Multiplication algorithm: Difference between revisions

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{{short description|Algorithm to multiply two numbers}}

{{Lead rewrite|date=August 2022}}

{{Use dmy dates|date=May 2019|cs1-dates=y}}

A '''multiplication algorithm''' is an [[algorithm]] (or method) to [[multiplication|multiply]] two numbers. Depending on the size of the numbers, different algorithms are more efficient than others. Numerous algorithms are known and there has been much research into the topic.

The oldest and simplest method, known since [[Ancient history|antiquity]] as '''long multiplication''' or '''grade-school multiplication''', consists of multiplying every digit in the first number by every digit in the second and adding the results. This has a [[time complexity]] of <math>O(n^2)</math>, where ''n'' is the number of digits. When done by hand, this may also be reframed as [[grid method multiplication]] or [[lattice multiplication]]. In software, this may be called "shift and add" due to [[bitshifts]] and addition being the only two operations needed.

In 1960, [[Anatoly Karatsuba]] discovered [[Karatsuba multiplication]], unleashing a flood of research into fast multiplication algorithms. This method uses three multiplications rather than four to multiply two two-digit numbers. (A variant of this can also be used to multiply [[complex numbers]] quickly.) Done [[recursively]], this has a time complexity of <math>O(n^{\log_2 3})</math>. Splitting numbers into more than two parts results in [[Toom-Cook multiplication]]; for example, using three parts results in the '''Toom-3''' algorithm. Using many parts can set the exponent arbitrarily close to 1, but the constant factor also grows, making it impractical.

In 1968, the [[Schönhage-StrasserStrassen algorithm]], which makes use of a [[Fourier transform]] over a [[Modulus (modular arithmetic)|modulus]], was discovered. It has a time complexity of <math>O(n\log n\log\log n)</math>. In 2007, [[Martin Fürer]] proposed an algorithm with complexity <math>O(n\log n 2^{\Theta(\log^* n)})</math>. In 2014, Harvey, [[Joris van der Hoeven]], and Lecerf proposed one with complexity <math>O(n\log n 2^{3\log^* n})</math>, thus making the [[implicit constant]] explicit; this was improved to <math>O(n\log n 2^{2\log^* n})</math> in 2018. Lastly, in 2019, Harvey and van der Hoeven came up with ana [[galactic algorithm]] with complexity <math>O(n\log n)</math>. This matches a guess by Schönhage and Strassen that this would be the optimal bound, although this remains a [[conjecture]] today.

Integer multiplication algorithms can also be used to multiply polynomials by means of the method of [[Kronecker substitution]].

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====Other notations====

In some countries such as [[Germany]], the above multiplication is depicted similarly but with the original product kept horizontal and computation starting with the first digit of the multiplier:<ref>{{Cite web |title=Multiplication |url=httphttps://www.mathematische-basteleien.de/multiplication.htm |access-date=2022-03-15 |website=www.mathematische-basteleien.de}}</ref>

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191665864

71874699

00000000

———————————————

139676498390

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|}

followed by addition to obtain 442, either in a single sum (see right), or through forming the row-by-row totals
: (300 + 40) + (90 + 12) = 340 + 102 = 442.

This calculation approach (though not necessarily with the explicit grid arrangement) is also known as the [[partial products algorithm]]. Its essence is the calculation of the simple multiplications separately, with all addition being left to the final gathering-up stage.

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====History of quarter square multiplication====

In prehistoric time, quarter square multiplication involved [[Floor and ceiling functions|floor function]]; that some sources<ref>{{citation |title= Quarter Tables Revisited: Earlier Tables, Division of Labor in Table Construction, and Later Implementations in Analog Computers |last=McFarland |first=David|url=httphttps://escholarship.org/uc/item/5n31064n |page=1 |year=2007}}</ref><ref>{{cite book| title=Mathematics in Ancient Iraq: A Social History |last=Robson |first=Eleanor |page=227 |year=2008 |publisher=Princeton University Press |isbn= 978-0691201405 }}</ref> attribute to [[Babylonian mathematics]] (2000–1600 BC).

