Spigot algorithm - Revision history

2600:1700:8990:9CF0:8180:2D70:A45D:AFEA: /* Example */

2023-07-29T02:44:26Z

Example

← Previous revision		Revision as of 02:44, 29 July 2023
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	We are only interested in the fractional part of this value, so we can replace each of the summands in the "head" by		We are only interested in the fractional part of this value, so we can replace each of the summands in the "head" by

	:<math>\frac{2^{7-k} \bmod k} k \, .</math>		:<math>\frac{2^{7-k} } k \bmod 1 = \frac{2^{7-k} \bmod k} k \, .</math>

	Calculating each of these terms and adding them to a running total where we again only keep the fractional part, we have:		Calculating each of these terms and adding them to a running total where we again only keep the fractional part, we have:

86.145.59.155 at 15:55, 13 May 2022

2022-05-13T15:55:27Z

← Previous revision		Revision as of 15:55, 13 May 2022
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	! ''A'' = 2<sup>7−''k''</sup>		! ''A'' = 2<sup>7−''k''</sup>
	! ''B'' = ''A'' mod ''k''		! ''B'' = ''A'' mod ''k''
	! ''C'' = ''B'' / ''k''		! ''C'' = {{sfrac\|''B''\|''k''}}
	! Sum of ''C'' mod 1		! Sum of ''C'' mod 1
	\|-		\|-
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	\| 16		\| 16
	\| 1		\| 1
	\| 1/3		\| {{sfrac\|1\|3}}
	\| 1/3		\| {{sfrac\|1\|3}}
	\|-		\|-
	\| 4		\| 4
Line 56:		Line 56:
	\| 0		\| 0
	\| 0		\| 0
	\| 1/3		\| {{sfrac\|1\|3}}
	\|-		\|-
	\| 5		\| 5
	\| 4		\| 4
	\| 4		\| 4
	\| 4/5		\| {{sfrac\|4\|5}}
	\| 2/15		\| {{sfrac\|2\|15}}
	\|-		\|-
	\| 6		\| 6
	\| 2		\| 2
	\| 2		\| 2
	\| 1/3		\| {{sfrac\|1\|3}}
	\| 7/15		\| {{sfrac\|7\|15}}
	\|-		\|-
	\| 7		\| 7
	\| 1		\| 1
	\| 1		\| 1
	\| 1/7		\| {{sfrac\|1\|7}}
	\| 64/105		\| {{sfrac\|64\|105}}
	\|}		\|}

Line 82:		Line 82:
	\|-		\|-
	! ''k''		! ''k''
	! ''D'' = 1/''k''2<sup>''k''−7</sup>		! ''D'' = {{sfrac\|1\|''k''2<sup>''k''−7</sup>}}
	! Sum of ''D''		! Sum of ''D''
	! Maximum error		! Maximum error
	\|-		\|-
	\| 8		\| 8
	\| 1/16		\| {{sfrac\|1\|16}}
	\| 1/16		\| {{sfrac\|1\|16}}
	\| 1/16		\| {{sfrac\|1\|16}}

	\|-		\|-
	\| 9		\| 9
	\| 1/36		\| {{sfrac\|1\|36}}
	\| 13/144		\| {{sfrac\|13\|144}}
	\| 1/36		\| {{sfrac\|1\|36}}
	\|-		\|-
	\| 10		\| 10
	\| 1/80		\| {{sfrac\|1\|80}}
	\| 37/360		\| {{sfrac\|37\|360}}
	\| 1/80		\| {{sfrac\|1\|80}}
	\|}		\|}

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	so the 8th to 11th binary digits in the binary expansion of ln(2) are 1, 0, 1, 1. Note that we have not calculated the values of the first seven binary digits – indeed, all information about them has been intentionally discarded by using [[modular arithmetic]] in the "head" sum.		so the 8th to 11th binary digits in the binary expansion of ln(2) are 1, 0, 1, 1. Note that we have not calculated the values of the first seven binary digits – indeed, all information about them has been intentionally discarded by using [[modular arithmetic]] in the "head" sum.

	The same approach can be used to calculate digits of the binary expansion of ln(2) starting from an arbitrary ''n''~~<sup>~~th~~</sup>~~ position. The number of terms in the "head" sum increases linearly with ''n'', but the complexity of each term only increases with the logarithm of ''n'' if an efficient method of [[modular exponentiation]] is used. The [[precision (arithmetic)\|precision]] of calculations and intermediate results and the number of terms taken from the "tail" sum are all independent of ''n'', and only depend on the number of binary digits that are being calculated – [[single precision]] arithmetic can be used to calculate around 12 binary digits, regardless of the starting position.		The same approach can be used to calculate digits of the binary expansion of ln(2) starting from an arbitrary ''n''th position. The number of terms in the "head" sum increases linearly with ''n'', but the complexity of each term only increases with the logarithm of ''n'' if an efficient method of [[modular exponentiation]] is used. The [[precision (arithmetic)\|precision]] of calculations and intermediate results and the number of terms taken from the "tail" sum are all independent of ''n'', and only depend on the number of binary digits that are being calculated – [[single precision]] arithmetic can be used to calculate around 12 binary digits, regardless of the starting position.

	==References==		==References==

Alcid1: Added the url to Abdali's paper

2022-02-27T17:38:17Z

Added the url to Abdali's paper

← Previous revision		Revision as of 17:38, 27 February 2022
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	{{Short description\|Algorithm for computing the value of a transcendental number}}A '''spigot algorithm''' is an [[algorithm]] for computing the value of a [[transcendental number]] (such as [[pi\|{{pi}}]] or [[e (mathematical constant)\|''e'']]) that generates the digits of the number sequentially from left to right providing increasing precision as the algorithm proceeds. Spigot algorithms also aim to minimize the amount of intermediate storage required. The name comes from the sense of the word "spigot" for a [[Tap (valve)\|tap or valve]] controlling the flow of a liquid. Spigot algorithms can be contrasted with algorithms that store and process complete numbers to produce successively more accurate approximations to the desired transcendental.		{{Short description\|Algorithm for computing the value of a transcendental number}}A '''spigot algorithm''' is an [[algorithm]] for computing the value of a [[transcendental number]] (such as [[pi\|{{pi}}]] or [[e (mathematical constant)\|''e'']]) that generates the digits of the number sequentially from left to right providing increasing precision as the algorithm proceeds. Spigot algorithms also aim to minimize the amount of intermediate storage required. The name comes from the sense of the word "spigot" for a [[Tap (valve)\|tap or valve]] controlling the flow of a liquid. Spigot algorithms can be contrasted with algorithms that store and process complete numbers to produce successively more accurate approximations to the desired transcendental.

	Interest in spigot algorithms was spurred in the early days of computational mathematics by extreme constraints on memory, and such an algorithm for calculating the digits of ''e'' appeared in a paper by Sale in 1968.<ref>{{cite journal \|author=Sale, AHJ \|year=1968 \|title=The calculation of ''e'' to many significant digits \|journal= The Computer Journal\|volume=11 \|issue=2 \|pages=229–230 \|doi=10.1093/comjnl/11.2.229\|doi-access=free }}</ref> In 1970, Abdali presented a more general algorithm to compute the sums of series in which the ratios of successive terms can be expressed as quotients of integer functions of term positions. This algorithm is applicable to many familiar series for trigonometric functions, logarithms, and transcendental numbers because these series satisfy the above condition.<ref>{{cite journal \|author=Abdali, S Kamal \|year=1970 \|title=Special Series Summation with Arbitrary Precision \|journal=Communications of the ACM \|volume=13 \|issue=9 \|page=570 \|doi=10.1145/362736.362756}}</ref> The name "spigot algorithm" seems to have been coined by [[Stanley Rabinowitz]] and [[Stan Wagon]], whose algorithm for calculating the digits of {{pi}} is sometimes referred to as "''the'' spigot algorithm for {{pi}}".<ref>		Interest in spigot algorithms was spurred in the early days of computational mathematics by extreme constraints on memory, and such an algorithm for calculating the digits of ''e'' appeared in a paper by Sale in 1968.<ref>{{cite journal \|author=Sale, AHJ \|year=1968 \|title=The calculation of ''e'' to many significant digits \|journal= The Computer Journal\|volume=11 \|issue=2 \|pages=229–230 \|doi=10.1093/comjnl/11.2.229\|doi-access=free }}</ref> In 1970, Abdali presented a more general algorithm to compute the sums of series in which the ratios of successive terms can be expressed as quotients of integer functions of term positions. This algorithm is applicable to many familiar series for trigonometric functions, logarithms, and transcendental numbers because these series satisfy the above condition.<ref>{{cite journal \|url=http://geomete.com/abdali/papers/algoCACM393.pdf \|author=Abdali, S Kamal \|year=1970 \|title=Special Series Summation with Arbitrary Precision \|journal=Communications of the ACM \|volume=13 \|issue=9 \|page=570 \|doi=10.1145/362736.362756}}</ref> The name "spigot algorithm" seems to have been coined by [[Stanley Rabinowitz]] and [[Stan Wagon]], whose algorithm for calculating the digits of {{pi}} is sometimes referred to as "''the'' spigot algorithm for {{pi}}".<ref>
	{{cite journal \|url=http://stanleyrabinowitz.com/bibliography/spigot.pdf \|title= A Spigot Algorithm for the Digits of Pi\|last1= Rabinowitz\|first1= Stanley\|last2=Wagon\|first2=Stan\|journal=American Mathematical Monthly\|volume=102\|year=1995\|pages=195–203\|accessdate=8 May 2013 \|doi=10.2307/2975006 \|issue=3\|jstor= 2975006}}</ref>		{{cite journal \|url=http://stanleyrabinowitz.com/bibliography/spigot.pdf \|title= A Spigot Algorithm for the Digits of Pi\|last1= Rabinowitz\|first1= Stanley\|last2=Wagon\|first2=Stan\|journal=American Mathematical Monthly\|volume=102\|year=1995\|pages=195–203\|accessdate=8 May 2013 \|doi=10.2307/2975006 \|issue=3\|jstor= 2975006}}</ref>

