On the Separation of All Real Roots of a POlynomial Equation on the Arbitrary Interval

This method finds all real roots of a polynomial equation on the arbitrary interval 〔a, b〕 such that a<b. Let f(x)=a_0x^n+a_1x^<n-1>+…+a_n=0 be a given polynomial whose coefficients are real but a_0〓0. From this, we construct a sequence of polynomial as follows. That is, f(x), f'(x), …, f^<(n-1)>(x). Now we can obtain the root x_<11> of f^<(n-1)>(x)=0 immediately because of f^<(n-1)>(x) being linear. If there is x_<11> on the interval 〔a, b〕, we divide the interval 〔a, b〕 into two parts 〔a, x_<11>〕 and 〔x_<11>, b〕 by using it. Next we consider whether there is or not a root of f^<(n-2)>(x)=0 on each part. If f^<(n-2)>(a)・f^<(n-2)>(x_<11>) is negative, then f^<(n-2)>(x)=0 has one root on the interval 〔a, x_<11>〕. If positive, then one has no root. Now suppose f^<(n-2)>(x)=0 has one root on the interval 〔a, x_<11>〕. Then by Newton-Raphson method starting with the first approximation x_1=(a+x_<11>)/2,we can obtain rapidly the root x_<21> of f^<(n-2)>(x)=0 on the interval 〔a, x_<11>〕. Similarly, on the other interval 〔x_<11>, b〕 we can obtain the root x_<22> of f^<(n-2)>(x)=0 if there is. By using x_<21> and x_<22>, we divide afresh the interval 〔a, b〕 into three parts 〔a, x_<21>〕, 〔x_<21>, x_<22>〕 and 〔x_<22>, b〕. Repeating the similar procedure for these parts, we can obtain the roots of f^<(n-3)>(x)=0 of degree greater than f^<(n-2)>(x). In general, on the interval 〔a, b〕 we can obtain the real roots of f^<(i-1)>(x)=0 by finding those of f^<(i)>(x)=0 of degree lower than f^<(i-1)>(x) where 1〓i〓n-1. Finally, containuing the method mentioned above, we can find the real roots of f(x)=0 on the interval 〔a, b〕, which is divided into some parts by those of f'(x)=0.