Antoine Voisin published a table of quarter squares from 1 to 1000 in 1817 as an aid in multiplication. A larger table of quarter squares from 1 to 100000 was published by Samuel Laundy in 1856,<ref>{{Citation |title=Reviews |journal=The Civil Engineer and Architect's Journal |year=1857 |pages=54–55 |url=https://books.google.com/books?id=gcNAAAAAcAAJ&pg=PA54 |postscript=.}}</ref> and a table from 1 to 200000 by Joseph Blater in 1888.<ref>{{Citation|title=Multiplying with quarter squares |first=Neville |last=Holmes| journal=The Mathematical Gazette |volume=87 |issue=509 |year=2003 |pages=296–299 |jstor=3621048|postscript=.|doi=10.1017/S0025557200172778 |s2cid=125040256 }}</ref>

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{{unsolved|computer science|What is the fastest algorithm for multiplication of two <math>n</math>-digit numbers?}}

A line of research in [[theoretical computer science]] is about the number of single-bit arithmetic operations necessary to multiply two <math>n</math>-bit integers. This is known as the [[computational complexity]] of multiplication. Usual algorithms done by hand have asymptotic complexity of <math>O(n^2)</math>, but in 1960 [[Anatoly Karatsuba]] discovered that better complexity was possible (with the [[Karatsuba algorithm]]).<ref>{{cite web | url=https://youtube.com/watch?v=AMl6EJHfUWo | title= The Genius Way Computers Multiply Big Numbers| website=[[YouTube]]| date= 2 January 2025}}</ref>

Currently, the algorithm with the best computational complexity is a 2019 algorithm of [[David Harvey (mathematician)|David Harvey]] and [[Joris van der Hoeven]], which uses the strategies of using [[number-theoretic transform]]s introduced with the [[Schönhage–Strassen algorithm]] to multiply integers using only <math>O(n\log n)</math> operations.<ref>{{cite journal | last1 = Harvey | first1 = David | last2 = van der Hoeven | first2 = Joris | author2-link = Joris van der Hoeven | doi = 10.4007/annals.2021.193.2.4 | issue = 2 | journal = [[Annals of Mathematics]] | mr = 4224716 | pages = 563–617 | series = Second Series | title = Integer multiplication in time <math>O(n \log n)</math> | volume = 193 | year = 2021| s2cid = 109934776 | url = https://hal.archives-ouvertes.fr/hal-02070778v2/file/nlogn.pdf }}</ref> This is conjectured to be the best possible algorithm, but lower bounds of <math>\Omega(n\log n)</math> are not known.

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By exploring patterns after expansion, one see following:

<math display="block">\begin{alignat}{5} (x_1 B^{ m} + x_0) (y_1 B^{m} + y_0) (z_1 B^{ m} + z_0) (a_1 B^{ m} + a_0) &= </math> <br>

<math> a_1 x_1 y_1 z_1 B^{4m4 m} &+ a_1 x_1 y_1 z_0 B^{3m} &+ a_1 x_1 y_0 z_1 B^{3 m} &+ a_1 x_0 y_1 z_1 B^{3 m} </math> <br>\\

<math>&+ a_0 x_1 y_1 z_1 B^{3 m} &+ a_1 x_1 y_0 z_0 B^{2 m} &+ a_1 x_0 y_1 z_0 B^{2 m} &+ a_0 x_1 y_1 z_0 B^{2 m}</math>\\ <br>

<math> &+ a_1 x_0 y_0 z_1 B^{2 m} &+ a_0 x_1 y_0 z_1 B^{2 m} &+ a_0 x_0 y_1 z_1 B^{2 m} &+ a_1 x_0 y_0 z_0 B^{ m\phantom{1}</math> <br>}\\

<math>&+ a_0 x_1 y_0 z_0 B^{m\phantom{1}} &+ a_0 x_0 y_1 z_0 B^{m\phantom{1}} &+ a_0 x_0 y_0 z_1 B^{ m\phantom{1}} &+ a_0 x_0 y_0 z_0 </math>\phantom{B^{1 m}}

\end{alignat}</math>

Each summand is associated to a unique binary number from 0 to

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If we express this in fewer terms, we get:

<math> display="block">\prod_{j=1}^N (x_{j,1} B^{ m} + x_{j,0}) = \sum_{i=1}^{2^{N+1}-1}\prod_{j=1}^N x_{j,c(i,j)}B^{m\sum_{j=1}^N c(i,j)} = \sum_{j=0}^{N}z_jB^{jm}

\prod_{j=1}^N x_{j,c(i,j)}B^{m\sum_{j=1}^N c(i,j)} = \sum_{j=0}^{N}z_jB^{jm}

</math>, where <math> c(i,j) </math> means digit in number i at position j. Notice that <math> c(i,j) \in \{0,1\} </math>

<math display="block">

\begin{align}

z_{0} &= \prod_{j=1}^N x_{j,0}

\\

</math><br>

z_{N} &= \prod_{j=1}^N x_{j,1}

<math>

\\

z_{N} = \prod_{j=1}^N x_{j,1}

z_{N-1} &= \prod_{j=1}^N (x_{j,0} + x_{j,1}) - \sum_{i \ne N-1}^{N} z_i

</math><br>

\end{align}

<math>

z_{N-1} = \prod_{j=1}^N (x_{j,0} + x_{j,1}) - \sum_{i \ne N-1}^{N} z_i

</math>

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{{Main|Schönhage–Strassen algorithm}}

[[File:Integer multiplication by FFT.svg|thumb|350px|Demonstration of multiplying 1234 × 5678 = 7006652 using fast Fourier transforms (FFTs). [[Number-theoretic transform]]s in the integers modulo 337 are used, selecting 85 as an 8th root of unity. Base 10 is used in place of base 2^''w'' for illustrative purposes.]]

