (math-add math-solve-rhs
(math-solve-get-int
(nth 2 math-solve-lhs)))))
((eq (car math-solve-lhs) 'calcFunc-log)
(cond ((not (math-expr-contains (nth 2 math-solve-lhs) math-solve-var))
(math-try-solve-for (nth 1 math-solve-lhs)
(math-pow (nth 2 math-solve-lhs) math-solve-rhs)))
((not (math-expr-contains (nth 1 math-solve-lhs) math-solve-var))
(math-try-solve-for (nth 2 math-solve-lhs) (math-pow
(nth 1 math-solve-lhs)
(math-div 1 math-solve-rhs))))))
((and (= (length math-solve-lhs) 2)
(symbolp (car math-solve-lhs))
(setq math-t1 (get (car math-solve-lhs) 'math-inverse))
(setq math-t2 (funcall math-t1 math-solve-rhs)))
(setq math-t1 (get (car math-solve-lhs) 'math-inverse-sign))
(math-try-solve-for (nth 1 math-solve-lhs) (math-normalize math-t2)
(and math-try-solve-sign math-t1
(if (integerp math-t1)
(* math-t1 math-try-solve-sign)
(funcall math-t1 math-solve-lhs
math-try-solve-sign)))))
((and (symbolp (car math-solve-lhs))
(setq math-t1 (get (car math-solve-lhs) 'math-inverse-n))
(setq math-t2 (funcall math-t1 math-solve-lhs math-solve-rhs)))
math-t2)
((setq math-t1 (math-expand-formula math-solve-lhs))
(math-try-solve-for math-t1 math-solve-rhs math-try-solve-sign))
(t
(calc-record-why "*No inverse known" math-solve-lhs)
nil))))
(defun math-try-solve-prod ()
(cond ((eq (car math-solve-lhs) '*)
(cond ((not (math-expr-contains (nth 1 math-solve-lhs) math-solve-var))
(math-try-solve-for (nth 2 math-solve-lhs)
(math-div math-solve-rhs (nth 1 math-solve-lhs))
(math-solve-sign math-try-solve-sign
(nth 1 math-solve-lhs))))
((not (math-expr-contains (nth 2 math-solve-lhs) math-solve-var))
(math-try-solve-for (nth 1 math-solve-lhs)
(math-div math-solve-rhs (nth 2 math-solve-lhs))
(math-solve-sign math-try-solve-sign
(nth 2 math-solve-lhs))))
((Math-zerop math-solve-rhs)
(math-solve-prod (let ((math-solve-ranges math-solve-ranges))
(math-try-solve-for (nth 2 math-solve-lhs) 0))
(math-try-solve-for (nth 1 math-solve-lhs) 0)))))
((eq (car math-solve-lhs) '/)
(cond ((not (math-expr-contains (nth 1 math-solve-lhs) math-solve-var))
(math-try-solve-for (nth 2 math-solve-lhs)
(math-div (nth 1 math-solve-lhs) math-solve-rhs)
(math-solve-sign math-try-solve-sign
(nth 1 math-solve-lhs))))
((not (math-expr-contains (nth 2 math-solve-lhs) math-solve-var))
(math-try-solve-for (nth 1 math-solve-lhs)
(math-mul math-solve-rhs (nth 2 math-solve-lhs))
(math-solve-sign math-try-solve-sign
(nth 2 math-solve-lhs))))
((setq math-t1 (math-try-solve-for (math-sub (nth 1 math-solve-lhs)
(math-mul (nth 2 math-solve-lhs)
math-solve-rhs))
0))
math-t1)))
((eq (car math-solve-lhs) '^)
(cond ((not (math-expr-contains (nth 1 math-solve-lhs) math-solve-var))
(math-try-solve-for
(nth 2 math-solve-lhs)
(math-add (math-normalize
(list 'calcFunc-log math-solve-rhs (nth 1 math-solve-lhs)))
(math-div
(math-mul 2
(math-mul '(var pi var-pi)
(math-solve-get-int
'(var i var-i))))
(math-normalize
(list 'calcFunc-ln (nth 1 math-solve-lhs)))))))
((not (math-expr-contains (nth 2 math-solve-lhs) math-solve-var))
(cond ((and (integerp (nth 2 math-solve-lhs))
(>= (nth 2 math-solve-lhs) 2)
(setq math-t1 (math-integer-log2 (nth 2 math-solve-lhs))))
(setq math-t2 math-solve-rhs)
(if (and (eq math-solve-full t)
(math-known-realp (nth 1 math-solve-lhs)))
(progn
(while (>= (setq math-t1 (1- math-t1)) 0)
(setq math-t2 (list 'calcFunc-sqrt math-t2)))
(setq math-t2 (math-solve-get-sign math-t2)))
(while (>= (setq math-t1 (1- math-t1)) 0)
(setq math-t2 (math-solve-get-sign
(math-normalize
(list 'calcFunc-sqrt math-t2))))))
(math-try-solve-for
(nth 1 math-solve-lhs)
(math-normalize math-t2)))
((math-looks-negp (nth 2 math-solve-lhs))
(math-try-solve-for
(list '^ (nth 1 math-solve-lhs)
(math-neg (nth 2 math-solve-lhs)))
(math-div 1 math-solve-rhs)))
((and (eq math-solve-full t)
(Math-integerp (nth 2 math-solve-lhs))
(math-known-realp (nth 1 math-solve-lhs)))
(setq math-t1 (math-normalize
(list 'calcFunc-nroot math-solve-rhs
