A Special Interval Newton Step for Solving Nonlinear Equations
IOANNIS A. NIKAS
Laboratory of Tourism Information Systems and Forecasting
Department of Tourism Management
University of Patras
M. Alexandrou 1, Patras
GREECE
Abstract: The problem of finding the zeros of a nonlinear equation has been extensively and thoroughly studied.
Point methods constitute a significant and extensive category of techniques, allowing for the efficient finding of
zeros with arbitrary precision under specific conditions. However, the limitation of these methods is that they
typically yield a single zero. An alternative approach employs Interval Analysis, leveraging its properties to
provide reliable and with certainty inclusions of all zeros within a given search interval. Interval methods, such as
the Interval Newton method, exhibit quadratic convergence to the corresponding inclusions when monotonicity
and simple zeros exist. Nonetheless, there exist pathological cases, like the existence of multiple zeros, where the
obtained inclusions cannot be bounded with arbitrary precision, necessitating the adoption of bisection schemes to
refine the search interval. These schemes not only increase computational time and cost but also result in a higher
number of enclosures, enclosing sometimes the same zero more than once. The main objective of this work is to
enhance the applicability of Interval Newton method in cases where no efficient alternative are available. Thus, in
this paper, the Interval Newton method is studied and an adjusted perturbation technique is proposed to address the
cases where multiple zeros exist. In particular, the given function is vertically shifted. Then, the Interval Newton
operator is applied once to this shifted function. The resulting enclosures are then used to efficiently partition
the search interval. The successful application of the Interval Newton method is expected to improve overall
performance and reduce reliance on bisection schemes. Experimental results on a set of problems demonstrate
the effectiveness of the proposed technique.
Key-Words: nonlinear equations, interval bisection, perturbation technique, pathological cases, bisection
schemes
Received: May 2, 2023. Revised: August 4, 2023. Accepted: August 25, 2023. Published: September 26, 2023.
1 Introduction
This paper addresses the problem of solving reliably
and with certainty the equation
(1)
within a closed interval R, where R
Ris a differentiable nonlinear function over the real
numbers.
The importance of (1), [1] arises in various numer-
ical analysis problems, including solving systems of
nonlinear equations and optimization problems, [2],
[3], as well as real-world applications in fields such as
chemical engineering (process design and flowsheet-
ing), computer graphics (render implicit surfaces),
robotics (determine the efficiency of the used motors),
and control theory (stability assessment of a linear
time-invariant system, robust mobile robot path plan-
ning), [4], [5], [6], [7], [8].
Several methods have been proposed to solve (1),
such as Newton-Raphson method, bisection method,
secant method etc., each having distinct characteris-
tics and advantages, [9]. Although certain techniques,
such as the Newton-Raphson method, demonstrate
favorable characteristics like quadratic convergence
when locating single or multiple zeros, their draw-
back resides in their capacity to approximate only a
single zero during each iteration. In addition, for the
efficient discovery of zeros with multiplicities greater
than one, a series of modifications has been proposed,
under the assumption that the multiplicity of the zero
being sought is already known, [10].
An alternative and efficient approach involves us-
ing Interval Arithmetic, initially introduced by Moore
in 1966, [11]. These methods reliably enclose all ze-
ros of (1) within a given search interval with arbi-
trary precision. Interval Newton is a typical and well-
known method from the class of interval methods.
The Interval Newton method exhibits appealing
properties, such as quadratic convergence and zero
uniqueness (Theorems 2 and 3) and, recently, some
improvements in the performance of the method have
been proposed, [12], [13]. Nevertheless, there exist
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certain cases in which the method fails to provide
sharp zero enclosures. One such case is the exis-
tence of multiple zeros. To address such pathologi-
cal cases, a bisection scheme is employed on the cur-
rent search interval. This approach successfully re-
duces the search interval, which is the primary ob-
jective of interval algorithms (Algorithm 1). How-
ever, it is important to note that the bisection scheme
may, at times, generate multiple sequences of inter-
vals that enclose the same zero. This can lead to an
increase in the number of final zero enclosures and,
consequently, a rise in the computational cost of the
method.
