Talk:Primorial numbers: Difference between revisions
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say "First ten primorials: ", join ", ", map { pn_primorial($_) } 0..9;</lang> and all is done ;-)
[[user#Horsth|Horsth]]
: I just added a version using the new vecprod command. It is quite a bit faster. It uses a product tree for efficient large vector products -- the old comment was misleading about using a factor tree. [[User:Danaj|Danaj]] ([[User talk:Danaj|talk]]) 16:50, 9 October 2018 (UTC)
::Once I figured out what a "product tree" is, simply multiplying in pairs (technically adjacent, not that my Phix entry does that) I was positively ''shocked'' to see a better than 50-fold performance improvement (6 mins => 6 secs). --[[User:Petelomax|Pete Lomax]] ([[User talk:Petelomax|talk]]) 04:28, 22 March 2020 (UTC)
== a faster-than-exponential growth rate in CPU time stated by dinosaur ==
===Story===
school - multiplication ist quadratic in runtime O(count of digits) ~ (count of digits)^2
<pre>Prime(100000) = 1299709, Primorial(100000) ~ 0.1909604E+563921 121.547 seconds.
Prime(1000000) = 15485863, Primorial(1000000) ~ 0.1147175E+6722809 17756.797 seconds.</pre>
10-times Numbers -> ~100-times runtime 121*100-> 12100
the digitcount of the numbers one multiplies grew be a factor of lg10(15.45Mio)/lg10(1.299Mio) =1,17
so you get 10-> (1.17*10) ^ 2 = 136,89*121,547= 16638 thats not far from 17756
Now take into account that the number gets so big, that it doesn't fit in level3 Cache ,which slow things down.Only 3% will be enough
121,547*(1.17*1.03*10)^2 s = 17651 s [[user#Horsth|Horsth]]
The implementation of primorial I am using, running on this laptop, took 0.002909 seconds for primorial 100000 and 0.049093 seconds for primorial 1000000. With digit counts, that became 0.004267 seconds and 0.04369 seconds. I guess that's faster than exponential growth rate if by fast you mean the execution time... but ... actually, I'm not sure where I'm going with this. I am mildly surprised that digit count could be faster than the raw primorial but that might just be random variation --[[User:Rdm|Rdm]] ([[User talk:Rdm|talk]]) 20:41, 8 April 2016 (UTC) Edit: System Info claims I'm working with an i7: 2.8GHz clock, 256KB L2, 6MB L3 - no mention of L1. --[[User:Rdm|Rdm]] ([[User talk:Rdm|talk]]) 09:26, 9 April 2016 (UTC)
:According to System Info, the computer has 64KB of L1 I-cache, 16KB L1 D-cache, and 2048KB of L2 cache, with further jargon about 2 way set associative, 64 byte line size, etc. and no mention of L3. Later, I see that the GIMP calculation gives 8MB for L3 on this AMD k10. However, I have no idea how much of each resource is devoted to the various activities such as running the GIMP crunch and windows itself. This is why experimental timings are tricky to perform as well as being a load on the patience. I am however surprised by the speeds quoted by some contributors, given that towards the end the number has millions of digits.
:As for the standard multiplication procedure of n digits by m digits consuming time proportional to n.m, remember that my version cheats by using a one digit multiplicand (err, the second number in a*b) even though it is much larger than the base of arithmetic. Computer crunching of integers is unaffected by whether x is multiplied by 12 or 1234567, except for overflows. Thus, its running time should be proportional to n, the number of digits in the big number. In principle, and ignoring caches. If successive primorials were attained by multiplying by a constant, then that number of digits would increase linearly, but of course the multiplicand is increasing in size itself. If that increase was the same as with factorials then the number of digits would increase according to log(x!) which is proportional to x*log(x) via Stirling's approximation, which is to say a linear growth (the x) multiplied by a factor that grows ever more slowly (the log(x))
:But Primorial(x) grows even faster than x! However, I didn't look and think long about the cpu timings before making the remark about "faster than exponential". I shall twiddle the prog. to produce more reports on timing... [[User:Dinosaur|Dinosaur]] ([[User talk:Dinosaur|talk]]) 08:51, 9 April 2016 (UTC)
:: On reading http://rosettacode.org/mw/index.php?title=Primorial_numbers&type=revision&diff=225184&oldid=225110 I think I'm beginning to understand your point of view. I guess the issue is "in proportion to what?" For example, if we consider resource consumption (quite a lot) in proportion to the number of arguments (always just one), resource growth is infinite. If we think of resource growth in proportion to the number of digits in the argument we get "resource consumption (time to complete) growth faster than exponential". If we think of resource growth in proportion to the numerical value of the argument time consumed growth grows faster than linear.