(nth 2 math-solve-lhs))))
(if (math-evenp (nth 2 math-solve-lhs))
(setq math-t1 (math-solve-get-sign math-t1)))
(math-try-solve-for
(nth 1 math-solve-lhs) math-t1
(and math-try-solve-sign
(math-oddp (nth 2 math-solve-lhs))
(math-solve-sign math-try-solve-sign
(nth 2 math-solve-lhs)))))
(t (math-try-solve-for
(nth 1 math-solve-lhs)
(math-mul
(math-normalize
(list 'calcFunc-exp
(if (Math-realp (nth 2 math-solve-lhs))
(math-div (math-mul
'(var pi var-pi)
(math-solve-get-int
'(var i var-i)
(and (integerp (nth 2 math-solve-lhs))
(math-abs
(nth 2 math-solve-lhs)))))
(math-div (nth 2 math-solve-lhs) 2))
(math-div (math-mul
2
(math-mul
'(var pi var-pi)
(math-solve-get-int
'(var i var-i)
(and (integerp (nth 2 math-solve-lhs))
(math-abs
(nth 2 math-solve-lhs))))))
(nth 2 math-solve-lhs)))))
(math-normalize
(list 'calcFunc-nroot
math-solve-rhs
(nth 2 math-solve-lhs))))
(and math-try-solve-sign
(math-oddp (nth 2 math-solve-lhs))
(math-solve-sign math-try-solve-sign
(nth 2 math-solve-lhs)))))))))
(t nil)))
(defun math-solve-prod (lsoln rsoln)
(cond ((null lsoln)
rsoln)
((null rsoln)
lsoln)
((eq math-solve-full 'all)
(cons 'vec (append (cdr lsoln) (cdr rsoln))))
(math-solve-full
(list 'calcFunc-if
(list 'calcFunc-gt (math-solve-get-sign 1) 0)
lsoln
rsoln))
(t lsoln)))
;;; This deals with negative, fractional, and symbolic powers of "x".
;; The variable math-solve-b is local to math-decompose-poly,
;; but is used by math-solve-poly-funny-powers.
(defvar math-solve-b)
(defun math-solve-poly-funny-powers (sub-rhs) ; uses "t1", "t2"
(setq math-t1 math-solve-lhs)
(let ((pp math-poly-neg-powers)
fac)
(while pp
(setq fac (math-pow (car pp) (or math-poly-mult-powers 1))
math-t1 (math-mul math-t1 fac)
math-solve-rhs (math-mul math-solve-rhs fac)
pp (cdr pp))))
(if sub-rhs (setq math-t1 (math-sub math-t1 math-solve-rhs)))
(let ((math-poly-neg-powers nil))
(setq math-t2 (math-mul (or math-poly-mult-powers 1)
(let ((calc-prefer-frac t))
(math-div 1 math-poly-frac-powers)))
math-t1 (math-is-polynomial
(math-simplify (calcFunc-expand math-t1)) math-solve-b 50))))
;;; This converts "a x^8 + b x^5 + c x^2" to "(a (x^3)^2 + b (x^3) + c) * x^2".
(defun math-solve-crunch-poly (max-degree) ; uses "t1", "t3"
(let ((count 0))
(while (and math-t1 (Math-zerop (car math-t1)))
(setq math-t1 (cdr math-t1)
count (1+ count)))
(and math-t1
(let* ((degree (1- (length math-t1)))
(scale degree))
(while (and (> scale 1) (= (car math-t3) 1))
(and (= (% degree scale) 0)
(let ((p math-t1)
(n 0)
(new-t1 nil)
(okay t))
(while (and p okay)
(if (= (% n scale) 0)
(setq new-t1 (nconc new-t1 (list (car p))))
(or (Math-zerop (car p))
(setq okay nil)))
(setq p (cdr p)
n (1+ n)))
(if okay
(setq math-t3 (cons scale (cdr math-t3))
math-t1 new-t1))))
(setq scale (1- scale)))
(setq math-t3 (list (math-mul (car math-t3) math-t2)
(math-mul count math-t2)))
(<= (1- (length math-t1)) max-degree)))))
(defun calcFunc-poly (expr var &optional degree)
(if degree
(or (natnump degree) (math-reject-arg degree 'fixnatnump))
(setq degree 50))
(let ((p (math-is-polynomial expr var degree 'gen)))
(if p
(if (equal p '(0))
(list 'vec)
(cons 'vec p))
(math-reject-arg expr "Expected a polynomial"))))
(defun calcFunc-gpoly (expr var &optional degree)
(if degree
(or (natnump degree) (math-reject-arg degree 'fixnatnump))
(setq degree 50))
(let* ((math-poly-base-variable var)
(d (math-decompose-poly expr var degree nil)))
(if d
(cons 'vec d)
(math-reject-arg expr "Expected a polynomial"))))
(defun math-decompose-poly (math-solve-lhs math-solve-var degree sub-rhs)
(let ((math-solve-rhs (or sub-rhs 1))
math-t1 math-t2 math-t3)
(setq math-t2 (math-polynomial-base
math-solve-lhs
(function
(lambda (math-solve-b)
(let ((math-poly-neg-powers '(1))
(math-poly-mult-powers nil)
(math-poly-frac-powers 1)
(math-poly-exp-base t))
(and (not (equal math-solve-b math-solve-lhs))
(or (not (memq (car-safe math-solve-b) '(+ -))) sub-rhs)
