Computational Complexity of Mathematical Operations - Arithmetic Functions

Arithmetic Functions

Operation Input Output Algorithm Complexity
Addition Two n-digit numbers One n+1-digit number Schoolbook addition with carry Θ(n)
Subtraction Two n-digit numbers One n+1-digit number Schoolbook subtraction with borrow Θ(n)
Multiplication Two n-digit numbers
One 2n-digit number Schoolbook long multiplication O(n2)
Karatsuba algorithm O(n1.585)
3-way Toom–Cook multiplication O(n1.465)
k-way Toom–Cook multiplication O(nlog (2k − 1)/log k)
Mixed-level Toom–Cook (Knuth 4.3.3-T) O(n 2√2 log n log n)
Schönhage–Strassen algorithm O(n log n log log n)
Fürer's algorithm O(n log n 2log* n)
Division Two n-digit numbers One n-digit number Schoolbook long division O(n2)
Newton–Raphson division O(M(n))
Square root One n-digit number One n-digit number Newton's method O(M(n))
Modular exponentiation Two n-digit numbers and a k-bit exponent One n-digit number Repeated multiplication and reduction O(2kM(n))
Exponentiation by squaring O(k M(n))
Exponentiation with Montgomery reduction O(k M(n))

Schnorr and Stumpf conjectured that no fastest algorithm for multiplication exists.

Read more about this topic:  Computational Complexity Of Mathematical Operations

Famous quotes containing the words arithmetic and/or functions:

    O! O! another stroke! that makes the third.
    He stabs me to the heart against my wish.
    If that be so, thy state of health is poor;
    But thine arithmetic is quite correct.
    —A.E. (Alfred Edward)

    Adolescents, for all their self-involvement, are emerging from the self-centeredness of childhood. Their perception of other people has more depth. They are better equipped at appreciating others’ reasons for action, or the basis of others’ emotions. But this maturity functions in a piecemeal fashion. They show more understanding of their friends, but not of their teachers.
    Terri Apter (20th century)