4.7 MANIPULATION OF POWER SERIES 515 12. [MZU] (Web hosting asp)
4.7 MANIPULATION OF POWER SERIES 515 12. [MZU] Find a connection between polynomial division and power series division: Given polynomials U(X) and w(z) of respective degrees m and n over a field, show how to find the polynomials q(x), r(z) such that U(X) = q(z)v(z) + r(z) and deg(r) < n, using only operations on power series. 13. [A&7] (Rational function approximation.) It is occasionally desirable to find polynomials whose quotient has the same initial terms as a given power series. For example, if W(z) = 1 + z + 32 + 7z3 f . , there are essentially four different ways to express W(z) as w~(z)/w~(z) + O(z4) w h ere wi(z) and ws(z) are polynomials with deg(wi) + deg(wz) < 4: (1 + z + 3z2 + 7z3) / 1 = 1 + z + 3z2 + 7z3 + o.z4 + . ) (3 -42 + 222) / (3 -72) = 1 + z + 3z2 + 72s + Yz + . , (1 -2) / (1 -22 -z ) = 1 + z + 3z2 + 7z3 + 17z4 + , l/(1-z-zz2-z+=1+Z+3z2+723+15z4+.... Rational functions of this kind are commonly called Pad6 approximations, since they were studied extensively by H. E. Pade [Annales Scient. de J&oJe Normale SupCrieure (3) 9 (1892), Sl-S93]. Show that all Pad6 approximations W(z) = w~(z)/w~(z) + O(zN) with deg(wi) f deg(ws) < N can be obtained by applying an extended Euclidean algorithm to the polynomials zN and WO+WI~+...+WN-IZ~- ; and design an all-integer algorithm for the case that each W, is an integer. [Hint: See exercise 4.6.1-26.1 b 14. [HMXI] Fill in the details of Brent and Traub s method for calculating U (z) when U(z) = z + Uk zk + . . , using (27) and (28). And it shall be, when thou hast made an end of reading this book, that thou shalt bind a stone to it, and cast it Into the midst of Euphrates. -Jeremiah 51:63
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