What Is Matrix Multiplication In Algorithm
2 Calculate following values recursively. Matrix Multiplication is one of the most fundamental operation in Machine Learning and optimizing it is the key to several optimizations.
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In the naïve method of matrix multiplication the number of iterating statements involved are 3 because of the presence of rows and columns.
What is matrix multiplication in algorithm. We rst cover a variant of the naive algorithmformulated in terms of block matrices and then parallelize it. Matrix multiplication is fairly simple. So the idea is- if we reduced the number of multiplications then that will make the matrix multiplication faster.
Ae bg af bh ce dg and cf dh. Strassens had given another algorithm for finding the matrix multiplication. MatrixMultiply A B.
Following is simple Divide and Conquer method to multiply two square matrices. Note that in the equation above n refers to the number of columns in matrix A. Since then we have come a long way to better and clever matrix multiplication algorithms.
Algorithm for Strassens matrix multiplication. Strassens Matrix Multiplication Algorithm The major work in matrix multiplication is multiplication only. Partition b into four sub matrices b11 b12 b21 b22.
Else Partition a into four sub matrices a11 a12 a21 a22. Do for i 1 to n-l 1 6. Do j i l -1 7.
In the above method we do 8 multiplications for matrices of size N2 x N2 and 4 additions. 3x3 3x1 and not 3x3 1x3 2 multiply the corresponding fields together and add to arrive at the final field. Thus the following pseudocode executes matrix multiplication.
For naive matrix multiply 2 on board. Multiplying two matrices A and B. What we are more interested in here is matrix multiplication ie.
Iterate through every row r Iterate through every column c. The computed value is placed in the new matrix Z. B2RnnandCABwherenis a power of two2We writeAandBas block matrices.
The element in each row of one matrix is multiplied with each element in the column of the second matrix. We like building things on level 3BLAS routines. For i 1 to n 3.
Idea - Block Matrix MultiplicationThe idea behind Strassens algorithm is in the formulationof matrix multiplication as a recursive problem. Mij 8. 1 Divide matrices A and B in 4 sub-matrices of size N2 x N2 as shown in the below diagram.
Do q m i k m k 1 j p i-1 p k p j 10. Example of Matrix multiplication. Scalar multiplication involves multiplying all elements of a matrix A by this number as per.
For k i to j-1 9. 2n2data 2n2flops These are examples of level 1 2 and 3 routines inBasic LinearAlgebra SubroutinesBLAS. Matrix multiplication involves multiplying each row by each column going from top to bottom for the rows and from left to right for the columns.
Scalar multiplication relates to multiplying a matrix by a number so 3A or more generically λA. For l 2 to n l is the chain length 5. 1 check that the dimensions agree eg.
Algorithm of Matrix Chain Multiplication MATRIX-CHAIN-ORDER p 1. Do m i i 0 4. Assume dimension of A is m x n dimension of B is p x q Begin if n is not same as p then exit otherwise define C matrix as m x q for i in range 0 to m - 1 do for j in range 0 to q 1 do for k in range 0 to p do C i j C i j A i k A k j done done done End.
In general multipling two matrices of size N X N takes N3 operations. Algorithm Strassenn a b d begin If n threshold then compute C a b is a conventional matrix.
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