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Graphs of functions commonly used in the analysis of algorithms, showing the number of operations versus input size for each function. The following tables list the computational complexity of various algorithms for common mathematical operations.
A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. [1]
Karatsuba's basic step works for any base B and any m, but the recursive algorithm is most efficient when m is equal to n/2, rounded up. In particular, if n is 2 k , for some integer k , and the recursion stops only when n is 1, then the number of single-digit multiplications is 3 k , which is n c where c = log 2 3.
For 8-bit integers the table of quarter squares will have 2 9 −1=511 entries (one entry for the full range 0..510 of possible sums, the differences using only the first 256 entries in range 0..255) or 2 9 −1=511 entries (using for negative differences the technique of 2-complements and 9-bit masking, which avoids testing the sign of ...
For example, when computing x 2 k −1, the binary method requires k−1 multiplications and k−1 squarings. However, one could perform k squarings to get x 2 k and then multiply by x −1 to obtain x 2 k −1. To this end we define the signed-digit representation of an integer n in radix b as
The Barrett multiplication previously described requires a constant operand b to pre-compute [] ahead of time. Otherwise, the operation is not efficient. Otherwise, the operation is not efficient. It is common to use Montgomery multiplication when both operands are non-constant as it has better performance.
The definition of matrix multiplication is that if C = AB for an n × m matrix A and an m × p matrix B, then C is an n × p matrix with entries = =. From this, a simple algorithm can be constructed which loops over the indices i from 1 through n and j from 1 through p, computing the above using a nested loop:
The Hadamard product operates on identically shaped matrices and produces a third matrix of the same dimensions. In mathematics, the Hadamard product (also known as the element-wise product, entrywise product [1]: ch. 5 or Schur product [2]) is a binary operation that takes in two matrices of the same dimensions and returns a matrix of the multiplied corresponding elements.