(setq math-t3 '(1 0) math-t2 1
math-t1 (math-is-polynomial math-solve-lhs
math-solve-b 50))
(if (and (equal math-poly-neg-powers '(1))
(memq math-poly-mult-powers '(nil 1))
(eq math-poly-frac-powers 1)
sub-rhs)
(setq math-t1 (cons (math-sub (car math-t1) math-solve-rhs)
(cdr math-t1)))
(math-solve-poly-funny-powers sub-rhs))
(math-solve-crunch-poly degree)
(or (math-expr-contains math-solve-b math-solve-var)
(math-expr-contains (car math-t3) math-solve-var))))))))
(if math-t2
(list (math-pow math-t2 (car math-t3))
(cons 'vec math-t1)
(if sub-rhs
(math-pow math-t2 (nth 1 math-t3))
(math-div (math-pow math-t2 (nth 1 math-t3)) math-solve-rhs))))))
(defun math-solve-linear (var sign b a)
(math-try-solve-for var
(math-div (math-neg b) a)
(math-solve-sign sign a)
t))
(defun math-solve-quadratic (var c b a)
(math-try-solve-for
var
(if (math-looks-evenp b)
(let ((halfb (math-div b 2)))
(math-div
(math-add
(math-neg halfb)
(math-solve-get-sign
(math-normalize
(list 'calcFunc-sqrt
(math-add (math-sqr halfb)
(math-mul (math-neg c) a))))))
a))
(math-div
(math-add
(math-neg b)
(math-solve-get-sign
(math-normalize
(list 'calcFunc-sqrt
(math-add (math-sqr b)
(math-mul 4 (math-mul (math-neg c) a)))))))
(math-mul 2 a)))
nil t))
(defun math-solve-cubic (var d c b a)
(let* ((p (math-div b a))
(q (math-div c a))
(r (math-div d a))
(psqr (math-sqr p))
(aa (math-sub q (math-div psqr 3)))
(bb (math-add r
(math-div (math-sub (math-mul 2 (math-mul psqr p))
(math-mul 9 (math-mul p q)))
27)))
m)
(if (Math-zerop aa)
(math-try-solve-for (math-pow (math-add var (math-div p 3)) 3)
(math-neg bb) nil t)
(if (Math-zerop bb)
(math-try-solve-for
(math-mul (math-add var (math-div p 3))
(math-add (math-sqr (math-add var (math-div p 3)))
aa))
0 nil t)
(setq m (math-mul 2 (list 'calcFunc-sqrt (math-div aa -3))))
(math-try-solve-for
var
(math-sub
(math-normalize
(math-mul
m
(list 'calcFunc-cos
(math-div
(math-sub (list 'calcFunc-arccos
(math-div (math-mul 3 bb)
(math-mul aa m)))
(math-mul 2
(math-mul
(math-add 1 (math-solve-get-int
1 3))
(math-half-circle
calc-symbolic-mode))))
3))))
(math-div p 3))
nil t)))))
(defun math-solve-quartic (var d c b a aa)
(setq a (math-div a aa))
(setq b (math-div b aa))
(setq c (math-div c aa))
(setq d (math-div d aa))
(math-try-solve-for
var
(let* ((asqr (math-sqr a))
(asqr4 (math-div asqr 4))
(y (let ((math-solve-full nil)
calc-next-why)
(math-solve-cubic math-solve-var
(math-sub (math-sub
(math-mul 4 (math-mul b d))
(math-mul asqr d))
(math-sqr c))
(math-sub (math-mul a c)
(math-mul 4 d))
(math-neg b)
1)))
(rsqr (math-add (math-sub asqr4 b) y))
(r (list 'calcFunc-sqrt rsqr))
(sign1 (math-solve-get-sign 1))
(de (list 'calcFunc-sqrt
(math-add
(math-sub (math-mul 3 asqr4)
(math-mul 2 b))
(if (Math-zerop rsqr)
(math-mul
2
(math-mul sign1
(list 'calcFunc-sqrt
(math-sub (math-sqr y)
(math-mul 4 d)))))
(math-sub
(math-mul sign1
(math-div
(math-sub (math-sub
(math-mul 4 (math-mul a b))
(math-mul 8 c))
(math-mul asqr a))
(math-mul 4 r)))
rsqr))))))
(math-normalize
(math-sub (math-add (math-mul sign1 (math-div r 2))
(math-solve-get-sign (math-div de 2)))
(math-div a 4))))
nil t))
(defvar math-symbolic-solve nil)
(defvar math-int-coefs nil)
;; The variable math-int-threshold is local to math-poly-all-roots,
;; but is used by math-poly-newton-root.
(defvar math-int-threshold)
;; The variables math-int-scale, math-int-factors and math-double-roots
;; are local to math-poly-all-roots, but are used by math-poly-integer-root.
(defvar math-int-scale)
(defvar math-int-factors)
(defvar math-double-roots)
(defun math-poly-all-roots (var p &optional math-factoring)
(catch 'ouch
(let* ((math-symbolic-solve calc-symbolic-mode)
(roots nil)
(deg (1- (length p)))
(orig-p (reverse p))
(math-int-coefs nil)
(math-int-scale nil)
(math-double-roots nil)
(math-int-factors nil)
(math-int-threshold nil)
(pp p))
;; If rational coefficients, look for exact rational factors.
