Number Theory
Algorithms for number theoretical calculations are studied in computational number theory.
Operation | Input | Output | Algorithm | Complexity |
---|---|---|---|---|
Greatest common divisor | Two n-digit numbers | One number with at most n digits | Euclidean algorithm | O(n2) |
Binary GCD algorithm | O(n2) | |||
Left/right k-ary binary GCD algorithm | O(n2/ log n) | |||
Stehlé–Zimmermann algorithm | O(log n M(n)) | |||
Schönhage controlled Euclidean descent algorithm | O(log n M(n)) | |||
Jacobi symbol | Two n-digit numbers | 0, −1, or 1 | ||
Schönhage controlled Euclidean descent algorithm | O(log n M(n)) | |||
Stehlé–Zimmermann algorithm | O(log n M(n)) | |||
Factorial | A fixed-size number m | One O(m log m)-digit number | Bottom-up multiplication | O(m2 log m) |
Binary splitting | O(log m M(m log m)) | |||
Exponentiation of the prime factors of m | O(log log m M(m log m)), O(M(m log m)) |
Read more about this topic: Computational Complexity Of Mathematical Operations
Famous quotes containing the words number and/or theory:
“There is something tragic about the enormous number of young men there are in England at the present moment who start life with perfect profiles, and end by adopting some useful profession.”
—Oscar Wilde (18541900)
“Could Shakespeare give a theory of Shakespeare?”
—Ralph Waldo Emerson (18031882)