This algorithm allows to find the distribution of the roots of a complex polynomialwith respect tothe unit circle in the complex plane. It is based on two auxiliary polynomials, introduced by Schur. For a complex polynomial of degree its reciprocal adjoint polynomial is defined by and its Schurtransform by where a bar denotes complex conjugation. So, if with, then with leading zero-terms, if any, removed. The coefficients of can therefore be directly expressed in those of and, since one or more leading coefficients cancel, has lower degree than. The roots of,, and are related as follows. ;Lemma Let be a complex polynomial and.
If, then, and share roots on the unit circle, including their multiplicities.
If, then and have the same number of roots inside the unit circle.
If, then and have the same number of roots inside the unit circle.
;Proof For we have and, in particular, for. Also implies. From this and the definitions above the first two statements follow. The other two statements are a consequence of Rouché's theorem applied on the unit circle to the functions and , where is a polynomial that has as its roots the roots of on the unit circle, with the same multiplicities. □ For a more accessible representation of the lemma, let, and denote the number of roots of inside, on, and outside the unit circle respectively and similarly for. Moreover let be the differencein degree of and. Then the lemma implies that if and if . Now consider the sequence of polynomials , where and. Application of the foregoing to each pair of consecutive members of this sequence gives the following result. ;Theorem Let be a complex polynomial with and let be the smallest number such that. Moreover let for and for.
All roots of lie outside the unit circle if and only if
for and.
If and if for and otherwise, then has no roots on the unit circle and the number of roots of inside the unit circle is
More generally, the distribution of the roots of a polynomial with respect to an arbitrary circle in the complex plane, say one with centre and radius, can be found by application of the Schur-Cohn test to the 'shifted and scaled' polynomial defined by. Not every scaling factor is allowed, however, for the Schur-Cohn test can be applied to the polynomial only if none of the following equalities occur: for some or while. Now, the coefficients of the polynomials are polynomials in and the said equalities result in polynomial equations for, which therefore hold for only finitely many values of. So a suitable scaling factor can always be found, even arbitrarily close to.
Lehmer's method
Lehmers method is as follows. For a given complex polynomial, with the Schur-Cohn test a circular disk can be found large enough to contain all roots of. Next this disk can be covered with a set of overlapping smaller disks, one of them placed concentrically and the remaining ones evenly spread over the annulus yet to be covered. From this set, using the test again, disks containing no root of can be removed. With each of the remaining disks this procedure of covering and removal can be repeated and so any number of times, resulting in a set of arbitrarily small disks that together contain all roots of. The merits of the method are that it consists of repetition of a single procedure and that all roots are found simultaneously, whether they are real or complex, single, multiple or clustered. Also deflation, i.e. removal of roots already found, is not needed and every test starts with the full-precision, original polynomial. And, remarkably, this polynomial has never to be evaluated. However, the smaller the disks become, the more the coefficients of the corresponding 'scaled' polynomials will differ in relative magnitude. This may cause overflow or underflow of computer computations, thus limiting the radii of the disks from below and thereby the precision of the computed roots. To avoid extreme scaling, or just for the sake of efficiency, one may start with testing a number of concentric disks for the number of included roots and thus reduce the region where roots occur to a number of narrow, concentric annuli. Repeating this procedure with another centre and combining the results, the said region becomes the union of intersections of such annuli. Finally, when a small disk is found that contains a single root, that root may be further approximated using other methods, e.g. Newton's method.