Every number in base B, can be written as a polynomial:

<math display="block"> X = \sum_{i=0}^N {x_iB^i} </math>

Furthermore, multiplication of two numbers could be thought of as a product of two polynomials:

<math display="block">XY = (\sum_{i=0}^N {x_iB^i})(\sum_{j=0}^N {y_iB^j}) </math>

Because,for <math> B^k </math>: <math>c_k =\sum_{(i,j):i+j=k} {a_ib_j} = \sum_{i=0}^k {a_ib_{k-i}} </math>,

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By using fft (fast fourier transformation) with convolution rule, we can get

<math display="block"> \hat{f}(a * b) = \hat{f}(\sum_{i=0}^k {a_ib_{k-i}}) = \hat{f}(a)</math> ● <math>\bullet \hat{f}(b) </math>. That is; <math> C_k = a_k </math> ● <math>\bullet b_k </math>, where <math> C_k </math>

is the corresponding coefficient in fourier space. This can also be written as: <math>\mathrm{fft}(a * b) = \mathrm{fft}(a) ●\bullet \mathrm{fft}(b)</math>.

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only consist of one unique term per coefficient:

<math display="block"> \hat{f}(x^n) = \left(\frac{i}{2\pi}\right)^n \delta^{(n)} </math> and

<math display="block"> \hat{f}(a\, X(\xi) + b\, Y(\xi)) = a\, \hat{X}(\xi) + b\, \hat{Y}(\xi)</math>

Convolution rule: <math> \hat{f}(X * Y) = \ \hat{f}(X) </math> ● <math> \hat{f}(Y) </math>

* Convolution rule: <math> \hat{f}(X * Y) = \ \hat{f}(X) </math> ● <math>\bullet \hat{f}(Y) </math>

We have reduced our convolution problem

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By finding ifft (polynomial interpolation), for each <math>c_k </math>, one get the desired coefficients.

Algorithm uses divide and conquer strategy, to divide problem to subproblems.

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==== History ====

The algorithm was invented by [[Volker Strassen|Strassen]] (1968). It was made practical and theoretical guarantees were provided in 1971 by [[Arnold Schönhage|Schönhage]] and Strassen resulting in the [[Schönhage–Strassen algorithm]].<ref name="schönhage">{{cite journal |first1=A. |last1=Schönhage |first2=V. |last2=Strassen |title=Schnelle Multiplikation großer Zahlen |journal=Computing |volume=7 |issue= 3–4|pages=281–292 |date=1971 |doi=10.1007/BF02242355 |s2cid=9738629 |url=https://link.springer.com/article/10.1007/BF02242355|url-access=subscription }}</ref>

=== Further improvements ===

In 2007 the [[asymptotic complexity]] of integer multiplication was improved by the Swiss mathematician [[Martin Fürer]] of Pennsylvania State University to ''<math display="inline">O(n''&nbsp; \log('' n'')&nbsp; \cdot {2<sup>Θ}^{\Theta([[iterated logarithm|\log^{^*}]](''n''))})</supmath> using Fourier transforms over [[complex number]]s,<ref name="fürer_1">{{cite book |first=M. |last=Fürer |chapter=Faster Integer Multiplication |chapter-url=https://ivv5hpp.uni-muenster.de/u/cl/WS2007-8/mult.pdf |doi=10.1145/1250790.1250800 |title=Proceedings of the thirty-ninth annual ACM symposium on Theory of computing, June 11–13, 2007, San Diego, California, USA |publisher= |___location= |date=2007 |isbn=978-1-59593-631-8 |pages=57–66 |s2cid=8437794 |url=}}</ref> where log^* denotes the [[iterated logarithm]]. Anindya De, Chandan Saha, Piyush Kurur and Ramprasad Saptharishi gave a similar algorithm using [[modular arithmetic]] in 2008 achieving the same running time.<ref>{{cite book |first1=A. |last1=De |first2=C. |last2=Saha |first3=P. |last3=Kurur |first4=R. |last4=Saptharishi |chapter=Fast integer multiplication using modular arithmetic |chapter-url= |doi=10.1145/1374376.1374447 |title=Proceedings of the 40th annual ACM Symposium on Theory of Computing (STOC) |publisher= |___location= |date=2008 |isbn=978-1-60558-047-0 |pages=499–506 |url= |arxiv=0801.1416|s2cid=3264828 }}</ref> In context of the above material, what these latter authors have achieved is to find ''N'' much less than 2^3''k'' + 1, so that ''Z''/''NZ'' has a (2''m'')th root of unity. This speeds up computation and reduces the time complexity. However, these latter algorithms are only faster than Schönhage–Strassen for impractically large inputs.

