# WHAT IS THE HOMOTOPY METHOD FOR A Chin-Hong Park and Hong-Tae Shim

```J. Appl. Math. & Computing Vol. 17(2005), No. 1 - 2 - 3, pp. 689 - 700
WHAT IS THE HOMOTOPY METHOD FOR A
SYSTEM OF NONLINEAR EQUATIONS(SURVEY) ?
Chin-Hong Park∗ and Hong-Tae Shim
Abstract.
In this paper we shall introduce the general idea, historical background and the reasons why we use it about the homotopy method for solving a
system of nonlinear systems.
AMS Mathematics Subject Classification : 65D10, 65D99
Key words and phrases : Homotopy, zero curve, regular imbedding, path-following
method, continuation method.
1. Introduction
We want to survey how this method called “ the homotopy method ” had been
used before 1990. When F (x) = 0 where F : Rn → Rn is a C 2 -map is a system
of nonlinear equations, we want to find the solution of F (x). Newton’s method
applied to nonlinear equations of the form F (x) = 0 in general will only converge
if the iteration is started near a root of the equation. If no approximation roots
are known, this method may be of little use. Many iterative techniques for the
solution of nonlinear equations have the drawback that convergence depends on
a good initial approximation to the solution.
The imbedding method has the advantages of producing solutions over a large
range of the independent variables, giving a complete description of the solution
behaviour and a substantial improvement to Newton’s method in terms of global
convergence. This idea is based on the homotopy concept in topology. Our aim
is to embed F (x) into a homotopy.
For example, if we have H(x, t) = F (x)+(t−1)F (x0 ) where x0 is a given point,
then we have H(x, 1) = F (x) when t = 1. Hence F (x) has been imbedded into a
homotopy. To find the solution of F (x), all we have to do is read the value of x
when t = 1. Here the only interesting curve is H(x, t) = 0, so-called zero curve.
Received March 10, 2004. Revised July 28, 2004. ∗ Corresponding author.
c 2005 Korean Society for Computational & Applied Mathematics and Korean SIGCAM.
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Chin-Hong Park, Jeong Keun Lee and Hong-Tae Shim
The first application of this tool to numerical solution of nonlinear equations
is attribute to E. Lahaye (1934, 1935) and D.F. Davidenko(1953). Lahaye’s
approach is a locally convergent, iterative continuation method while that of
Davidenko is continuation method by differentiation. The more references before
1970 can be found in [2] but most of them have not discussed about algorithm
in detail to implement the imbedding method. After 1970 the notable facts
that can be discussed can be found in [6], [7], [10], [8], [12], [13], [9], [5], [3],
[14], [4] and [11] etc,.... Moreover, D. Leder, R. Menzel and H. Schwetlick have
discussed the algorithm and theoretical aspect in detail about both singular and
nonsingular cases.
2. Historical background
The first application of this tool to numerical solution of nonlinear equations
is attribute to E. Lahaye (1934, 1935) and D.F. Davidenko (1953). The more
references before 1970 can be found in J.M. Ortega and W.C. Rheinboldt but
most of them are not discussed about algorithm in detail to implement the
continuation method.
After 1970, the notable facts that can be discussed can be found in W. C.
Rheinboldt, H. B. Keller, Shui-Nee Chow, John Mallet-Paret, J. A. Yorke, L.
T. Watson, C. B. Garcia, F. J. Gould, S. Smale, D. Leder, R. Menzel and H.
Schwetlick, etc,... We may summarize the background briefly as follows :
1934, 1935 : E. Lahaye used the basic numerical continuation process for a
single equation. This was first to us as a numerical tool.
1948 : E. Lahaye used it for systems of equations.
1951 : F. Ficken used the contracting mapping theorem.
1953 : D. F. Davidenko used differential equation underlying a related
homotopy
1965 : Yakovlev - application of these techniques to Banach spaces.
1966 : Moore and Anselone had used this method.
1967 : Deist, Sefor and Bittner.
1968 : G. H. Meyer - application to Banach spaces.
1974 : D. Leder
1975 : W.C. Rheinboldt, H. Schwetlick
1976 : S. Smale
1978 : S. N. Chow and J. A. Yorke - applications with fixed points. H. B.
Keller,
C. B. Garcia, F. J. Gould.
1979 : L. T. Watson - applications with fixed points.