It is worth noting that Interval methods generally
efficiently handle the problem of enclosing zeros with
multiplicity greater than one, [14]. Although the com-
putational cost increases, the method will eventually
provide enclosures for these zeros with arbitrary pre-
cision.
The aforementioned pathological cases have been
acknowledged within the scholarly discourse. In
some instances, no proposals for solutions are given.
In others, directions for a possible search strategy are
provided, depending on the zero multiplicity, [13].
Alternatively, more precise interval arithmetics may
be adopted to obtain sharp enclosures, but these ap-
proaches do not clearly avoid bisection schemes in
these cases, [12].
In this work, the behavior of the Interval Newton
method is studied for cases where a bisection scheme
is required in the execution flow of the algorithm, due
to the existence of multiple zero. Such pathologi-
cal cases occur when there exists a possible multiple
zero located in the middle point of the search interval.
The inability of Interval Newton to derive better en-
closures leads to the adoption of a bisection scheme.
This increases the computational cost of the method,
especially given the lack of an efficient alternative.
Targeting this weakness of the method, a technique
is proposed to directly enclose a multiple zero. In par-
ticular, a temporary perturbation of the initial function
is considered, and then the Interval Newton operator
is applied once to it to enclose its zeros. These ze-
ros, referred to as pseudo-zeros, are not the actual ze-
ros of . The resulting ’pseudo-enclosures’
are employed to deliver an efficient partition of the
search interval. The proposed technique enables the
inclusion of a multiple zero within a single sequence
of intervals, significantly enhancing the overall per-
formance of the method. The given numerical results
support the efficiency of the proposed technique.
The paper’s structure is as follows: The next sec-
tion briefly introduces the interval methods frame-
work. Subsequently, the motivation behind this work
is discussed, and the proposed idea, along with an al-
gorithm, is presented. Following that, numerical re-
sults are provided, and finally, the paper concludes
with a brief discussion of future research directions.
2 Interval Methods
Following are some definitions regarding the concept
of intervals and their corresponding arithmetic that
are used in this work. An extensive study of Inter-
val arithmetic and interval methods can be found in
[11], [15], [16], [17], [18], [19].
2.1 Notation and basic concepts
A closed, compact interval, denoted as , is the set
of all points
R
where ,denotes the lower and upper bound of in-
terval , respectively, and the set of all closed an
compact intervals is denoted as IR. The intersection
of two intervals , is defined as the common
points between the two intervals, while the empty in-
tersection, the lack of common points is defined as
. The width of an interval is defined by
while the midpoint of an interval is defined as
For operations between two intervals ,
IR, a corresponding interval arithmetic was defined,
which, in general, is described as a set extension of
real arithmetic, using the elementary operations
The above definition is equivalent to the following
simpler rules, [20],
min max
min
max
where for the case of interval division. For
the case where an extended interval arithmetic
have been proposed, [21].
Let RRbe a real function. An interval
extension of is an interval function IRIR
such that for each IRand for each
.is inclusion isotonic if
implies , where IR.
The following theorem, known as the Fundamen-
tal Theorem of Interval Analysis, [11], [15], can be
used to bound the exact range of a given expression.
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Theorem 1 If is an inclusion isotonic interval ex-
tension of , then
The resulting enclosure of the range is, in gen-
eral, overestimated due to the phenomenon of depen-
dency, as described in [15]. Specifically, the more oc-
currences a variable has in an expression, the higher
the overestimation will be.
2.2 The form of Interval algorithms
The fundamental principle behind interval algorithms
employed to solve (1) is centered on generating se-
quences of nested intervals, [11]. The primary goal is
to iteratively produce narrower intervals in each iter-
ation, converging eventually to a zero enclosure. The
conceptual framework of these methods is illustrated
in Algorithm 1. The basic implementations of this
approach occurs when a simple bisection scheme is
adopted (Algorithm 1 without Lines 7-11 and 14-15).
