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The method for general multiplication is a method to achieve multiplications with low space complexity, i.e. as few temporary results as possible to be kept in memory. . This is achieved by noting that the final digit is completely determined by multiplying the last digit of the multiplic
In arbitrary-precision arithmetic, it is common to use long multiplication with the base set to 2 w, where w is the number of bits in a word, for multiplying relatively small numbers. To multiply two numbers with n digits using this method, one needs about n 2 operations.
The square of mn, mn being a two-digit integer, can be calculated as 10 × m(mn + n) + n 2. Meaning the square of mn can be found by adding n to mn, multiplied by m, adding 0 to the end and finally adding the square of n. For example, 23 2: 23 2 = 10 × 2(23 + 3) + 3 2 = 10 × 2(26) + 9 = 520 + 9 = 529. So 23 2 = 529.
Since the additions, subtractions, and digit shifts (multiplications by powers of B) in Karatsuba's basic step take time proportional to n, their cost becomes negligible as n increases. More precisely, if T(n) denotes the total number of elementary operations that the algorithm performs when multiplying two n-digit numbers, then
A grid is drawn up, and each cell is split diagonally. The two multiplicands of the product to be calculated are written along the top and right side of the lattice, respectively, with one digit per column across the top for the first multiplicand (the number written left to right), and one digit per row down the right side for the second multiplicand (the number written top-down).
The simplest sort of multiplication, a number with multiple digits by a number with a single digit, is done by placing rods representing the multi-digit number in the frame against the left edge. The answer is read off the row corresponding to the single-digit number which is marked on the left of the frame, with a small amount of addition ...