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2037:"Though the BBP formula can directly calculate the value of any given digit of π with less computational effort than formulas that must calculate all intervening digits, BBP remains linearithmic whereby successively larger values of n require increasingly more time to calculate; that is, the "further out" a digit is, the longer it takes BBP to calculate it, just like the standard π-computing algorithms."
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The section "BBP compared to other methods of computing π" states that using bbp to calculate individual digits is 'the fastest way to compute all the digits from 1 to n". This claim seems suspicious to me, because bbp is not the most efficient method for calculating digits of pi on a single (serial)
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Not at all obvious. I mean, all the pop-lit descriptions of it makes it sound like its constant time. You have to not only see the formula, but also think about it for a while. And even then, its not clear: is it time complexity O(N) or O(Nlog N)? I guess maybe its not O(N^2) but that's not obvious.
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In the "The BBP formula for π" section, it claims: "Once this is done, the four summations are put back into the sum to π". We already have been presented a handful of summations in this section. Which "four summations" is it talking about? Seems confusing. Why not write the whole formula at once?
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Borwein et al. (2004) have recently shown that has no Machin-type BBP arctangent formula that is not binary, although this does not rule out a completely different scheme for digit-extraction algorithms in other bases. A beautiful example of a BBP-type formula in a non-integer base is  (34)
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where and are integer polynomials in (Bailey 2000; Borwein and Bailey 2003, pp. 54 and 128-129). Bailey (2000) and
Borwein and Bailey (2003, pp. 128-129) give a collection of such formulas:    (2)    (3)    (4)    (5)    (6)    (7)    (8)    (9)    (10)
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the constant characters in the above render as "Unicode
Character 'OBJECT REPLACEMENT CHARACTER' (U+FFFC)" on my system (MacOS X 10.13.4 Beta 17E197a). This is in both the page-view and edit-view. Is there a way to extract the Unicode code point hex from the page, or was this lost when
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I added a citation and removed the "citations-needed" (refimprove) message. I've studied this area at least a small amount, and do not think that its a requirement that further citations be added. Yes, of course there are likely more, but I think the current amount is adequate.
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When I first heard about this it was presented as if you could calculate the n:th digit in constant time which would have been truly amazing. Maybe it is immediately obvious to some that is not the case(?) but i doubt it is to everyone not familiar with the subject.
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The whole article seems to be about finding the nth digit of pi, yet doesn't even once state the formula for it. It has a formula for pi, and an explanation of digit-extraction--but never states what the digit extraction formula actually is, as a whole.
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where is
Catalan's constant, is the hyperbolic volume of the figure eight knot complement, is Clausen's integral, and is also the hyperbolic volume of the knot complement of the figure eight knot. Another example is the Dirichlet L-series  (32)
174:   (11)    (12)    (13)    (14)    (15)    (16)    (17)    (18)    (19)    (20)    (21)    (22)    (23)    (24)    (25)    (26)    (27)    (28)    (29)    (30)    (31)
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Near the top the article says "This does not compute the nth decimal of π (i.e., in base 10).", which I think is true but could be taken as meaning, it is impossible to calculate the nth decimal digit by itself, whereas mathworld says it is. See
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181:(Bailey and Borwein 2005; Bailey et al. 2007, pp. 5 and 62). Note that this sort of sum is closely related to the polygamma function since, for example, the above sum can also be written  (33)
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I for one would be very grateful if someone could post a piece of pseudocode for how the n:th digit of pi is calculated. Please, do add this. Earl of
Warwick 11:03, 28 February 2012 (UTC)
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Reference seems to made to a formula that, for any positive integer n, computes the n-th binary digit of pi, but no such formula is shown. I expect to see something like
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computer. Furthermore, the source cited at the end of the paragraph does not corroborate said statement. Is there an alternative source for it that i missed?
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The proof begins by finding the arctangent and Ln components for the expression below for g=(1,2,3,4,5,6,7)and solving various linear equations
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The "citation-needed" that has appeared is slightly idiotic. A citation is hardly possible for a general assumption.
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where is the golden ratio, found by B. Cloitre (Cloitre; Borwein and
Chamberland 2005; Bailey et al. 2007, p. 277).