:: Anyways, I guess a general issue here is that we tend to think in "shorthand" and when relating those thought to other people it often takes some extra work to make sure they understand our thinking. --[[User:Rdm|Rdm]] ([[User talk:Rdm|talk]]) 01:58, 10 April 2016 (UTC)
:Our edits clashed! In the article I have rephrased the usage of "exponential" to note that it is the task that is growing exponentially (n = 10^i for i = 1, 2, 3, ...) so that even for a linear exercise, the work would grow exponentially. But here there is a further growth factor beyond the linear, because the number size is growing faster than linear, because n is growing... The supposition being that a multiply is linear in (number of digits) given that the multiplicand is one (embolismic) digit only. Except, we know that sooner or later, its digit count will grow. The unit of "sizeness" is not always obvious or convenient. To calculate Primorial(n) the task size is obviously determined by the value of n, from which is deduced log(n) for the number of digits both of n itself and Primorial(n) and one argues about the time for a multiply with those sizes of digit strings, and then perhaps the integration of those times so as to reach n from 1. All of which depends on n. But, since digit string length is so important, one might choose to talk about log(n) as being the unit of size. In the factoring of numbers, for example, I (and Prof. Knuth) speak of n and sqrt(n), but, with ever larger n, it seems now it is fashionable to use log(n) as the unit, and speak of "polynomial time in N" or similar, when N is actually log(n)
:Righto, another overnight run, this time with a report every thousand for n, d, and t where t is the cpu time to attain n after reaching n - 1000. So, from 99000 to attain 100000 the digit count went from 557809 to 563921 and the cpu time was 2·625. For 999000 to 1000000 the digit count went from 6715619 to 6722809 and the cpu time was 33·969. Plotting showed almost linear growth rates ''across this limited span'':
Correlation
0·99990 t = d*5·636E-6 -0·41458
0·99973 d = n*6·6983 -125797·8
0·99943 t = n*3·7744E-5 - 1·1205
:These were for data only up to n = 800,000 because there was a change in slope subsequently, probably because I stopped using the computer for other minor activity and thus didn't disrupt its various buffers. The Mersenne prime calculations continued. The points wobble above and below the fitted line across the span, and there is a bend visible at the start where the overall line is too low. That is, at the start the calculation is going a little faster and presumably, one could talk about not yet overflowing some on-chip memory buffer as another reason for non-linearity. Similarly, at the high end the points start to rise above the fitted line, and one could talk about a growth factor such as log(n) rather than a constant. This is obscured by the greater speed later on when I had walked away from the toiling computer. Nevertheless, one can see that a slight curve is being approximated by a straight line over a limited span, which means that using the start to attempt to identify the effect of a cpu buffer size will be tricky. Even so, a log(n) vs. log(t) plot shows a change of slope at the start - except that is for only the first few points. Alas, with "only" a few thousand digits to the big number, the resolution of the CPU timing function is insufficient to show finer details, and repeating the calculation a hundred times (say) will require auxiliary storage to enable the repetition and this will change the workings of the presumed buffering processes away from what was intended to be investigated. A frustrating business.
:Unfortunately, there is no provision for posting images, and anyway I've had a lot of trouble with Octave producing damaged plot files (.svg, .png, etc.) having to resort to screen image captures, so instead, here are the data:
:: You should be able to post images now. Look for an "Upload File" link at the bottom of the page. Also, you might want to look at where the time is going in your implementation - I am going to guess that it's in your code to generate the prime numbers. --[[User:Rdm|Rdm]] ([[User talk:Rdm|talk]]) 03:00, 10 April 2016 (UTC)
:::Pox! I have just created file Primorial.cpuTime.svg, but, "Error creating thumbnail: /bin/bash: rsvg-convert: command not found" results. Inkscape however draws the plot without difficulty, though at 657KB the file is a bit large for just a thousand points, a line, and some annotation. And now that it is uploaded, I can't see how to delete it. I shall attempt to replace it by a screen capture. As for Octave, it won't "Print" the plot as either a .png nor a .jpg due to some complaint of its own "'C:\Program' is not recognized as an internal or external command, operable program or batch file." so a tar pit there.
