Matrix Multiplication Strassen Algorithm
Since then we have come a long way to better and clever matrix multiplication algorithms. A Show that the exact solution is T N a 3 1 N log 2 7 2-a 3 N.
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B Find an approximation to N 0 such that T N 0 T DEF N 0 when a 15.
Matrix multiplication strassen algorithm. Following is simple Divide and Conquer method to multiply two square matrices. Use the strassens formulae. Strassens Matrix Multiplication Algorithm.
P 1 XQ S P 2 X YS P 3 Z WP P 4 WRP P 5 X WP S P 6 Y WR S P 7 X ZP Q. Strassens algorithm isnt optimal though. It was the first algorithm to prove that the basic O n3 runtime was not optiomal.
Order of both of the matrices are n n. Next we display the final matrix. Algorithm for Strassens matrix multiplication.
The Algorithm multiplied by the Strassen matrix is compared to a normal algorithm but only one multiplication there are many time complexity. In this context using Strassens Matrix multiplication algorithm the time consumption can be improved a little bit. Algorithm Strassenn a b d begin If n threshold then compute C a b is a conventional matrix.
We have discussed Strassens Algorithm here. Strassens algorithm gives a performance improvement for large-ishN depending on the architecture eg. Partition b into four sub matrices b11 b12 b21 b22.
Review Strassens sequential algorithm for matrix multiplication which requires Onlog 2 7 On281 operations. Rows and columns of both the elements Check ifthe number of columns of first matrix is same as the rows of second matrix condition for matrix multiplication. The algorithm is amenable to parallelizable4 A variant of Strassens sequential algorithm was developed by Coppersmith and Winograd they achieved a run time of On23753.
Summary Strassen rst to show matrix multiplication can be done faster thanON3 time. First compute the following seven matrices. In the above method we do 8 multiplications for matrices of size N2 x N2 and 4 additions.
In general multipling two matrices of size N X N takes N3 operations. Strassen found a way to get all the required information with only 7 matrix multiplications instead of 8. 2 Calculate following values recursively.
The usual matrix multiplication of two nn n n matrices has a time-complexity of On3 O n 3. Think instead of multiplying a k1 vector by a 1k vector. Part I was about simple matrix multiplication algorithms and Part II was about the Strassen algorithm.
Volker Strassen first published his algorithm in 1969. Strassens Algorithm is based on observing that XP YR XQ YS ZP WR and ZQ WS can be computed with only seven instead of eight as in Algorithm MMDC matrix multiplication operations as follows. Recurrence for new algorithm is Tn 7Tn2 Θn2 Solving the Recurrence Relation Applying the Master Theorem to Tn a Tnb fn with a7 b2 and fnΘn2.
Matrices Strassens multiplication algorithm leads to a recurrence of the form T N 7 T N 4 aN 4 N 1 T 1 1 N 1 where N n 2 is the number of entries of the matrices. It can be seen that a small difference may lead to a. Else Partition a into four sub matrices a11 a12 a21 a22.
Over the years its beenimproved. The Strassens method of matrix multiplication is a typical divide and conquer algorithm. This means if n n doubles the time for the computation increases by a factor of 8.
Strassens Matrix multiplication can be performed only on square matrices where n is a power of 2. Strassen algorithm is a recursive method for matrix multiplication where we divide the matrix into 4 sub-matrices of dimensions n2 x n2 in each. This preview shows page 19 - 22 out of 31 pages.
Ae bg af bh ce dg and cf dh. Divide X Y and Z into four n2 n2 matrices as represented below. The matrix multiplication algorithm with best asymptotic complexity runs in O n23728596 time given by Josh Alman and Virginia Vassilevska Williams however this algorithm is a galactic algorithm because of the large constants and cannot be realized practically.
This needs k2 multiplications and you get a kk matrix out at the end. However lets get again on whats behind the divide and conquer approach and implement it. Part III is about parallel matrix multiplication.
Feeding the values in the final matrix. It is required to see this post before further understanding. 1 Divide matrices A and B in 4 sub-matrices of size N2 x N2 as shown in the below diagram.
The only thing thats different here is that the elements of your vector are nn matrices so youll wind up doing O k2 n log 7 scalar multiplications if you use Strassens algorithm to multiply nn matrices.
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