Beloochee at 10:49, 22 February 2022

2022-02-22T10:49:46Z

← Previous revision		Revision as of 10:49, 22 February 2022
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	{{Short description\|Algorithm for computing the value of a transcendental number}}A '''spigot algorithm''' is an [[algorithm]] for computing the value of a [[transcendental number]] (such as [[pi\|{{pi}}]] or [[e (mathematical constant)\|''e'']]) that generates the digits of the number sequentially from left to right providing increasing precision as the algorithm proceeds. Spigot algorithms also aim to minimize the amount of intermediate storage required. The name comes from the sense of the word "spigot" for a [[Tap (valve)\|tap or valve]] controlling the flow of a liquid. Spigot algorithms can be contrasted with algorithms that store and process complete numbers to produce successively more accurate approximations to the desired transcendental.		{{Short description\|Algorithm for computing the value of a transcendental number}}A '''spigot algorithm''' is an [[algorithm]] for computing the value of a [[transcendental number]] (such as [[pi\|{{pi}}]] or [[e (mathematical constant)\|''e'']]) that generates the digits of the number sequentially from left to right providing increasing precision as the algorithm proceeds. Spigot algorithms also aim to minimize the amount of intermediate storage required. The name comes from the sense of the word "spigot" for a [[Tap (valve)\|tap or valve]] controlling the flow of a liquid. Spigot algorithms can be contrasted with algorithms that store and process complete numbers to produce successively more accurate approximations to the desired transcendental.

	Interest in spigot algorithms was spurred in the early days of computational mathematics by extreme constraints on memory, and such an algorithm for calculating the digits of ''e'' appeared in a paper by Sale in 1968.<ref>{{cite journal \|author=Sale, AHJ \|year=1968 \|title=The calculation of ''e'' to many significant digits \|journal= The Computer Journal\|volume=11 \|issue=2 \|pages=229–230 \|doi=10.1093/comjnl/11.2.229\|doi-access=free }}</ref> In 1970, Abdali presented a more general algorithm to compute the sums of series in which the ratios of successive terms can be expressed as quotients of integer functions of term positions. This algorithm is applicable to many familiar series for trigonometric functions, logarithms, and transcendental numbers because these series satisfy the above condition. <ref>{{cite journal \|author=Abdali, S Kamal \|year=1970 \|title=Special Series Summation with Arbitrary Precision \|journal=Communications of the ACM \|volume=13 \|issue=9 \|page=570 \|doi=10.1145/362736.362756}}</ref>The name "spigot algorithm" seems to have been coined by [[Stanley Rabinowitz]] and [[Stan Wagon]], whose algorithm for calculating the digits of {{pi}} is sometimes referred to as "''the'' spigot algorithm for {{pi}}".<ref>		Interest in spigot algorithms was spurred in the early days of computational mathematics by extreme constraints on memory, and such an algorithm for calculating the digits of ''e'' appeared in a paper by Sale in 1968.<ref>{{cite journal \|author=Sale, AHJ \|year=1968 \|title=The calculation of ''e'' to many significant digits \|journal= The Computer Journal\|volume=11 \|issue=2 \|pages=229–230 \|doi=10.1093/comjnl/11.2.229\|doi-access=free }}</ref> In 1970, Abdali presented a more general algorithm to compute the sums of series in which the ratios of successive terms can be expressed as quotients of integer functions of term positions. This algorithm is applicable to many familiar series for trigonometric functions, logarithms, and transcendental numbers because these series satisfy the above condition.<ref>{{cite journal \|author=Abdali, S Kamal \|year=1970 \|title=Special Series Summation with Arbitrary Precision \|journal=Communications of the ACM \|volume=13 \|issue=9 \|page=570 \|doi=10.1145/362736.362756}}</ref> The name "spigot algorithm" seems to have been coined by [[Stanley Rabinowitz]] and [[Stan Wagon]], whose algorithm for calculating the digits of {{pi}} is sometimes referred to as "''the'' spigot algorithm for {{pi}}".<ref>
	{{cite journal \|url=http://stanleyrabinowitz.com/bibliography/spigot.pdf \|title= A Spigot Algorithm for the Digits of Pi\|last1= Rabinowitz\|first1= Stanley\|last2=Wagon\|first2=Stan\|journal=American Mathematical Monthly\|volume=102\|year=1995\|pages=195–203\|accessdate=8 May 2013 \|doi=10.2307/2975006 \|issue=3\|jstor= 2975006}}</ref>		{{cite journal \|url=http://stanleyrabinowitz.com/bibliography/spigot.pdf \|title= A Spigot Algorithm for the Digits of Pi\|last1= Rabinowitz\|first1= Stanley\|last2=Wagon\|first2=Stan\|journal=American Mathematical Monthly\|volume=102\|year=1995\|pages=195–203\|accessdate=8 May 2013 \|doi=10.2307/2975006 \|issue=3\|jstor= 2975006}}</ref>

Beloochee at 10:44, 22 February 2022

2022-02-22T10:44:08Z

← Previous revision		Revision as of 10:44, 22 February 2022
Line 1:		Line 1:
	{{Short description\|Algorithm for computing the value of a transcendental number}}A '''spigot algorithm''' is an [[algorithm]] for computing the value of a [[transcendental number]] (such as [[pi\|{{pi}}]] or [[e (mathematical constant)\|''e'']]) that generates the digits of the number sequentially from left to right providing increasing precision as the algorithm proceeds. Spigot algorithms also aim to minimize the amount of intermediate storage required. The name comes from the sense of the word "spigot" for a [[Tap (valve)\|tap or valve]] controlling the flow of a liquid. Spigot algorithms can be contrasted with algorithms that store and process complete numbers to produce successively more accurate approximations to the desired transcendental.		{{Short description\|Algorithm for computing the value of a transcendental number}}A '''spigot algorithm''' is an [[algorithm]] for computing the value of a [[transcendental number]] (such as [[pi\|{{pi}}]] or [[e (mathematical constant)\|''e'']]) that generates the digits of the number sequentially from left to right providing increasing precision as the algorithm proceeds. Spigot algorithms also aim to minimize the amount of intermediate storage required. The name comes from the sense of the word "spigot" for a [[Tap (valve)\|tap or valve]] controlling the flow of a liquid. Spigot algorithms can be contrasted with algorithms that store and process complete numbers to produce successively more accurate approximations to the desired transcendental.