W. F. Schmidt - the use of contractive mapping theorem
( International J. for Numerical Method in Engineering).
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H. Schwetlick.
1980 : R. Menzel, C. B. Garcia and F. J. Gould.
(1) Drawback of Newton’s method or iterative method :
(i) Newton’s method applied to nonlinear equations of the form F (x) = 0 in
general will only converge if the iteration is started near a root of the equation.
If no approximate roots are known, this method may be of little use.
(ii) Many iterative techniques for the solution of nonlinear equations have the
drawback that convergence depends on a good initial value approximation to
the solution. Correspondingly, most convengence results only guarantee the
existence of a well-defined convergent sequence of iterates for very restricted
sets of starting points.
(2) Lahaye’s iterative continuation approach(1934, 1935 and 1948):
It uses a locally convergent iterative method for solving H(x, t) = 0, t ∈ [0, 1]
with t0 < t1 < t2 < ... < tn = 1. The last iteration at tk is equal to the initial
approximation for the iteration at tk+1 . The main delimiter of the t-steps is
the dependence of the local method on the initial data. That is to say, larger
steps are possible when the convergent domains increase. If a step-algorithm
is used which adjusts to the convergent behavior of local iterations, this is
a suitable method. “ Unfortunately, such algorithms do not appear to be
available ”.
(3) Davidenko’s approach (1953) :
It has been based on the observation that under the suitable differentialbility
conditions, the unknown continuation curve is a solution of the initial value
problem.
∂1 H(x, t)
dx
+ ∂2 H(x, t) = 0, t ∈ [ 0, 1], x(0) = x0 .
dt
Accordingly, we may approximate this curve by applying some discrete variable method to ∂1 Hx0 + ∂2 H = 0 ( W.C. Rheinboldt ). By nature, numerical
method for solving intial value problems such as
∂1 Hx0 + ∂2 H = 0
are designed to approximate the entire solution curve within a given, potentially small, error tolerance. This is a continuation method by differentiation.
The entire curve is of interest and is to be approximated closely.
∂1 H(x, t)
dx
+ ∂2 H(x, t) = 0, t ∈ [ 0, 1], x(0) = x0 .
dt
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Chin-Hong Park, Jeong Keun Lee and Hong-Tae Shim
is called Davidenko differential equation. In connection with Lahaye’s approach, this may waste effort since more information is obtained than required.
(4) Common features for F (x) = 0 in many practical finite element applications
(W.C. Rheinboldt, 1977) :
(i) F : D ⊂ Rn −→ Rn given by F (x) = (f1 (x), ..., fn (x))T , x = (x1 , ..., xn )T
: n-dimensional nonlinear equation F (x) = 0.
(ii) the dimension n is large and the operator F is sparse(i.e., each component
function fi depends only on some of the variables xj ).
(iii) computational evaluation of F (x) and F 0 (x) is inherently costly.
(iv) in general, there are many solutions of F (x) = 0.
These methods is to overcome that convergence depends on only good intial
value approximation.
(i) it does not depend on good initial value - supper advantages ( It is to
overcome that the convergence depends on only good initial approximation Schwetlick and H.B. Keller )
(ii) it is producing solutions over a large range of the indenpendent variables.
(iii) this is a tool in overcoming the local convengence of iterative processes H. B. Keller and Rheinboldt.
(iv) this is to widen the domain of convergence or a procedure to obtain the
good starting points - J. W. Schmidt.
(v) an attempt to solve the global convergence of Newton’s method which is
an open mathematical problem.
(6) Problems :
One of the crucial problems encountered when we use an imbedding method(
continuation method) is the selection of the stepsize. A step that is too large
may result in the initial estimate being outside the convergent region of the
iterative process and result in a failure of the process or may pass over critical
points in the solution.
In H. Schwetlick and R. Menzel, the automatic variable step-sizes have been
used. In W.C. Rheinboldt, the following problems were pointed out : Efficient
design of step-length selection, analysis and control of the accuracy, stability
of the computational solution, and control of the computational cost. That
is to say, to design algorithms, we have to consider the followings :
(i) most of numerous iterative process are designed to gererate one sequence
of points of R which converges to some solution of the equation.
(ii) particular limit point depends strongly on the choice of the starting data
of the iteration.
Accordingly, the efficient design of overall algorithm which allows for the
desired computational analysis of the solution field( for example, computation
of F (x) and F 0 (x) ) is a relatively open problem.
