Einstein–Hilbert action: Difference between revisions

Revision as of 16:13, 13 December 2013

In mathematics and computer science, optimal addition-chain exponentiation is a method of exponentiation by positive integer powers that requires a minimal number of multiplications. It works by creating a shortest addition chain that generates the desired exponent. Each exponentiation in the chain can be evaluated by multiplying two of the earlier exponentiation results. More generally, addition-chain exponentiation may also refer to exponentiation by non-minimal addition chains constructed by a variety of algorithms (since a shortest addition chain is very difficult to find).

The shortest addition-chain algorithm requires no more multiplications than binary exponentiation and usually less. The first example of where it does better is for a¹⁵, where the binary method needs six multiplies but a shortest addition chain requires only five:

a^{15} = a \times (a \times [a \times a^{2}]^{2})^{2}

(binary, 6 multiplications)

a^{15} = a^{3} \times ([a^{3}]^{2})^{2}

(shortest addition chain, 5 multiplications).

Table demonstrating how to do *Exponentiation* using *Addition Chains*
Number of Multiplications	Actual Exponentiation	Specific implementation of Addition Chains to do Exponentiation
0	a¹	a
1	a²	a × a
2	a³	a × a × a
2	a⁴	(a × a→b) × b
3	a⁵	(a × a→b) × b × a
3	a⁶	(a × a→b) × b × b
4	a⁷	(a × a→b) × b × b × a
3	a⁸	((a × a→b) × b→d) × d
4	a⁹	(a × a × a→c) × c × c
4	a¹⁰	((a × a→b) × b→d) × d × b
5	a¹¹	((a × a→b) × b→d) × d × b × a
4	a¹²	((a × a→b) × b→d) × d × d
5	a¹³	((a × a→b) × b→d) × d × d × a
5	a¹⁴	((a × a→b) × b→d) × d × d × b
5	a¹⁵	((a × a→b) × b × a→e) × e × e
4	a¹⁶	(((a × a→b) × b→d) × d→h) × h

On the other hand, the addition-chain method is much more complicated, since the determination of a shortest addition chain seems quite difficult: no efficient optimal methods are currently known for arbitrary exponents, and the related problem of finding a shortest addition chain for a given set of exponents has been proven NP-complete.^[1] Even given a shortest chain, addition-chain exponentiation requires more memory than the binary method, because it must potentially store many previous exponents from the chain simultaneously. In practice, therefore, shortest addition-chain exponentiation is primarily used for small fixed exponents for which a shortest chain can be precomputed and is not too large.

However, there are also several methods to approximate a shortest addition chain, and which often require fewer multiplications than binary exponentiation. Indeed, binary exponentiation itself is a suboptimal addition-chain algorithm. The optimal algorithm choice depends on the context (such as the relative cost of the multiplication and the number of times a given exponent is re-used).^[2]

Note that the problem of finding the shortest addition chain cannot be solved by dynamic programming, because it does not satisfy the assumption of optimal substructure. That is, it is not sufficient to decompose the power into smaller powers, each of which is computed minimally, since the addition chains for the smaller powers may be related (to share computations). For example, in the shortest addition chain for a¹⁵ above, the subproblem for a⁶ must be computed as (a³)² since a³ is re-used (as opposed to, say, a⁶ = a²(a²)², which also requires three multiplies).

Addition-subtraction–chain exponentiation

If both multiplication and division are allowed, then an addition-subtraction chain may be used to obtain even fewer total multiplications+divisions (where subtraction corresponds to division). However, the slowness of division compared to multiplication makes this technique unattractive in general. For exponentiation to negative integer powers, on the other hand, since one division is required anyway, an addition-subtraction chain is often beneficial. One such example is a⁻³¹, where computing 1/a³¹ by a shortest addition chain for a³¹ requires 7 multiplications and one division, whereas the shortest addition-subtraction chain requires 5 multiplications and one division:

a^{- 31} = a / ((((a^{2})^{2})^{2})^{2})^{2}

(addition-subtraction chain, 5 mults + 1 div).