	Interest in spigot algorithms was spurred in the early days of computational mathematics by extreme constraints on memory, and such an algorithm for calculating the digits of ''e'' appeared in a paper by Sale in 1968.<ref>{{cite journal \|author=Sale, AHJ \|year=1968 \|title=The calculation of ''e'' to many significant digits \|journal= The Computer Journal\|volume=11 \|issue=2 \|pages=229–230 \|doi=10.1093/comjnl/11.2.229\|doi-access=free }}</ref> In 1970, Abdali presented a more general algorithm to compute the sums of series in which the ratios of successive terms can be expressed as quotients of integer functions of term positions. This algorithm is applicable to many familiar series for trigonometric functions, logarithms, and transcendental numbers because these series satisfy the above condition. <ref>{{cite journal \|author=Abdali, S Kamal \|year=1970 \|title=Special Series Summation with Arbitrary Precision \|journal=Communications of the ACM \|volume=13 \|issue=9 \|page=570 ~~\|accessdate=8 May 2013~~ \|doi=10.~~2307~~/~~2975006doi=10~~.~~2307/2975006 \|issue=3\|jstor= 2975006~~}}</ref>The name "spigot algorithm" seems to have been coined by [[Stanley Rabinowitz]] and [[Stan Wagon]], whose algorithm for calculating the digits of {{pi}} is sometimes referred to as "''the'' spigot algorithm for {{pi}}".<ref>		Interest in spigot algorithms was spurred in the early days of computational mathematics by extreme constraints on memory, and such an algorithm for calculating the digits of ''e'' appeared in a paper by Sale in 1968.<ref>{{cite journal \|author=Sale, AHJ \|year=1968 \|title=The calculation of ''e'' to many significant digits \|journal= The Computer Journal\|volume=11 \|issue=2 \|pages=229–230 \|doi=10.1093/comjnl/11.2.229\|doi-access=free }}</ref> In 1970, Abdali presented a more general algorithm to compute the sums of series in which the ratios of successive terms can be expressed as quotients of integer functions of term positions. This algorithm is applicable to many familiar series for trigonometric functions, logarithms, and transcendental numbers because these series satisfy the above condition. <ref>{{cite journal \|author=Abdali, S Kamal \|year=1970 \|title=Special Series Summation with Arbitrary Precision \|journal=Communications of the ACM \|volume=13 \|issue=9 \|page=570 \|doi=10.1145/362736.362756}}</ref>The name "spigot algorithm" seems to have been coined by [[Stanley Rabinowitz]] and [[Stan Wagon]], whose algorithm for calculating the digits of {{pi}} is sometimes referred to as "''the'' spigot algorithm for {{pi}}".<ref>
	{{cite journal \|url=http://stanleyrabinowitz.com/bibliography/spigot.pdf \|title= A Spigot Algorithm for the Digits of Pi\|last1= Rabinowitz\|first1= Stanley\|last2=Wagon\|first2=Stan\|journal=American Mathematical Monthly\|volume=102\|year=1995\|pages=195–203\|accessdate=8 May 2013 \|doi=10.2307/2975006 \|issue=3\|jstor= 2975006}}</ref>		{{cite journal \|url=http://stanleyrabinowitz.com/bibliography/spigot.pdf \|title= A Spigot Algorithm for the Digits of Pi\|last1= Rabinowitz\|first1= Stanley\|last2=Wagon\|first2=Stan\|journal=American Mathematical Monthly\|volume=102\|year=1995\|pages=195–203\|accessdate=8 May 2013 \|doi=10.2307/2975006 \|issue=3\|jstor= 2975006}}</ref>

Beloochee at 10:40, 22 February 2022

2022-02-22T10:40:43Z

← Previous revision		Revision as of 10:40, 22 February 2022
Line 1:		Line 1:
	{{Short description\|Algorithm for computing the value of a transcendental number}}A '''spigot algorithm''' is an [[algorithm]] for computing the value of a [[transcendental number]] (such as [[pi\|{{pi}}]] or [[e (mathematical constant)\|''e'']]) that generates the digits of the number sequentially from left to right providing increasing precision as the algorithm proceeds. Spigot algorithms also aim to minimize the amount of intermediate storage required. The name comes from the sense of the word "spigot" for a [[Tap (valve)\|tap or valve]] controlling the flow of a liquid. Spigot algorithms can be contrasted with algorithms that store and process complete numbers to produce successively more accurate approximations to the desired transcendental.		{{Short description\|Algorithm for computing the value of a transcendental number}}A '''spigot algorithm''' is an [[algorithm]] for computing the value of a [[transcendental number]] (such as [[pi\|{{pi}}]] or [[e (mathematical constant)\|''e'']]) that generates the digits of the number sequentially from left to right providing increasing precision as the algorithm proceeds. Spigot algorithms also aim to minimize the amount of intermediate storage required. The name comes from the sense of the word "spigot" for a [[Tap (valve)\|tap or valve]] controlling the flow of a liquid. Spigot algorithms can be contrasted with algorithms that store and process complete numbers to produce successively more accurate approximations to the desired transcendental.

	Interest in spigot algorithms was spurred in the early days of computational mathematics by extreme constraints on memory, and such an algorithm for calculating the digits of ''e'' appeared in a paper by Sale in 1968.<ref>{{cite journal \|author=Sale, AHJ \|year=1968 \|title=The calculation of ''e'' to many significant digits \|journal= The Computer Journal\|volume=11 \|issue=2 \|pages=229–230 \|doi=10.1093/comjnl/11.2.229\|doi-access=free }}</ref> In 1970, Abdali presented a more general algorithm to compute the sums of series in which the ratios of successive terms can be expressed as quotients of integer functions of term positions. This algorithm is applicable to many familiar series for trigonometric functions, logarithms, and transcendental numbers because these series satisfy the above condition. <ref>{{cite journal \|author=Abdali, S Kamal \|year=1970 \|title=Special Series Summation with Arbitrary Precision \|journal=Communications of the ACM \|volume=13 \|issue=9 \|page=570 ~~\|doi=10.2307/2975006~~\|accessdate=8 May 2013 \|doi=10.2307/2975006 \|issue=3\|jstor= 2975006}}</ref>The name "spigot algorithm" seems to have been coined by [[Stanley Rabinowitz]] and [[Stan Wagon]], whose algorithm for calculating the digits of {{pi}} is sometimes referred to as "''the'' spigot algorithm for {{pi}}".<ref>		Interest in spigot algorithms was spurred in the early days of computational mathematics by extreme constraints on memory, and such an algorithm for calculating the digits of ''e'' appeared in a paper by Sale in 1968.<ref>{{cite journal \|author=Sale, AHJ \|year=1968 \|title=The calculation of ''e'' to many significant digits \|journal= The Computer Journal\|volume=11 \|issue=2 \|pages=229–230 \|doi=10.1093/comjnl/11.2.229\|doi-access=free }}</ref> In 1970, Abdali presented a more general algorithm to compute the sums of series in which the ratios of successive terms can be expressed as quotients of integer functions of term positions. This algorithm is applicable to many familiar series for trigonometric functions, logarithms, and transcendental numbers because these series satisfy the above condition. <ref>{{cite journal \|author=Abdali, S Kamal \|year=1970 \|title=Special Series Summation with Arbitrary Precision \|journal=Communications of the ACM \|volume=13 \|issue=9 \|page=570 \|accessdate=8 May 2013 \|doi=10.2307/2975006doi=10.2307/2975006 \|issue=3\|jstor= 2975006}}</ref>The name "spigot algorithm" seems to have been coined by [[Stanley Rabinowitz]] and [[Stan Wagon]], whose algorithm for calculating the digits of {{pi}} is sometimes referred to as "''the'' spigot algorithm for {{pi}}".<ref>
	{{cite journal \|url=http://stanleyrabinowitz.com/bibliography/spigot.pdf \|title= A Spigot Algorithm for the Digits of Pi\|last1= Rabinowitz\|first1= Stanley\|last2=Wagon\|first2=Stan\|journal=American Mathematical Monthly\|volume=102\|year=1995\|pages=195–203\|accessdate=8 May 2013 \|doi=10.2307/2975006 \|issue=3\|jstor= 2975006}}</ref>		{{cite journal \|url=http://stanleyrabinowitz.com/bibliography/spigot.pdf \|title= A Spigot Algorithm for the Digits of Pi\|last1= Rabinowitz\|first1= Stanley\|last2=Wagon\|first2=Stan\|journal=American Mathematical Monthly\|volume=102\|year=1995\|pages=195–203\|accessdate=8 May 2013 \|doi=10.2307/2975006 \|issue=3\|jstor= 2975006}}</ref>

Beloochee at 10:38, 22 February 2022

2022-02-22T10:38:58Z

← Previous revision		Revision as of 10:38, 22 February 2022
Line 1:		Line 1:
	{{Short description\|Algorithm for computing the value of a transcendental number}}A '''spigot algorithm''' is an [[algorithm]] for computing the value of a [[transcendental number]] (such as [[pi\|{{pi}}]] or [[e (mathematical constant)\|''e'']]) that generates the digits of the number sequentially from left to right providing increasing precision as the algorithm proceeds. Spigot algorithms also aim to minimize the amount of intermediate storage required. The name comes from the sense of the word "spigot" for a [[Tap (valve)\|tap or valve]] controlling the flow of a liquid. Spigot algorithms can be contrasted with algorithms that store and process complete numbers to produce successively more accurate approximations to the desired transcendental.		{{Short description\|Algorithm for computing the value of a transcendental number}}A '''spigot algorithm''' is an [[algorithm]] for computing the value of a [[transcendental number]] (such as [[pi\|{{pi}}]] or [[e (mathematical constant)\|''e'']]) that generates the digits of the number sequentially from left to right providing increasing precision as the algorithm proceeds. Spigot algorithms also aim to minimize the amount of intermediate storage required. The name comes from the sense of the word "spigot" for a [[Tap (valve)\|tap or valve]] controlling the flow of a liquid. Spigot algorithms can be contrasted with algorithms that store and process complete numbers to produce successively more accurate approximations to the desired transcendental.

