how to optimize matrix multiplication in cpp in terms of time complexity?

fastest matrix multiplication algorithm
strassen algorithm
quick matrix multiplication
strassen matrix multiplication for 4x4 in c++
advantages and disadvantages of strassen's matrix multiplication
limitations of strassen's matrix multiplication
strassen matrix multiplication wiki
application of strassen's matrix multiplication

given any 2 matrics a and b (which don't have special properties) do we have a better way of computing the multiplication than this:?

for(i=0; i<r1; ++i)
for(j=0; j<c2; ++j)
for(k=0; k<c1; ++k)
{
    mult[i][j]+=a[i][k]*b[k][j];
}

If you are curious if they exist in theory, then yes. For example, Strassen algorithm (see https://en.wikipedia.org/wiki/Strassen_algorithm). And it's not even the fastest we know. As far as I'm concerned the best for now is Coppersmith–Winograd algorithm (see https://en.wikipedia.org/wiki/Coppersmith%E2%80%93Winograd_algorithm) and it is something like O(n^{2.37}) (Strassen's time complexity is something like O(n^{2.8}).

But in practice they are much harder to implement than the one you wrote and also they have pretty large time constant hidden under O() so O(n^3) algorithm you wrote is even better on low values of n and much easier to implement.

Also there is a Strassen's hypothesis which claims that for every eps > 0 there is an algorithm which multiplies two matrixes with time complexity O(n^{2 + eps}). But as you might have noticed it is just an hypothesis for now.

Matrix multiplication algorithm, Because matrix multiplication is such a central operation in many numerical algorithms, much work has been invested in making matrix multiplication algorithms efficient. Applications of matrix multiplication in computational problems are found in This algorithm transmits O(n2/p2/3) words per processor​, which is  Number of slow memory references on unblocked matrix multiply m = n3 to read each column of B n times + n2 to read each row of A once + 2n2 to read and write each element of C once = n3 + 3n2 So q = f / m = 2n3 / (n3 + 3n2) » 2 for large n, no improvement over matrix-vector multiply Inner two loops are just matrix-vector multiply, of row i of

As a very easy solution you can transpose the second matrix before multiplication, so your code will get much less processor cache misses. The complexity will be the same but it may improve a time constant a bit.

Foundations of Algorithms Using C++ Pseudocode, Table 2.3 compares the time complexities of the standard algorithm and Strassen's recursive calls, Strassen's algorithm is always more efficient in terms of multiplications, and for efficient as Strassen's algorithm for matrix multiplication. In 2012, she developed a new algorithm that was faster than the Coppersmith–Winograd algorithm, which had reigned supreme in matrix multiplication since the 1980s. Williams’ method reduced the number of steps required to multiply matrices. Her algorithm is only slightly slower than the current record-holder. Dealing with complexity

These are the problems that many bright souls in this world have solved before you. Do not torture yourself and use BLAS ?GEMM.

http://www.netlib.org/blas/#_level_3

Computational complexity of mathematical operations, The following tables list the computational complexity of various algorithms for common Note: Due to the variety of multiplication algorithms, M(n) below stands in for Optimized CW-like algorithms, O(n2.373) invert a matrix runs with the same time complexity as the matrix multiplication algorithm that is used internally. There is another form of matrix multiplication: A multiplied with the transpose of B, i.e., C = A BT (2) where BT is the transpose of B, BT ji = B ij and B 2Rn k. In this paper we call Equation 2 NT operation (T means transpose). The time complexity of schoolbook matrix multiplication is O(m k n), which makes it very time-consuming for large

This is a good question that deserves a more complete answer than "use a library".

Of course, if you want to do a good job, you probably should not try to write it yourself. But if this question is about learning how to do matrix multiplication faster, here is a complete answer.

  1. As a practical matter, the code you show writes to memory too much. If the inner loop adds the dot product in a scalar variable, then only write at the end, the code will be faster. Most compilers are not smart enough to understand this.

    double dot = 0; for(k=0; k

This also improves multi-core performance, since if you use multiple cores they have to share memory bandwidth. If you are using an array of rows, switch your representation to a single block of memory.

  1. As mentioned by someone above, you can do a transpose so the matrix traversals are both in sequential order. Memory is designed to efficiently read in sequentially, but your b[k][j] is jumping around, so this is about 3x faster typically as the size gets big (on the order of 1000x1000, the cost of the initial transpose is negligable).

  2. When the matrix gets large enough, Strassen and Coppersmith-Winograd are faster ways of multiplying that fundamentally change the rules, but they do so by cleverly rearranging terms to achieve the same theoretical result with a lower complexity bound. In practice, they change the answer because roundoff error is different and for large matrices, the answers produced by these algorithms is likely to be far worse than the brute force multiplication.