(while (and pp (Math-ratp (car pp)))
(setq pp (cdr pp)))
(if pp
(if (or math-factoring math-symbolic-solve)
(throw 'ouch nil))
(let ((lead (car orig-p))
(calc-prefer-frac t)
(scale (apply 'math-lcm-denoms p)))
(setq math-int-scale (math-abs (math-mul scale lead))
math-int-threshold (math-div '(float 5 -2) math-int-scale)
math-int-coefs (cdr (math-div (cons 'vec orig-p) lead)))))
(if (> deg 4)
(let ((calc-prefer-frac nil)
(calc-symbolic-mode nil)
(pp p)
(def-p (copy-sequence orig-p)))
(while pp
(if (Math-numberp (car pp))
(setq pp (cdr pp))
(throw 'ouch nil)))
(while (> deg (if math-symbolic-solve 2 4))
(let* ((x (math-poly-any-root def-p '(float 0 0) nil))
b c pp)
(if (and (eq (car-safe x) 'cplx)
(math-nearly-zerop (nth 2 x) (nth 1 x)))
(setq x (calcFunc-re x)))
(or math-factoring
(setq roots (cons x roots)))
(or (math-numberp x)
(setq x (math-evaluate-expr x)))
(setq pp def-p
b (car def-p))
(while (setq pp (cdr pp))
(setq c (car pp))
(setcar pp b)
(setq b (math-add (math-mul x b) c)))
(setq def-p (cdr def-p)
deg (1- deg))))
(setq p (reverse def-p))))
(if (> deg 1)
(let ((math-solve-var '(var DUMMY var-DUMMY))
(math-solve-sign nil)
(math-solve-ranges nil)
(math-solve-full 'all))
(if (= (length p) (length math-int-coefs))
(setq p (reverse math-int-coefs)))
(setq roots (append (cdr (apply (cond ((= deg 2)
'math-solve-quadratic)
((= deg 3)
'math-solve-cubic)
(t
'math-solve-quartic))
math-solve-var p))
roots)))
(if (> deg 0)
(setq roots (cons (math-div (math-neg (car p)) (nth 1 p))
roots))))
(if math-factoring
(progn
(while roots
(math-poly-integer-root (car roots))
(setq roots (cdr roots)))
(list math-int-factors (nreverse math-int-coefs) math-int-scale))
(let ((vec nil) res)
(while roots
(let ((root (car roots))
(math-solve-full (and math-solve-full 'all)))
(if (math-floatp root)
(setq root (math-poly-any-root orig-p root t)))
(setq vec (append vec
(cdr (or (math-try-solve-for var root nil t)
(throw 'ouch nil))))))
(setq roots (cdr roots)))
(setq vec (cons 'vec (nreverse vec)))
(if math-symbolic-solve
(setq vec (math-normalize vec)))
(if (eq math-solve-full t)
(list 'calcFunc-subscr
vec
(math-solve-get-int 1 (1- (length orig-p)) 1))
vec))))))
(defun math-lcm-denoms (&rest fracs)
(let ((den 1))
(while fracs
(if (eq (car-safe (car fracs)) 'frac)
(setq den (calcFunc-lcm den (nth 2 (car fracs)))))
(setq fracs (cdr fracs)))
den))
(defun math-poly-any-root (p x polish) ; p is a reverse poly coeff list
(let* ((newt (if (math-zerop x)
(math-poly-newton-root
p '(cplx (float 123 -6) (float 1 -4)) 4)
(math-poly-newton-root p x 4)))
(res (if (math-zerop (cdr newt))
(car newt)
(if (and (math-lessp (cdr newt) '(float 1 -3)) (not polish))
(setq newt (math-poly-newton-root p (car newt) 30)))
(if (math-zerop (cdr newt))
(car newt)
(math-poly-laguerre-root p x polish)))))
(and math-symbolic-solve (math-floatp res)
(throw 'ouch nil))
res))
(defun math-poly-newton-root (p x iters)
(let* ((calc-prefer-frac nil)
(calc-symbolic-mode nil)
(try-integer math-int-coefs)
(dx x) b d)
(while (and (> (setq iters (1- iters)) 0)
(let ((pp p))
(math-working "newton" x)
(setq b (car p)
d 0)
(while (setq pp (cdr pp))
(setq d (math-add (math-mul x d) b)
b (math-add (math-mul x b) (car pp))))
(not (math-zerop d)))
(progn
(setq dx (math-div b d)
x (math-sub x dx))
(if try-integer
(let ((adx (math-abs-approx dx)))
(and (math-lessp adx math-int-threshold)
(let ((iroot (math-poly-integer-root x)))
(if iroot
(setq x iroot dx 0)
(setq try-integer nil))))))
(or (not (or (eq dx 0)
(math-nearly-zerop dx (math-abs-approx x))))
(progn (setq dx 0) nil)))))
(cons x (if (math-zerop x)
1 (math-div (math-abs-approx dx) (math-abs-approx x))))))
(defun math-poly-integer-root (x)
(and (math-lessp (calcFunc-xpon (math-abs-approx x)) calc-internal-prec)
math-int-coefs
(let* ((calc-prefer-frac t)
(xre (calcFunc-re x))
(xim (calcFunc-im x))
(xresq (math-sqr xre))
(ximsq (math-sqr xim)))
(if (math-lessp ximsq (calcFunc-scf xresq -1))
;; Look for linear factor
(let* ((rnd (math-div (math-round (math-mul xre math-int-scale))
math-int-scale))
(icp math-int-coefs)
(rem (car icp))
(newcoef nil))
(while (setq icp (cdr icp))
(setq newcoef (cons rem newcoef)
rem (math-add (car icp)
(math-mul rem rnd))))
(and (math-zerop rem)
(progn
(setq math-int-coefs (nreverse newcoef)
math-int-factors (cons (list (math-neg rnd))
math-int-factors))
rnd)))
;; Look for irreducible quadratic factor
(let* ((rnd1 (math-div (math-round
(math-mul xre (math-mul -2 math-int-scale)))
math-int-scale))
(sqscale (math-sqr math-int-scale))
(rnd0 (math-div (math-round (math-mul (math-add xresq ximsq)
sqscale))
sqscale))
(rem1 (car math-int-coefs))
(icp (cdr math-int-coefs))
(rem0 (car icp))
(newcoef nil)
(found (assoc (list rnd0 rnd1 (math-posp xim))
math-double-roots))
this)
(if found
(setq math-double-roots (delq found math-double-roots)
rem0 0 rem1 0)
(while (setq icp (cdr icp))
(setq this rem1
newcoef (cons rem1 newcoef)
rem1 (math-sub rem0 (math-mul this rnd1))
rem0 (math-sub (car icp) (math-mul this rnd0)))))
(and (math-zerop rem0)
(math-zerop rem1)
(let ((aa (math-div rnd1 -2)))
(or found (setq math-int-coefs (reverse newcoef)
math-double-roots (cons (list
(list
rnd0 rnd1
(math-negp xim)))
math-double-roots)
math-int-factors (cons (cons rnd0 rnd1)
math-int-factors)))
(math-add aa
(let ((calc-symbolic-mode math-symbolic-solve))
(math-mul (math-sqrt (math-sub (math-sqr aa)
rnd0))
(if (math-negp xim) -1 1)))))))))))
;;; The following routine is from Numerical Recipes, section 9.5.
