In mathematics, the splitting circle method is a numerical algorithm for the numerical factorization of a polynomial and, ultimately, for finding its complex roots. It was introduced by Arnold Schönhage in his 1982 paper The fundamental theorem of algebra in terms of computational complexity (Technical report, Mathematisches Institut der Universität Tübingen). A revised algorithm was presented by Victor Pan in 1998. An implementation was provided by Xavier Gourdon in 1996 for the Magma and PARI/GP computer algebra systems.
General description The fundamental idea of the splitting circle method is to use methods of complex analysis, more precisely the residue theorem, to construct factors of polynomials. With those methods it is possible to construct a factor of a given polynomial p ( x ) = x n + p n − 1 x n − 1 + ⋯ + p 0 {\displaystyle p(x)=x^{n}+p_{n-1}x^{n-1}+\cdots +p_{0}} for any region of the complex plane with a piecewise smooth boundary. Most of those factors will be trivial, that is constant polynomials. Only regions that contain roots of p(x) result in nontrivial factors that have exactly those roots of p(x) as their own roots, preserving multiplicity. In the numerical realization of this method one uses disks D(c,r) (center c, radius r) in the complex plane as regions. The boundary circle of a disk splits the set of roots of p(x) in two parts, hence the name of the method. To a given disk one computes approximate factors following the analytical theory and refines them using Newton's method. To avoid numerical instability one has to demand that all roots are well separated from the boundary circle of the disk. So to obtain a good splitting circle it should be embedded in a root free annulus A(c,r,R) (center c, inner radius r, outer radius R) with a large relative width R/r. Repeating this process for the factors found, one finally arrives at an approximative factorization of the polynomial at a required precision. The factors are either linear polynomials representing well isolated zeros or higher order polynomials representing clusters of zeros.
Details of the analytical construction Newton's identities are a bijective relation between the elementary symmetric polynomials of a tuple of complex numbers and its sums of powers. Therefore, it is possible to compute the coefficients of a polynomial
p ( x ) = x n + p n − 1 x n − 1 + ⋯ + p 0 = ( x − z 1 ) ⋯ ( x − z n ) {\displaystyle p(x)=x^{n}+p_{n-1}x^{n-1}+\cdots +p_{0}=(x-z_{1})\cdots (x-z_{n})}
(or of a factor of it) from the sums of powers of its zeros
t m = z 1 m + ⋯ + z n m , m = 0 , 1 , … , n {\displaystyle t_{m}=z_{1}^{m}+\cdots +z_{n}^{m}\,,\quad m=0,1,\dots ,n}
by solving the triangular system that is obtained by comparing the powers of u in the following identity of formal power series
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