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The cache miss rate of recursive matrix multiplication is the same as that of a tiled iterative version, but unlike that algorithm, the recursive algorithm is cache-oblivious: [9] there is no tuning parameter required to get optimal cache performance, and it behaves well in a multiprogramming environment where cache sizes are effectively ...
The left column visualizes the calculations necessary to determine the result of a 2x2 matrix multiplication. Naïve matrix multiplication requires one multiplication for each "1" of the left column. Each of the other columns (M1-M7) represents a single one of the 7 multiplications in the Strassen algorithm. The sum of the columns M1-M7 gives ...
In theoretical computer science, the computational complexity of matrix multiplication dictates how quickly the operation of matrix multiplication can be performed. Matrix multiplication algorithms are a central subroutine in theoretical and numerical algorithms for numerical linear algebra and optimization, so finding the fastest algorithm for matrix multiplication is of major practical ...
The three parts generated at each recursive step are apparent in the structure of the final result. Applying the function derived from Q3 to the same argument multiple times gives different results because the pivots are chosen at random. In-order traversal of the results does yield the same sorted array.
Done recursively, this has a time complexity of (). Splitting numbers into more than two parts results in Toom-Cook multiplication; for example, using three parts results in the Toom-3 algorithm. Using many parts can set the exponent arbitrarily close to 1, but the constant factor also grows, making it impractical.
For the example below, there are four sides: A, B, C and the final result ABC. A is a 10×30 matrix, B is a 30×5 matrix, C is a 5×60 matrix, and the final result is a 10×60 matrix. The regular polygon for this example is a 4-gon, i.e. a square: The matrix product AB is a 10x5 matrix and BC is a 30x60 matrix.
If for example PE(0,0) calculates in the first step, PE(0,1) chooses first. The selection of k := (i + j) mod n for PE(i,j) satisfies this constraint for the first step. In the first step we distribute the input matrices between the processors based on the previous rule.
Folds can be regarded as consistently replacing the structural components of a data structure with functions and values. Lists, for example, are built up in many functional languages from two primitives: any list is either an empty list, commonly called nil ([]), or is constructed by prefixing an element in front of another list, creating what is called a cons node ( Cons(X1,Cons(X2,Cons ...