	Interest in spigot algorithms was spurred in the early days of computational mathematics by extreme constraints on memory, and such an algorithm for calculating the digits of ''e'' appeared in a paper by Sale in 1968.<ref>{{cite journal \|author=Sale, AHJ \|year=1968 \|title=The calculation of ''e'' to many significant digits \|journal= The Computer Journal\|volume=11 \|issue=2 \|pages=229–230 \|doi=10.1093/comjnl/11.2.229\|doi-access=free }}</ref> In 1970, Abdali presented a more general algorithm to compute the sums of series in which the ratios of successive terms can be expressed as quotients of integer functions of term positions. This algorithm is applicable to many familiar series for trigonometric functions, logarithms, and transcendental numbers because these series satisfy the above condition. <ref>{{cite journal \|author=Abdali, S Kamal \|year=1970 \|title=Special Series Summation with Arbitrary Precision \|journal=Communications of the ACM \|volume=13 \|issue=9 \|page=570 \|doi=10.2307/2975006}}</ref>The name "spigot algorithm" seems to have been coined by [[Stanley Rabinowitz]] and [[Stan Wagon]], whose algorithm for calculating the digits of {{pi}} is sometimes referred to as "''the'' spigot algorithm for {{pi}}".<ref>		Interest in spigot algorithms was spurred in the early days of computational mathematics by extreme constraints on memory, and such an algorithm for calculating the digits of ''e'' appeared in a paper by Sale in 1968.<ref>{{cite journal \|author=Sale, AHJ \|year=1968 \|title=The calculation of ''e'' to many significant digits \|journal= The Computer Journal\|volume=11 \|issue=2 \|pages=229–230 \|doi=10.1093/comjnl/11.2.229\|doi-access=free }}</ref> In 1970, Abdali presented a more general algorithm to compute the sums of series in which the ratios of successive terms can be expressed as quotients of integer functions of term positions. This algorithm is applicable to many familiar series for trigonometric functions, logarithms, and transcendental numbers because these series satisfy the above condition. <ref>{{cite journal \|author=Abdali, S Kamal \|year=1970 \|title=Special Series Summation with Arbitrary Precision \|journal=Communications of the ACM \|volume=13 \|issue=9 \|page=570 \|doi=10.2307/2975006\|accessdate=8 May 2013 \|doi=10.2307/2975006 \|issue=3\|jstor= 2975006}}</ref>The name "spigot algorithm" seems to have been coined by [[Stanley Rabinowitz]] and [[Stan Wagon]], whose algorithm for calculating the digits of {{pi}} is sometimes referred to as "''the'' spigot algorithm for {{pi}}".<ref>
	{{cite journal \|url=http://stanleyrabinowitz.com/bibliography/spigot.pdf \|title= A Spigot Algorithm for the Digits of Pi\|last1= Rabinowitz\|first1= Stanley\|last2=Wagon\|first2=Stan\|journal=American Mathematical Monthly\|volume=102\|year=1995\|pages=195–203\|accessdate=8 May 2013 \|doi=10.2307/2975006 \|issue=3\|jstor= 2975006}}</ref>		{{cite journal \|url=http://stanleyrabinowitz.com/bibliography/spigot.pdf \|title= A Spigot Algorithm for the Digits of Pi\|last1= Rabinowitz\|first1= Stanley\|last2=Wagon\|first2=Stan\|journal=American Mathematical Monthly\|volume=102\|year=1995\|pages=195–203\|accessdate=8 May 2013 \|doi=10.2307/2975006 \|issue=3\|jstor= 2975006}}</ref>

Beloochee at 10:33, 22 February 2022

2022-02-22T10:33:51Z

← Previous revision		Revision as of 10:33, 22 February 2022
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	{{Short description\|Algorithm for computing the value of a transcendental number}}A '''spigot algorithm''' is an [[algorithm]] for computing the value of a [[transcendental number]] (such as [[pi\|{{pi}}]] or [[e (mathematical constant)\|''e'']]) that generates the digits of the number sequentially from left to right providing increasing precision as the algorithm proceeds. Spigot algorithms also aim to minimize the amount of intermediate storage required. The name comes from the sense of the word "spigot" for a [[Tap (valve)\|tap or valve]] controlling the flow of a liquid. Spigot algorithms can be contrasted with algorithms that store and process complete numbers to produce successively more accurate approximations to the desired transcendental.		{{Short description\|Algorithm for computing the value of a transcendental number}}A '''spigot algorithm''' is an [[algorithm]] for computing the value of a [[transcendental number]] (such as [[pi\|{{pi}}]] or [[e (mathematical constant)\|''e'']]) that generates the digits of the number sequentially from left to right providing increasing precision as the algorithm proceeds. Spigot algorithms also aim to minimize the amount of intermediate storage required. The name comes from the sense of the word "spigot" for a [[Tap (valve)\|tap or valve]] controlling the flow of a liquid. Spigot algorithms can be contrasted with algorithms that store and process complete numbers to produce successively more accurate approximations to the desired transcendental.

	Interest in spigot algorithms was spurred in the early days of computational mathematics by extreme constraints on memory, and such an algorithm for calculating the digits of ''e'' appeared in a paper by Sale in 1968.<ref>{{cite journal \|author=Sale, AHJ \|year=1968 \|title=The calculation of ''e'' to many significant digits \|journal= The Computer Journal\|volume=11 \|issue=2 \|pages=229–230 \|doi=10.1093/comjnl/11.2.229\|doi-access=free }}</ref> In 1970, Abdali presented a more general algorithm to compute the sums of series in which the ratios of successive terms can be expressed as quotients of integer functions of term positions. This algorithm is applicable to many familiar series for trigonometric functions, logarithms, and transcendental numbers because these series satisfy the above condition. <ref>{{cite journal \|author=Abdali, S Kamal \|year=1970 \|title=Special Series Summation with Arbitrary Precision \|journal=Communications of the ACM \|volume=13 \|issue=9 \|page=570 \|doi=10.2307/2975006~~.\|doi-access=JSTOR 2975006. Retrieved 8 May 2013~~}}</ref>The name "spigot algorithm" seems to have been coined by [[Stanley Rabinowitz]] and [[Stan Wagon]], whose algorithm for calculating the digits of {{pi}} is sometimes referred to as "''the'' spigot algorithm for {{pi}}".<ref>		Interest in spigot algorithms was spurred in the early days of computational mathematics by extreme constraints on memory, and such an algorithm for calculating the digits of ''e'' appeared in a paper by Sale in 1968.<ref>{{cite journal \|author=Sale, AHJ \|year=1968 \|title=The calculation of ''e'' to many significant digits \|journal= The Computer Journal\|volume=11 \|issue=2 \|pages=229–230 \|doi=10.1093/comjnl/11.2.229\|doi-access=free }}</ref> In 1970, Abdali presented a more general algorithm to compute the sums of series in which the ratios of successive terms can be expressed as quotients of integer functions of term positions. This algorithm is applicable to many familiar series for trigonometric functions, logarithms, and transcendental numbers because these series satisfy the above condition. <ref>{{cite journal \|author=Abdali, S Kamal \|year=1970 \|title=Special Series Summation with Arbitrary Precision \|journal=Communications of the ACM \|volume=13 \|issue=9 \|page=570 \|doi=10.2307/2975006}}</ref>The name "spigot algorithm" seems to have been coined by [[Stanley Rabinowitz]] and [[Stan Wagon]], whose algorithm for calculating the digits of {{pi}} is sometimes referred to as "''the'' spigot algorithm for {{pi}}".<ref>
	{{cite journal \|url=http://stanleyrabinowitz.com/bibliography/spigot.pdf \|title= A Spigot Algorithm for the Digits of Pi\|last1= Rabinowitz\|first1= Stanley\|last2=Wagon\|first2=Stan\|journal=American Mathematical Monthly\|volume=102\|year=1995\|pages=195–203\|accessdate=8 May 2013 \|doi=10.2307/2975006 \|issue=3\|jstor= 2975006}}</ref>		{{cite journal \|url=http://stanleyrabinowitz.com/bibliography/spigot.pdf \|title= A Spigot Algorithm for the Digits of Pi\|last1= Rabinowitz\|first1= Stanley\|last2=Wagon\|first2=Stan\|journal=American Mathematical Monthly\|volume=102\|year=1995\|pages=195–203\|accessdate=8 May 2013 \|doi=10.2307/2975006 \|issue=3\|jstor= 2975006}}</ref>

