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.

The partial-fractions expansion of a function F(s)/(s-a)m, m > 1, involves the computation of m coefficients, namely (1 /i!)(diF(a)/dsi), 0 ≤ i ≤ m-1. Wehrhahn [1] and Karni [3] have provided a method for computing these coefficients algebraically. A new approach is taken here which involves approximating a multiple pole by neighboring simple poles. The theory developed turns out to have a very interesting resemblance to the FFT algorithm. The algorithm is illustrated by several examples. Typical applications are finding the inverse Laplace transform of a function having multiple poles and the evaluation of higher order derivatives of an arbitrary function H(z) at some arbitrary z=z0.