In 2014, Harvey, [[Joris van der Hoeven]] and Lecerf<ref>{{cite journal

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===Lower bounds===

There is a trivial lower bound of [[Big O notation#Family of Bachmann–Landau notations|Ω]](''n'') for multiplying two ''n''-bit numbers on a single processor; no matching algorithm (on conventional machines, that is on Turing equivalent machines) nor any sharper lower bound is known. The [[Hartmanis–Stearns conjecture]] would imply that <math>O(n)</math> cannot be achieved. Multiplication lies outside of [[ACC0|AC⁰[''p'']]] for any prime ''p'', meaning there is no family of constant-depth, polynomial (or even subexponential) size circuits using AND, OR, NOT, and MOD_''p'' gates that can compute a product. This follows from a constant-depth reduction of MOD_''q'' to multiplication.<ref>{{cite book |first1=Sanjeev |last1=Arora |first2=Boaz |last2=Barak |title=Computational Complexity: A Modern Approach |publisher=Cambridge University Press |date=2009 |isbn=978-0-521-42426-4 |url={{GBurl|8Wjqvsoo48MC|pg=PR7}}}}</ref> Lower bounds for multiplication are also known for some classes of [[branching program]]s.<ref>{{cite journal |first1=F. |last1=Ablayev |first2=M. |last2=Karpinski |title=A lower bound for integer multiplication on randomized ordered read-once branching programs |journal=Information and Computation |volume=186 |issue=1 |pages=78–89 |date=2003 |doi=10.1016/S0890-5401(03)00118-4 |url=https://core.ac.uk/download/pdf/82445954.pdf}}</ref>

==Complex number multiplication==

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This algorithm uses only three multiplications, rather than four, and five additions or subtractions rather than two. If a multiply is more expensive than three adds or subtracts, as when calculating by hand, then there is a gain in speed. On modern computers a multiply and an add can take about the same time so there may be no speed gain. There is a trade-off in that there may be some loss of precision when using floating point.

For [[fast Fourier transform]]s (FFTs) (or any [[Linear map|linear transformation]]) the complex multiplies are by constant coefficients ''c''&nbsp;+&nbsp;''di'' (called [[twiddle factor]]s in FFTs), in which case two of the additions (''d''−''c'' and ''c''+''d'') can be precomputed. Hence, only three multiplies and three adds are required.<ref>{{cite journal |first1=P. |last1=Duhamel |first2=M. |last2=Vetterli |title=Fast Fourier transforms: A tutorial review and a state of the art |journal=Signal Processing |volume=19 |issue=4 |pages=259–299 See Section 4.1 |date=1990 |doi=10.1016/0165-1684(90)90158-U |bibcode=1990SigPr..19..259D |url=https://core.ac.uk/download/pdf/147907050.pdf}}</ref> However, trading off a multiplication for an addition in this way may no longer be beneficial with modern [[floating-point unit]]s.<ref>{{cite journal |first1=S.G. |last1=Johnson |first2=M. |last2=Frigo |title=A modified split-radix FFT with fewer arithmetic operations |journal=IEEE Trans. Signal Process. |volume=55 |issue= 1|pages=111–9 See Section IV |date=2007 |doi=10.1109/TSP.2006.882087 |bibcode=2007ITSP...55..111J |s2cid=14772428 |url=https://www.fftw.org/newsplit.pdf }}</ref>

==Polynomial multiplication==

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* [[Horner scheme]] for evaluating of a polynomial

* [[Logarithm]]

* [[Matrix multiplication algorithm]]

* [[Mental calculation]]

* [[Number-theoretic transform]]

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===Basic arithmetic===

* [httphttps://www.nychold.com/em-arith.html The Many Ways of Arithmetic in UCSMP Everyday Mathematics]

* [httphttps://math.widulski.net/slides/CH05_MustAllGoodThings.ppt A Powerpoint presentation about ancient mathematics]

* [httphttps://www.pedagonet.com/maths/lattice.htm Lattice Multiplication Flash Video]

===Advanced algorithms===

* [httphttps://gmplib.org/manual/Multiplication-Algorithms.html#Multiplication%20Algorithms Multiplication Algorithms used by GMP]

{{Number-theoretic algorithms}}