Beloochee at 10:33, 22 February 2022

2022-02-22T10:33:04Z

← Previous revision		Revision as of 10:33, 22 February 2022
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	{{Short description\|Algorithm for computing the value of a transcendental number}}A '''spigot algorithm''' is an [[algorithm]] for computing the value of a [[transcendental number]] (such as [[pi\|{{pi}}]] or [[e (mathematical constant)\|''e'']]) that generates the digits of the number sequentially from left to right providing increasing precision as the algorithm proceeds. Spigot algorithms also aim to minimize the amount of intermediate storage required. The name comes from the sense of the word "spigot" for a [[Tap (valve)\|tap or valve]] controlling the flow of a liquid. Spigot algorithms can be contrasted with algorithms that store and process complete numbers to produce successively more accurate approximations to the desired transcendental.		{{Short description\|Algorithm for computing the value of a transcendental number}}A '''spigot algorithm''' is an [[algorithm]] for computing the value of a [[transcendental number]] (such as [[pi\|{{pi}}]] or [[e (mathematical constant)\|''e'']]) that generates the digits of the number sequentially from left to right providing increasing precision as the algorithm proceeds. Spigot algorithms also aim to minimize the amount of intermediate storage required. The name comes from the sense of the word "spigot" for a [[Tap (valve)\|tap or valve]] controlling the flow of a liquid. Spigot algorithms can be contrasted with algorithms that store and process complete numbers to produce successively more accurate approximations to the desired transcendental.

	Interest in spigot algorithms was spurred in the early days of computational mathematics by extreme constraints on memory, and such an algorithm for calculating the digits of ''e'' appeared in a paper by Sale in 1968.<ref>{{cite journal \|author=Sale, AHJ \|year=1968 \|title=The calculation of ''e'' to many significant digits \|journal= The Computer Journal\|volume=11 \|issue=2 \|pages=229–230 \|doi=10.1093/comjnl/11.2.229\|doi-access=free }}</ref> In 1970, Abdali presented a more general algorithm to compute the sums of series in which the ratios of successive terms can be expressed as quotients of integer functions of term positions. This algorithm is applicable to many familiar series for trigonometric functions, logarithms, and transcendental numbers because these series satisfy the above condition. <ref>{{cite journal \|author=Abdali, S Kamal \|year=1970 \|title=Special Series Summation with Arbitrary Precision \|journal=Communications of the ACM \|volume=13 \|issue=9 \|page=570 \|doi:10.2307/2975006.\|doi-JSTOR 2975006. Retrieved 8 May 2013}}</ref>The name "spigot algorithm" seems to have been coined by [[Stanley Rabinowitz]] and [[Stan Wagon]], whose algorithm for calculating the digits of {{pi}} is sometimes referred to as "''the'' spigot algorithm for {{pi}}".<ref>		Interest in spigot algorithms was spurred in the early days of computational mathematics by extreme constraints on memory, and such an algorithm for calculating the digits of ''e'' appeared in a paper by Sale in 1968.<ref>{{cite journal \|author=Sale, AHJ \|year=1968 \|title=The calculation of ''e'' to many significant digits \|journal= The Computer Journal\|volume=11 \|issue=2 \|pages=229–230 \|doi=10.1093/comjnl/11.2.229\|doi-access=free }}</ref> In 1970, Abdali presented a more general algorithm to compute the sums of series in which the ratios of successive terms can be expressed as quotients of integer functions of term positions. This algorithm is applicable to many familiar series for trigonometric functions, logarithms, and transcendental numbers because these series satisfy the above condition. <ref>{{cite journal \|author=Abdali, S Kamal \|year=1970 \|title=Special Series Summation with Arbitrary Precision \|journal=Communications of the ACM \|volume=13 \|issue=9 \|page=570 \|doi=10.2307/2975006.\|doi-access=JSTOR 2975006. Retrieved 8 May 2013}}</ref>The name "spigot algorithm" seems to have been coined by [[Stanley Rabinowitz]] and [[Stan Wagon]], whose algorithm for calculating the digits of {{pi}} is sometimes referred to as "''the'' spigot algorithm for {{pi}}".<ref>
	{{cite journal \|url=http://stanleyrabinowitz.com/bibliography/spigot.pdf \|title= A Spigot Algorithm for the Digits of Pi\|last1= Rabinowitz\|first1= Stanley\|last2=Wagon\|first2=Stan\|journal=American Mathematical Monthly\|volume=102\|year=1995\|pages=195–203\|accessdate=8 May 2013 \|doi=10.2307/2975006 \|issue=3\|jstor= 2975006}}</ref>		{{cite journal \|url=http://stanleyrabinowitz.com/bibliography/spigot.pdf \|title= A Spigot Algorithm for the Digits of Pi\|last1= Rabinowitz\|first1= Stanley\|last2=Wagon\|first2=Stan\|journal=American Mathematical Monthly\|volume=102\|year=1995\|pages=195–203\|accessdate=8 May 2013 \|doi=10.2307/2975006 \|issue=3\|jstor= 2975006}}</ref>

Beloochee at 10:29, 22 February 2022

2022-02-22T10:29:34Z

← Previous revision		Revision as of 10:29, 22 February 2022
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	{{Short description\|Algorithm for computing the value of a transcendental number}}A '''spigot algorithm''' is an [[algorithm]] for computing the value of a [[transcendental number]] (such as [[pi\|{{pi}}]] or [[e (mathematical constant)\|''e'']]) that generates the digits of the number sequentially from left to right providing increasing precision as the algorithm proceeds. Spigot algorithms also aim to minimize the amount of intermediate storage required. The name comes from the sense of the word "spigot" for a [[Tap (valve)\|tap or valve]] controlling the flow of a liquid. Spigot algorithms can be contrasted with algorithms that store and process complete numbers to produce successively more accurate approximations to the desired transcendental.		{{Short description\|Algorithm for computing the value of a transcendental number}}A '''spigot algorithm''' is an [[algorithm]] for computing the value of a [[transcendental number]] (such as [[pi\|{{pi}}]] or [[e (mathematical constant)\|''e'']]) that generates the digits of the number sequentially from left to right providing increasing precision as the algorithm proceeds. Spigot algorithms also aim to minimize the amount of intermediate storage required. The name comes from the sense of the word "spigot" for a [[Tap (valve)\|tap or valve]] controlling the flow of a liquid. Spigot algorithms can be contrasted with algorithms that store and process complete numbers to produce successively more accurate approximations to the desired transcendental.

	Interest in spigot algorithms was spurred in the early days of computational mathematics by extreme constraints on memory, and such an algorithm for calculating the digits of ''e'' appeared in a paper by Sale in 1968.<ref>{{cite journal \|author=Sale, AHJ \|year=1968 \|title=The calculation of ''e'' to many significant digits \|journal= The Computer Journal\|volume=11 \|issue=2 \|pages=229–230 \|doi=10.1093/comjnl/11.2.229\|doi-access=free }}</ref> In 1970, Abdali presented a more general algorithm to compute the sums of series in which the ratios of successive terms can be expressed as quotients of integer functions of term positions. This algorithm is applicable to many familiar series for trigonometric functions, logarithms, and transcendental numbers because these series satisfy the above condition. <ref>{{cite journal \|author=Abdali, S Kamal \|year=1970 \|title=Special Series Summation with Arbitrary Precision \|journal=Communications of the ACM \|volume=13 \|issue=9 \|page=570 \|doi:10.2307/2975006.}}</ref>The name "spigot algorithm" seems to have been coined by [[Stanley Rabinowitz]] and [[Stan Wagon]], whose algorithm for calculating the digits of {{pi}} is sometimes referred to as "''the'' spigot algorithm for {{pi}}".<ref>		Interest in spigot algorithms was spurred in the early days of computational mathematics by extreme constraints on memory, and such an algorithm for calculating the digits of ''e'' appeared in a paper by Sale in 1968.<ref>{{cite journal \|author=Sale, AHJ \|year=1968 \|title=The calculation of ''e'' to many significant digits \|journal= The Computer Journal\|volume=11 \|issue=2 \|pages=229–230 \|doi=10.1093/comjnl/11.2.229\|doi-access=free }}</ref> In 1970, Abdali presented a more general algorithm to compute the sums of series in which the ratios of successive terms can be expressed as quotients of integer functions of term positions. This algorithm is applicable to many familiar series for trigonometric functions, logarithms, and transcendental numbers because these series satisfy the above condition. <ref>{{cite journal \|author=Abdali, S Kamal \|year=1970 \|title=Special Series Summation with Arbitrary Precision \|journal=Communications of the ACM \|volume=13 \|issue=9 \|page=570 \|doi:10.2307/2975006.\|doi-JSTOR 2975006. Retrieved 8 May 2013}}</ref>The name "spigot algorithm" seems to have been coined by [[Stanley Rabinowitz]] and [[Stan Wagon]], whose algorithm for calculating the digits of {{pi}} is sometimes referred to as "''the'' spigot algorithm for {{pi}}".<ref>
	{{cite journal \|url=http://stanleyrabinowitz.com/bibliography/spigot.pdf \|title= A Spigot Algorithm for the Digits of Pi\|last1= Rabinowitz\|first1= Stanley\|last2=Wagon\|first2=Stan\|journal=American Mathematical Monthly\|volume=102\|year=1995\|pages=195–203\|accessdate=8 May 2013 \|doi=10.2307/2975006 \|issue=3\|jstor= 2975006}}</ref>		{{cite journal \|url=http://stanleyrabinowitz.com/bibliography/spigot.pdf \|title= A Spigot Algorithm for the Digits of Pi\|last1= Rabinowitz\|first1= Stanley\|last2=Wagon\|first2=Stan\|journal=American Mathematical Monthly\|volume=102\|year=1995\|pages=195–203\|accessdate=8 May 2013 \|doi=10.2307/2975006 \|issue=3\|jstor= 2975006}}</ref>

