## Efficient recursive methods for partial fraction expansion of general rational functions.(English)Zbl 1442.65013

Summary: Partial fraction expansion (pfe) is a classic technique used in many fields of pure or applied mathematics. The paper focuses on the pfe of general rational functions in both factorized and expanded form. Novel, simple, and recursive formulas for the computation of residues and residual polynomial coefficients are derived. The proposed pfe methods require only simple pure-algebraic operations in the whole computation process. They do not involve derivatives when tackling proper functions and require no polynomial division when dealing with improper functions. The methods are efficient and very easy to apply for both computer and manual calculation. Various numerical experiments confirm that the proposed methods can achieve quite desirable accuracy even for pfe of rational functions with multiple high-order poles or some tricky ill-conditioned poles.

### MSC:

 65D15 Algorithms for approximation of functions
Full Text:

### References:

 [1] Fateman, R. J., Rational Function Computing with Poles and Residues (2010), Berkeley, Calif, USA: University of California, Berkeley, Berkeley, Calif, USA [2] Simons, F. O.; Harden, R. C., Purely real arithmetic algorithms optimized for the analytical and computational evaluation of partial fraction expansions, Proceedings of the 30th Southeastern Symposium on System Theory [3] Linner, L. J. P., The computation of the kth derivative of polynomials and rational functions in factored form and related matters, IEEE Transactions on Circuits and Systems, 21, 2, 233-236 (1974) [4] Brugia, O., A noniterative method for the partial fraction expansion of a rational function with high order poles, SIAM Review, 7, 381-387 (1965) · Zbl 0132.12001 [5] Mahoney, J. F.; Sivazlian, B. D., Partial fractions expansion: a review of computational methodology and efficiency, Journal of Computational and Applied Mathematics, 9, 3, 247-269 (1983) · Zbl 0521.65014 [6] Karni, S., Easy partial fraction expansion with multiple poles, Proceedings of the IEEE, 57, 231-232 (1969) [7] Moad, M. F.; Karni, S., On partial fraction expansion with multiple poles through derivatives, Proceedings of the IEEE [8] Chin, F. Y.; Steiglitz, K., An $$O(N^2)$$ algorithm for partial fraction expansion, IEEE Transactions on Circuits and Systems, 24, 1, 42-45 (1977) · Zbl 0346.65011 [9] Westreich, D., Partial fraction expansion without derivative evaluation, IEEE Transactions on Circuits and Systems, 38, 6, 658-660 (1991) · Zbl 0747.65007 [10] Pottle, C., On the partial fraction expansion of a rational function with multiple poles by digital computer, IEEE Transactions on Circuit Theory, CT-11, 161-162 (1964) [11] Uraz, A.; N.-Nagy, F. L., Matrix formulation for partial-fraction expansion of transfer functions, Journal of the Franklin Institute, 297, 81-87 (1974) · Zbl 0307.93013 [12] Kung, S. H., Partial fraction decomposition by division, The College Mathematics Journal, 37, 132-134 (2006) [13] Witula, R.; Slota, D., Partial fractions decompositions of some rational functions, Applied Mathematics and Computation, 197, 1, 328-336 (2008) · Zbl 1135.65314 [14] Özyapici, A.; Pintea, C. S., Complex partial fraction decompositions of rational functions, Journal of Computational and Applied Mathematics, 1 (2012) [15] Garcia-Planas, M. I.; Dominguez, J. L., A general approach for computing residues of partial-fraction expansion of transfer matrices, WSEAS Transactions on Mathematics, 12, 647-756 (2013) [16] Man, Y. K., A cover-up approach to partial fractions with linear or irreducible quadratic factors in the denominators, Applied Mathematics and Computation, 219, 8, 3855-3862 (2012) · Zbl 1311.26005 [17] Ma, Y. N.; Yu, J. H.; Wang, Y. Y., An easy pure algebraic method for partial expansion of rational functions with multiple high-order poles, IEEE Transactions on Circuits and Systems I, 61, 803-810 (2014) [18] Chin, F. Y., The partial fraction expansion problem and its inverse, SIAM Journal on Computing, 6, 3, 554-562 (1977) · Zbl 0357.68037 [19] Kung, H. T.; Tong, D. M., Fast algorithms for partial fraction decomposition, SIAM Journal on Computing, 6, 3, 582-593 (1977) · Zbl 0357.68036 [20] Calvetti, D.; Gallopoulos, E.; Reichel, L., Incomplete partial fractions for parallel evaluation of rational matrix functions, Journal of Computational and Applied Mathematics, 59, 3, 349-380 (1995) · Zbl 0839.65054
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.