J. Inst. Maths Applies (1976) 17,99-110
A Consistently Rapid Algorithm for Solving Polynomial Equations ~
J. B. MOORE
Department
of Electrical Engineering, University of Newcastle,
New South Wales, 2308, Australia
[Received 25 March 1974 and in revised form 20 September 1974]
Novel approaches are used to ensure consistently rapid convergence of an algorithm,
based on Newton’s method, for the solution of polynomial equations with real or
complex coefficients.lt appears that in terms of algorithm complexity and calculation
time, the new algorithm represents a considerable improvement on the various always
convergent algorithms in the literature. In terms of algorithm accuracy no improvements
are claimed.
1. Introduction
SINCEthe number of algorithms available for solving polynomial equations is legion
(indicating perhaps the unsatisfactory performance of existing algorithms) there
must be strong justification for introducing yet another such algorithm. To grasp the
signiiicanee of the algorithm introduced in this paper, a broad classification of the
various existing algorithms is in order.
First, there are the simple algorithms which usually converge for initial approximations in the neighborhood
of a polynomial zero but may not converge, or may
converge very slowly, when the initial approximation is not in a neighborhood of a
zero (Wilkinson, 1963). These algorithms fall naturally into three categories. There are
those which, when convergent, converge linearly (Lin’s algorithm) or superlinearly
(Muller’s algorithm) in the neighbourhood of a zero and require only the evaluation
of the polynomial itself at each iteration. Then there are the more popular quadratically
convergent (when convergent) algorithms of Newton and of Bairstow. For these,
both the polynomial and its derivative are evaluated at each iteration. Bairstow’s
method is restricted to polynomials with real coefficients and is marginally more
efficient than Newton’s method for these polynomials. Then again there are the
cubically convergent methods such as those of Laguerre. For these, the polynomial and
both its first and second derivative are evaluated at each iteration. These latter
algorithms give the least trouble. In fact, it is possible to arrive at a good universai
polynomial solver by first using Laguerre’s method and then, if convergence is not
reached in say thirty iterations, switching to Bairstow’s method. More reeently, simple
algorithms which converge to all zeros simultaneously either quadratically or cubically
have been developed as in Alberth (1973).
A second group of polynomial solving algorithms is the one where convergence is
guaranteed (Jenkins & Traub, 1970, 1972; Dejon & Nickel, 1969; Moore, 1967, 1970;
Grant & Hitchins, 1971; Ward, 1957; Householder, 1971; Stewart, 1969; Lehmer,
1961; Back, 1969). These algorithms are more sophisticated, particularly those that
f Work supportedby the Australian Resemch Grants Committee.
99
100
J. B. MOORE
guarantee a rapid rate of convergence. See, for example, the three stage algorithm of
Jenkins & Traub (1970).
In a previous paper by the author (Moore, 1967) a simple always convergent
algorithm is described. (This algorithm has been rediscovered recently in Grant &
Hitchins (1971)). The algorithm consists of Newton’s algorithm augmented with a
few logic instructions and a minimal number of calculations to ensure convergence of
the method. The resulting algorithm is simple enough, but there is still the difficulty
that for some polynomials and for some initial approximations convergence may be
quite slow. An attempt to accelerate this algorithm (Moore, 1970) using standard
type accelerating techniques has been reasonably successful for low order (less than
tenth order) polynomials, but these techniques result in slow convergence for certain
high order polynomials and certain initial conditions.
From the above sketchy review of available methods, one point emerges. There is
still the need for a simple always-convergent algorithm which gives consistently rapid
convergence irrespective of the polynomial equation considered and irrespective of
the first approximation to a root of this equation. The algorithm of this paper is
designed to supply this need.
Novel approaches are employed in an attempt to minimize the number of iterations
required for polynomial zero evaluations subject to the constraint that the calculation
time for each iteration be of the same order as that of the particular simple algorithm
on which it is based. The paper will be concerned with algorithms based on the
familiar quadratically convergent (when convergent) Newton’s method.