:::The calculation of the primes is not a problem, because all million are prepared in array PRIME before any Primorial attempts are made, and the time for this is somewhere in the blink of starting the run and the first Primorial results appearing even as the screen window manifests. [[User:Dinosaur|Dinosaur]] ([[User talk:Dinosaur|talk]]) 06:30, 10 April 2016 (UTC)
::::And I can't replace the .svg file by a .jpg because I can't change the name of the file. So, a new file (only 38KB in this format), and damnit, I forgot to give the name the prefix "Primorial.cpu" [[File:Time.jpg]]
And here is a plot of cpu time against the number of digits in the big number, which is successively being multiplied by a ''single'' digit value. [[File:Primorial.CPUtime.digits.jpg]]
:::::Only admins are allowed to delete files. I've removed "Primorial.cpuTime.svg". --[[User:AndiPersti|Andreas Perstinger]] ([[User talk:AndiPersti|talk]]) 18:36, 10 April 2016 (UTC)
===Grist===
<pre>
1000, 3393, 0.000
2000, 7483, 0.016
3000, 11829, 0.047
4000, 16340, 0.047
5000, 20975, 0.078
6000, 25706, 0.094
7000, 30519, 0.109
8000, 35400, 0.141
9000, 40342, 0.156
10000, 45337, 0.172
11000, 50381, 0.219
12000, 55468, 0.203
13000, 60595, 0.250
14000, 65759, 0.266
15000, 70957, 0.250
16000, 76187, 0.328
17000, 81447, 0.344
18000, 86735, 0.359
19000, 92050, 0.391
20000, 97389, 0.406
21000, 102753, 0.422
22000, 108139, 0.469
23000, 113547, 0.484
24000, 118975, 0.500
25000, 124424, 0.531
26000, 129891, 0.562
27000, 135377, 0.578
28000, 140881, 0.609
29000, 146401, 0.625
30000, 151937, 0.656
31000, 157490, 0.703
32000, 163057, 0.688
33000, 168640, 0.750
34000, 174237, 0.766
35000, 179849, 0.750
36000, 185473, 0.797
37000, 191111, 0.812
38000, 196762, 0.859
39000, 202425, 0.891
40000, 208100, 0.906
41000, 213787, 0.938
42000, 219486, 0.953
43000, 225196, 1.000
44000, 230916, 1.016
45000, 236648, 1.047
46000, 242390, 1.062
47000, 248143, 1.094
48000, 253905, 1.156
49000, 259678, 1.156
50000, 265460, 1.156
51000, 271251, 1.203
52000, 277052, 1.234
53000, 282862, 1.234
54000, 288681, 1.297
55000, 294508, 1.297
56000, 300345, 1.328
57000, 306189, 1.359
58000, 312042, 1.406
59000, 317903, 1.422
60000, 323772, 1.453
61000, 329649, 1.484
62000, 335534, 1.516
63000, 341426, 1.531
64000, 347326, 1.562
65000, 353233, 1.594
66000, 359148, 1.609
67000, 365069, 1.641
68000, 370998, 1.688
69000, 376933, 1.719
70000, 382876, 1.719
71000, 388825, 1.719
72000, 394780, 1.766
73000, 400743, 1.812
74000, 406711, 1.844
75000, 412686, 1.875
76000, 418668, 1.922
77000, 424656, 1.938
78000, 430649, 1.938
79000, 436649, 1.969
80000, 442655, 2.016
81000, 448667, 2.047
82000, 454684, 2.062
83000, 460708, 2.109
84000, 466737, 2.141
85000, 472772, 2.156
86000, 478812, 2.281
87000, 484857, 2.234
88000, 490908, 2.281
89000, 496965, 2.297
90000, 503026, 2.328
91000, 509093, 2.359
92000, 515165, 2.391
93000, 521242, 2.422
94000, 527324, 2.438
95000, 533411, 2.500
96000, 539504, 2.516
97000, 545601, 2.547
98000, 551703, 2.578
99000, 557809, 2.594
100000, 563921, 2.625
101000, 570037, 2.656
102000, 576158, 2.672
103000, 582283, 2.719
104000, 588413, 2.750
105000, 594548, 2.797
106000, 600687, 2.812
107000, 606831, 2.859
108000, 612978, 2.891
109000, 619131, 2.922
110000, 625287, 2.969
111000, 631448, 2.984
112000, 637613, 3.047
113000, 643782, 3.094
114000, 649956, 3.094
115000, 656133, 3.156
116000, 662315, 3.156
117000, 668500, 3.203
118000, 674690, 3.281
119000, 680884, 3.297
120000, 687081, 3.344
121000, 693283, 3.359
122000, 699488, 3.422
123000, 705698, 3.453
124000, 711911, 3.500
125000, 718128, 3.516
126000, 724349, 3.547