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(iii) Design of algorithms.
Two objectives of algorithm design :
(a) The endpoint of the curve(zero curve) is of more interest while the
curve itself is of lesser interest - based on Lahaye’s iterative continuation method.
(b) The entire curve is of interest and is to be approximated closely Davidenko approach.
(iv) selection of the stepsizes.
(v) control of the accuracy and stability of the computational solution.
(vi) control of the computational cost ( unexplored part).
(vii) computation of turning points - G. Ponisch and H. Schwetlick (Computing 26, 1981)
(viii) analysis of stability - E. Allgower and K. Georg ( SIAM Review, 1980 ).
4. General ideas and how to find the solution of F (x)
This idea is based on the homotopy concept in topology. The names in numerical analysis are known variously as follows :
(1) contonuous Newton’s method,
(4) Imbedding method,
(2) continuation method,
(5) path-following method.
(3) Davidenko’s method,
Here we use a homotopy H. Let f, g : X −→ Y be continuous functions where
X, Y are any spaces. We call H a homotopy from g to f if H : X × I −→ Y is
a continuous map such that
H(x, 0) = g(x), H(x, 1) = f (x), ∀x ∈ X.
We denote H : g ' f where I = [0, 1]. Let αt = H( , t), i.e., αt (x) = H(x, t), x ∈
X, t ∈ I. Then the homotopy H is seen to represent a family {αt : t ∈ I} of
maps from X to Y , varying continuously with t such that α0 = g, α1 = f. This
means that H gives a continuous deformation of g into f . Intuitively we can say
that g can be continuously transformed into f .
For example, let g(x) = x and f (x) = 0 in Rn . Then g is homotopic to f .
i.e., we define H : Rn × I −→ Rn by
H(x, t) = (1 − t)x.
Then it is clear. So our aim is to embed F (x) into a homotopy where F (x) is a
nonlinear system of equations.
As we mentioned above, the procedure to solve F (x) = 0 is as follows :
(i) we define
G(x, t) = tF (x) + (1 − t)(x − x0 )
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Chin-Hong Park, Jeong Keun Lee and Hong-Tae Shim
or
G(x, t) = F (x) + (t − 1)F (x0 )
where G : Rn × Rn → Rn is a C 2 -map in nbd U of G−1 (0)(zero curve) and
x0 is an arbitrary assigned value. These are the most common homotopies
used.
(ii) start at (x0 , 0) :
We start at (x0 , 0) and follow the curve of G(x, t) = 0 until t = 1. Here the
curve of G(x, t) = 0 means the zero curve. Finally we read the value of x
when t = 1. This is the desired solution x∗ of F (x) = 0.
(iii) tools used :
we use “ tengent vectors ”, “ normal planes ” and
kG(un+1 )k ≤ kG(un )k + τn
where un = (xn , tn ), is an assigned accuracy, is a variable step-size. The
essence of the imbedding method is “ path-following ”. Theoretically this is
very simple but the problem is how a method is implemented by computer
program efficiently. i.e., how do we follow the zero curve ? Because the end
point of the zero curve is of more interest while the curve itself is of lesser
interest, we think that Lahaye’s approach is more appropriate. In general,
there are many solutions of F (x) = 0. The details for this can be found in
H.B. Keller or C.B. Garcia and F. J. Gould.
We need to note the following important points :
(a) The use of contractive mapping theorem : W. F. Schmit ( International
J for Numerical Method in Engineering, 1979 ).
(b) Computation of turning points : G. Ponisch and H. Schwetlick ( Computing 26, 1981).
(c) Analysis of stability : E. Allgower and K. Georg ( SIAM Review, 1980 ).
5. Assumptions
The following Theorems come from the fact that F : Rn −→ Rn and G :
R × R −→ Rn are C 2 -maps from the assumptions of H. Schwetlick and R.
Menzel. The assumptions with the slight modifications are as follows :
n
Assumption.
(1) (i) G is defined by a homotopy H. i.e., G(x, t) = H(x, t) for all (x, t) ∈
Rn × R.
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(ii) G0 (u) satisfies the Lipschitz condition on a nbd U of Z0 where Z0 =
W ([ 0, 1]). i.e., for some L > 0,
kG0 (u1 ) − G0 (u2 )k ≤ Lku1 − u2 k
for all u1 , u2 ∈ U.
