Large Integer Methods
Methods designed for hardware implementation generally do not scale to integers with thousands or millions of decimal digits; these frequently occur, for example, in modular reductions in cryptography. For these large integers, more efficient division algorithms transform the problem to use a small number of multiplications, which can then be done using an asymptotically efficient multiplication algorithm such as the Karatsuba algorithm, Toom–Cook multiplication or the Schönhage–Strassen algorithm. It results that the computational complexity of the division is of the same order (up a multiplicative constant) as that of the multiplication. Examples include reduction to multiplication by Newton's method as described above as well as the slightly faster Barrett reduction algorithm. Newton's method's is particularly efficient in scenarios where one must divide by the same divisor many times, since after the initial Newton inversion only one (truncated) multiplication is needed for each division.
Read more about this topic: Division Algorithm
Famous quotes containing the words large and/or methods:
“In looking back over the college careers of those who for various reasons have been prominent in undergraduate life ... one cannot help noticing that these men have nearly always shown from the start an interest in the lives of their fellow students. A large acquaintance means that many persons are dependent on a man and conversely that he himself is dependent on many. Success necessarily means larger responsibilities, and responsibilities mean many friends.”
—Franklin D. Roosevelt (1882–1945)
“A writer who writes, “I am alone” ... can be considered rather comical. It is comical for a man to recognize his solitude by addressing a reader and by using methods that prevent the individual from being alone. The word alone is just as general as the word bread. To pronounce it is to summon to oneself the presence of everything the word excludes.”
—Maurice Blanchot (b. 1907)