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 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 by solving the triangular system that is obtained by comparing the powers of u in the following identity of formal power series

is a domain with piecewise smooth boundary C and if the zeros of p(x) are pairwise distinct and not on the boundary C, then from the residue theorem

of p(x) corresponding to the zeros of p(x) inside G. By polynomial division one also obtains the second factor g(x) in p(x) = f(x)g(x).

The commonly used regions are circles in the complex plane. Each circle gives raise to a split of the polynomial p(x) in factors f(x) and g(x). Repeating this procedure on the factors using different circles yields finer and finer factorizations. This recursion stops after a finite number of proper splits with all factors being nontrivial powers of linear polynomials.

The challenge now consists in the conversion of this analytical procedure into a numerical algorithm with good running time. The integration is approximated by a finite sum of a numerical integration method, making use of the fast Fourier transform for the evaluation of the polynomials p(x) and p(x). The polynomial f(x) that results will only be an approximate factor. To ensure that its zeros are close to the zeros of p inside G and only to those, one must demand that all zeros of p are far away from the boundary C of the region G.

Basic numerical observation

be a polynomial of degree n which has k zeros inside the circle of radius 1/2 and the remaining n-k zeros outside the circle of radius 2. With N=O(k) large enough, the approximation of the contour integrals using N of the factor f where the norm of a polynomial is the sum of the moduli of its coefficients.

as close as wanted to the zeros of f by choosing N large enough. However, one can improve this approximation faster using a Newton method. Division of p of the remaining factor g using any variant of the extended Euclidean algorithm. This is repeated until the increments are zero relative to the chosen precision.

Graeffe iteration

The crucial step in this method is to find an annulus of relative width 4 in the complex plane that contains no zeros of p and contains approximately as many zeros of p inside as outside of it. Any annulus of this characteristic can be transformed, by translation and scaling of the polynomial, into the annulus between the radii 1/2 and 2 around the origin. But, not every polynomial admits such a splitting annulus.

To remedy this situation, the Graeffe iteration-th dyadic powers of the roots of the initial polynomial p and thus tend to infinity. Choosing j large enough one finally finds a splitting annulus of relative width 4 around the origin.

is now to be lifted back to the original polynomial. To this end an alternation of Newton steps and Padé approximations holds. The polynomials on the left side are known in step j, the polynomials on the right side can be obtained as Padé approximants of the corresponding degrees for the power series expansion of the fraction on the left side.

Finding a good circle

Making use of the Graeffe iteration and any known estimate for the absolute value of the largest root one can find estimates R of any root of p(x) to any of the five center points 0, 2R, −2R, 2Ri, −2Ri Graeffe iterations, the corresponding annulus of the iterated polynomial has a relative width greater than 11 > 4, as required for the initial splitting described above , allowing a much simplified initial splitting

To locate the best root-free annulus one uses a consequence of the Rouché theorem: For k = 1, ..., n− u > 0, has, by Descartes' rule of signs. In the latter case, there are exactly k is a root-free (open) annulus.

References

• Magma documentation. Real and Complex Fields: Element Operations.