A brief outline of the remainder of the paper is as follows. Section 2 reviews the
derivation of the ideas and results of Moore (1967) and Grant & Hitchins (1971) and
develops concurrently additional results and insights necessary for understanding the
algorithms of the paper. Section 3 introduces the novel approaches taken in this paper
to achieve a consistently rapid polynomial zero finding algorithm, while comparative
performance data and concluding remarks are the material of the final Section 4.
2. Theory and Background
Consider the polynomial equation
f(z) =
,$0akz”-’= o,
(7D =
t.
(t)
where z = x + iy k the complex variable and a’ are the coefficients which may be real
or complex. We seek an iterative technique for updating an estimate of a real or
complex solution to this equation.
Consider now the algorithm
AZj = –f(zj) f,- 1(zj)
Zj = zj_l +aAzj_l,
where
AZ = AX+ iAY,
ZOgiven,
(2)
forj=l,2,
. . . where zj is the .jth estimate of a zero off(z). The notation ~z(z) is
used to indicate the derivative f(z). The possibly complex quantity u is termed the
step size scale factor. When u = 1, the algorithm (2) is simply Newton’s algorithm
which when convergent is quadratically convergent—at least to a simple zero off(z).
For some initial approximations ZO, Newton’s algorithm may not converge. For
COIWSTENTLY
RAPIDALGORITHM
101
example, if~(z) has no real roots and ZOis real, then Zj is real for all j and the algorithm
will not converge.
Straightforward extensions to Newton’s algorithm ensure its convergence (Moore,
1967; Grant & Hitchins, 1971). Some of the ideas of these papers are restated with
additional insights into the problem included in the restatement.
Consider the functions (norms off(z) and Az = –~(z)~z- l(z) respectively)
(3)
F(z) = l~(z)l’ and D(z) = l~(z)~~ ‘(z){’.
lt is not difficult to show that these functions have the following properties.
(a) The functions are non-negative for all z and become infinite as z becomes infiniteq
(b) The functions and their inverses are continuous and differentiable in the complex
plane except possibly at isolated points ~(z) is analytic for all z, ,~- 1(z) and
Az- ‘(z) are analytic for all z except the zeros of ~(z), and Az(z) is analytic for all
: except for the zeros of~,(z)].
(c) The zeros of each of the functions are the zeros off(z).
(d) The only minima of the functions are the zeros off(z).
(e) In any closed domain of the z-plane [not containing a zero of~z(z)], the function
F(z)[D(z)] cannot have a maximum.
(f) Extrema of F(z) that are not zeros of F(z) are saddle points. Saddle points of F(z)
are poles of D(z) or equivalently zeros of F=(z).
Notice that (d) and (e) do not exclude the existence of saddle points of F(z) and D(z).
As a consequence of properties (a)-(f), the problem of finding a zero of ~(z) is
equivalent to the problem of finding the location of minima (or equivalently zeros) of
F(z) [or of D(z)].
The algorithms of Moore and Grant & Hitchins incorporate modifications of
Newton’s algorithm to ensure that F(z) is decreased at each step. In particular, for the
algorithm (2), the step size scale factor a is selected as x = 1, ~, ~, . . . until there is a
decrease in F(z). The key result of Grant & Hitchins is that if saddle points of F(z)
are not encountered (that is ]~:(zj) I > c for somes > 0 then the percentage decrease
in F(z) is bounded below and thus the algorithm converges to a zero of F(z), or
equivalently to a zero off(z). An important intermediate result is that the directions
of the increments z (for the cmes~(z) and~,(z) non-zero) are in the direction of steepest
descent of F(z). The steepest descent directions are denoted 8.
