An investigation of Lehmer's method for finding the roots of polynomial equations using the Royal-McBee LGP-30
“The solution of the general polynomial equation f (x) = O, where f(x) = anxn + an-1xn-1 + … + a1x = ao, has received the attention of many mathematicians for hundreds of years and is at present in a very highly developed state. Even a cursory examination of the literature will reveal many volumes on this subject. However, this study is concerned primarily with the numerical methods for solving polynomial equations, hence the classical methods will be treated here only as they contribute to this field.
Virtually all of these methods possess serious drawbacks to being programmed for a computer. Two of the more common disadvantages are: 1. uncertainty of the convergence of the iterative scheme and 2. the necessity for making decisions based on judgements which are difficult to incorporate in a program for a digital computer.
The first attempts at numerical solutions involved hand computation or desk calculators, and a human being was in control at each step of the process. If at any time some difficulty arose, he could make the proper adjustments, and possibly change to a different technique which would get him out of the trouble. If the programmer were always operating the digital computer himself, he could perform the same type of function. However, for an automatic computer, one must incorporate into his program routines to deal with any situation, so that the machine will take the data and, without any interference by the operator, produce the desired results.
The first difficulty, i.e. the question of convergence, can often be overcome if the initial values used in starting the iterative scheme are close enough to the true roots.
There are many theorems for isolating the roots and finding appropriate starting values. One of these is the well known Descartes’ rule of signs. Other schemes were developed by Newton and Budan. One of the most dependable, involving Sturm’s functions, will be discussed in the next chapter.
The second difficulty, that of making the decisions, arises from the fact that in many of the older methods, special facilities must be devised to handle the cases of multiple roots, complex and imaginary roots, and roots of the same or nearly equal magnitudes. These questions can usually be resolved only by very complex programs.
The method presented here is designed to overcome both of these difficulties. It was designed especially for use on a digital computer instead of being adapted from some method originally developed for hand computation.
The main reference used is Lehmer. This method has been coded successfully for large computers, but to the author’s knowledge has never been adapted to smaller computers such as the Royal-McBee L.G.P.-30.
In review of Lehmer’s article, Ralston has suggested the need for further investigation of this method. It is hoped that this document will make some contribution to this investigation"--Justification, pages 1-3.
Lee, Ralph E., 1921-2010
Govier, John P., 1913-1998
Erkiletian, Dickran Hagop, Jr.
Nelson, John August
Mathematics and Statistics
M.S. in Mathematics
Missouri School of Mines and Metallurgy
v, 41 pages
Note about bibliography
Includes bibliographical references (page 36).
© 1962 James Walter Joiner, All rights reserved.
Thesis - Open Access
Equations -- Numerical solutions -- Computer programs
Equations -- Numerical solutions -- Computer simulation
Equations -- Numerical solutions -- Mathematical models
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Joiner, James W., "An investigation of Lehmer's method for finding the roots of polynomial equations using the Royal-McBee LGP-30" (1962). Masters Theses. 2699.