← Previous revision		Revision as of 17:38, 27 February 2022
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	{{Short description\|Algorithm for computing the value of a transcendental number}}A '''spigot algorithm''' is an [[algorithm]] for computing the value of a [[transcendental number]] (such as [[pi\|{{pi}}]] or [[e (mathematical constant)\|''e'']]) that generates the digits of the number sequentially from left to right providing increasing precision as the algorithm proceeds. Spigot algorithms also aim to minimize the amount of intermediate storage required. The name comes from the sense of the word "spigot" for a [[Tap (valve)\|tap or valve]] controlling the flow of a liquid. Spigot algorithms can be contrasted with algorithms that store and process complete numbers to produce successively more accurate approximations to the desired transcendental.		{{Short description\|Algorithm for computing the value of a transcendental number}}A '''spigot algorithm''' is an [[algorithm]] for computing the value of a [[transcendental number]] (such as [[pi\|{{pi}}]] or [[e (mathematical constant)\|''e'']]) that generates the digits of the number sequentially from left to right providing increasing precision as the algorithm proceeds. Spigot algorithms also aim to minimize the amount of intermediate storage required. The name comes from the sense of the word "spigot" for a [[Tap (valve)\|tap or valve]] controlling the flow of a liquid. Spigot algorithms can be contrasted with algorithms that store and process complete numbers to produce successively more accurate approximations to the desired transcendental.

	Interest in spigot algorithms was spurred in the early days of computational mathematics by extreme constraints on memory, and such an algorithm for calculating the digits of ''e'' appeared in a paper by Sale in 1968.<ref>{{cite journal \|author=Sale, AHJ \|year=1968 \|title=The calculation of ''e'' to many significant digits \|journal= The Computer Journal\|volume=11 \|issue=2 \|pages=229–230 \|doi=10.1093/comjnl/11.2.229\|doi-access=free }}</ref> In 1970, Abdali presented a more general algorithm to compute the sums of series in which the ratios of successive terms can be expressed as quotients of integer functions of term positions. This algorithm is applicable to many familiar series for trigonometric functions, logarithms, and transcendental numbers because these series satisfy the above condition.<ref>{{cite journal \|author=Abdali, S Kamal \|year=1970 \|title=Special Series Summation with Arbitrary Precision \|journal=Communications of the ACM \|volume=13 \|issue=9 \|page=570 \|doi=10.1145/362736.362756}}</ref> The name "spigot algorithm" seems to have been coined by [[Stanley Rabinowitz]] and [[Stan Wagon]], whose algorithm for calculating the digits of {{pi}} is sometimes referred to as "''the'' spigot algorithm for {{pi}}".<ref>		Interest in spigot algorithms was spurred in the early days of computational mathematics by extreme constraints on memory, and such an algorithm for calculating the digits of ''e'' appeared in a paper by Sale in 1968.<ref>{{cite journal \|author=Sale, AHJ \|year=1968 \|title=The calculation of ''e'' to many significant digits \|journal= The Computer Journal\|volume=11 \|issue=2 \|pages=229–230 \|doi=10.1093/comjnl/11.2.229\|doi-access=free }}</ref> In 1970, Abdali presented a more general algorithm to compute the sums of series in which the ratios of successive terms can be expressed as quotients of integer functions of term positions. This algorithm is applicable to many familiar series for trigonometric functions, logarithms, and transcendental numbers because these series satisfy the above condition.<ref>{{cite journal \|url=http://geomete.com/abdali/papers/algoCACM393.pdf \|author=Abdali, S Kamal \|year=1970 \|title=Special Series Summation with Arbitrary Precision \|journal=Communications of the ACM \|volume=13 \|issue=9 \|page=570 \|doi=10.1145/362736.362756}}</ref> The name "spigot algorithm" seems to have been coined by [[Stanley Rabinowitz]] and [[Stan Wagon]], whose algorithm for calculating the digits of {{pi}} is sometimes referred to as "''the'' spigot algorithm for {{pi}}".<ref>
	{{cite journal \|url=http://stanleyrabinowitz.com/bibliography/spigot.pdf \|title= A Spigot Algorithm for the Digits of Pi\|last1= Rabinowitz\|first1= Stanley\|last2=Wagon\|first2=Stan\|journal=American Mathematical Monthly\|volume=102\|year=1995\|pages=195–203\|accessdate=8 May 2013 \|doi=10.2307/2975006 \|issue=3\|jstor= 2975006}}</ref>		{{cite journal \|url=http://stanleyrabinowitz.com/bibliography/spigot.pdf \|title= A Spigot Algorithm for the Digits of Pi\|last1= Rabinowitz\|first1= Stanley\|last2=Wagon\|first2=Stan\|journal=American Mathematical Monthly\|volume=102\|year=1995\|pages=195–203\|accessdate=8 May 2013 \|doi=10.2307/2975006 \|issue=3\|jstor= 2975006}}</ref>

← Previous revision		Revision as of 10:49, 22 February 2022
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	{{Short description\|Algorithm for computing the value of a transcendental number}}A '''spigot algorithm''' is an [[algorithm]] for computing the value of a [[transcendental number]] (such as [[pi\|{{pi}}]] or [[e (mathematical constant)\|''e'']]) that generates the digits of the number sequentially from left to right providing increasing precision as the algorithm proceeds. Spigot algorithms also aim to minimize the amount of intermediate storage required. The name comes from the sense of the word "spigot" for a [[Tap (valve)\|tap or valve]] controlling the flow of a liquid. Spigot algorithms can be contrasted with algorithms that store and process complete numbers to produce successively more accurate approximations to the desired transcendental.		{{Short description\|Algorithm for computing the value of a transcendental number}}A '''spigot algorithm''' is an [[algorithm]] for computing the value of a [[transcendental number]] (such as [[pi\|{{pi}}]] or [[e (mathematical constant)\|''e'']]) that generates the digits of the number sequentially from left to right providing increasing precision as the algorithm proceeds. Spigot algorithms also aim to minimize the amount of intermediate storage required. The name comes from the sense of the word "spigot" for a [[Tap (valve)\|tap or valve]] controlling the flow of a liquid. Spigot algorithms can be contrasted with algorithms that store and process complete numbers to produce successively more accurate approximations to the desired transcendental.

