# matrix_chain_dynamic

matrix_chain_dynamic, a Python code which finds the cost of the most efficient ordering to use when multiplying a sequence of matrices, using dynamic programming.

We are given a sequence of n matrices of conformable dimensions. Matrix Ai has D(i) rows and D(i+1) columns. The product A1 * A2 * ... * An is to be computed. In terms of scalar multiplications, the cost of computing A(i) * A(i+1) is D(i) * D(i+1) * D(i+2). We may carry out the pairs of multiplication in any order we wish. The total cost of the computation will, in general, vary depending on the order in which we compute the individual matrix multiplications. The goal of the algorithm is simply to determine the cost of the most efficient ordering.

Given n matrices, there are (n-1)! possible orderings for the sequence of matrix multiplications. One approach to determining the minimum cost would be to consider each of these orderings, but this is considered a wasteful and inefficient approach for large values of n. The code given here depends instead on a dynamic programming approach. As such, it requires several n by n arrays in order to store intermediate information. Thus it needs n^2 space, and the algorithm takes on the order of n^3 operations to determine the result.

### Licensing:

The information on this web page is distributed under the MIT license.

### Languages:

matrix_chain_dynamic is available in a C version and a C++ version and a Fortran90 version and a MATLAB version and an Octave version and a Python version.

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### Reference:

1. Thomas Cormen, Charles Leiserson, Ronald Rivest, Clifford Stein,
Introduction to Algorithms,
MIT Press, 2001,
ISBN: 978-0-262-03293-3,
LC: QA76.C662.
2. Matrix Chain Multiplication,
Wikipedia page,
https://en.wikipedia.org/wiki/matrix_chain_multiplication

### Source Code:

Last revised on 30 December 2023.