(defun math-poly-laguerre-root (p x polish)
(let* ((calc-prefer-frac nil)
(calc-symbolic-mode nil)
(iters 0)
(m (1- (length p)))
(try-newt (not polish))
(tried-newt nil)
b d f x1 dx dxold)
(while
(and (or (< (setq iters (1+ iters)) 50)
(math-reject-arg x "*Laguerre's method failed to converge"))
(let ((err (math-abs-approx (car p)))
(abx (math-abs-approx x))
(pp p))
(setq b (car p)
d 0 f 0)
(while (setq pp (cdr pp))
(setq f (math-add (math-mul x f) d)
d (math-add (math-mul x d) b)
b (math-add (math-mul x b) (car pp))
err (math-add (math-abs-approx b) (math-mul abx err))))
(math-lessp (calcFunc-scf err (- -2 calc-internal-prec))
(math-abs-approx b)))
(or (not (math-zerop d))
(not (math-zerop f))
(progn
(setq x (math-pow (math-neg b) (list 'frac 1 m)))
nil))
(let* ((g (math-div d b))
(g2 (math-sqr g))
(h (math-sub g2 (math-mul 2 (math-div f b))))
(sq (math-sqrt
(math-mul (1- m) (math-sub (math-mul m h) g2))))
(gp (math-add g sq))
(gm (math-sub g sq)))
(if (math-lessp (calcFunc-abssqr gp) (calcFunc-abssqr gm))
(setq gp gm))
(setq dx (math-div m gp)
x1 (math-sub x dx))
(if (and try-newt
(math-lessp (math-abs-approx dx)
(calcFunc-scf (math-abs-approx x) -3)))
(let ((newt (math-poly-newton-root p x1 7)))
(setq tried-newt t
try-newt nil)
(if (math-zerop (cdr newt))
(setq x (car newt) x1 x)
(if (math-lessp (cdr newt) '(float 1 -6))
(let ((newt2 (math-poly-newton-root
p (car newt) 20)))
(if (math-zerop (cdr newt2))
(setq x (car newt2) x1 x)
(setq x (car newt))))))))
(not (or (eq x x1)
(math-nearly-equal x x1))))
(let ((cdx (math-abs-approx dx)))
(setq x x1
tried-newt nil)
(prog1
(or (<= iters 6)
(math-lessp cdx dxold)
(progn
(if polish
(let ((digs (calcFunc-xpon
(math-div (math-abs-approx x) cdx))))
(calc-record-why
"*Could not attain full precision")
(if (natnump digs)
(let ((calc-internal-prec (max 3 digs)))
(setq x (math-normalize x))))))
nil))
(setq dxold cdx)))
(or polish
(math-lessp (calcFunc-scf (math-abs-approx x)
(- calc-internal-prec))
dxold))))
(or (and (math-floatp x)
(math-poly-integer-root x))
x)))
(defun math-solve-above-dummy (x)
(and (not (Math-primp x))
(if (and (equal (nth 1 x) '(var SOLVEDUM SOLVEDUM))
(= (length x) 2))
x
(let ((res nil))
(while (and (setq x (cdr x))
(not (setq res (math-solve-above-dummy (car x))))))
res))))
(defun math-solve-find-root-term (x neg) ; sets "t2", "t3"
(if (math-solve-find-root-in-prod x)
(setq math-t3 neg
math-t1 x)
(and (memq (car-safe x) '(+ -))
(or (math-solve-find-root-term (nth 1 x) neg)
(math-solve-find-root-term (nth 2 x)
(if (eq (car x) '-) (not neg) neg))))))
(defun math-solve-find-root-in-prod (x)
(and (consp x)
(math-expr-contains x math-solve-var)
(or (and (eq (car x) 'calcFunc-sqrt)
(setq math-t2 2))
(and (eq (car x) '^)
(or (and (memq (math-quarter-integer (nth 2 x)) '(1 2 3))
(setq math-t2 2))
(and (eq (car-safe (nth 2 x)) 'frac)
(eq (nth 2 (nth 2 x)) 3)
(setq math-t2 3))))
(and (memq (car x) '(* /))
(or (and (not (math-expr-contains (nth 1 x) math-solve-var))
(math-solve-find-root-in-prod (nth 2 x)))
(and (not (math-expr-contains (nth 2 x) math-solve-var))
(math-solve-find-root-in-prod (nth 1 x))))))))
;; The variable math-solve-vars is local to math-solve-system,
;; but is used by math-solve-system-rec.