	Interest in spigot algorithms was spurred in the early days of computational mathematics by extreme constraints on memory, and such an algorithm for calculating the digits of ''e'' appeared in a paper by Sale in 1968.<ref>{{cite journal \|author=Sale, AHJ \|year=1968 \|title=The calculation of ''e'' to many significant digits \|journal= The Computer Journal\|volume=11 \|issue=2 \|pages=229–230 \|doi=10.1093/comjnl/11.2.229\|doi-access=free }}</ref> In 1970, Abdali presented a more general algorithm to compute the sums of series in which the ratios of successive terms can be expressed as quotients of integer functions of term positions. This algorithm is applicable to many familiar series for trigonometric functions, logarithms, and transcendental numbers because these series satisfy the above condition. <ref>{{cite journal \|author=Abdali, S Kamal \|year=1970 \|title=Special Series Summation with Arbitrary Precision \|journal=Communications of the ACM \|volume=13 \|issue=9 \|page=570 \|doi=10.1145/362736.362756}}</ref>The name "spigot algorithm" seems to have been coined by [[Stanley Rabinowitz]] and [[Stan Wagon]], whose algorithm for calculating the digits of {{pi}} is sometimes referred to as "''the'' spigot algorithm for {{pi}}".<ref>		Interest in spigot algorithms was spurred in the early days of computational mathematics by extreme constraints on memory, and such an algorithm for calculating the digits of ''e'' appeared in a paper by Sale in 1968.<ref>{{cite journal \|author=Sale, AHJ \|year=1968 \|title=The calculation of ''e'' to many significant digits \|journal= The Computer Journal\|volume=11 \|issue=2 \|pages=229–230 \|doi=10.1093/comjnl/11.2.229\|doi-access=free }}</ref> In 1970, Abdali presented a more general algorithm to compute the sums of series in which the ratios of successive terms can be expressed as quotients of integer functions of term positions. This algorithm is applicable to many familiar series for trigonometric functions, logarithms, and transcendental numbers because these series satisfy the above condition.<ref>{{cite journal \|author=Abdali, S Kamal \|year=1970 \|title=Special Series Summation with Arbitrary Precision \|journal=Communications of the ACM \|volume=13 \|issue=9 \|page=570 \|doi=10.1145/362736.362756}}</ref> The name "spigot algorithm" seems to have been coined by [[Stanley Rabinowitz]] and [[Stan Wagon]], whose algorithm for calculating the digits of {{pi}} is sometimes referred to as "''the'' spigot algorithm for {{pi}}".<ref>
	{{cite journal \|url=http://stanleyrabinowitz.com/bibliography/spigot.pdf \|title= A Spigot Algorithm for the Digits of Pi\|last1= Rabinowitz\|first1= Stanley\|last2=Wagon\|first2=Stan\|journal=American Mathematical Monthly\|volume=102\|year=1995\|pages=195–203\|accessdate=8 May 2013 \|doi=10.2307/2975006 \|issue=3\|jstor= 2975006}}</ref>		{{cite journal \|url=http://stanleyrabinowitz.com/bibliography/spigot.pdf \|title= A Spigot Algorithm for the Digits of Pi\|last1= Rabinowitz\|first1= Stanley\|last2=Wagon\|first2=Stan\|journal=American Mathematical Monthly\|volume=102\|year=1995\|pages=195–203\|accessdate=8 May 2013 \|doi=10.2307/2975006 \|issue=3\|jstor= 2975006}}</ref>

← Previous revision		Revision as of 10:44, 22 February 2022
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	{{Short description\|Algorithm for computing the value of a transcendental number}}A '''spigot algorithm''' is an [[algorithm]] for computing the value of a [[transcendental number]] (such as [[pi\|{{pi}}]] or [[e (mathematical constant)\|''e'']]) that generates the digits of the number sequentially from left to right providing increasing precision as the algorithm proceeds. Spigot algorithms also aim to minimize the amount of intermediate storage required. The name comes from the sense of the word "spigot" for a [[Tap (valve)\|tap or valve]] controlling the flow of a liquid. Spigot algorithms can be contrasted with algorithms that store and process complete numbers to produce successively more accurate approximations to the desired transcendental.		{{Short description\|Algorithm for computing the value of a transcendental number}}A '''spigot algorithm''' is an [[algorithm]] for computing the value of a [[transcendental number]] (such as [[pi\|{{pi}}]] or [[e (mathematical constant)\|''e'']]) that generates the digits of the number sequentially from left to right providing increasing precision as the algorithm proceeds. Spigot algorithms also aim to minimize the amount of intermediate storage required. The name comes from the sense of the word "spigot" for a [[Tap (valve)\|tap or valve]] controlling the flow of a liquid. Spigot algorithms can be contrasted with algorithms that store and process complete numbers to produce successively more accurate approximations to the desired transcendental.

	Interest in spigot algorithms was spurred in the early days of computational mathematics by extreme constraints on memory, and such an algorithm for calculating the digits of ''e'' appeared in a paper by Sale in 1968.<ref>{{cite journal \|author=Sale, AHJ \|year=1968 \|title=The calculation of ''e'' to many significant digits \|journal= The Computer Journal\|volume=11 \|issue=2 \|pages=229–230 \|doi=10.1093/comjnl/11.2.229\|doi-access=free }}</ref> In 1970, Abdali presented a more general algorithm to compute the sums of series in which the ratios of successive terms can be expressed as quotients of integer functions of term positions. This algorithm is applicable to many familiar series for trigonometric functions, logarithms, and transcendental numbers because these series satisfy the above condition. <ref>{{cite journal \|author=Abdali, S Kamal \|year=1970 \|title=Special Series Summation with Arbitrary Precision \|journal=Communications of the ACM \|volume=13 \|issue=9 \|page=570 ~~\|accessdate=8 May 2013~~ \|doi=10.~~2307~~/~~2975006doi=10~~.~~2307/2975006 \|issue=3\|jstor= 2975006~~}}</ref>The name "spigot algorithm" seems to have been coined by [[Stanley Rabinowitz]] and [[Stan Wagon]], whose algorithm for calculating the digits of {{pi}} is sometimes referred to as "''the'' spigot algorithm for {{pi}}".<ref>		Interest in spigot algorithms was spurred in the early days of computational mathematics by extreme constraints on memory, and such an algorithm for calculating the digits of ''e'' appeared in a paper by Sale in 1968.<ref>{{cite journal \|author=Sale, AHJ \|year=1968 \|title=The calculation of ''e'' to many significant digits \|journal= The Computer Journal\|volume=11 \|issue=2 \|pages=229–230 \|doi=10.1093/comjnl/11.2.229\|doi-access=free }}</ref> In 1970, Abdali presented a more general algorithm to compute the sums of series in which the ratios of successive terms can be expressed as quotients of integer functions of term positions. This algorithm is applicable to many familiar series for trigonometric functions, logarithms, and transcendental numbers because these series satisfy the above condition. <ref>{{cite journal \|author=Abdali, S Kamal \|year=1970 \|title=Special Series Summation with Arbitrary Precision \|journal=Communications of the ACM \|volume=13 \|issue=9 \|page=570 \|doi=10.1145/362736.362756}}</ref>The name "spigot algorithm" seems to have been coined by [[Stanley Rabinowitz]] and [[Stan Wagon]], whose algorithm for calculating the digits of {{pi}} is sometimes referred to as "''the'' spigot algorithm for {{pi}}".<ref>
	{{cite journal \|url=http://stanleyrabinowitz.com/bibliography/spigot.pdf \|title= A Spigot Algorithm for the Digits of Pi\|last1= Rabinowitz\|first1= Stanley\|last2=Wagon\|first2=Stan\|journal=American Mathematical Monthly\|volume=102\|year=1995\|pages=195–203\|accessdate=8 May 2013 \|doi=10.2307/2975006 \|issue=3\|jstor= 2975006}}</ref>		{{cite journal \|url=http://stanleyrabinowitz.com/bibliography/spigot.pdf \|title= A Spigot Algorithm for the Digits of Pi\|last1= Rabinowitz\|first1= Stanley\|last2=Wagon\|first2=Stan\|journal=American Mathematical Monthly\|volume=102\|year=1995\|pages=195–203\|accessdate=8 May 2013 \|doi=10.2307/2975006 \|issue=3\|jstor= 2975006}}</ref>

← Previous revision		Revision as of 10:40, 22 February 2022
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	{{Short description\|Algorithm for computing the value of a transcendental number}}A '''spigot algorithm''' is an [[algorithm]] for computing the value of a [[transcendental number]] (such as [[pi\|{{pi}}]] or [[e (mathematical constant)\|''e'']]) that generates the digits of the number sequentially from left to right providing increasing precision as the algorithm proceeds. Spigot algorithms also aim to minimize the amount of intermediate storage required. The name comes from the sense of the word "spigot" for a [[Tap (valve)\|tap or valve]] controlling the flow of a liquid. Spigot algorithms can be contrasted with algorithms that store and process complete numbers to produce successively more accurate approximations to the desired transcendental.		{{Short description\|Algorithm for computing the value of a transcendental number}}A '''spigot algorithm''' is an [[algorithm]] for computing the value of a [[transcendental number]] (such as [[pi\|{{pi}}]] or [[e (mathematical constant)\|''e'']]) that generates the digits of the number sequentially from left to right providing increasing precision as the algorithm proceeds. Spigot algorithms also aim to minimize the amount of intermediate storage required. The name comes from the sense of the word "spigot" for a [[Tap (valve)\|tap or valve]] controlling the flow of a liquid. Spigot algorithms can be contrasted with algorithms that store and process complete numbers to produce successively more accurate approximations to the desired transcendental.