Insight into the algorithms of Moore, Grant & Hitchins is gained if one notes the
property of the contours of constant [) = Az, namely that these contours change
direction dramatically in the vicinity of a saddle point region of F(z) or pass through
the saddle points (see Fig. 1 for an example). Now since for a real, the algorithm (2)
gives increments tangent to the constant-O contours, and since the magnitude of these
increments increases as saddle point regions of F(z) (zeros of ~Z(z))are approached,
it is clear that, to allow navigations of the sharp direction changes of the 8 contours,
the step size alAzl must be decreased, and many iterations consumed. The restriction
of a to the set of real numbers promotes slow convergence in the saddle point regions
and allows the possibility of convergence to a saddle point of F(z). This latter possibilityy can be avoided by the introduction of certain heuristics to the algorithm which
will not be discussed here.
102
J. B. MOORE
The algorithm of this paper is a variation of the algorithms of Moore and Grant &
Hitchins. The variations appear minor at first glance but the impact of the variations
is very significant. The variations are now summarized qualitatively.
–5=
x
iy
7’
IIero
t
mod I
Olgor
2
(b)
FIG.1,(a) Constant Fand Ocontours for Z8+ 1 = O.(b) Algorithm trajectories against a background
of constant D contours for Z8+ 1 = O.
lt is not difficult to establish that an increasein f)(z) at any jttxaljon of the standard
Newton algorithm (2) indicates that the iterations are heading for a saddle point region
of F or equivalentlyfor the region of sharp direction changes of the steepest descent
~ CO1’ltOUl’$,
~etreathg
parameciatdike
totheprevious
zero
approximation and then
venturing forward ag~ln v~~. a ide stepping kelahn <complex a), k is pos<~ble to
avoid the saddle point region of F and thereby gain proximity to a polynomial zero.
In othel WO~dSthe ~eu~~sticsof the algorithm of this paper take note of any increase
\n D of the ~ewton a\gorithm to determine complex a in order to ik3e step saddle
@n~ Iegiions of F(z) or equivalentlyto navigate efhi~entlysharp &rection changes in
~,e steegest deswnt Mection contours 0. The heuilstlcs ensure that any side step is
CONSISTENTLY
RAPID
103
ALOORITHM
to the left (right) of the steepest descent direction if any change in this direction at
each iteration is to the left (right).
The heuristics also accelerate Newton’s algorithm by a selection of the step size
scale factor as a = 1, 2, . . . . provided that the following three conditions are satisfied.
(a) Convergence is slow.
(b) There is no increase in D.
(c) Any change in direction is less than (say) 12°.
The acceleration steps are invaluable when converging to multiple zeros or when the
initial approximation is considerably in error.
A further variation is to set up the algorithm as a minimization of D(z) rather than
of F(z) as in Moore and Grant & Hitchins. The reason for this variation is to simplify
the algorithm and also experience indicates that it works somewhat better. Convergence proofs are less satisfactory than those in Grant & Hitchins but since they
parallel these and tell us nothing of the rate of convergence except in the vicinity of a
zero off(z), they will not be included here. Suffice it to say that with a few additional
instructions, it is possible to ensure that at (say) every twentieth iteration F(z) is
decreased using the ideas of Grant & Hitchins. Such an algorithm is guaranteed to
converge using the proofs of Grant & Hitchins. Our experience is that such modifications are superfluous,
3. A Consistently Rapid Algorithm Based on Newton’s Method
In constructing the polynomial equation solving algorithm of this paper, algorithm
complexity is kept to a minimum while still achieving good computational efficiency.
This is achieved by a judicious selection of permissible scale factors a. A classification
of permissible increments or steps that can be taken at any iteration, is as follows.
(1) Newton step. Here u = 1.
(2) Short Newton step. Here a = 1 but Zj = zj_l +AzJDJDj
rather than Zj =
Zj.l + Az as in (2) where D~ is a maximum allowable step size.
(3) Side step. Here a = ~ 1“5/45”.The particular sign of the 45° direction change
is chosen to anticipate th~direction change of the constant f3 contours. It is
calculated using 0 information as follows.
sgn[Sin(Oj– ‘j-~)] = sgn(sin Oj
.
sgn(Ayjbj.l
COS oj-~
– COS Oj
sin ‘j-~)
– AxjAyj-l).