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link). I changed the formulas a bit and think I discovered an error (multiplication by 16 instead of 16^n). --
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994:{\displaystyle {\frac {\pi }{4}}=2\arctan \left({\frac {1}{2}}\right)-\arctan \left({\frac {1}{7}}\right)}
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01:47, 7 Jan 2005 (UTC) A base- BBP-type formula is a convergent series formula of the type  (1)
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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It should be pretty obvious. From the formula you can already see that you would need at least a
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and the other part increases, which slows down the convergence rate (i.e. causes
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907:{\displaystyle \arctan \left({\frac {1}{7}}\right)=\sum _{k=0}^{\infty }\left}
597:{\displaystyle \arctan \left({\frac {1}{3}}\right)=\sum _{k=0}^{\infty }\left}
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Could the text be rephrased? Or perhaps just delete the sentence? Bill
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One is left feeling that maybe there's some trick one didn't think of.
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At first, I was confused, too, but then I followed the link to
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https://mathworld.wolfram.com/Digit-ExtractionAlgorithm.html
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already present as the first equation in the entire article
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when you wrote this comment, and it is still there now.
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Using the later identity and this arctangent relation
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Most efficient method for parallel computation of pi?
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n-th digit of pi = ... some formula involving n ...
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115:Knowledge:WikiProject Mathematics
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109:and see a list of open tasks.
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1844:This can be simplified to
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2:
2370:
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2228:
2224:
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2216:
2212:
2211:
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2208:
2204:
2200:
2196:
2188:
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2178:
2174:
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2161:
2155:
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2147:
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2132:
2128:
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2086:
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2036:
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2028:
2026:
2020:
2018:
2004:
1999:
1992:
1989:
1986:
1983:
1979:
1974:
1968:
1965:
1962:
1959:
1955:
1950:
1944:
1941:
1938:
1935:
1931:
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1893:
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1024:
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1013:
1005:
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987:
982:
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826:
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764:
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749:
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734:
730:
724:
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715:
712:
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689:
681:
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670:
664:
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541:
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482:
476:
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454:
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427:
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408:
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399:
393:
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386:
379:
371:
368:
364:
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343:
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329:
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308:
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294:
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278:
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208:
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142:
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108:
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100:
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91:
85:
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78:
75:
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70:
66:
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57:
54:
50:
45:
41:
35:
27:
23:
18:
17:
2328:
2293:31.48.26.170
2287:— Preceding
2271:31.48.26.170
2265:— Preceding
2262:
2243:
2236:
2219:Double sharp
2193:— Preceding
2189:
2185:
2165:
2146:67.198.37.16
2126:
2106:
2102:
2040:
2032:
2024:
1843:
1668:
1656:
1561:
1558:
1006:
1003:
915:
608:
605:
298:
258:213.122.8.48
255:
249:
206:
202:chad miller
190:
187:
183:
180:
176:
172:
163:
161:
158:Backporting?
137:Mid-priority
136:
96:
62:Mid‑priority
40:WikiProjects
212:Chadman8000
192:Chadman8000
112:Mathematics
103:mathematics
59:Mathematics
30:Start-class
2347:Categories
2021:Pseudocode
2331:kencf0618
2131:Bdijkstra
230:bdijkstra
2309:Sanpitch
2289:unsigned
2267:unsigned
2195:unsigned
1658:Source:
281:Adrianwn
2246:W102102
1001:obtain
216:Jimw338
164:Deutsch
139:on the
968:arctan
941:arctan
616:arctan
306:arctan
36:scale.