(iii)G0 (u) has full-rank on Z0 . i.e., rank G0 (u) = n for all u ∈ Z0 .
(2) There exists a C 2 -map W : (−, 1 + ) → Rn × R such that the following
conditions are satisfied :
(i)W (0) = (x0 , 0)t , W (1) = (x∗ , 1)t , t0 (0) > 0 and t(s) < 1 for all s ∈ [
0, 1].
(ii)W 0 (s) = dW
ds 6= 0, GW (s) = 0 for all s ∈ [ 0, 1] where > 0 and x0
are the given values and x∗ is the value of x with H(x, 1) = 0.
Note. For (ii) of (1) we can assume that G : Rn × R → Rn is a C 2 -map in a
nbd U of Z0 .
We can see that the following Theorems and Lemmas hold under the above
assumptions. Moreover, we note that Rn is locally compact.
Lemma 1. (1) there is δ0 > 0 such that
Z(δ0 ) = {u ∈ U :k u − u0 k< δ0 , u0 ∈ Z0 } ⊂ U
¯ 0 ) ⊂ U where U is an open nbd of Z0
is a compact nbd of Z0 with Z(δ0 ) ⊂ Z(δ
and Z0 = W ([0, 1]).
(2) k G0 (u)+ k ≤ M for some M > 0 for every u ∈ Z0 where G0 (u)+ is
pseudo-inverse to G0 (u).
The following Lemma and Theorem hold with δ0 obtained from Lemma 1.
Lemma 2. Let Z0 = W ([0, 1]). For u0 ∈ Z0 and G0 (u0 )v0 = 0 with kv0 k = 1,
¯ 0 ) → L(Rn+1 ) as follows :
we define a map h : Z(δ
h(u) =
G0 (u)
v0t
.
Then ∃ δ1 , η, c with δ0 ≥ δ1 > 0, η > 0 and c ∈ (0, 1) such that for every
¯ 0 ; δ1 ) = {u : ku − u0 k ≤ δ1 },
u ∈ S(u
0
G (u)
(i) h(u) =
is regular ;
v0t
(ii) rank G0 (u) = n and ∃!v ∈ Rn+1 with G0 (u)v = 0, kvk = 1 and (v0 )t v ≥
c>0;
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Chin-Hong Park, Jeong Keun Lee and Hong-Tae Shim
(iii) kv − v0 k ≤ η ku − u0 k.
We note that from Lemma 2(ii) since G0 (u) has full rank on Z(δ1 ), the tangent
direction v ∈ Rn+1 is apart from sign uniquely determined by G0 (u)v = 0, kvk =
1. The following Theorem is trivial from the above Lemma, letting
[
¯ 0 ; δ1 ).
Z(δ1 ) =
S(u
u0 ∈Z0
Also from the above Lemma 2 we know that G must be an C 2 -map to prove
rank G0 (u) = n.
Theorem 3. For the compact nbd Z(δ0 ) of Z0 , there is a full rank nbd Z(δ1 ) ⊂
¯ 0 ) of Z0 in the sense that rank T 0 (u) = n for every u ∈ Z(δ1 ).
Z(δ
6. The reasons why we need homotopy method
Many iterative techniques and Newton method for the solution of nonlinear
equations have the drawback that the convergence depends on only good approximation. If no approximation roots are known, most of the classical method
may be of little use.
The method for following the path is a substantial global convergence. But
Newton method could cycle or could blow up when the Jacobian matrix at some
point is singular. The imbedding method has the advantages of producing solutions over a large range of the independent variables, thereby giving a complete
description of the solution behaviour and is used as a tool in overcoming the
local convergence of iterative processes.
The imbedding methods may be considered as a possibility to widen the
domain of convergence or from another point of view, as a procedure to obtain
sufficiently close starting points.
7. The existence of solution curve
The existence of solution curve with H(x, t) = 0 has been discussed in J.M.
Ortega and W.C. Rheinboldt(1970), H.B. Keller(1978), S.N. Chow et al.(1978)
and H. Schwetlick(1975, 1976). In J.M. Ortega and W.C. Rheinboldt(1970) the
existence has been described as follows:
Theorem 4. Let F : Rn −→ Rn be a C 1 -map on Rn . Assume that F 0 (x) is
nonsingular for all x ∈ Rn and kF 0 (x)−1 k ≤ M for some M > 0 and all x ∈ Rn .