If 8 = 8j_l then an arbitrary sgn is used.
step. Here u = 2,3,...
(5) Deceleration step. Here l~jl = ~laj-l 1,/aj
——= /~j-l + 131°.
(4) Acceleration
A number of tests are employed to determine when each of the above steps is to be
used. The tests are now listed.
(1) (Dj – Dj-l)—t@s whether D is decreasing at each stage.
(2) (Dj – D~)—tests whether D is less than some maximum allowable step size.
(3) (Y–P) where (Y–P) = ~DjDj-l cos(oj– @j-l) = (A~jAXj.l+ A~jAyj-~)–
0.98 JDjDj-l (tests direction changes).
1. B.
104
MOORE
(4) (la]Sj – Sj-l) where S’j = Fj/Fj-l (tests whether convergence is better than
linear).
(5) (la! –0.96)—tests magnitude of Ial. The value 0.96 is a flag or indicator.
The flow chart of Fig. 2 indicates an algorithm which is considered to be a reasonable compromise on complexity and calculation efficiency.
A number of comments are now offered on this algorithm.
(1) Although at first glance the algorithm appears to be more complicated than the
algorithms of Moore and Grant & Hitchins in which a is always real, in fact the
Select 2., set a ‘ Q95, L-=19-7, G = Fo=1035, ,9~=05
I
I
I
c!=Colculote
f(zj),
fz(fi),
w = I
f’=(zj)lz
I
I
I
i
4
=Fj/Fj_l
~ =lf(~)lz, S’/.
1
I
1
Logic
for
selection
of a [see
Fig.
2(b)]
J
I
1
I
~
Zj=
~-l+aA+l, j=j+t
1
FIG. 2(a), Algorithm flow chart.
Accept
-?j as a zero
of f(z)
CONSISTENTLY RAPID ALGORITHM
105
computer program requires less than twenty additional FORTRANstatements. The gain,
as far as computational efficiency is concerned, is considerable as the next section
indicates.
(2) Notice that acceleration is only permitted when Dj < Dj_l, lA6jl <12°
(i.e. Y > P) and ]til~j > Sj-l or on a side step (see (5) above). The first requirement
needs no explanation. The second requirement is simply that the direction change at
an iteration is small. Clearly it is unwise to consider acceleration when the direction
change at an iteration is more than say 12°. The third requirement is that the algorithm
without acceleration is achieving no better than linear convergence of F. Acceleration
is achieved by incrementing a by unity.
——
____ ————
___
r
1,
a
~
,,
r+l---l
I
u
i
ii
Y
( ? H
i
......—.. __.J
<, ~–
s
L-l
I
IN
—,,
i
III-’--I”’
u
~-----”--’
!
.;“l;__:{
(“m‘“---T
4,
u
I
L .—
___
—
___
—’
--w--”’—-
J
!
106
J. B. MoORE
(3) The short Newton step is taken whenever Dj > D~ where D~ is a maximum
allowed step size. Whenever Dj < D~, then DM is set equal to Dj. The initial value of
D~ is set to be the suspected diameter of the smallest region containing but one zero.
After a zero removal of nonzero magnitude, D~ is set to be the magnitude of the
previously found zero.
(4) A side stepping (acceleration) iteration is taken whenever Dj ~ Dj_l and
Ial >1. In other words, whenever a saddle point region of F(z) is suspected, an
attempt is made to side step this region. Such an iteration rarely fails to reduce D,
but if it does not reduce D, a decderation step is used.
(5) The algorithm is designed so that rarely is Dj > Dj_l after one side stepping
iteration. In this event a deceleration step is Used. It may be possible to construct a
polynomial and an initial value such that in some very complicated saddle point
regions of F(z), D does not decrease for the deceleration steps prescribed by the
algorithm. Such nonconvergent or its cousin S1OWconvergen~ is easily detected and
a new starting point can be chosen to avoid such difficulties. This possible improvement is not incorporated into the flow chart of Fig. 2 to avoid overcomplication.
