On Efficient Computation of Matrix Chain Products

It is pointed out that the number of scalar multiplications (additions) required to evaluate a matrix chain product depends on the sequence in which the associative law of matrix multiplication is applied. An algorithm is developed to find the optimum sequence that minimizes the number of scalar multiplications. A program is written for use on the CDC 6600 computer to implement this algorithm and also to carry out the chain product according to the optimum sequence. Several examples are included to illustrate the algorithm. The saving in computation and improvement in accuracy that can result from the use of this algorithm can be quite significant for chain products of large arrays and in iterative solutions of matrix equations involving chain products.