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Matrix chain multiplication (or the matrix chain ordering problem [1]) is an optimization problem concerning the most efficient way to multiply a given sequence of matrices. The problem is not actually to perform the multiplications, but merely to decide the sequence of the matrix multiplications involved.
For example, (2 + 3) × 4 = 20 forces addition to precede multiplication, while (3 + 5) 2 = 64 forces addition to precede exponentiation. If multiple pairs of parentheses are required in a mathematical expression (such as in the case of nested parentheses), the parentheses may be replaced by other types of brackets to avoid confusion, as in [2 ...
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 the same result as the full matrix multiplication on the left.
Matrix chain multiplication is a well-known example that demonstrates utility of dynamic programming. For example, engineering applications often have to multiply a chain of matrices. It is not surprising to find matrices of large dimensions, for example 100×100. Therefore, our task is to multiply matrices ,,....
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 ...
If x 2 − y 2 is evaluated as ((x × x) − y × y) (following Kahan's suggested notation in which redundant parentheses direct the compiler to round the (x × x) term first) using fused multiply–add, then the result may be negative even when x = y due to the first multiplication discarding low significance bits. This could then lead to an ...
Even though the parentheses were rearranged on each line, the values of the expressions were not altered. Since this holds true when performing addition and multiplication on any real numbers, it can be said that "addition and multiplication of real numbers are associative operations".
[19] [20] Victor Pan proposed so-called feasible sub-cubic matrix multiplication algorithms with an exponent slightly above 2.77, but in return with a much smaller hidden constant coefficient. [21] [22] Freivalds' algorithm is a simple Monte Carlo algorithm that, given matrices A, B and C, verifies in Θ(n 2) time if AB = C.