	Interest in spigot algorithms was spurred in the early days of computational mathematics by extreme constraints on memory, and such an algorithm for calculating the digits of ''e'' appeared in a paper by Sale in 1968.<ref>{{cite journal \|author=Sale, AHJ \|year=1968 \|title=The calculation of ''e'' to many significant digits \|journal= The Computer Journal\|volume=11 \|issue=2 \|pages=229–230 \|doi=10.1093/comjnl/11.2.229\|doi-access=free }}</ref> In 1970, Abdali presented a more general algorithm to compute the sums of series in which the ratios of successive terms can be expressed as quotients of integer functions of term positions. This algorithm is applicable to many familiar series for trigonometric functions, logarithms, and transcendental numbers because these series satisfy the above condition. <ref>{{cite journal \|author=Abdali, S Kamal \|year=1970 \|title=Special Series Summation with Arbitrary Precision \|journal=Communications of the ACM \|volume=13 \|issue=9 \|page=570 ~~\|doi=10.2307/2975006~~\|accessdate=8 May 2013 \|doi=10.2307/2975006 \|issue=3\|jstor= 2975006}}</ref>The name "spigot algorithm" seems to have been coined by [[Stanley Rabinowitz]] and [[Stan Wagon]], whose algorithm for calculating the digits of {{pi}} is sometimes referred to as "''the'' spigot algorithm for {{pi}}".<ref>		Interest in spigot algorithms was spurred in the early days of computational mathematics by extreme constraints on memory, and such an algorithm for calculating the digits of ''e'' appeared in a paper by Sale in 1968.<ref>{{cite journal \|author=Sale, AHJ \|year=1968 \|title=The calculation of ''e'' to many significant digits \|journal= The Computer Journal\|volume=11 \|issue=2 \|pages=229–230 \|doi=10.1093/comjnl/11.2.229\|doi-access=free }}</ref> In 1970, Abdali presented a more general algorithm to compute the sums of series in which the ratios of successive terms can be expressed as quotients of integer functions of term positions. This algorithm is applicable to many familiar series for trigonometric functions, logarithms, and transcendental numbers because these series satisfy the above condition. <ref>{{cite journal \|author=Abdali, S Kamal \|year=1970 \|title=Special Series Summation with Arbitrary Precision \|journal=Communications of the ACM \|volume=13 \|issue=9 \|page=570 \|accessdate=8 May 2013 \|doi=10.2307/2975006doi=10.2307/2975006 \|issue=3\|jstor= 2975006}}</ref>The name "spigot algorithm" seems to have been coined by [[Stanley Rabinowitz]] and [[Stan Wagon]], whose algorithm for calculating the digits of {{pi}} is sometimes referred to as "''the'' spigot algorithm for {{pi}}".<ref>
	{{cite journal \|url=http://stanleyrabinowitz.com/bibliography/spigot.pdf \|title= A Spigot Algorithm for the Digits of Pi\|last1= Rabinowitz\|first1= Stanley\|last2=Wagon\|first2=Stan\|journal=American Mathematical Monthly\|volume=102\|year=1995\|pages=195–203\|accessdate=8 May 2013 \|doi=10.2307/2975006 \|issue=3\|jstor= 2975006}}</ref>		{{cite journal \|url=http://stanleyrabinowitz.com/bibliography/spigot.pdf \|title= A Spigot Algorithm for the Digits of Pi\|last1= Rabinowitz\|first1= Stanley\|last2=Wagon\|first2=Stan\|journal=American Mathematical Monthly\|volume=102\|year=1995\|pages=195–203\|accessdate=8 May 2013 \|doi=10.2307/2975006 \|issue=3\|jstor= 2975006}}</ref>

← Previous revision		Revision as of 10:38, 22 February 2022
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	{{Short description\|Algorithm for computing the value of a transcendental number}}A '''spigot algorithm''' is an [[algorithm]] for computing the value of a [[transcendental number]] (such as [[pi\|{{pi}}]] or [[e (mathematical constant)\|''e'']]) that generates the digits of the number sequentially from left to right providing increasing precision as the algorithm proceeds. Spigot algorithms also aim to minimize the amount of intermediate storage required. The name comes from the sense of the word "spigot" for a [[Tap (valve)\|tap or valve]] controlling the flow of a liquid. Spigot algorithms can be contrasted with algorithms that store and process complete numbers to produce successively more accurate approximations to the desired transcendental.		{{Short description\|Algorithm for computing the value of a transcendental number}}A '''spigot algorithm''' is an [[algorithm]] for computing the value of a [[transcendental number]] (such as [[pi\|{{pi}}]] or [[e (mathematical constant)\|''e'']]) that generates the digits of the number sequentially from left to right providing increasing precision as the algorithm proceeds. Spigot algorithms also aim to minimize the amount of intermediate storage required. The name comes from the sense of the word "spigot" for a [[Tap (valve)\|tap or valve]] controlling the flow of a liquid. Spigot algorithms can be contrasted with algorithms that store and process complete numbers to produce successively more accurate approximations to the desired transcendental.

	Interest in spigot algorithms was spurred in the early days of computational mathematics by extreme constraints on memory, and such an algorithm for calculating the digits of ''e'' appeared in a paper by Sale in 1968.<ref>{{cite journal \|author=Sale, AHJ \|year=1968 \|title=The calculation of ''e'' to many significant digits \|journal= The Computer Journal\|volume=11 \|issue=2 \|pages=229–230 \|doi=10.1093/comjnl/11.2.229\|doi-access=free }}</ref> In 1970, Abdali presented a more general algorithm to compute the sums of series in which the ratios of successive terms can be expressed as quotients of integer functions of term positions. This algorithm is applicable to many familiar series for trigonometric functions, logarithms, and transcendental numbers because these series satisfy the above condition. <ref>{{cite journal \|author=Abdali, S Kamal \|year=1970 \|title=Special Series Summation with Arbitrary Precision \|journal=Communications of the ACM \|volume=13 \|issue=9 \|page=570 \|doi=10.2307/2975006}}</ref>The name "spigot algorithm" seems to have been coined by [[Stanley Rabinowitz]] and [[Stan Wagon]], whose algorithm for calculating the digits of {{pi}} is sometimes referred to as "''the'' spigot algorithm for {{pi}}".<ref>		Interest in spigot algorithms was spurred in the early days of computational mathematics by extreme constraints on memory, and such an algorithm for calculating the digits of ''e'' appeared in a paper by Sale in 1968.<ref>{{cite journal \|author=Sale, AHJ \|year=1968 \|title=The calculation of ''e'' to many significant digits \|journal= The Computer Journal\|volume=11 \|issue=2 \|pages=229–230 \|doi=10.1093/comjnl/11.2.229\|doi-access=free }}</ref> In 1970, Abdali presented a more general algorithm to compute the sums of series in which the ratios of successive terms can be expressed as quotients of integer functions of term positions. This algorithm is applicable to many familiar series for trigonometric functions, logarithms, and transcendental numbers because these series satisfy the above condition. <ref>{{cite journal \|author=Abdali, S Kamal \|year=1970 \|title=Special Series Summation with Arbitrary Precision \|journal=Communications of the ACM \|volume=13 \|issue=9 \|page=570 \|doi=10.2307/2975006\|accessdate=8 May 2013 \|doi=10.2307/2975006 \|issue=3\|jstor= 2975006}}</ref>The name "spigot algorithm" seems to have been coined by [[Stanley Rabinowitz]] and [[Stan Wagon]], whose algorithm for calculating the digits of {{pi}} is sometimes referred to as "''the'' spigot algorithm for {{pi}}".<ref>
	{{cite journal \|url=http://stanleyrabinowitz.com/bibliography/spigot.pdf \|title= A Spigot Algorithm for the Digits of Pi\|last1= Rabinowitz\|first1= Stanley\|last2=Wagon\|first2=Stan\|journal=American Mathematical Monthly\|volume=102\|year=1995\|pages=195–203\|accessdate=8 May 2013 \|doi=10.2307/2975006 \|issue=3\|jstor= 2975006}}</ref>		{{cite journal \|url=http://stanleyrabinowitz.com/bibliography/spigot.pdf \|title= A Spigot Algorithm for the Digits of Pi\|last1= Rabinowitz\|first1= Stanley\|last2=Wagon\|first2=Stan\|journal=American Mathematical Monthly\|volume=102\|year=1995\|pages=195–203\|accessdate=8 May 2013 \|doi=10.2307/2975006 \|issue=3\|jstor= 2975006}}</ref>