“Karatsuba Multiplication”的概念、定义、习题解题方法及翻译-数学辞典

单词 Karatsuba Multiplication

释义

Karatsuba Multiplication

It is possible to perform Multiplication of Large Numbers in (many) fewer operations than theusual brute-force technique of ``long multiplication.'' As discovered by Karatsuba and Ofman (1962), Multiplication oftwo -Digit numbers can be done with a Bit Complexity of less than using identities of the form

(1)

Proceeding recursively then gives Bit Complexity

, where

(Borwein et al. 1989). The best known bound is

steps for

(Schönhage and Strassen 1971, Knuth1981). However, this Algorithm is difficult to implement, but a procedure based on the Fast Fourier Transform isstraightforward to implement and gives Bit Complexity

(Brigham 1974, Borodin andMunro 1975, Knuth 1981, Borwein et al. 1989).

As a concrete example, consider Multiplication of two numbers each just two ``digits'' long in base ,

			(2)
			(3)

then their Product is



			(4)

Instead of evaluating products of individual digits, now write

			(5)
			(6)
			(7)

The key term is

, which can be expanded, regrouped, and written in terms of the

(8)

However, since

, and

, it immediately follows that

			(9)
			(10)
			(11)

so the three ``digits'' of

have been evaluated using three multiplications rather than four. The technique can begeneralized to multidigit numbers, with the trade-off being that more additions and subtractions are required.

Now consider four-``digit'' numbers

(12)

which can be written as a two-``digit'' number represented in the base

(13)

The ``digits'' in the new base are now

			(14)
			(15)

and the Karatsuba algorithm can be applied to

and

in this form. Therefore, the Karatsuba algorithm is notrestricted to multiplying two-digit numbers, but more generally expresses the multiplication of two numbers in terms ofmultiplications of numbers of half the size. The asymptotic speed the algorithm obtains by recursive applicationto the smaller required subproducts is

(Knuth 1981).

When this technique is recursively applied to multidigit numbers, a point is reached in the recursion when the overhead ofadditions and subtractions makes it more efficient to use the usual Multiplication algorithm to evaluatethe partial products. The most efficient overall method therefore relies on a combination of Karatsuba and conventionalmultiplication.

See also Complex Multiplication, Multiplication, Strassen Formulas

References

Borodin, A. and Munro, I. The Computational Complexity of Algebraic and Numeric Problems. New York: American Elsevier, 1975.

Borwein, J. M.; Borwein, P. B.; and Bailey, D. H. ``Ramanujan, Modular Equations, and Approximations to Pi, or How to Compute One Billion Digits of Pi.'' Amer. Math. Monthly 96, 201-219, 1989.

Brigham, E. O. The Fast Fourier Transform. Englewood Cliffs, NJ: Prentice-Hall, 1974.

Brigham, E. O. Fast Fourier Transform and Applications. Englewood Cliffs, NJ: Prentice-Hall, 1988.

Cook, S. A. On the Minimum Computation Time of Functions. Ph.D. Thesis. Cambridge, MA: Harvard University, pp. 51-77, 1966.

Hollerbach, U. ``Fast Multiplication & Division of Very Large Numbers.'' sci.math.research posting, Jan. 23, 1996.

Karatsuba, A. and Ofman, Yu. ``Multiplication of Many-Digital Numbers by Automatic Computers.'' Doklady Akad. Nauk SSSR 145, 293-294, 1962. Translation in Physics-Doklady 7, 595-596, 1963.

Knuth, D. E. The Art of Computing, Vol. 2: Seminumerical Algorithms, 2nd ed. Reading, MA: Addison-Wesley, pp. 278-286, 1981.

Schönhage, A. and Strassen, V. ``Schnelle Multiplikation Grosser Zahlen.'' Computing 7, 281-292, 1971.

Toom, A. L. ``The Complexity of a Scheme of Functional Elements Simulating the Multiplication of Integers.'' Dokl. Akad. Nauk SSSR 150, 496-498, 1963. English translation in Soviet Mathematics 3, 714-716, 1963.

Zuras, D. ``More on Squaring and Multiplying Large Integers.'' IEEE Trans. Comput. 43, 899-908, 1994.

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