This method, known as Interval Bisection, remains a
reliable and widely used technique for determining all
solutions of a non-differentiable equation to this day.
1Function genIZF (,,)
2if then
3if then
4Accept as a zero-enclosure
5end
6else
7Reduce the search interval - find a
new
8if then
9Initiate the method on
10 end
11 else
12 Bisect the search interval
13 Initiate the method on both of
them
14 end
15 end
16 end
17 else
18 Discard search interval - contains
no zeros
19 end
20 end
Algorithm 1: General Zero-Finding Interval Al-
gorithm
If the reduction of the search interval is done using
the following Interval Newton operator
(2)
then the Interval Newton method will be obtained.
The basic properties of this method are described by
the following Theorems.
Theorem 2 (Proof can be found in [11] and [18])
Let RRbe a continuously differentiable
function and let IR be a given search interval.
Then Interval Newton operator given by (2) has the
following properties.
1. If is a zero of then
.
2. If then there exist no zero of
in .
3. If then there exist a unique zero of
in and therefore in .
Theorem 3 (Proof can be found in [14] and [22])
Let RRbe a continuously differentiable
function and let IR be a given search interval.
1. If for some then
for all .
2. If there exists a constant such that
.
The above theorems tell us that, on the one hand,
if there is no zero of the equation, the algorithm will
yield this result. On the other hand, if the function
is monotonic within a search interval, the algorithm
can computationally prove both the monotonicity of
the function and the uniqueness of the enclosing zero.
Additionally, it will converge to the zero enclosure
very fast, even if the search interval is very wide. The
following Algorithm 2 describes the Interval Newton
method employing extended interval arithmetic.
2.3 Pathological cases
In certain instances, the Interval Newton method fails
to return a narrower interval. This occurs when both
the numerator and denominator of the operator are
zero or contain zero:
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1Function inewton (,,,)
2if then
3if then
4Accept as a zero-enclosure
5end
6else
7
8if then
9Discard search interval
10 end
11 else
12 if then
13 Initiate the method on
14 end
15 else
16 Bisect the search interval
17 Initiate the method on both
of them
18 end
19 end
20 end
21 end
22 else
23 Discard search interval - contains
no zeros
24 end
25 end
Algorithm 2: Interval Newton Algorithm
This pathological case may occur when the method
attempts to enclose a possible multiple zero and the
midpoint of search interval coincides with a zero of
, i.e. (Figure 1). At this point, a
bisection scheme is applied to the current search in-
terval:
It is clear that both of the generated interval sequences
containing and , respectively, will converge
to at least an enclosure containing , a zero of
. For example, consider the equation
defined over the interval with
and . The application of
Interval Newton on this problem, given a solution tol-
erance of , will perform as it is shown in Ta-
ble 1. The application of the method in the third ex-
ample results in two enclosures of a single zero, due
to the aforementioned pathological case.
Again, a bisection scheme will be needed when
there is an overestimation in the enclosure of the
derivative of , but is actually monotone on the cur-
rent search interval, as shown in Figure 2. For ex-
ample, consider, now, the equation defined
over the interval with
Figure 1: An example of pathological case with even
multiplicity.
Table 1: Application of Interval Newton [ ,
]
SI NZ ENCL ITS FE DE BS
1120 39 19 0
1120 39 19 0
1241 80 19 1
SI: Search Interval, NZ: Number of zeros, ENCL: Enclosures found,
ITS: Iterations, FE/DE: Function and Derivative evaluations,
BS: Needed bisections.
and . The application of Interval
Newton on this equation, given a solution tolerance
of , will perform as it is shown in Table 2.
The application of the method in the third example
results in two enclosures of a single zero, due to the
aforementioned pathological case.
Table 2: Application of Interval Newton [ ,
]
SI NZ ENCL ITS FE DE BS
1125 49 24 0
1125 49 24 0
1251 100 49 1
SI: Search Interval, NZ: Number of zeros, ENCL: Enclosures found,
ITS: Iterations, FE/DE: Function and Derivative evaluations,
BS: Needed bisections.
