The document discusses Strassen's algorithm for matrix multiplication. Strassen's algorithm improves upon the standard O(n^3) matrix multiplication algorithm by reducing the number of multiplication operations from eight to seven when multiplying 2x2 matrices. It achieves this reduction using a divide and conquer approach along with seven specialized formulas. Pseudocode is provided to demonstrate how the algorithm recursively breaks down matrices and calculates the matrix product. The time complexity of Strassen's algorithm is O(n^2.8074), an improvement over the standard algorithm.

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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