For exponentiation on elliptic curves, the inverse of a point (x, y) is available at no cost, since it is simply (x, −y), and therefore addition-subtraction chains are optimal in this context even for positive integer exponents.^[3]

References

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Donald E. Knuth, The Art of Computer Programming, Volume 2: Seminumerical Algorithms, 3rd edition, §4.6.3 (Addison-Wesley: San Francisco, 1998).
Daniel J. Bernstein, "Pippenger's Algorithm," to be incorporated into author's High-speed cryptography book. (2002)

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The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

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In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
↑ François Morain and Jorge Olivos, "Speeding up the computations on an elliptic curve using addition-subtraction chains," RAIRO Informatique théoretique et application 24, pp. 531-543 (1990).

[1] One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

[2] One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

[3] François Morain and Jorge Olivos, "Speeding up the computations on an elliptic curve using addition-subtraction chains," RAIRO Informatique théoretique et application 24, pp. 531-543 (1990).

[1]

[2]

[3]

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+In [[mathematics]] and [[computer science]], optimal '''addition-chain exponentiation''' is a method of [[exponentiation]] by positive [[integer]] powers that requires a minimal number of multiplications. It works by creating a shortest [[addition chain]] that generates the desired exponent. Each exponentiation in the chain can be evaluated by multiplying  two of the earlier exponentiation results.  More generally, ''addition-chain exponentiation'' may also refer to exponentiation by non-minimal addition chains constructed by a variety of algorithms (since a shortest addition chain is very difficult to find).
+The shortest addition-chain [[algorithm]] requires no more multiplications than [[binary exponentiation]] and usually less. The first example of where it does better is for ''a''<sup>15</sup>, where the binary method needs six multiplies but a shortest addition chain requires only five:
+:<math>a^{15} = a \times (a \times [a \times a^2]^2)^2  \!</math>  (binary, 6 multiplications)
+:<math>a^{15} = a^3 \times ([a^3]^2)^2  \!</math> (shortest addition chain, 5 multiplications).
+{|class=wikitable
+|+Table demonstrating how to do ''Exponentiation'' using ''Addition Chains''
+|-
+!Number of<br>Multiplications || Actual<br>Exponentiation || Specific implementation of<br>''Addition Chains'' to do Exponentiation
+|-
+|0|| a<sup>1</sup> || a
+|-
+|1|| a<sup>2</sup> || a × a
+|-
+|2|| a<sup>3</sup> || a × a × a
+|-
+|2|| a<sup>4</sup> || (a × a→b) × b
+|-
+|3|| a<sup>5</sup> || (a × a→b) × b × a
+|-
+|3|| a<sup>6</sup> || (a × a→b) × b × b
+|-
+|4|| a<sup>7</sup> || (a × a→b) × b × b × a
+|-
+|3|| a<sup>8</sup> || ((a × a→b) × b→d) × d
+|-
+|4|| a<sup>9</sup> || (a × a × a→c) × c × c
+|-
+|4|| a<sup>10</sup> || ((a × a→b) × b→d) × d × b
+|-
+|5|| a<sup>11</sup> || ((a × a→b) × b→d) × d × b × a
+|-
+|4|| a<sup>12</sup> || ((a × a→b) × b→d) × d × d
+|-
+|5|| a<sup>13</sup> || ((a × a→b) × b→d) × d × d × a
+|-
+|5|| a<sup>14</sup> || ((a × a→b) × b→d) × d × d × b
+|-
+|5|| a<sup>15</sup> || ((a × a→b) × b × a→e) × e × e
+|-
+|4|| a<sup>16</sup> || (((a × a→b) × b→d) × d→h) × h
+|}
