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:
“O thou undaunted daughter of desires!
By all thy dower of lights and fires;
By all the eagle in thee, all the dove;
By all thy lives and deaths of love;
By thy large draughts of intellectual day,
And by thy thirsts of love more large then they;
By all thy brim-fill’d Bowls of fierce desire,
By thy last Morning’s draught of liquid fire;
By the full kingdom of that final kiss
That seiz’d thy parting Soul, and seal’d thee his;”
—Richard Crashaw (1613?–1649)
“Cold and hunger seem more friendly to my nature than those methods which men have adopted and advise to ward them off.”
—Henry David Thoreau (1817–1862)