Monimatriisimenetelmät

Monimatriisimenetelmät, known in English as matrix multiplication algorithms, are a class of computational techniques used to perform the multiplication of two matrices. The standard algorithm for matrix multiplication, often taught in introductory linear algebra, has a time complexity of O(n^3) for multiplying two n x n matrices. This means that as the size of the matrices increases, the computational effort grows cubically.

For many applications, particularly in scientific computing and machine learning, matrices can be very large, making the standard O(n^3) algorithm computationally expensive. This has driven the development of more efficient algorithms. Strassen's algorithm, discovered in 1969, was a significant breakthrough, reducing the complexity to approximately O(n^2.807). While theoretically faster, Strassen's algorithm can be more complex to implement and may have higher constant factors, meaning it's not always faster in practice for smaller matrices.

Further research has led to even more asymptotically faster algorithms, such as the Coppersmith-Winograd algorithm and its subsequent improvements, which have theoretical complexities approaching O(n^2.37). However, these algorithms are extremely complex and their practical advantage is limited to exceptionally large matrices. In practice, the choice of algorithm often depends on the size of the matrices, the available hardware, and specific implementation details, with optimized versions of the standard algorithm or Strassen's algorithm often being favored for moderate to large-sized matrices.