+On the other hand, the addition-chain method is much more complicated, since the determination of a shortest addition chain seems quite difficult: no efficient optimal methods are currently known for arbitrary exponents, and the related problem of finding a shortest addition chain for a given set of exponents has been proven [[NP-complete]].<ref>{{Cite journal|first1=Peter|last1=Downey|first2=Benton|last2=Leong|first3=Ravi|last3=Sethi|title=Computing sequences with addition chains|journal=SIAM Journal on Computing|volume=10|issue=3|year=1981|pages=638–646|doi=10.1137/0210047}}</ref> Even given a shortest chain, addition-chain exponentiation requires more memory than the binary method, because it must potentially store many previous exponents from the chain simultaneously.  In practice, therefore, shortest addition-chain exponentiation is primarily used for small fixed exponents for which a shortest chain can be precomputed and is not too large.
+However, there are also several methods to ''approximate'' a shortest addition chain, and which often require fewer multiplications than binary exponentiation. Indeed, binary exponentiation itself is a suboptimal addition-chain algorithm. The optimal algorithm choice depends on the context (such as the relative cost of the multiplication and the number of times a given exponent is re-used).<ref>{{cite journal |author=Gordon, D. M. |year=1998 |title=A survey of fast exponentiation methods |journal=J. Algorithms |volume=27 |pages=129–146 |doi=10.1006/jagm.1997.0913 |url=http://saluc.engr.uconn.edu/refs/sidechannel/gordon97asurvey.pdf}}</ref>
+Note that the problem of finding the shortest addition chain cannot be solved by [[dynamic programming]], because it does not satisfy the assumption of [[optimal substructure]].  That is, it is not sufficient to decompose the power into smaller powers, each of which is computed minimally, since the addition chains for the smaller powers may be related (to share computations).  For example, in the shortest addition chain for ''a''<sup>15</sup> above, the subproblem for ''a''<sup>6</sup> must be computed as (''a''<sup>3</sup>)<sup>2</sup> since ''a''<sup>3</sup> is re-used (as opposed to, say, ''a''<sup>6</sup>&nbsp;=&nbsp;''a''<sup>2</sup>(''a''<sup>2</sup>)<sup>2</sup>, which also requires three multiplies).
+==Addition-subtraction–chain exponentiation==
+If both multiplication and division are allowed, then an [[addition-subtraction chain]] may be used to obtain even fewer total multiplications+divisions (where subtraction corresponds to division). However, the slowness of division compared to multiplication makes this technique unattractive in general.  For exponentiation to [[negative number|negative]] integer powers, on the other hand, since one division is required anyway, an addition-subtraction chain is often beneficial.  One such example is ''a''<sup>&minus;31</sup>, where computing 1/''a''<sup>31</sup> by a shortest addition chain for ''a''<sup>31</sup> requires 7 multiplications and one division, whereas the shortest addition-subtraction chain requires 5 multiplications and one division:
+:<math>a^{-31} = a / ((((a^2)^2)^2)^2)^2 \!</math> (addition-subtraction chain, 5 mults + 1 div).
+For exponentiation on [[elliptic curve]]s, the inverse of a point (''x'',&nbsp;''y'') is available at no cost, since it is simply (''x'',&nbsp;&minus;''y''), and therefore addition-subtraction chains are optimal in this context even for positive integer exponents.<ref>François Morain and Jorge Olivos, "[ftp://ftp.inria.fr/INRIA/publication/Theses/TU-0144/ch4.ps Speeding up the computations on an elliptic curve using addition-subtraction chains]," ''RAIRO Informatique théoretique et application'' '''24''', pp. 531-543 (1990).</ref>
+==References==
+{{reflist}}
+* [[Donald E. Knuth]], ''The Art of Computer Programming, Volume 2: Seminumerical Algorithms'', 3rd edition, §4.6.3 (Addison-Wesley: San Francisco, 1998).
+* Daniel J. Bernstein, "[http://cr.yp.to/papers/pippenger.pdf Pippenger's Algorithm]," to be incorporated into author's ''High-speed cryptography'' book. (2002)
+[[Category:Addition chains]]
+[[Category:Exponentials]]
+[[Category:Computer arithmetic algorithms]]

Einstein–Hilbert action: Difference between revisions

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Addition-subtraction–chain exponentiation

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