Strassen's algorithm is an efficient method for multiplying matrices that reduces the number of multiplications from eight in standard methods to seven, enhancing performance through a divide and conquer approach. It is implemented using specific formulas and a pseudo code algorithm that recursively divides matrices until 2x2 matrices are achieved. The time complexity of Strassen's algorithm is O(n^2.8074), showcasing its advantage as the matrix size grows.

1. Strassen’s
Algorithm
PR ES EN TATIO N BY,
ABDOURAZAK H. GUEDI I M S C ( I T)
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2. content
 Introduction
 Procedure
 Formulas
Pseudo code
Conclusion
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3. Introduction
The Strassen algorithm, also known as Strassen's matrix multiplication
algorithm.
Strassen in 1969 which gives an overview that how we can find the
multiplication of two matrix 2*2 dimension by the brute-force
algorithm.
But by using divide and conquer technique the overall complexity
for multiplication of two matrices is reduced.
This happens by decreasing the total number of multiplication
performed at the expenses of a slight increase in the number of
addition.
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4. For multiplying the two 2*2 dimension matrices Strassen's used
some formulas in which there are eight multiplication and eighteen
addition, subtraction, but in brute force algorithm, there is seven
multiplication and four addition.
The utility of Strassen's formula is shown by its asymptotic superiority
when order n of matrix reaches infinity.
Let us consider two matrices A and B, n*n dimension, where n is a
power of two. It can be observed that we can contain four n/2*n/2
submatrices from A, B and their product C. C is the resultant matrix of
A and B.
Introduction
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5. Procedure for Strassen matrix
multiplication
There are some procedures:
1.Divide a matrix of order of 2*2 recursively till we get the matrix of 2*2.
2.Use the previous set of formulas to carry out 2*2 matrix multiplication.
3.In this seven multiplication and four additions, subtraction are performed.
4.Combine the result of two matrixes to find the final product or final matrix.
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6. Formulas for Strassen matrix
multiplication
In Strassen's matrix multiplication there are seven multiplication and four addition, subtraction in
total.
D1 = (a11 + a22).(b11 + b22)
D2 = (a21 + a22).b11
D3 = (b12 – b22).a11
D4 = (b21 – b11).a22
D5 = (a11 + a12).b22
D6 = (a21 – a11).(b11 + b12)
D7 = (a12 – a22).(b21 + b22)
C11 = d1 + d4 – d5 + d7
C12 = d3 + d5
C21 = d2 + d4
C22 = d1 - d2 + d3 + d6
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7. Pseudo code
Algorithm Strassen(n, a, b, d)
1. If n = threshold
2. C = a * b
Else
3. Partition a into four sub matrices a11,a12,a21,a22.
4. Partition b into four sub matrices b11,b12,b21,b22.
5. Strassen ( n/2, a11 + a22, b11 + b22, d1)
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8. 6. Strassen ( n/2, a21 + a22, b11, d2)
7. Strassen ( n/2, a11, b12 – b22, d3)
8. Strassen ( n/2, a22, b21 – b11, d4)
9. Strassen ( n/2, a11 + a12, b22, d5)
10.Strassen (n/2, a21 – a11, b11 + b12, d6)
11.Strassen (n/2, a12 – a22, b21 + b22, d7)
12.C = d1+d4-d5+d7 d3+d5 d2+d4 D1-d2+d3+d6
end if
return (C)
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9. Code:
#include <stdio.h>
int main()
{
int a[2][2],b[2][2],c[2][2],i,j;
int m1,m2,m3,m4,m5,m6,m7;
printf("Enter the 4 elements of first matrix: ");
for(i=0;i<2;i++)
for(j=0;j<2;j++)
scanf("%d",&a[i][j]);
printf("Enter the 4 elements of second matrix: "); for(i=0;i<2;i++)
for(j=0;j<2;j++)
scanf("%d",&b[i][j]);
printf("nThe first matrix isn");
for(i=0;i<2;i++)
{ printf("n");
for(j=0;j<2;j++)
printf("%dt",a[i][j]); }
printf("nThe second matrix isn");
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10. for(i=0;i<2;i++) {
printf("n");
for(j=0;j<2;j++)
printf("%dt",b[i][j]); }
m1= (a[0][0] + a[1][1])*(b[0][0]+b[1][1]);
m2= (a[1][0]+a[1][1])*b[0][0];
m3= a[0][0]*(b[0][1]-b[1][1]);
m4= a[1][1]*(b[1][0]-b[0][0]);
m5= (a[0][0]+a[0][1])*b[1][1];
m6= (a[1][0]-a[0][0])*(b[0][0]+b[0][1]);
m7= (a[0][1]-a[1][1])*(b[1][0]+b[1][1]);
c[0][0]=m1+m4-m5+m7;
c[0][1]=m3+m5;
c[1][0]=m2+m4;
c[1][1]=m1-m2+m3+m6;
printf("nAfter multiplication using n");
for(i=0;i<2;i++) {
printf("n");
for(j=0;j<2;j++)
printf("%dt",c[i][j]); }
return 0;
}
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Output:
Complexity: The time complexity is O(N2.8074).
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11. Conclusion
The Strassen algorithm is a fast matrix multiplication algorithm that reduces
the number of basic multiplications required for matrix multiplication.
It reduced the number of multiplications from 8 (in the standard matrix
multiplication) to 7, which results in a slightly lower overall time complexity.
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12. Thank You
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