代数方程式の数値解法

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In this paper author describes a method for finding roots of a polynomial. Let ƒ(z)=a_0z^n+a_1z^<n-1>+…+a_n be a polynomial whose coefficients are complex. Using Lehmer's algorithm for ƒ(z), we can construct a polynomial sequence such that ƒ_0(z), ƒ_1(z), ……, ƒ_<k-1>(z), ƒ_k. Besides, we calculate P_<i-1>=|P_i|(|ƒ_<i-1>(0)|^2-1)/(|ƒ_<i-1>(0)|^2+1) where i=1,2,……, k and P_0=1. Let m be the maximum i such that |P_i|>d^<-t> where d (=2 or 10) is the base of the computer's number system and t is the number of digits in the mantissa of the float ng point system. Then we can obtain the number M of roots which ƒ(z) has in the unit circle by the recurrence formula so that M_<i-1>=1(-P_i)n_<i-1>+sgn(P_i) M_i where 1(x) is unit function, sgn(x) is sign function, i=m, m-1,……, 1 and M_m=0. Furthermore, we can know easily the number of roots of ƒ(z) in a circle of radius R by transforming ƒ(z) to ƒ(Rz). From this we can find as follows : (1) the circle Γ having one or more roots of ƒ(z) on itself, (2) the places of roots of ƒ(z) lying on Γ. The process for finding (1) and (2) is a second order. Also, this method can obtain double roots and roots of greater multiplicity.
山口大学の論文