  3. If you have a truly parallel computer, you can copy the matrix to multiple CPUs and have them work in parallel on the answer.

  4. You can put the code onto your video card and use the far more parallel CPUs there which have far more memory bandwidth. That's probably the most effective way to get a real speedup on your computer (assuming you have a graphics card). See CUDA or Vulkan.

The fundamental problem is that multiple cores don't help much for matrix multiplication because you are limited by memory bandwidth. That's why doing it on a video card is so good, because bandwidth there is far higher.

[PDF] Ultra-Fast Matrix Multiplication, The development of high-performance matrix multiplication algorithms is important in the and Strassen's algorithm), we derive their theoretical run-time complexity and then compare these optimized vector processing unit developed by Apple in conjunction with Motorola and IBM. terms of the number of additions and. Later on in the course, when we deal with the problem of computing the price of an European Option efficiently using the method of Dynamic Programming, we will have to deal with the problem of taking an n × n matrix A and raising it to a large number. That is, we will have to compute A k for large values of k.

You could use multiple threads by dividing the multiplication to them. So divide the lines/columns of the first dimension of the first matrix or the last dimension of the last into a number of tasks equal to the cores you have in your processor. If these aren't evenly divisible, some cores will have to do an extra cycle. But any way, the idea is give the multiplication to more cores and divide e.g. the first matrix in 4 parts ( I have 4 cores), do the multiplication with 4 tasks, and reassemble (that isn't necessary as the cores may work on the same data).

(PDF) Optimizing the Matrix Multiplication Using Strassen and , We propose an efficient hybrid matrix multiplication implementation based on Strassen and Winograd algorithms (S-MM and W-MM) on many-core. This very demanding computational task involves dense and sparse linear al- The complexity of S-MM algorithm in terms of mented using C++ with OpenMp on MIC. Multiplication of matrix does take time surely. Time complexity of matrix multiplication is O(n^3) using normal matrix multiplication. And Strassen algorithm improves it and its time complexity is O(n^(2.8074)). But, Is there any way to improve the performance of matrix multiplication using the normal method. Multi-threading can be done to

Matrix Chain Multiplication, C#, C# Programs, C++, C++ Programs, C++ Quiz, CAT, CAT Quiz, Combinatorial Given a sequence of matrices, find the most efficient way to multiply these In other words, no matter how we parenthesize the product, the result will be the same. Time complexity of the above naive recursive approach is exponential. Time Complexity: O(n^3) Auxiliary Space: O(n^2) Matrix Chain Multiplication (A O(N^2) Solution) Printing brackets in Matrix Chain Multiplication Problem. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above. Applications: Minimum and Maximum values of an expression with * and +

Divide and Conquer, No, there are no algorithms that are optimized for 4x4 matrix multiplication because the The basic cubic-time complexity algorithm tends to fare quite well, and  Iterative algorithm. The definition of matrix multiplication is that if C = AB for an n × m matrix A and an m × p matrix B, then C is an n × p matrix with entries = ∑ =. From this, a simple algorithm can be constructed which loops over the indices i from 1 through n and j from 1 through p, computing the above using a nested loop:

Which algorithm is performant for matrix multiplication of 4x4 , Time and space complexity. 2. repeated matrix multiplication,. 2. What is the (​time) complexity of the original recursive definition as a means of computing What does this result say about the efficient use of recursion? complexity in terms of the algorithm's input size and should reflect where the real work is con-. The time complexity of the above program is O(n 3). It can be optimized using Strassen’s Matrix Multiplication. This article is contributed by Aditya Ranjan. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your

Comments
  • It very much depends on the size really. If you have a large matrix you could parallelize it with e.g. threads or other similar techniques.
  • Use a library like Eigen or MKL where they have vectorized the math for you.
  • This answer might be helpful: stackoverflow.com/a/54546544/3234205
  • Note that Strassen's algorithm uses matrix addition as a subroutine, which is relatively easy to parallelize even on one processor using SIMD. Strassen's algorithm, if implemented well,can perform better than the naive method for n=16 or n=32, which I consider a relatively small size.
  • @Codor, do you have a source for that? I have seen some hybrid methods which uses Strassen for n>1000 large but n=32 or smaller is very small and I doubt Strassen helps for that.
  • Perhaps you are right; this Paper seems to conclude that Strassen's algorithm is competitive for n=512, see Table 1. However, apparently neither SIMD for addition nor an improved memory locality via so-called Morton layout is used.
  • Please check these two papers if you are interested in the high-performance implementation of Strassen's algorithm (SIMD for addition / Morton layout for multi-level Strassen are used): 1. "Strassen's Algorithm Reloaded", in SC16. 2. "Generating Families of Practical Fast Matrix Multiplication Algorithms", in IPDPS17.