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From the definition of the permutation P above, every other element x of the cycle is obtained by repeatedly multiplying s by N modulo MN−1, and therefore every other element is divisible by d. But, since N and MN − 1 are coprime, x cannot be divisible by any factor of MN − 1 larger than d , and hence d = gcd ( x , M N − 1 ...
As exchanging the indices of an array is the essence of array transposition, an array stored as row-major but read as column-major (or vice versa) will appear transposed. As actually performing this rearrangement in memory is typically an expensive operation, some systems provide options to specify individual matrices as being stored transposed.
The correct number of sections for a fence is n − 1 if the fence is a free-standing line segment bounded by a post at each of its ends (e.g., a fence between two passageway gaps), n if the fence forms one complete, free-standing loop (e.g., enclosure accessible by surmounting, such as a boxing ring), or n + 1 if posts do not occur at the ends ...
In array languages, operations are generalized to apply to both scalars and arrays. Thus, a+b expresses the sum of two scalars if a and b are scalars, or the sum of two arrays if they are arrays. An array language simplifies programming but possibly at a cost known as the abstraction penalty.
The QR decomposition via Givens rotations is the most involved to implement, as the ordering of the rows required to fully exploit the algorithm is not trivial to determine. However, it has a significant advantage in that each new zero element affects only the row with the element to be zeroed (i) and a row above (j). This makes the Givens ...
var c = 0.0 // The array input has elements indexed for i = 1 to input.length do // c is zero the first time around. var y = input[i] + c // sum + c is an approximation to the exact sum. (sum,c) = Fast2Sum(sum,y) // Next time around, the lost low part will be added to y in a fresh attempt. next i return sum
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.
The empty relation R (defined so that aRb is never true) on a set X is vacuously symmetric and transitive; however, it is not reflexive (unless X itself is empty). The relation "is approximately equal to" between real numbers, even if more precisely defined, is not an equivalence relation, because although reflexive and symmetric, it is not ...