Here, we describe a remarkable algorithm for matrix...

Here, we describe a remarkable algorithm for matrix multiplication discovered by Strassen, [82]. Let $A=\left(\begin{array}{ll}A_1 & A_2 \\ A_3 & A_4\end{array}\right), B=\left(\begin{array}{ll}B_1 & B_2 \\ B_3 & B_4\end{array}\right)$, and $C=\left(\begin{array}{ll}C_1 & C_2 \\ C_3 & C_4\end{array}\right)=A B$ be block matrices of size $n=2 m$, where all blocks are of size $m \times m$. (a) Let $D_1=$ $\left(A_1+A_4\right)\left(B_1+B_4\right), D_2=\left(A_1-A_3\right)\left(B_1+B_2\right), D_3=\left(A_2-A_4\right)\left(B_3+B_4\right)$, $D_4=\left(A_1+A_2\right) B_4, D_5=\left(A_3+A_4\right) B_1, D_6=A_4\left(B_1-B_3\right), D_7=A_1\left(B_2-B_4\right)$. Show that $C_1=D_1+D_3-D_4-D_6, C_2=D_4+D_7, C_3=D_5-D_6, C_4=D_1-D_2-D_5+D_7$. (b) How many arithmetic operations are required when $A$ and $B$ are $2 \times 2$ matrices? How does this compare with the usual method of multiplying $2 \times 2$ matrices? (c) In the general case, suppose we use standard matrix multiplication for the matrix products in $D_1, \ldots, D_7$. Prove that Strassen's Method is faster than the direct algorithm for computing $A B$ by a factor of $\approx \frac{7}{8}$. (d) When $A$ and $B$ have size $n \times n$ with $n=2^r$, we can recursively apply Strassen's Method to multiply the $2^{r-1} \times 2^{r-1}$ blocks $A_i, B_i$. Prove that the resulting algorithm requires a total of $7^r=n^{\log _2 7}=n^{2.80735}$ multiplications

Here, we describe a remarkable algorithm for matrix multiplication discovered by Strassen, [82]. Let $A=\left(\begin{array}{ll}A_1 & A_2 \\ A_3 & A_4\end{array}\right), B=\left(\begin{array}{ll}B_1 & B_2 \\ B_3 & B_4\end{array}\right)$, and $C=\left(\begin{array}{ll}C_1 & C_2 \\ C_3 & C_4\end{array}\right)=A B$ be block matrices of size $n=2 m$, where all blocks are of size $m \times m$. (a) Let $D_1=$ $\left(A_1+A_4\right)\left(B_1+B_4\right), D_2=\left(A_1-A_3\right)\left(B_1+B_2\right), D_3=\left(A_2-A_4\right)\left(B_3+B_4\right)$, $D_4=\left(A_1+A_2\right) B_4, D_5=\left(A_3+A_4\right) B_1, D_6=A_4\left(B_1-B_3\right), D_7=A_1\left(B_2-B_4\right)$. Show that $C_1=D_1+D_3-D_4-D_6, C_2=D_4+D_7, C_3=D_5-D_6, C_4=D_1-D_2-D_5+D_7$.
(b) How many arithmetic operations are required when $A$ and $B$ are $2 \times 2$ matrices? How does this compare with the usual method of multiplying $2 \times 2$ matrices?
(c) In the general case, suppose we use standard matrix multiplication for the matrix products in $D_1, \ldots, D_7$. Prove that Strassen's Method is faster than the direct algorithm for computing $A B$ by a factor of $\approx \frac{7}{8}$. (d) When $A$ and $B$ have size $n \times n$ with $n=2^r$, we can recursively apply Strassen's Method to multiply the $2^{r-1} \times 2^{r-1}$ blocks $A_i, B_i$. Prove that the resulting algorithm requires a total of $7^r=n^{\log _2 7}=n^{2.80735}$ multiplications

Key Concepts

Divide and Conquer Recurrence

The divide and conquer paradigm recursively breaks down a problem into smaller subproblems, solves these subproblems independently, and then combines their solutions to form the final result. Strassen's algorithm applies this strategy by partitioning matrices into blocks and multiplying them with reduced operations. This recursive process is characterized by a recurrence relation that, in the case of Strassen's method, yields an overall time complexity expressed as n^(log2 7), which is asymptotically faster than the standard cubic time algorithm.

Arithmetic Complexity

Arithmetic complexity refers to the total count of fundamental operations, such as additions, subtractions, and multiplications, required to complete an algorithm. In the context of matrix multiplication, reducing the number of multiplications can significantly lower the computational cost. Strassen’s algorithm demonstrates this by performing fewer multiplications compared to the classical method, thereby improving computational efficiency.

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra where two matrices are combined to produce a third matrix. The standard definition involves taking the dot product of rows of the first matrix with columns of the second, resulting in an algorithm with a cubic time complexity when performed using the most straightforward approach.

Block Matrix Partitioning

Block matrix partitioning involves dividing a larger matrix into smaller submatrices or blocks. This approach is often used in the design of algorithms to simplify operations on large matrices by reducing them into operations on smaller matrices, which can lead to more efficient computation and facilitate parallel processing.

Strassen's Algorithm

Strassen's Algorithm is a groundbreaking method for matrix multiplication that reduces the number of necessary multiplicative operations. Instead of performing the standard eight multiplications for multiplying two 2 × 2 matrices, Strassen’s method uses only seven multiplications along with additional matrix additions and subtractions. This clever reorganization leads to a reduction in the overall arithmetic operations needed and underpins the development of faster algorithms for large matrices.

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