Then for any fixed x0 ∈ Rn there is a unique C 1 -map z : [0, 1] −→ Rn such that
H(z(t), t) = 0, ∀ t ∈ [ 0, 1].
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697
Moreover,
z 0 (t) = −F 0 (z(t))−1 F (x0 ), ∀ t ∈ [ 0, 1], x(0) = x0
where H : D × [0, 1] → Rn is a homotopy given by
H(x, t) = F (x) + (t − 1)F (x0 ) and D ⊂ Rn .
The conditions of F for the existence have been replaced by the conditions of
H in H. Schwetlick(1975). i.e., if H satisfies the above conditions, there exists a
unique C 1 -map z : [0, 1] → Rn such that
H(z(t), t) = 0, ∀ t ∈ [ 0, 1].
Also it is called a regular imbedding for F to x0 ∈ Rn .
The most important thing in a regular imbedding is that ∂1 H(x, t) is nonsingular. If ∂1 H(x, t) is singular at some point, it is called a singular imbedding
for F to x0 ∈ Rn with some conditions in R. Menzel(1980). In that case,
another map G : Rn × R → Rn is defined by G(x, t) = H(x, t) and the requirement to regularity of ∂1 H is replaced by the condition of linearly independent
decompositions from
G0 (u) = (∂1 H, ∂2 H).
The another representation of existence can be expressed by means of regular
value as follows :
Theorem 5. Let H : Rn+1 −→ Rn be a C 1 -map and let 0 be a regular value
of H. Then H −1 (0) is a C 1 -submanifold with dimension 1. i.e., this means the
existence of zero curves.
In S.N. Chow et al.(1978) this has been expressed by the more general,
parametrized Sard’s theorem. The theorem is described as follows :
Theorem 6 (Chow-Paret-Yorke). Let U ⊂ Rm and V ⊂ Rp be open sets and
let f : U × V → Rn be C r -map, r > max{0, m − n}. If 0 ∈ Rn is a regular value
of f , then for almost all a ∈ V 0 is a regular value of fa where fa (x) = f (x, a).
In general we can state this theorem as a parametrized Sard’s Theorem on
manifolds. Also this is described as follows :
Theorem 7 (parametrized Sard’s Theorem on manifolds). Let M, S and N be
manifolds with dimensions m, p and n respectively. Let U ⊂ M and V ⊂ S be
open sets. Let f : U × V → N be a C r -map, r > max{0, m − n}. If 0 ∈ N is a
regular value of f , then for almost every a ∈ V , 0 is a regular value of fa where
fa (x) = f (x, a).
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Chin-Hong Park, Jeong Keun Lee and Hong-Tae Shim
For simplicity, let B n = {x ∈ Rn : kxk < 1} and we will consider the
regularizising homotopy
H(x, t) = tF (x) + (1 − t)(x − x0 ).
¯ n −→ B
¯ n be a C 2 -map and T : B
¯n ×
Theorem 8 (Chow-Yorke). Let F : B
n
n
2
(0, 1) × B −→ R be a C -map given by
T (x, t, a) = tF (x) + (1 − t)(x − a).
Let Za be the component of Ta −1 (0) ∩ B n × (0, 1) whose closure contains (a,0)
where Ta (x, t) = T (x, t, a). Then
(1) 0 ∈ Rn is a regular value of T . i.e., the Jacobian matrix of T (x, t, a) has
full-rank on T −1 (0). Note that T −1 (0) is a closed set;
(2) 0 ∈ Rn is a regular value of T for almost every a ∈ B n ;
(3) Ta −1 (0) is a C 2 -submanifold(C 2 -curve) of B n × (0, 1) with dimension 1
for almost every a ∈ B n ;
(4) for almost every a ∈ B n , Za is a smooth curve in B n × (0, 1) connecting
between (a, 0) and a zero point of F (x) at t = 1. This means that Za ⊂
¯ n × (0, 1) is not diffeomorphic to a circle and has no limit points on
B
∂B n × (0, 1);
(5) If F 0 (x) is nonsingular for every zero point x of F (x), Z¯a is s smmooth
¯ n × [0, 1] and has finite arc length;
curve in B
(6) For summary of Ta −1 (0), Ta −1 (0) consists of
¯ n × (0, 1),
(a) a finite number of closed loops in B
n
¯
¯ n ×{1},
(b) a finite of number of arcs in B ×(0, 1) with endpoints in B
(c) Za .