(defvar math-solve-vars)
;; The variable math-solve-simplifying is local to math-solve-system
;; and math-solve-system-rec, but is used by math-solve-system-subst.
(defvar math-solve-simplifying)
(defun math-solve-system (exprs math-solve-vars math-solve-full)
(setq exprs (mapcar 'list (if (Math-vectorp exprs)
(cdr exprs)
(list exprs)))
math-solve-vars (if (Math-vectorp math-solve-vars)
(cdr math-solve-vars)
(list math-solve-vars)))
(or (let ((math-solve-simplifying nil))
(math-solve-system-rec exprs math-solve-vars nil))
(let ((math-solve-simplifying t))
(math-solve-system-rec exprs math-solve-vars nil))))
;;; The following backtracking solver works by choosing a variable
;;; and equation, and trying to solve the equation for the variable.
;;; If it succeeds it calls itself recursively with that variable and
;;; equation removed from their respective lists, and with the solution
;;; added to solns as well as being substituted into all existing
;;; equations. The algorithm terminates when any solution path
;;; manages to remove all the variables from var-list.
;;; To support calcFunc-roots, entries in eqn-list and solns are
;;; actually lists of equations.
;; The variables math-solve-system-res and math-solve-system-vv are
;; local to math-solve-system-rec, but are used by math-solve-system-subst.
(defvar math-solve-system-vv)
(defvar math-solve-system-res)
(defun math-solve-system-rec (eqn-list var-list solns)
(if var-list
(let ((v var-list)
(math-solve-system-res nil))
;; Try each variable in turn.
(while
(and
v
(let* ((math-solve-system-vv (car v))
(e eqn-list)
(elim (eq (car-safe math-solve-system-vv) 'calcFunc-elim)))
(if elim
(setq math-solve-system-vv (nth 1 math-solve-system-vv)))
;; Try each equation in turn.
(while
(and
e
(let ((e2 (car e))
(eprev nil)
res2)
(setq math-solve-system-res nil)
;; Try to solve for math-solve-system-vv the list of equations e2.
(while (and e2
(setq res2 (or (and (eq (car e2) eprev)
res2)
(math-solve-for (car e2) 0
math-solve-system-vv
math-solve-full))))
(setq eprev (car e2)
math-solve-system-res (cons (if (eq math-solve-full 'all)
(cdr res2)
(list res2))
math-solve-system-res)
e2 (cdr e2)))
(if e2
(setq math-solve-system-res nil)
;; Found a solution. Now try other variables.
(setq math-solve-system-res (nreverse math-solve-system-res)
math-solve-system-res (math-solve-system-rec
(mapcar
'math-solve-system-subst
(delq (car e)
(copy-sequence eqn-list)))
(delq (car v) (copy-sequence var-list))
(let ((math-solve-simplifying nil)
(s (mapcar
(function
(lambda (x)
(cons
(car x)
(math-solve-system-subst
(cdr x)))))
solns)))
(if elim
s
(cons (cons
math-solve-system-vv
(apply 'append math-solve-system-res))
s)))))
(not math-solve-system-res))))
(setq e (cdr e)))
(not math-solve-system-res)))
(setq v (cdr v)))
math-solve-system-res)
;; Eliminated all variables, so now put solution into the proper format.
(setq solns (sort solns
(function
(lambda (x y)
(not (memq (car x) (memq (car y) math-solve-vars)))))))
(if (eq math-solve-full 'all)
(math-transpose
(math-normalize
(cons 'vec
(if solns
(mapcar (function (lambda (x) (cons 'vec (cdr x)))) solns)
(mapcar (function (lambda (x) (cons 'vec x))) eqn-list)))))
(math-normalize
(cons 'vec
(if solns
(mapcar (function (lambda (x) (cons 'calcFunc-eq x))) solns)
(mapcar 'car eqn-list)))))))
(defun math-solve-system-subst (x) ; uses "res" and "v"
(let ((accum nil)
(res2 math-solve-system-res))
(while x
(setq accum (nconc accum
(mapcar (function
(lambda (r)
(if math-solve-simplifying
(math-simplify
(math-expr-subst
(car x) math-solve-system-vv r))
(math-expr-subst
(car x) math-solve-system-vv r))))
(car res2)))
x (cdr x)
res2 (cdr res2)))
accum))
;; calc-command-flags is declared in calc.el
(defvar calc-command-flags)
(defun math-get-from-counter (name)
(let ((ctr (assq name calc-command-flags)))
(if ctr
(setcdr ctr (1+ (cdr ctr)))
(setq ctr (cons name 1)
calc-command-flags (cons ctr calc-command-flags)))
(cdr ctr)))
(defvar var-GenCount)
(defun math-solve-get-sign (val)
(setq val (math-simplify val))
(if (and (eq (car-safe val) '*)
(Math-numberp (nth 1 val)))
(list '* (nth 1 val) (math-solve-get-sign (nth 2 val)))
(and (eq (car-safe val) 'calcFunc-sqrt)
(eq (car-safe (nth 1 val)) '^)
