3 October, Day 11
Chapter 25 – Fast Multiplication
How do computers actually multiply numbers together? We know the
hardware performs the processing, but how? People memorize rules to
help us with multiplication, but what is the actual algorithm?
Consider “standard multiplication” in decimal, eg 123 * 57. To
calculate must a) break up the problem by column, b) perform
standard rules of multiplication (eg, 7*3, 7*2, 7*1, etc) to form partial
results, and then c) add the partial results together. As the size of the
multiplicands increases, so too does the time to solve (but not just
linearly as we will see).
As multiplication problem (input) increases in size, the time to solve
increases too. But is it linear? no. It is O(n2). Why … consider:
123 vs 743,116
*57 * 4,199
----- -------
861 6,688,044
+615 66,880,440
----- 74,311,600
7011 +2,972,464,000
-------------
3,120,344,084
Note that second problem is just twice as big (n=6) compared to first
problem (n=3); yet the effort is much greater than twice (ie, not O(n) –
linear)
{remember n is the input size}
{note that both “dimensions” of problem doubled – twice as many
columns and twice as many partial results}
binary arithmetic
Consider addition: regardless of the “length” or size (number of bits,
n), we will perform n (2 bit addition operations). If we have two 10 bit
words to add, it will require 10 (2 bit addition operations). Thus, time
for add is O(n)
1010
+1011
-----
CSC 490 Course Notes and Outline, © Dr. Gary Locklair, Fall 2006
