Chapter 10 - Bernoulli, Quotient-Difference, and Integral Methods

Abstract

This chapter treats several topics, starting with Bernoulli’s method. This method iteratively solves a linear difference equation whose coefficients are the same as those of the polynomial. The ratios of successive iterates tends to the root of largest magnitude. Special versions are used for complex and/or multiple roots. The iteration may be accelerated, and Aitken’s variation finds all the roots simultaneously. The Quotient-Difference algorithm uses two sequences qk(m+1)=qk(m)+ek(m)-ek-1(m+1) (with a similar one for ek(m)). Then, if the roots are well separated, qk(m)→1kthroot. Special techniques are used for roots of equal modulus. The Lehmer–Schur method uses a test to determine whether a given circle contains a root or not. Using this test we find an annulus R<|z|<2R which contains a root, whereas the circle |z|<R does not. We cover the annulus with 8 smaller circles and test which one contains the roots. We repeat the process until a sufficiently small circle is known to contain the root. We also consider methods using integration, such as by Delves–Lyness and Kravanja et al.