Lattice multiplication instructions mod#

Lattice multiplication instructions mod#

Multiply ( a, b, base ) // Operands containing rightmost digits at index 1 tot = 0 for ri = 1 to p + q - 1 //For each digit of result for bi = MAX ( 1, ri - p + 1 ) to MIN ( ri, q ) //Digits from b that need to be considered ai = ri − bi + 1 //Digits from a follow 'symmetry' tot = tot + ( a * b ) product = tot mod base tot = floor ( tot / base ) product = tot mod base //Last digit of the result comes from last carry return product Usage in computers

In pseudocode, the log-space algorithm is: Thus both these values can be stored in O(log n) digits.

The sum is at most D * n, and the carry to the next column is at most D * n / D, or n. R i = ( c i − 1 + ∑ j + k = i a j b k ) mod D c i = ⌊ ( c i − 1 + ∑ j + k = i a j b k ) / D ⌋ c 0 = 0 Ī simple inductive argument shows that the carry can never exceed n and the total sum for r i can never exceed D * n: the carry into the first column is zero, and for all other columns, there are at most n digits in the column, and a carry of at most n from the previous column (by the induction hypothesis). Let a i and b i be the i-th digit of the operand, r i be the i-th digit of the result and c i be the carry generated for r i (i=1 is the right most digit) then The method is based on the observation that each digit of the result can be computed from right to left with only knowing the carry from the previous step. Note that operands themselves still need to be kept in memory and their Θ( n) space is not considered in this analysis. This is the double logarithm of the numbers being multiplied themselves (log log N). In these scenarios, long multiplication has the advantage that it can easily be formulated as a log space algorithm that is, an algorithm that only needs working space proportional to the logarithm of the number of digits in the input (Θ(log n)). However, in certain applications, the entire result need not be kept in memory and instead the digits of the result can be streamed out as they are computed (for example, to system console or file). If the result must be kept in memory then the space complexity is trivially Θ( n). Let n be the total number of digits in the two input numbers in base D. ( September 2012) ( Learn how and when to remove this template message) Unsourced material may be challenged and removed. Please help improve this section by adding citations to reliable sources. Note that the '+=' operator is used to denote sum to existing value and store operation (akin to languages such as Java and C) for compactness. It keeps only one row to maintain the sum which finally becomes the result. This example uses long multiplication to multiply 23,958,233 (multiplicand) by 5,830 (multiplier) and arrives at 139,676,498,390 for the result (product).īelow pseudocode describes the process of above multiplication. A person doing long multiplication on paper will write down all the products and then add them together an abacus-user will sum the products as soon as each one is computed. Computers initially used a very similar shift and add algorithm in base 2, but modern processors have optimized circuitry for fast multiplications using more efficient algorithms, at the price of a more complex hardware realization. This is the usual algorithm for multiplying larger numbers by hand in base 10. It requires memorization of the multiplication table for single digits. Multiply the multiplicand by each digit of the multiplier and then add up all the properly shifted results. If a positional numeral system is used, a natural way of multiplying numbers is taught in schoolsĪs long multiplication, sometimes called grade-school multiplication, sometimes called Standard Algorithm: Nevertheless, it is seen as a usefully explicit method to introduce the idea of multiple-digit multiplications and, in an age when most multiplication calculations are done using a calculator or a spreadsheet, it may in practice be the only multiplication algorithm that some students will ever need. The grid method can in principle be applied to factors of any size, although the number of sub-products becomes cumbersome as the number of digits increases. Its essence is the calculation of the simple multiplications separately, with all addition being left to the final gathering-up stage. This calculation approach (though not necessarily with the explicit grid arrangement) is also known as the partial products algorithm. Followed by addition to obtain 442, either in a single sum (see right), or through forming the row-by-row totals (300 + 40) + (90 + 12) = 340 + 102 = 442.