(6) Mult@e zeros off(z). For multiple roots, acceleration steps are required in
order to give rapid convergence. Accuracy may be a problem as pointed out in
discussing iteration termination.
(7) Calculation time for each iteration. The increase in calculation time for each
iteration over that of a regular Newton iteration is at most the equivalent of (say)
twelve multiplications. This is negligible compared to the 8n or so multiplications
required for the regular Newton iteration once n is four or greater. The decision
network to select a suitable a consists of about four or so decisions for each path
through the network and thus represents a negligible cost.
(8) It might be thought that since saddle points of F cause trouble in algorithms of
Moore and Grant & Hitchins, saddle points of D may muse trouble in the algorithm
of Fig. 2. This is not the case. On the contrary, it is in these regions that the side step
iteration is most efficient. Moreover, the algorithm is not attempting to decrease D
along the direction of steepest descent of D. (To calculate this direction, higher
derivatives would be needed.) Actually it would be convenient to have available the
steepest descent direction of D, for then a composite direction of steepest descent of F
and this direction would result in a highly efficient algorithm. In essence, the side
stepping iteration of the present algorithm is a crude attempt to attain this composite
direction when it is needed most.
(9) Slow convergence or nonconvergence may sometimes occur in practice due to
poor scaling and a poor initial zero estimate combined with a selection of too small a
maximum allowable step size, computer overflow, etc. However, since convergence is
normally consistently rapid, such abnormal situations are quickly detected. If the
&~gOli~hmhas not converged to a zero in <say) 20+-~ n iterations then an abnormal
situation has arisen.
Some additional comments concerning the well-researched problems of termination
of the iteration process, estimation of calculation accuracy, polynomial deflation and
the selection of an initial zero estimate are given in the Appendix. A good reference on
these topics is Peters & Wilkinson (1971).
107
CONSISTENTLY
RAPID ALGORITHM
4. ComparativePerformance
Any realistic comparison must consider the three factors, accuracy, calculation
time, and program complexity.
In the first instince it is reasonable to compare the present algorithm with other
algorithms based on Newton’s method. Now since most algorithms based on Newton’s
method use the same recursive equation in the final iterations for finding a polynomial
zero, the accuracy of all the various algorithms based on Newton’s method is of the
same order onm the precision of the calculations is specified. The calculation time for
each iteration is approximately the same for the two cases studied in this section.
The complexity of the present algorithms require the addition of about 30 FORTRAN
statements to the simplest Newton algorithm. We suggest that the iteration number is
a reasonable criterion for comparison.
Figure 3(a) presents comparative information on the number of iterations to find a
zero of z“+ 1 = Ofor n = 3 to 100 with a fist approximation ZO= 1+ il. Figure 3(b)
indicates the average number of iterations to solve for all the zeros of z“+ 1 = O
using deflation and using the deflation zero as the starting point for the next zero
w
Mdafk?d Newton
algor, Ihm
)
AlgOrlthm of F!g z
,~
DegreeN ofFolymmIol
z“+ I=0
10
6 –
76–
54–
3–
~–
I
o
10
30
20
40
I
5(
n
FIG.
3.
(a) Performance data. (b) Solution of z“+ 1 = Ofor all zeros.
108
J. B. MOORE
calculation. Complex zeros are extracted in pairs and the iteration number for the
extraction of real roots is weighted by one half since real arithmetic is involved.
Double precision arithmetic is used, but even so ill-conditioning becomes a difficulty
for some of the polynomials above order 60. Notice that the algorithm of Moore and
Grant & Hitchins is very inefficient for some n. In fact, convergence is so slow in
some instances as to be just as annoying as an algorithm which fails to converge for
some initial conditions. The algorithm of this paper clearly represents a con.~iderable
improvement in consistency of convergence rate. (Almost
identical
results are obtained
for other first approximations with /zOI < l—at least with the maximum allowable
step size set initially as D~ = 0.4.)
