Multiplication algorithm: Difference between revisions

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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-Strassen 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 [[gallacticgalactic 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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71874699

00000000

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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}}