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Algorithm for integer multiplication
The Karatsuba algorithm is a fast multiplication algorithm for integers. It was discovered by Anatoly Karatsuba in 1960 and published in 1962. It is a
Karatsuba_algorithm
Algorithms which recursively solve subproblems
efficient algorithms for many problems, such as sorting (e.g., quicksort, merge sort), multiplying large numbers (e.g., the Karatsuba algorithm), finding
Divide-and-conquer_algorithm
Russian mathematician (1937–2008)
went to two editions, 1975 and 1983. The Karatsuba algorithm is the earliest known divide and conquer algorithm for multiplication and lives on as a special
Anatoly_Karatsuba
Algorithm to multiply two numbers
In 1960, Anatoly Karatsuba discovered Karatsuba multiplication, unleashing a flood of research into fast multiplication algorithms. This method uses
Multiplication_algorithm
Recursive algorithm for matrix multiplication
multiplication algorithm multiplies two complex numbers using 3 real multiplications instead of 4 Toom-Cook algorithm, a faster generalization of the Karatsuba algorithm
Strassen_algorithm
finding algorithm Cipolla's algorithm Tonelli–Shanks algorithm Multiplication algorithms: fast multiplication of two numbers Karatsuba algorithm Schönhage–Strassen
List_of_algorithms
Overview of and topical guide to algorithms
test Modular exponentiation Fast Fourier transform Karatsuba algorithm Schönhage–Strassen algorithm Gaussian elimination LU decomposition QR decomposition
Outline_of_algorithms
Method for division with remainder
efficient multiplication algorithm such as the Karatsuba algorithm, Toom–Cook multiplication or the Schönhage–Strassen algorithm. The result is that the
Division_algorithm
Algorithm for multiplying large numbers
additions and multiplication by small constants (Knuth, p. 296). The Karatsuba algorithm is equivalent to Toom-2, where the number is split into two smaller
Toom–Cook_multiplication
Multiplication algorithm
1971 until 2007. It is asymptotically faster than older methods such as Karatsuba and Toom–Cook multiplication, and starts to outperform them in practice
Schönhage–Strassen_algorithm
Number, approximately 3.14
They include the Karatsuba algorithm, Toom–Cook multiplication, and Fourier transform-based methods. The Gauss–Legendre iterative algorithm: Initialize a
Pi
Calculations where numbers' precision is only limited by computer memory
{\displaystyle \mathbb {Z} } . Fürer's algorithm Karatsuba algorithm Mixed-precision arithmetic Schönhage–Strassen algorithm Toom–Cook multiplication Little
Arbitrary-precision arithmetic
Arbitrary-precision_arithmetic
Algorithmic runtime requirements for common math procedures
The following tables list the computational complexity of various algorithms for common mathematical operations. Here, complexity refers to the time complexity
Computational complexity of mathematical operations
Computational_complexity_of_mathematical_operations
Decomposition of a number into a product
efficient non-quantum integer factorization algorithm is known. However, it has not been proven that such an algorithm does not exist. The presumed difficulty
Integer_factorization
Greatest integer less than or equal to square root
an example. The Karatsuba square root algorithm applies the same divide-and-conquer principle as the Karatsuba multiplication algorithm to compute integer
Integer_square_root
Branch of elementary mathematics
multiplication algorithms with a low computational complexity to be able to efficiently multiply very large integers, such as the Karatsuba algorithm, the Schönhage–Strassen
Arithmetic
Arithmetical operation
Multiplication algorithm Karatsuba algorithm, for large numbers Toom–Cook multiplication, for very large numbers Schönhage–Strassen algorithm, for huge numbers
Multiplication
Algorithmic technique
Transm. 27, No.4, 339-360 (1991); translation from Probl. Peredachi Inf. 27, No.4, 76–99 (1991). Ekatherina Karatsuba. Fast Algorithms and the FEE method
Binary_splitting
Exponent of a power of two
divide and conquer algorithms, such as the Karatsuba algorithm for multiplying n-bit numbers in time O(nlog2 3), and the Strassen algorithm for multiplying
Binary_logarithm
Prize in Economics winner Anatoly Karatsuba, developed the Karatsuba algorithm (the first fast multiplication algorithm) David Kazhdan, Soviet, American
List of Russian mathematicians
List_of_Russian_mathematicians
Multiplication: Multiplication algorithm — general discussion, simple methods Karatsuba algorithm — the first algorithm which is faster than straightforward
List of numerical analysis topics
List_of_numerical_analysis_topics
Anatoly Karatsuba, developed the Karatsuba algorithm (the first fast multiplication algorithm) Leonid Khachiyan, developed the Ellipsoid algorithm for linear
List_of_Russian_scientists
Anatoly Karatsuba, developed the Karatsuba algorithm (the first fast multiplication algorithm) Leonid Khachiyan, developed the Ellipsoid algorithm for linear
List_of_Russian_IT_developers
converting NFA into DFA published by Michael O. Rabin and Dana Scott 1960 – Karatsuba multiplication 1961 – CRC (Cyclic redundancy check) invented by W. Wesley
Timeline_of_algorithms
Analytic function in mathematics
S2CID 250796539. Karatsuba, A. A. (1996). "Density theorem and the behavior of the argument of the Riemann zeta function". Mat. Zametki (60): 448–449. Karatsuba, A
Riemann_zeta_function
first personal computer MIR Anatoly Karatsuba, developed the Karatsuba algorithm (the first fast multiplication algorithm) Yevgeny Kaspersky, developer of
List_of_Russian_people
Branch of pure mathematics
equation f ( x , y ) = 0 {\displaystyle f(x,y)=0} . Long 1972, p. 1. Karatsuba, A.A. (2020). "Number theory". Encyclopedia of Mathematics. Springer.
