Matrix Multiplication Order Algorithm

30 Mar, 2021

The minimum number of multiplications are obtained by putting parenthesis in following way A BCD -- 203010 402010 401030 Input. There is one more hypercube matrix-multiplication algorithm.

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Example of Matrix multiplication.

Matrix multiplication order algorithm. Matrix C mh kh matrix A and kh nh matrix B are all stored in level h of the memory hierarchy then forming C AB C costs time 2mhnhkhγh. When two matrices are of order m x p and n x m the order of product will be n x p. 5 A big difference.

To go faster than the last algorithm it assumes that all log p wires connected to each processor can be used simultaneously to get log p parallelism in communication. Suppose two matrices are A and B and their dimensions are A m x n and B p x q the resultant matrix can be found if and only if n p. The multiplication is possible if and only if the number of columns of first matrix is equal to the number of rows of second matrix.

Use the previous set of formulas to carry out 22 matrix multiplication. Divide a matrix of order of 22 recursively till we get the matrix of 22. Matrix Multiplication Let A be an n x m matrix B an m x p matrix The product of A and B is n x p matrix AB whose ij-th entry is k1 m a ik b kj In other words we multiply the entries of the i-th row of A with the entries of the j-th column of B and add them up.

For l 2 to n l is the chain length 5. 5 When 1 and then. Algorithm of Matrix Chain Multiplication MATRIX-CHAIN-ORDER p 1.

Mij 8. The matrix multiplication can only be performed if it satisfies this condition. For i 1 to n 3.

Do for i 1 to n-l 1 6. Idea - Block Matrix MultiplicationThe idea behind Strassens algorithm is in the formulationof matrix multiplication as a recursive problem. An so as to minimize the total cost assuming that the cost of multiplying an pi 1 pi matrix by a pi pi 1 matrix using the naive algorithm is pi 1 pi pi 1.

The order of product of two matrices is distinct. P 10 20 30 40 30 Output. Do j i l -1 7.

For k i to j-1 9. Then the order of the resultant matrix C will be m x q. A B C AB AC.

Do m i i 0 4. Notice that γh will depend on mh nh and kh. Direct Matrix multiplication of Given a matrix a matrix and a matrix then can be computed in two ways and.

Let the input 4 matrices be A B C and D. Chain Matrix multiplication is an associative operation the chain matrix multiplication order does not affect the final result but it can affect the total number of performed operations as shown in Figs. This corresponds to the hardware available on the CM-2.

3 Building-Blocks for Matrix Multiplication Consider the matrix multiplication C AB C where mh1 nh1 matrix C mh1 kh1 matrix A and kh1 nh1 matrix B are all stored in Lh1. B2RnnandCABwherenis a power of two2We writeAandBas block matrices. We rst cover a variant of the naive algorithmformulated in terms of block matrices and then parallelize it.

The multiplication sequence parenthesization is important. Matrix multiplication follows distributive rule over matrix addition. The number of multiplications needed are.

Do q m i k m k 1 j p i-1 p k p j 10. Procedure of Strassen matrix multiplication There are some procedures. Outline 1 Matrix operations Importance Dense and sparse matrices Matrices and arrays 2 Matrix-vector multiplication Row-sweep algorithm Column-sweep algorithm 3 Matrix-matrix multiplication Standard algorithm ijk-forms CPS343 Parallel and HPC Matrix Multiplication Spring 2020 332.

Note that this algorithm does not perform the multiplications it just figures out the best order in which to perform the multiplication operations. In this section we will see how to multiply two matrices. 30000 There are 4 matrices of.

Right Distribution A B C AC AC.

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