(setq val (math-normalize (list '^
(nth 1 (nth 1 val))
(math-div (nth 2 (nth 1 val)) 2)))))
(if math-solve-full
(if (and (calc-var-value 'var-GenCount)
(Math-natnump var-GenCount)
(not (eq math-solve-full 'all)))
(prog1
(math-mul (list 'calcFunc-as var-GenCount) val)
(setq var-GenCount (math-add var-GenCount 1))
(calc-refresh-evaltos 'var-GenCount))
(let* ((var (concat "s" (int-to-string (math-get-from-counter 'solve-sign))))
(var2 (list 'var (intern var) (intern (concat "var-" var)))))
(if (eq math-solve-full 'all)
(setq math-solve-ranges (cons (list var2 1 -1)
math-solve-ranges)))
(math-mul var2 val)))
(calc-record-why "*Choosing positive solution")
val)))
(defun math-solve-get-int (val &optional range first)
(if math-solve-full
(if (and (calc-var-value 'var-GenCount)
(Math-natnump var-GenCount)
(not (eq math-solve-full 'all)))
(prog1
(math-mul val (list 'calcFunc-an var-GenCount))
(setq var-GenCount (math-add var-GenCount 1))
(calc-refresh-evaltos 'var-GenCount))
(let* ((var (concat "n" (int-to-string
(math-get-from-counter 'solve-int))))
(var2 (list 'var (intern var) (intern (concat "var-" var)))))
(if (and range (eq math-solve-full 'all))
(setq math-solve-ranges (cons (cons var2
(cdr (calcFunc-index
range (or first 0))))
math-solve-ranges)))
(math-mul val var2)))
(calc-record-why "*Choosing 0 for arbitrary integer in solution")
0))
(defun math-solve-sign (sign expr)
(and sign
(let ((s1 (math-possible-signs expr)))
(cond ((memq s1 '(4 6))
sign)
((memq s1 '(1 3))
(- sign))))))
(defun math-looks-evenp (expr)
(if (Math-integerp expr)
(math-evenp expr)
(if (memq (car expr) '(* /))
(math-looks-evenp (nth 1 expr)))))
(defun math-solve-for (lhs rhs math-solve-var math-solve-full &optional sign)
(if (math-expr-contains rhs math-solve-var)
(math-solve-for (math-sub lhs rhs) 0 math-solve-var math-solve-full)
(and (math-expr-contains lhs math-solve-var)
(math-with-extra-prec 1
(let* ((math-poly-base-variable math-solve-var)
(res (math-try-solve-for lhs rhs sign)))
(if (and (eq math-solve-full 'all)
(math-known-realp math-solve-var))
(let ((old-len (length res))
new-len)
(setq res (delq nil
(mapcar (function
(lambda (x)
(and (not (memq (car-safe x)
'(cplx polar)))
x)))
res))
new-len (length res))
(if (< new-len old-len)
(calc-record-why (if (= new-len 1)
"*All solutions were complex"
(format
"*Omitted %d complex solutions"
(- old-len new-len)))))))
res)))))
(defun math-solve-eqn (expr var full)
(if (memq (car-safe expr) '(calcFunc-neq calcFunc-lt calcFunc-gt
calcFunc-leq calcFunc-geq))
(let ((res (math-solve-for (cons '- (cdr expr))
0 var full
(if (eq (car expr) 'calcFunc-neq) nil 1))))
(and res
(if (eq math-solve-sign 1)
(list (car expr) var res)
(if (eq math-solve-sign -1)
(list (car expr) res var)
(or (eq (car expr) 'calcFunc-neq)
(calc-record-why
"*Can't determine direction of inequality"))
(and (memq (car expr) '(calcFunc-neq calcFunc-lt calcFunc-gt))
(list 'calcFunc-neq var res))))))
(let ((res (math-solve-for expr 0 var full)))
(and res
(list 'calcFunc-eq var res)))))
(defun math-reject-solution (expr var func)
(if (math-expr-contains expr var)
(or (equal (car calc-next-why) '(* "Unable to find a symbolic solution"))
(calc-record-why "*Unable to find a solution")))
(list func expr var))
(defun calcFunc-solve (expr var)
(or (if (or (Math-vectorp expr) (Math-vectorp var))
(math-solve-system expr var nil)
(math-solve-eqn expr var nil))
(math-reject-solution expr var 'calcFunc-solve)))
(defun calcFunc-fsolve (expr var)
(or (if (or (Math-vectorp expr) (Math-vectorp var))
(math-solve-system expr var t)
(math-solve-eqn expr var t))
(math-reject-solution expr var 'calcFunc-fsolve)))
(defun calcFunc-roots (expr var)
(let ((math-solve-ranges nil))
(or (if (or (Math-vectorp expr) (Math-vectorp var))
(math-solve-system expr var 'all)
(math-solve-for expr 0 var 'all))
(math-reject-solution expr var 'calcFunc-roots))))
(defun calcFunc-finv (expr var)
(let ((res (math-solve-for expr math-integ-var var nil)))
(if res
(math-normalize (math-expr-subst res math-integ-var var))
(math-reject-solution expr var 'calcFunc-finv))))
(defun calcFunc-ffinv (expr var)
(let ((res (math-solve-for expr math-integ-var var t)))
(if res
(math-normalize (math-expr-subst res math-integ-var var))