127000, 730573, 3.625
128000, 736801, 3.672
129000, 743033, 3.703
130000, 749268, 3.734
131000, 755507, 3.797
132000, 761750, 3.828
133000, 767996, 3.875
134000, 774246, 3.922
135000, 780499, 3.953
136000, 786756, 3.984
137000, 793016, 4.047
138000, 799280, 4.062
139000, 805547, 4.141
140000, 811817, 4.172
141000, 818091, 4.188
142000, 824368, 4.250
143000, 830648, 4.281
144000, 836932, 4.312
145000, 843219, 4.375
146000, 849509, 4.391
147000, 855802, 4.422
148000, 862099, 4.438
149000, 868398, 4.500
150000, 874701, 4.547
151000, 881007, 4.578
152000, 887316, 4.641
153000, 893628, 4.688
154000, 899943, 4.688
155000, 906261, 4.750
156000, 912583, 4.766
157000, 918907, 4.781
158000, 925234, 4.859
159000, 931564, 4.875
160000, 937898, 4.922
161000, 944234, 4.984
162000, 950573, 4.984
163000, 956915, 5.016
164000, 963259, 5.078
165000, 969607, 5.094
166000, 975958, 5.109
167000, 982311, 5.188
168000, 988667, 5.203
169000, 995026, 5.156
170000,1001388, 5.219
171000,1007752, 5.312
172000,1014119, 5.328
173000,1020489, 5.375
174000,1026862, 5.391
175000,1033238, 5.484
176000,1039616, 5.500
177000,1045997, 5.484
178000,1052380, 5.562
179000,1058766, 5.594
180000,1065155, 5.656
181000,1071546, 5.656
182000,1077940, 5.703
183000,1084336, 5.688
184000,1090735, 5.750
185000,1097137, 5.812
186000,1103541, 5.875
187000,1109948, 5.906
188000,1116357, 5.922
189000,1122768, 5.891
190000,1129182, 6.000
191000,1135599, 6.016
192000,1142018, 6.047
193000,1148439, 6.109
194000,1154863, 6.094
195000,1161290, 6.172
196000,1167718, 6.188
197000,1174149, 6.250
198000,1180583, 6.250
199000,1187018, 6.297
200000,1193457, 6.344
201000,1199897, 6.359
202000,1206340, 6.406
203000,1212785, 6.438
204000,1219233, 6.469
205000,1225682, 6.516
206000,1232134, 6.547
207000,1238588, 6.578
208000,1245045, 6.609
209000,1251504, 6.609
210000,1257965, 6.703
211000,1264428, 6.750
212000,1270893, 6.734
213000,1277361, 6.828
214000,1283831, 6.859
215000,1290303, 6.859
216000,1296777, 6.938
217000,1303254, 6.953
218000,1309732, 7.000
219000,1316213, 7.047
220000,1322696, 7.047
221000,1329181, 7.141
222000,1335668, 7.125
223000,1342158, 7.188
224000,1348649, 7.203
225000,1355143, 7.281
226000,1361638, 7.266
227000,1368136, 7.328
228000,1374635, 7.344
229000,1381137, 7.406
230000,1387641, 7.406
231000,1394147, 7.484
232000,1400654, 7.516
233000,1407164, 7.531
234000,1413676, 7.594
235000,1420189, 7.609
236000,1426705, 7.609
237000,1433223, 7.703
238000,1439742, 7.719
239000,1446264, 7.734
240000,1452788, 7.812
241000,1459313, 7.828
242000,1465841, 7.891
243000,1472370, 7.938
244000,1478902, 7.875
245000,1485435, 7.938
246000,1491970, 8.031
247000,1498507, 8.078
248000,1505046, 8.094
249000,1511587, 8.094
250000,1518130, 8.203
251000,1524675, 8.219
252000,1531221, 8.250
253000,1537770, 8.266
254000,1544320, 8.328
255000,1550873, 8.344
256000,1557427, 8.391
257000,1563983, 8.422
258000,1570540, 8.484
259000,1577100, 8.500
260000,1583661, 8.531
261000,1590224, 8.562
262000,1596789, 8.609
263000,1603355, 8.656
264000,1609924, 8.672
265000,1616494, 8.719
266000,1623066, 8.672
267000,1629640, 8.750
268000,1636215, 8.812
269000,1642792, 8.875
270000,1649371, 8.859
271000,1655952, 8.906
272000,1662534, 8.984
273000,1669118, 9.031
274000,1675704, 9.047
275000,1682291, 9.109
276000,1688881, 9.109
277000,1695472, 9.141
278000,1702064, 9.188
279000,1708658, 9.250
280000,1715254, 9.297
281000,1721852, 9.297
282000,1728451, 9.312
283000,1735052, 9.375
284000,1741655, 9.406
285000,1748259, 9.391
286000,1754865, 9.516
287000,1761472, 9.516
288000,1768082, 9.578
289000,1774692, 9.594
290000,1781305, 9.625
291000,1787919, 9.703
