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Karatsuba multiplication of az+b and cz+d (boxed), and 1234 and 567 with z=100. Magenta arrows denote multiplication, amber denotes addition, silver denotes subtraction and cyan denotes left shift. (A), (B) and (C) show recursion with z=10 to obtain intermediate values. The Karatsuba algorithm is a fast multiplication algorithm.
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.)
The run-time bit complexity to multiply two n-digit numbers using the algorithm is ( ) in big O notation. The Schönhage–Strassen algorithm was the asymptotically fastest multiplication method known from 1971 until 2007.
Trachtenberg defined this algorithm with a kind of pairwise multiplication where two digits are multiplied by one digit, essentially only keeping the middle digit of the result. By performing the above algorithm with this pairwise multiplication, even fewer temporary results need to be held. Example:
Two -digit numbers One +-digit number Schoolbook addition with carry Subtraction: Two -digit numbers One + ... Fast matrix multiplication [30] ...
Another method is to simply multiply the number by 10, and add the original number to the result. For example: 17 × 11 17 × 10 = 170 170 + 17 = 187 17 × 11 = 187 One last easy way: If one has a two-digit number, take it and add the two numbers together and put that sum in the middle, and one can get the answer.