Number_theory
Mathematical operation on points on an elliptic curve
providing security just over 200 bits was proposed in which a variant of Karatsuba strategy was used to implement the field multiplication needed for the
Elliptic curve point multiplication
Elliptic_curve_point_multiplication
Conjecture on zeros of the zeta function
conjecture. The estimates of Selberg and Karatsuba can not be improved in respect of the order of growth as T → ∞. Karatsuba (1992) proved that an analog of the
Riemann_hypothesis
Division with remainder of integers
Haining Fan; Ming Gu; Jiaguang Sun; Kwok-Yan Lam (2012). "Obtaining More Karatsuba-Like Formulae over the Binary Field". IET Information Security. 6 (1):
Euclidean_division
Finite-state machine whose output values are determined only by its current state
N.J.(1956). Karatsuba A. A. Solution of one problem from the theory of finite automata. Usp. Mat. Nauk, 15:3, 157–159 (1960). Karatsuba A. A. Experimente
Moore_machine
possible to adapt fast integer multiplication algorithms such as the Karatsuba and Toom-Cook algorithms to work with carry-less multiplications. The definition
Carry-less_product
Fast summation method in mathematics
of series of a special form. It was constructed in 1990 by Ekaterina Karatsuba and is so-named because it makes fast computations of the Siegel E-functions
FEE_method
Constants of the mathematical zeta function
2006) below. A fast algorithm for the calculation of Riemann's zeta function for any integer argument is given by E. A. Karatsuba. In general, for negative
Particular values of the Riemann zeta function
Particular_values_of_the_Riemann_zeta_function
polynomial Karatsuba multiplication Lenstra–Lenstra–Lovász lattice basis reduction algorithm (for polynomial factorization) Lindsey–Fox algorithm Remez algorithm
List_of_polynomial_topics
converse is not necessarily true. Grantham's stated goal when developing the algorithm was to provide a test that primes would always pass and composites would
Quadratic_Frobenius_test
Variant of fast Fourier transform
accelerate eliptic curve cryptography over F(2521-1), the P-521. This is a Karatsuba-like technique featuring a cyclic convolution similar to IBDWT. For examples
Irrational base discrete weighted transform
Irrational_base_discrete_weighted_transform
Mathematical problem in number theory
A. A. Karatsuba, "Trigonometric sums in number theory and analysis". Berlin–New-York: Walter de Gruyter, (2004). G. I. Arkhipov, A. A. Karatsuba, V. N
Waring's_problem
Russian mathematician (born 1939)
589.{{cite journal}}: CS1 maint: untitled periodical (link) Anatolii A. Karatsuba and Yu. P. Ofman (1962), "Умножение многозначных чисел на автоматах" ("Multiplication
Yuri_Ofman
Extension of the factorial function
A fast algorithm for calculation of the Euler gamma function for any algebraic argument (including rational) was constructed by E.A. Karatsuba. For arguments
Gamma_function
Form of interpolation
essential to perform sub-quadratic multiplication and squaring, such as Karatsuba multiplication and Toom–Cook multiplication, where interpolation through
Polynomial_interpolation
ISSN 0022-2488. Karatsuba, Ekatherina A. (2007-07-20). "On an approach to the study of the Jaynes–Cummings sum in quantum optics". Numerical Algorithms. 45 (1–4)
ATS_theorem
Number, approximately 0.916
Ramanujan, for the second formula. The algorithms for fast evaluation of the Catalan constant were constructed by E. Karatsuba. Using these series, calculating
Catalan's_constant
Difference between logarithm and harmonic series
"Euler–Mascheroni constant". MathWorld. Jonathan Sondow. Fast Algorithms and the FEE Method, E.A. Karatsuba (2005) Further formulae which make use of the constant:
Euler's_constant
Approximation for factorials
Notebook and Other Unpublished Papers, p. 339 – via Internet Archive Karatsuba, Ekatherina A. (2001), "On the asymptotic representation of the Euler
Stirling's_approximation
KARATSUBA ALGORITHM
KARATSUBA ALGORITHM
KARATSUBA ALGORITHM
KARATSUBA ALGORITHM
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KARATSUBA ALGORITHM
KARATSUBA ALGORITHM
KARATSUBA ALGORITHM
KARATSUBA ALGORITHM
KARATSUBA ALGORITHM
n.
Alt. of Algorithm
n.
The art of calculating with any species of notation; as, the algorithms of fractions, proportions, surds, etc.
n.
The art of calculating by nine figures and zero.