Finally, due to the nature of certain functions,
there is often a significant overestimation of their
range, and Interval Newton method cannot provide
sharp bounds for the zero enclosures, returning
. These cases arise when the search interval is
of short width, and as in the earlier cases, the method
proceeds using a bisection scheme.
The issue of overestimation is addressed by choos-
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Figure 2: An example of pathological case with odd
multiplicity.
ing appropriate interval extensions, [11], [16], which
is beyond the scope of this paper.
2.4 Function perturbation
In this paper, a new technique is introduced to ad-
dress pathological cases that may arise during the ap-
plication of the Interval Newton method. The pro-
posed technique is described as follows: If function
perturbed by a small amount, denoted as , a new
function is derived. Consequently, the midpoint
of will no longer be a zero of , result-
ing in . Additionally, the equation
will exhibit either two zeros or a single
one, all different from those of , for the cases
of even or odd multiplicity respectively. These zeros
will be referred to as pseudo-zeros (Figure 3 and Fig-
ure 4).
Figure 3: Function perturbation - Case with even mul-
tiplicity.
Figure 4: Function perturbation - Case with odd mul-
tiplicity.
By employing the Interval Newton method on the
function , we can obtain enclosures for these
pseudo-zeros. It is essential to emphasize that our pri-
mary objective is to provide zero-enclosures for the
function , rather than for . These
resulting enclosures serve as an efficient way to par-
tition the search interval. For the case of even mul-
tiplicity the search interval will be partitioned as fol-
lows:
where
,
denotes the zero enclosures of
and the mid-interval. For the case of
odd multiplicity the partition of the search interval is
given as follows:
or
where
denotes the zero enclosure of
and the complementary interval. Subsequently,
the next step for the method is to search for zeros
within these intervals.
Remark 1 It is clear that in the case of a multiple
zero of even multiplicity, the proposed technique can
yield its existence, given that it can be proved that the
pseudo-zeros enclosures contain simple zeros. This
can be achieved, for instance, utilizing the method’s
property . This holds since, as the per-
turbed function tends to the original one, the pseudo-
zeros gradually shift towards the endpoints of the mid-
interval. Consequently, the mid-interval will contain
at least one zero of .
Remark 2 For the case of a multiple zero of odd mul-
tiplicity the same result can be obtained by simply
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choosing an alternative to the midpoint. Assuming
that and it is not possible to
determine the type of multiplicity. Thus, choosing a
different point for Interval Newton method is not ef-
fective in all cases.
2.5 The Special Interval Newton step
The implementation of the proposed technique modi-
fies the Lines 16-17 of Algorithm 2 and it is described
by the following Algorithm 3.
1Function sinewton (,,,)
2if then
3Define
4
5Partition based on
6Initiate inewton method on each
non-empty interval
7if pathology exists then
8Bisect the search interval
9Initiate inewton method on both of
them
10 end
11 end
12 end
Algorithm 3: Special Interval Newton Step
The amount of perturbation is chosen heuristi-
cally such that it satisfies the condition , where
is the given tolerance for the problem (1). In Line 7,
the statement ”if the pathology exists”, indicates that,
due to overestimation issues, both enclosures
and
may result in an empty intersection with ,
leading to the mid-interval being equal to the
current search interval. In this case, the use of a bi-
section scheme becomes unavoidable.
3 Numerical Results
This section presents the application of the proposed
step in different instances. To evaluate the perfor-
mance of the method using the proposed step a set of
eight test-functions was considered containing simple
and multiple zeros (Table 3). The experiments were
performed using MATLAB and intlab toolbox, [23],
requiring an accuracy of and a perturbation
size of . The results of the
experiments in this work are summarized in the fol-
lowing Table 4. Based on the numerical findings, it is
clear that when the equation has a unique zero, a sig-
nificant improvement can be achieved by choosing an
appropriate amount of perturbation. Even in the cases
where multiple zeros exists and there is a substan-
tial overestimation, employing an appropriate amount
Table 3: Test Functions
PR f SI NZ ML
1 2 1
2 1 1
3 1 2
4 2 2
5 3 2
6 1 3
7 1 6
8 3 1&2
PR: Problem, f: Test function, SI: Search Interval,
NZ: Number of zeros, ML: Multiplicity.
of perturbation appears to reduce the reliance on bi-
section schemes, making the Interval Newton method
more efficient and reliable.
