Following this conclusion, after using partial fractions, we may write the r-th (1 [less than or equal to] r [less than or equal to] m) component of [W.sup.*.sub.A](s) (in (4) which also has factor [[pi].sup.-*]([[phi].sub.1](s))) as follows:
It maybe remarked here that while making partial fractions, we have assumed that the roots [[bar.h].sub.i] and [[bar.s].sub.ij] are distinct.
After computation of the required roots, we construct partial fractions for each component of [W.sup.*.sub.A](s) as given in (10).
The determined dynamic characteristic [Z.sub.U](s) = 1/[Y.sub.U](s) is decomposed into partial fractions, obtaining
Finally, the distribution of impedance function equation (4) into partial fractions taking into account (9) in (6) takes the form
Such obtained mechanical impedance [Z.sub.U](s), in the form of "partial fractions," could be represented as the sum of any rational functions.
Man, "A cover-up approach to partial fractions with linear or irreducible quadratic factors in the denominators," Applied Mathematics and Computation, vol.
Reichel, "Incomplete partial fractions for parallel evaluation of rational matrix functions," Journal of Computational and Applied Mathematics, vol.
Partial fraction expansion (pfe) has been a powerful tool widely used in the field of calculus, differential equations, control theory, and some other areas of pure or applied mathematics.
In view of (3.8), L applied to (3.7) yields (L[y.sub.n-1])(s) = p[(s).sup.-1] which can be decomposed into partial fractions as in (3.5).
Together with some elementary facts of linear algebra, we finally arrive at a first inversion algorithm which requires the computation of the partial fraction decomposition of [(det([lambda]I - A)).sup.-1].
1) Compute the coefficients [p.sub.ij] of the partial fraction expansion (3.5) and form the vector [w.sub.n-1] as in (3.6).