About Theorem 8 it is necessary to look at the following Note:
Note.
1. If condition of (5) is satisfied, the curves of (a), (b) and (c) have finite
lengths. Moreover these curves are disjoint and C 1 -curves. The proof
of this theorem can be found in S.N. Chow et al. and L.T. Watson but
in S.N. Chow and L. T. Watson the fixed points were discussed. In the
same way, this theorem can be proved for F (x) = 0.
2. The more references for this theorem can be found in H. B. Keller and C.
B. Garcia. The main theorem of H. B. Keller and C. B. Garcia is almost
same under Smale’s boundary condition. In C.B. Garcia the details in
What is the homotopy method ?
699
algorithms is not there and in H. B. Keller the main idea of algorithm is
the same as that of R. Menzel.
3. In S. N. Chow et al. they say that if we choose a point a at random
from V or B n , the probability is one that each component of Ta −1 (0) is
a smooth curve.
4. If we let G = Tx0 for x0 ∈ B n , we will get Z0 = Zx0 and the justification
of assumptions turns out to be clear where x0 ∈ B n − A, A is the set of
measure zero with respect to n-dimensional Lebesgue measure.
5 In practice, for most F the curve Za forms a smooth arc leading to a
fixed point of F . But possibilities remains that Za will not actually meet
the fixed point but will converge to a set of fixed points of F . Hence
Ta−1 (0) is smooth when restricted to (0, 1) × B n or [0, 1) × B n but not
¯n.
necessarily in [0, 1] × B
8. Implementable algorithms
The algorithm for finding a root of F (x) = 0 is as follows : Start with
t = 0, x0 ∈ Rn and follow the zero curve Z0 of Gx0 (x, t) = G(x, t) emanating
from (x0 , 0). By the theorems, Z0 reaches a zero x∗ of F . The algorithm
considered here is based on the algorithm of H. Schwetlick and R. Menzel. But
there may be much modifications and developments.
In imbedding methods, the algorithms can be carried out without using the
global constants such as Lipschitzian and regularity constants. The implementation is not as easy as it appears. Here mainly Lahay’s approach will be considerable for algorithm because Davidenko’s approach requires the closer approximation of zero curve and hence the more computing time. But in case of the
turning points we have to follow the zero curve very closely. The reference for
the failure of Menzel algorithm may be found in R. Menzel.
When the first predictor u0 k is approached too closely to some unwanted point
of zero curve, the failure may occure. In fact we think that our aim is x∗ rather
than the curve itself. The concept of parametrized homotopy has been used
for the algorithm called “ the basic tangent algorithm” in H. Schwetlick and R.
Menzel. The method using the tangent vector, tangent plane and normal plane
have also been discussed in H. B. Keller and there to estimate the stepsize, he
has used Newton-Kantorovich theorem. The more details for the implementable
algorithms can be found in Park[10].
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points and solutions to systems of equations, Siam Review Vol.22(1980), No.1, 28-85.
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Chin-Hong Park, Jeong Keun Lee and Hong-Tae Shim
2. Shui-Nee Chow, John Mallet-Paret and J.A. Yorke, Finding zeros of maps : Homotopy
methods that are constructive with probability one, Math. of Comput. Vol.32(1978), 887899.
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nonlinear equations depending on a parameter, Computing 26(1981), 107-121.
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Chin-Hong Park received his BS in Mathematics Education and MS degrees in Mathematics from Seoul National University. Also he received MS degree in Computer Engineering from University of Michigan and Ph.D in Mathematics from University of Iowa. He
was an Assistant Professor at Blackburn University(U.S.A.) and Fayetteville State University(U.S.A.) as a tenure track position. Now he is a full Professor at Sunmoon University.
His research interests focus on the Structure Theory of Automata and related Modules,
Computer Algorithms and Computer Hardware Design.
Department of Mathematics, Sunmoon University, Asan, ChungNam 336-708, Korea
e-mail : [email protected] or [email protected]
Hong-Tae Shim received Ph. D from the University of Wisconsin-Milwaukee under the
supervision of Professor Gilbert G. Walter. His research interests focus on wavelet theories,
Sampling theories and Gibbs’ phenomenon for series of special functions.
Department of Mathematics, Sunmoon University, Asan, Choongnam 336-708, Korea
e-mail: [email protected]
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