Remark 3 The numerical results show that the
smaller the perturbation, the smaller the required bi-
sections.
4 Conclusions - Further Work
In this paper, we have considered the problem of
finding all zeros of a differentiable function within
a closed search interval. Specifically, we have stud-
ied the pathological cases that arise from the exis-
tence of multiple zeros within a given search interval.
In these cases, the efficiency of the Interval Newton
method decreases due to the employment of bisection
schemes. This is because the Interval Newton opera-
tor is unable to reduce the search interval, and there-
fore cannot improve the bounds of zero enclosures.
The proposed Special Newton Step, temporarily
transforms a function with a multiple zero into a
function where is not a zero. This transformation
is achieved by vertically shifting the original func-
tion by an appropriate amount, effectively reducing
the limitations of the Interval Newton operator. The
application of this step creates more favorable condi-
tions for the method, significantly enhancing its effi-
ciency. Our numerical results show that it is feasible
to avoid the use of bisection schemes in cases where
the application point of the method coincides with a
multiple zero. Moreover, we demonstrate that a mul-
tiple zero is not only enclosed faster but, also, large
intervals surrounding the zero can also be discarded.
Additionally, it is evident from the results that the ef-
ficiency and effectiveness of the method depend sig-
nificantly on the amount of the perturbation applied
to the function.
The rapid inclusion of a multiple zero raises ques-
tions about existence and uniqueness, which, along
with the problem of finding the optimal perturbation,
should be topics for a future research.
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Table 4: Numerical Results [ ]
PR Interval Newton Interval Newton+
ENCL FE+DE BS ENCL FE+DE BS
1 2 35 0 2 35 0
2 2 32 1 1 17 0
3 2 179 1 2 88 1
4 6 265 3 4 208 1
5 170 2177 169 148 2046 161
6 2 227 1 1 119 0
7 2 179 1 1 59 0
8 4 256 1 3 203 0
PR Interval Newton Interval Newton+
ENCL FE+DE BS ENCL FE+DE BS
1 2 35 0 2 35 0
2 2 32 1 1 17 0
3 2 179 1 2 58 1
4 6 265 3 4 196 1
5 170 2177 169 148 1956 143
6 2 227 1 1 119 0
7 2 179 1 1 36 0
8 4 256 1 3 197 0
PR Interval Newton Interval Newton+
ENCL FE+DE BS ENCL FE+DE BS
1 2 35 0 2 35 0
2 2 32 1 1 17 0
3 2 179 1 1 6 0
4 6 265 3 3 192 0
5 170 2177 169 135 1880 130
6 2 227 1 1 116 0
7 2 179 1 1 6 0
8 4 256 1 3 203 0
Interval Newton: Interval Newton Method,
Interval Newton+: Interval Newton method using Special Interval Newton
Step, SI: Search Interval, NZ: Number of zeros, ENCL: Enclosures found,
ITS: Iterations, FE/DE: Function and Derivative evaluations,
BS: Needed bisections.
References:
[1] Joel Dahne, Marcelo F. Ciappina, and Warwick
Tucker. Enclosing all zeros of a system of ana-
lytic functions. Applied Mathematics and Com-
putation, 348:513–522, 2019.
[2] Theodoula N. Grapsa and Michael N. Vrahatis.
The implicit function theorem for solving sys-
tems of nonlinear equations in R.Interna-
tional Journal of Computer Mathematics, 28(1–
4):171–181, 1989.
[3] Theodoula N. Grapsa and Michael N. Vra-
hatis. A dimension-reducing method for uncon-
strained optimizatio. J. Comput. Appl. Math.,
66(1–2):239–253, January 1996.