The same problem is solved by an always convergent algorithm in Back (1969)
where each iteration requires three function evaluations as well as a derivative
evaluation (that is each iteration involves twice the amount of calculations as the
algorithms of Sections 2 and 3). @ the basis of the results reported in Back, his
algorithm requires at best twice as many iterations as that of the algorithm of this
paper and at worst ten times as many iterations. (That is, the present algori!hm is
from four to twenty times faster than that of the algorithm of Back and only a little
(if any) more complicated.) This comparison points up the consistency quality of the
present algorithm.
Of course the polynomial~(z) = z“ + 1 is not the only standard difficult poiy nomial
to solve but the results for this problem are in fact almost identical to those obtained
with other difficult polynomials.
For ten polynomials each with [1O, 20, 30, 40, 50, 60] random zeros (Izl < 2), the
average number of iterations to solve for all zeros of each polynomial (extracting
complex zeros in pairs, weighting real zero iterations by one half, and using double
precision arithmetic) is in the range [5f- 1, 5++ 1, 6f 1, 6++ 1, 7*1, 8+1] confirming
the statement above.
For polynomials with low order multiple zeros there is little or no variation from
the results indicated so far. For the case of high order multiple zeros inherent accuracy
limitations are encountered as illustrated in for example the solution of (z + 1~ = O
with first estimate [–0.5, 0,01]. Here with n = 3, 4, 5, 6, 7, 8, 9 the number of
iterations to yield a first zero is 22, 24, 27, 27, 27 and the average number of iterations
to yield all zeros is 7, 6, 7, 6, 8, 6, 9 respectively. For higher order polynomkds the
inaccuracies using double precision arithmetic are intolerable. The methods of this
paper have not improved the accuracy of the basic Newton algorithm-only
speeded
it up and avoided poor convergence characteristics.
To compare the present algorithm with those of Jenkins & Traub, Dejon & Nickel
and Lehmer is best done first on the basis of complexity. These algorithms involve
considerably more computation for each iteration than the present algorithm.
For example, the algorithm of Dejon & Nickel requires all derivatives off(z) to be
calculated. Thus the calculation time for each iteration goes up in proportion to
n
~ i as compared to
II
~
i as in the present algorithm. It is not surprising therefore
i=]
i=~—1
that the number of iterations is less than that of the algorithm of Section 3. Even so,
the complexity and calculation times appear to be of an order of magnitude greater
than that of the algorithm of this paper. The same may be said of many of the always
109
CONSISTENTLY
RAPIDALGORITHM
convergent algorithms in the literature. We shall not give a detailed comparison for
each of’these methods.
To conclude, it is perhaps worthwhile to mention that investigations so far carried
out in applying the techniques of this paper to Bairstow’s method and Laguerre’s
method have also been partially successful and fully successful respectively. As for
Newton’s method, the first indication that convergence may not be proceeding
satisfldctorily in either of the above methods is that the step size increases at a particular
iteration. Also, as for Newton’s method, the application of ideas of Section 3, namely
to introduce a direction change when step size increases, helps considerably. However,
with second derivatives information available as in Laguerre’s method, more efficient
ways of dealing with the problem mentioned are available. Even so, the algorithm of
the present paper is more eflicient in terms of calculation time and program complexity.
With the modified Bairstow’s method, due to the existence of local minima of D in
some polynomials, convergence cannot be guaranteed and frequently slow convergence
occurs. Even when the modified Bairstow’s method converges its performance on the
average is not as good as for the modified Newton method.
REFERENCES
D. A. 1967 Communs A. C.&l. 10, 655-658.
ALBERTH,
O. 1973 Maths Cornput. 27, 339-344.
ADAMS,
BACK,
H, 1969 Communs A. C.&l. 12, 675-684.
1969In Constructive aspects of the flmdamenta[ theorem of algebra;
DEJON, B. & NICKEL, K.
proceedings of the symposium conducted at the IBM Research Laboratory, ZurichRuscldikon, Switzerland, June 5-7, 1967 (eds B. Dejon & P. Henrici). London: WileyInterscience, pp. 1-35,
GRANT, J. A. & HITCHINS, G. D. 1971.1. Inst. Maths AppIics 8, 122-129.
