Is there any method for multiplying matrices having O(n) complexity?

Question

I want to multiply two matrices but the triple loop has O(n³) complexity. Is there any algorithm in dynamic programming to multiply two matrices with O(n) complexity?

Answer 1

Short answer: No

Long answer: There are ways if you have special kinds of matricies (for instance a diagonal matrix). The better matrix multiplication algorithms out there can pare you down to something like O(n^2.4) (http://en.wikipedia.org/wiki/Coppersmith-Winograd_algorithm). The major one I am somewhat familiar with uses a divide and conquer algorithm to split up the workload (not the one I linked to).

I hope this helps!

Answer 2

If

• your matrices are large
• they have a lot of zeroes
• you are willing to store them in strange formats

you can devise algorithms whose complexities depend only on the number of non-zero elements. This can be mandatory for some problems (eg. finite element methods).

Answer 3

If the matrices are known to be diagonal, you can multiply them in O(N) operations. But in general, you cannot.

Answer 4

There is a theoretical lower bound for matrix multiplication at O(n^2) as you have to touch that many memory locations to do the multiplication. As others have said, there are algorithms that drop us below O(n^3), but are usually impractical in real use.

If you need to speed it up, you might want to look at Cache Oblivious Algorithms, such as this one (http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.44.5650) that accelerate performance by performing operations in a cache cohesive way, ensuring that data is in the cache when needed.

Answer 5

The best Matrix Multiplication Algorithm known so far is the "Coppersmith-Winograd algorithm" with O(n^2.38 ) complexity but it is not used for practical purposes.

However you can always use "Strassen's algorithm" which has O(n^2.81 ) complexity but there is no such known algorithm for matrix multiplication with O(n) complexity.

Answer 6

Matrices have O(n²) elements, and every element must be considered at least once for the result, so there is no possible way for a matrix multiplication algorithm to run in less than O(n²) operations.