2335:talk
2313:talk
2297:talk
2275:talk
2250:talk
2223:talk
2203:talk
2173:talk
2150:talk
2135:talk
2048:talk
285:talk
279:. –
262:talk
234:talk
220:talk
196:talk
207:All
131:Mid
2349::
2337:)
2325:0x
2315:)
2299:)
2277:)
2259:CN
2252:)
2225:)
2205:)
2175:)
2152:)
2137:)
2120:16
2066:∑
2050:)
1975:−
1951:−
1927:−
1890:16
1874:∞
1859:∑
1852:π
1799:−
1775:−
1751:−
1714:16
1698:∞
1683:∑
1676:π
1585:∞
1570:∑
1517:16
1514:−
1483:11
1480:−
1446:−
1436:−
1412:−
1402:−
1396:19
1387:20
1381:16
1378:−
1368:−
1362:17
1353:20
1347:14
1344:−
1328:15
1319:20
1313:14
1310:−
1294:13
1285:20
1279:10
1276:−
1260:11
1251:20
1242:−
1232:−
1217:20
1208:−
1183:20
1174:−
1164:−
1149:20
1140:−
1130:−
1115:20
1099:−
1084:20
1056:20
1036:∞
1021:∑
1014:π
971:
965:−
944:
927:π
874:18
871:−
861:−
840:15
837:−
827:−
806:13
803:−
793:−
769:−
735:−
701:−
679:20
659:∞
644:∑
619:
564:11
561:−
551:−
527:−
517:−
493:−
483:−
459:−
425:−
391:−
369:12
349:∞
334:∑
309:
287:)
264:)
236:)
222:)
198:)
2333:(
2311:(
2295:(
2273:(
2248:(
2221:(
2201:(
2171:(
2148:(
2133:(
2127:C
2123:
2117:/
2114:1
2111:
2107:n
2103:n
2087:C
2084:+
2081:n
2076:n
2073:=
2070:k
2046:(
2005:]
2000:)
1993:6
1990:+
1987:k
1984:8
1980:1
1969:5
1966:+
1963:k
1960:8
1956:1
1945:2
1942:+
1939:k
1936:4
1932:1
1921:1
1918:+
1915:k
1912:8
1908:4
1902:(
1894:k
1886:1
1880:[
1869:0
1866:=
1863:k
1855:=
1829:]
1824:)
1817:6
1814:+
1811:k
1808:8
1804:1
1793:5
1790:+
1787:k
1784:8
1780:1
1769:4
1766:+
1763:k
1760:8
1756:2
1745:1
1742:+
1739:k
1736:8
1732:4
1726:(
1718:k
1710:1
1704:[
1693:0
1690:=
1687:k
1679:=
1642:]
1637:)
1631:g
1628:+
1625:k
1622:8
1616:g
1612:a
1606:(
1600:k
1596:m
1591:[
1580:0
1577:=
1574:k
1544:]
1539:)
1532:7
1529:+
1526:k
1523:8
1510:2
1504:+
1498:5
1495:+
1492:k
1489:8
1476:2
1470:+
1464:3
1461:+
1458:k
1455:8
1449:6
1442:2
1430:1
1427:+
1424:k
1421:8
1415:1
1408:2
1393:+
1390:k
1374:2
1359:+
1356:k
1340:2
1334:+
1325:+
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1306:2
1300:+
1291:+
1288:k
1272:2
1266:+
1257:+
1254:k
1245:8
1238:2
1226:9
1223:+
1220:k
1211:6
1204:2
1198:+
1192:7
1189:+
1186:k
1177:4
1170:2
1158:5
1155:+
1152:k
1143:4
1136:2
1124:3
1121:+
1118:k
1109:0
1105:2
1093:1
1090:+
1087:k
1078:2
1074:2
1067:(
1059:k
1052:2
1048:1
1042:[
1031:0
1028:=
1025:k
1017:=
988:)
983:7
980:1
975:(
961:)
956:2
953:1
948:(
938:2
935:=
930:4
901:]
896:)
889:7
886:+
883:k
880:8
867:2
855:6
852:+
849:k
846:8
833:2
821:5
818:+
815:k
812:8
799:2
787:3
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781:k
778:8
772:8
765:2
759:+
753:2
750:+
747:k
744:8
738:5
731:2
725:+
719:1
716:+
713:k
710:8
704:3
697:2
690:(
682:k
675:2
671:1
665:[
654:0
651:=
648:k
640:=
636:)
631:7
628:1
623:(
591:]
586:)
579:7
576:+
573:k
570:8
557:2
545:6
542:+
539:k
536:8
530:9
523:2
511:5
508:+
505:k
502:8
496:8
489:2
477:3
474:+
471:k
468:8
462:5
455:2
449:+
443:2
440:+
437:k
434:8
428:3
421:2
415:+
409:1
406:+
403:k
400:8
394:2
387:2
380:(
372:k
365:2
361:1
355:[
344:0
341:=
338:k
330:=
326:)
321:3
318:1
313:(
283:(
260:(
232:(
218:(
194:(
143:.
42::
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
