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The software for the Jenkins–Traub algorithm was published as Jenkins and Traub [http://portal.acm.org/citation.cfm?id=361262&coll=portal&dl=ACM Algorithm 419: Zeros of a Complex Polynomial].<ref>Jenkins, M. A. and Traub, J. F. (1972), [http://portal.acm.org/citation.cfm?id=361262&coll=portal&dl=ACM Algorithm 419: Zeros of a Complex Polynomial], Comm. ACM, 15, 97–99.</ref> The software for the real algorithm was published as Jenkins [http://portal.acm.org/citation.cfm?id=355643&coll=ACM&dl=ACM Algorithm 493: Zeros of a Real Polynomial].<ref>Jenkins, M. A. (1975), [http://portal.acm.org/citation.cfm?id=355643&coll=ACM&dl=ACM Algorithm 493: Zeros of a Real Polynomial], ACM TOMS, 1, 178–189.</ref>
The methods have been extensively tested by many people.{{Who|date=December 2021}} As predicted they enjoy faster than quadratic convergence for all distributions of zeros.
However, there are polynomials which can cause loss of precision<ref>{{Cite web|date=8 August 2005|title=William Kahan Oral history interview by Thomas Haigh|url=http://history.siam.org/oralhistories/kahan.htm|url-status=live|access-date=2021-12-03|website=The History of Numerical Analysis and Scientific Computing|publication-place=Philadelphia, PA}}</ref> as illustrated by the following example. The polynomial has all its zeros lying on two half-circles of different radii. [[James H. Wilkinson|Wilkinson]] recommends that it is desirable for stable deflation that smaller zeros be computed first. The second-stage shifts are chosen so that the zeros on the smaller half circle are found first. After deflation the polynomial with the zeros on the half circle is known to be ill-conditioned if the degree is large; see Wilkinson,<ref>Wilkinson, J. H. (1963), Rounding Errors in Algebraic Processes, Prentice Hall, Englewood Cliffs, N.J.</ref> p. 64. The original polynomial was of degree 60 and suffered severe deflation instability.
==References==
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