On optimizing computations for transition matrices

For the special case where the coefficient matrix is in standard companion form, all of the elements of the transition matrix may be obtained recursively from a single row of elements. The number of computational steps necessary to generate this required row is an order of magnitude less than that required for general coefficient matrices. Also the forced-response coefficients are shown to require negligible additional calculations for the common problem with a single input and a zero-order data hold. These computational savings enable the typical, modest-sized digital computer to address the previously formidable problem of non-stationary, high-order transfer functions in real time.