HOUSEHOLDER, A. S. 1971
JENKINS, M.
A.
Num. Math. 16, 375-382.
1970 Num. Math. 14, 252–263. (See
& TRAUB, J. F.
also 1972 Communs
A. C./W. 15, 97-99.)
LEIiMER, D.
H. 1961J. Ass. cornput. Mach. 8, 151-162.
MOORE, J. B. 1967 f. Ass. comput. Mach. 14, 311-315.
MOORE, J. B. 1970 IEEE Trans. on Computers C-19(1), 79-80.
PETERS,G. & WILKINSON, J. H. 1971 J. Inst. Maths Applies 8,
STEWART, G. W. 1969 Num. Math. 13, 458-471.
WARD, J. A. 1957 J. Ass. comput. Mach. 4, 148-150.
WILKINSON,
J. H.
1963
16-35.
Rounding errors in algebraic processes. Englewood Cliffs, N.J.:
Prentice-Hall.
Appendix
(1) Evaluation off(z) andfJz) for the algorithm (2) requires 2n complex (8n real)
multiplications for the case when ak are complex and half this amount when the
coeficien~s a~ a~e reaL Perhaps the Qmplest method to evaluate these quantities is by
nested rnuhip~~cation as follows. Two sequences bk and c~ defined by
bO = 1,
b,+, = z“ b~+a~
c~+~= Z“Ck+bk
co = 1,
are calculated simultaneously for k = 1 to n to obtain ~(z) = b.+l and
f.(z)
=
co.
110
J. B. MOORE
After a zero of ~(z) has been accepted and divided out from the polynomial, the
n- 1
reduced order polynomial is ~ bkzn-1 -‘.
k=o
(2) Termination of the algorithm. Notice that the algorithm of Fig. 2 accepts a zero
of~(z) when D has reached its minimum value achievable using the Newton algorithm
(2). This is detected when D increases at an iteration and yet ~D is less than a small
value E which is initially set for any zero removal as a conservative estimate of a
round-off error bound. Notice also that in the event that the initial estimate E is set
too low, as will usually be the case when converging to a high order multiple zero for
example, the actual round-off errors cause oscillations in D. These oscillations in D
are detected and cause an increase in E so that the algorithm will terminate. For real
polynomials an alternative approach is to use a round-off error bound for E (Adams,
1967).
(3) Estimate of calculation accuracy. The value of ~Dj at the final iteration is a
reasonable estimate of the computation accuracy and should be printed out along
with the value of zj at the final iteration.
(4) Polynomial defiation. Here the results of Peters & Wilkinson are relevant if full
accuracy is required in a composite deflation process. With the simpler forward
deflation process, zeros should be found in increasing magnitude. This presents no
special difficulty since with the initial estimate of [0, O],the algorithm usually finds the
zero of minimum magnitude. Once zeros are found using a deflation technique (with
high precision arithmetic in at least the last four iterations), improvement in accuracy
can be achieved using the undefeated polynomial. If ill-conditioning is suspected, the
zeros found using the techniques above could be chosen as starting points for the
algorithms of which find all roots simultaneously. The combination of the methods of
this paper with the methods of Alberth (1973) will yield more rapid solutions than
the methods of Alberth alone—at least when there is little a priori information concerning zero locations.
(5) Selection of initial zero estimate Zo. It is perhaps worthwhile for the algorithm to
select an initial approximation to a zero off(z) from two or three tentative estimates of
the minimum magnitude zero by choosing the one with the minimum value of D.
Suggestionsfor the various initial estimates are: (a) standard initial estimates such as
[O@05,O]or random zeros in the unit square, (b) the conjugate value of the previously
calculated root (for the case of real coefficients this is a good check on accuracy
involving only a few extra iterations), (c) the previously calculated zero value (in
case there is a multipIe root or cluster of roots).