[4] Gopalan V. Balaji and J.D. Seader. Appli-
cation of interval newton’s method to chemi-
cal engineering problems. Reliable Computing,
1(3):215–223, 1995.
[5] Dragoslav D. Šiljak and Dusan M. Stipanović.
Robust -stability via positivity. Automatica,
35:1477–1484, 1999.
[6] Ole Carpani, Lars Hvidegaard, Mikkel
Mortensen, and Thomas Schneider. Ro-
bust and efficient ray intersection of implicit
surfaces. Reliable Computing, 6:9–21, 2000.
[7] The Coprin Project. Minimal and maximal real
roots of parametric polynomials using interval
analysis. In C. Bliek, C. Jermann, and A. Neu-
maier, editors, Global Optimization and Con-
straint Satisfaction, First International Work-
shop Global Constraint Optimization and Con-
straint Satisfaction, COCOS 2002, Valbonne-
Sophia Antipolis, France, October 2-4, 2002,
Revised Selected Papers, volume 2861 of Lec-
ture Notes in Computer Science, pages 71–86.
Springer, 2003.
[8] Matthew J. Keeter. Massively parallel rendering
of complex closed-form implicit surfaces. ACM
Trans. Graph., 39(4), aug 2020.
[9] William H. Press, Saul A. Teukolsky, William T.
Vetterling, and Brian P. Flannery. Numerical
Recipes 3rd Edition: The Art of Scientific Com-
puting. Cambridge University Press, 3 edition,
2007.
[10] Juan Ramon Torregrosa, Alicia Cordero, Neus
Garrido, Neus Garrido, and Paula Triguero-
Navarro. Modifying kurchatov’s method to find
multiple roots of nonlinear equations, October
21, 2022.
[11] Ramon E. Moore. Interval Analysis. Auto-
matic Computation. Prentice-Hall, Inc., Engle-
wood Cliffs, New Jersey, 1966.
[12] Chin-Yun Chen. Extended interval newton
method based on the precise quotient set. Com-
puting, 92:297–315, 2011.
[13] Walter F. Mascarenhas. Root finding with inter-
val arithmetic, 2021. 2110.07126, arXiv.
[14] Eldon Hansen. A globally convergent interval
method for computing and bounding real roots.
BIT, 18:415–424, 1978.
[15] Ramon E. Moore, R. Baker Kearfott, and
Michael J. Cloud. Introduction to interval anal-
ysis. Society for Industrial and Applied Mathe-
matics, 2009.
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.74
Ioannis A. Nikas
E-ISSN: 2224-2880
680
Volume 22, 2023
[16] Arnold Neumaier. Interval methods for systems
of equations. Encyclopedia of mathematics and
its applications/37. Cambridge University Press,
Cambridge, 1990.
[17] Eldon R. Hansen and G. William Walster. Sharp
bounds in interval polynomial roots. Reliable
Computing, 8(2):115–122, April 2002.
[18] Eldon Hansen. Global optimization using inter-
val analysis. Monographs and textbooks in pure
and applied mathematics/165. Marcel Dekker,
New York, 1992.
[19] Götz Alefeld and Jürgen Herzberger. Intro-
duction to Interval Computations. Computer
science and applied mathematics series. Aca-
demic Press, New York, 1983. Translated by Jon
Rokne.
[20] Helmut Ratschek and Jon Rokne. Computer
Methods for the Range of Functions. Mathemat-
ics and Its Applications. Ellis Horwood, Chich-
ester, West Sussex, England, 1984.
[21] Dietmar Ratz. Inclusion isotone extended in-
terval arithmetic. Technical report, Universität
Karlsruhe, 1996.
[22] Eldon Hansen and G. William Walster. Global
Optimization using Interval analysis. Pure
and Applied Mathematics. Marcel Dekker, New
York, USA/Basel, Switzerland, 2nd edition,
2004.
[23] Siegfried M. Rump. Fast and parallel interval
arithmetic. BIT, 39(3):534–554, 1999.
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