(math-reject-solution expr var 'calcFunc-finv))))
(put 'calcFunc-inv 'math-inverse
(function (lambda (x) (math-div 1 x))))
(put 'calcFunc-inv 'math-inverse-sign -1)
(put 'calcFunc-sqrt 'math-inverse
(function (lambda (x) (math-sqr x))))
(put 'calcFunc-conj 'math-inverse
(function (lambda (x) (list 'calcFunc-conj x))))
(put 'calcFunc-abs 'math-inverse
(function (lambda (x) (math-solve-get-sign x))))
(put 'calcFunc-deg 'math-inverse
(function (lambda (x) (list 'calcFunc-rad x))))
(put 'calcFunc-deg 'math-inverse-sign 1)
(put 'calcFunc-rad 'math-inverse
(function (lambda (x) (list 'calcFunc-deg x))))
(put 'calcFunc-rad 'math-inverse-sign 1)
(put 'calcFunc-ln 'math-inverse
(function (lambda (x) (list 'calcFunc-exp x))))
(put 'calcFunc-ln 'math-inverse-sign 1)
(put 'calcFunc-log10 'math-inverse
(function (lambda (x) (list 'calcFunc-exp10 x))))
(put 'calcFunc-log10 'math-inverse-sign 1)
(put 'calcFunc-lnp1 'math-inverse
(function (lambda (x) (list 'calcFunc-expm1 x))))
(put 'calcFunc-lnp1 'math-inverse-sign 1)
(put 'calcFunc-exp 'math-inverse
(function (lambda (x) (math-add (math-normalize (list 'calcFunc-ln x))
(math-mul 2
(math-mul '(var pi var-pi)
(math-solve-get-int
'(var i var-i))))))))
(put 'calcFunc-exp 'math-inverse-sign 1)
(put 'calcFunc-expm1 'math-inverse
(function (lambda (x) (math-add (math-normalize (list 'calcFunc-lnp1 x))
(math-mul 2
(math-mul '(var pi var-pi)
(math-solve-get-int
'(var i var-i))))))))
(put 'calcFunc-expm1 'math-inverse-sign 1)
(put 'calcFunc-sin 'math-inverse
(function (lambda (x) (let ((n (math-solve-get-int 1)))
(math-add (math-mul (math-normalize
(list 'calcFunc-arcsin x))
(math-pow -1 n))
(math-mul (math-half-circle t)
n))))))
(put 'calcFunc-cos 'math-inverse
(function (lambda (x) (math-add (math-solve-get-sign
(math-normalize
(list 'calcFunc-arccos x)))
(math-solve-get-int
(math-full-circle t))))))
(put 'calcFunc-tan 'math-inverse
(function (lambda (x) (math-add (math-normalize (list 'calcFunc-arctan x))
(math-solve-get-int
(math-half-circle t))))))
(put 'calcFunc-arcsin 'math-inverse
(function (lambda (x) (math-normalize (list 'calcFunc-sin x)))))
(put 'calcFunc-arccos 'math-inverse
(function (lambda (x) (math-normalize (list 'calcFunc-cos x)))))
(put 'calcFunc-arctan 'math-inverse
(function (lambda (x) (math-normalize (list 'calcFunc-tan x)))))
(put 'calcFunc-sinh 'math-inverse
(function (lambda (x) (let ((n (math-solve-get-int 1)))
(math-add (math-mul (math-normalize
(list 'calcFunc-arcsinh x))
(math-pow -1 n))
(math-mul (math-half-circle t)
(math-mul
'(var i var-i)
n)))))))
(put 'calcFunc-sinh 'math-inverse-sign 1)
(put 'calcFunc-cosh 'math-inverse
(function (lambda (x) (math-add (math-solve-get-sign
(math-normalize
(list 'calcFunc-arccosh x)))
(math-mul (math-full-circle t)
(math-solve-get-int
'(var i var-i)))))))
(put 'calcFunc-tanh 'math-inverse
(function (lambda (x) (math-add (math-normalize
(list 'calcFunc-arctanh x))
(math-mul (math-half-circle t)
(math-solve-get-int
'(var i var-i)))))))
(put 'calcFunc-tanh 'math-inverse-sign 1)
(put 'calcFunc-arcsinh 'math-inverse
(function (lambda (x) (math-normalize (list 'calcFunc-sinh x)))))
(put 'calcFunc-arcsinh 'math-inverse-sign 1)
(put 'calcFunc-arccosh 'math-inverse
(function (lambda (x) (math-normalize (list 'calcFunc-cosh x)))))
(put 'calcFunc-arctanh 'math-inverse
(function (lambda (x) (math-normalize (list 'calcFunc-tanh x)))))
(put 'calcFunc-arctanh 'math-inverse-sign 1)
(defun calcFunc-taylor (expr var num)
(let ((x0 0) (v var))
(if (memq (car-safe var) '(+ - calcFunc-eq))
(setq x0 (if (eq (car var) '+) (math-neg (nth 2 var)) (nth 2 var))
v (nth 1 var)))
(or (and (eq (car-safe v) 'var)
(math-expr-contains expr v)
(natnump num)
(let ((accum (math-expr-subst expr v x0))
(var2 (if (eq (car var) 'calcFunc-eq)
(cons '- (cdr var))
var))
(n 0)
(nfac 1)
(fprime expr))
(while (and (<= (setq n (1+ n)) num)
(setq fprime (calcFunc-deriv fprime v nil t)))
(setq fprime (math-simplify fprime)
nfac (math-mul nfac n)
accum (math-add accum
(math-div (math-mul (math-pow var2 n)
(math-expr-subst
fprime v x0))
nfac))))
(and fprime
(math-normalize accum))))
(list 'calcFunc-taylor expr var num))))
(provide 'calcalg2)
;;; calcalg2.el ends here
|