292000,1794534, 9.734
293000,1801151, 9.750
294000,1807770, 9.781
295000,1814391, 9.828
296000,1821013, 9.859
297000,1827636, 9.844
298000,1834261, 9.906
299000,1840888, 9.938
300000,1847516, 9.953
301000,1854146, 9.969
302000,1860777,10.047
303000,1867410,10.094
304000,1874045,10.109
305000,1880681,10.203
306000,1887318,10.047
307000,1893958,10.250
308000,1900598,10.281
309000,1907240,10.250
310000,1913884,10.328
311000,1920529,10.422
312000,1927176,10.438
313000,1933824,10.500
314000,1940474,10.547
315000,1947125,10.516
316000,1953777,10.359
317000,1960431,10.547
318000,1967087,10.578
319000,1973744,10.719
320000,1980402,10.688
321000,1987062,10.781
322000,1993724,10.719
323000,2000387,10.828
324000,2007051,10.875
325000,2013717,10.594
326000,2020384,10.859
327000,2027052,11.016
328000,2033722,11.062
329000,2040394,11.031
330000,2047067,11.156
331000,2053741,11.078
332000,2060417,11.125
333000,2067094,11.188
334000,2073772,11.297
335000,2080452,11.297
336000,2087133,11.344
337000,2093816,11.297
338000,2100500,11.359
339000,2107186,11.391
340000,2113873,11.453
341000,2120561,11.516
342000,2127250,11.562
343000,2133941,11.438
344000,2140633,11.625
345000,2147327,11.609
346000,2154022,11.656
347000,2160718,11.688
348000,2167416,11.641
349000,2174115,11.781
350000,2180815,11.859
351000,2187517,11.812
352000,2194220,11.938
353000,2200924,11.953
354000,2207630,11.984
355000,2214337,12.062
356000,2221045,12.078
357000,2227755,12.109
358000,2234466,11.953
359000,2241178,12.172
360000,2247891,12.141
361000,2254606,12.219
362000,2261322,12.344
363000,2268040,12.156
364000,2274758,12.281
365000,2281478,12.375
366000,2288199,12.406
367000,2294922,12.250
368000,2301646,12.484
369000,2308371,12.500
370000,2315097,12.641
371000,2321825,12.484
372000,2328553,12.453
373000,2335283,12.766
374000,2342015,12.734
375000,2348747,12.750
376000,2355481,12.844
377000,2362216,12.812
378000,2368952,12.750
379000,2375690,12.859
380000,2382429,12.875
381000,2389169,12.906
382000,2395910,13.031
383000,2402652,13.031
384000,2409396,13.094
385000,2416141,13.125
386000,2422887,13.250
387000,2429635,13.250
388000,2436383,13.219
389000,2443133,13.312
390000,2449884,13.312
391000,2456636,13.203
392000,2463389,13.469
393000,2470144,13.344
394000,2476900,13.453
395000,2483657,13.547
396000,2490415,13.594
397000,2497174,13.609
398000,2503935,13.641
399000,2510696,13.672
400000,2517459,13.781
401000,2524223,13.766
402000,2530988,13.859
403000,2537755,13.828
404000,2544522,13.812
405000,2551291,13.797
406000,2558061,13.906
407000,2564832,13.812
408000,2571604,14.000
409000,2578377,14.047
410000,2585151,14.016
411000,2591927,14.125
412000,2598703,14.094
413000,2605481,14.125
414000,2612260,14.234
415000,2619040,14.250
416000,2625821,14.266
417000,2632603,14.375
418000,2639387,14.312
419000,2646171,14.375
420000,2652957,14.500
421000,2659743,14.484
422000,2666531,14.438
423000,2673320,14.578
424000,2680110,14.531
425000,2686901,14.578
426000,2693694,14.641
427000,2700487,14.719
428000,2707281,14.781
429000,2714077,14.750
430000,2720873,14.812
431000,2727671,14.844
432000,2734470,14.766
433000,2741270,14.906
434000,2748071,14.969
435000,2754873,14.969
436000,2761676,14.938
437000,2768480,14.984
438000,2775285,15.000
439000,2782091,15.266
440000,2788899,15.219
441000,2795707,15.359
442000,2802516,15.312
443000,2809327,15.375
444000,2816139,15.406
445000,2822951,15.422
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1000000,6722809,33.969
</pre>
:Happy plotting! [[User:Dinosaur|Dinosaur]] ([[User talk:Dinosaur|talk]]) 02:45, 10 April 2016 (UTC)
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