J. KSIAM Vol.6 , No.1, 1-15, 2002 RELATIONSHIPS ......J. KSIAM Vol.6 , No.1, 1-15, 2002 RELATIONSHIPS AMONG CHARACTERISTIC FINITE ELEMENT METHODS FOR ADVECTION-DIFFUSION PROBLEMS ZHANGXIN

J. KSIAM Vol.6, No.1, 1-15, 2002

RELATIONSHIPS AMONG CHARACTERISTIC FINITE ELEMENTMETHODS FOR ADVECTION-DIFFUSION PROBLEMS

ZHANGXIN CHEN

Abstract. Advection-dominated transport problems possess difficulties in the de-sign of numerical methods for solving them. Because of the hyperbolic nature of ad-vective transport, many characteristic numerical methods have been developed suchas the classical characteristic method, the Eulerian-Lagrangian method, the trans-port diffusion method, the modified method of characteristics, the operator splittingmethod, the Eulerian-Lagrangian localized adjoint method, the characteristic mixedmethod, and the Eulerian-Lagrangian mixed discontinuous method. In this paperrelationships among these characteristic methods are examined. In particular, weshow that these sometimes diverse methods can be given a unified formulation. Thispaper focuses on characteristic finite element methods. Similar examination can bepresented for characteristic finite difference methods.

1. Introduction

Advection-diffusion transport problems arise in many areas of engineering and ap-plied sciences [16, 25, 30]. These problems have a nondissipative (hyperbolic) advectiveterm and a dissipative (parabolic) part. When the parabolic part dominates, all reason-able numerical methods perform well. When the hyperbolic part dominates, however,strictly parabolic numerical methods do not perform well; they exhibit excessive non-physical oscillations or excessive numerical diffusions. Although extremely fine meshrefinement is possible to overcome some of the difficulties, it is not a feasible approachdue to excessive computational efforts.

Many classes of numerical methods have been developed for solving advection-diffusiontransport problems [16, 27, 30]. One of them is the class of characteristic methods.

AMS subject classification: 35K60, 35K65, 65N30, 65N22Key words: Characteristic methods, Eulerian-Lagrangian, mixed methods, finite elements, advection-diffusion problems, transport diffusion, operator splitting, discontinuous methodsThis work is supported in part by National Science Foundation grants DMS-9626179, DMS-9972147,and INT-9901498, and by a gift grant from Mobil Technology Company

1

2 ZHANGXIN CHEN

Because of the hyperbolic nature of advective transport, it is natural to look to a char-acteristic treatment in solving these problems. There is a rich family of characteristicmethods in the literature, which bear a variety of names, the method of characteristics(MOC) [28, 34, 40], the modified method of characteristics (MMOC) [23], the transportdiffusion method (TDM) [5, 29, 35], the Eulerian-Lagrangian method (ELM) [32, 33],the operator splitting method (OSM) [24, 44], the Eulerian-Lagrangian localized ad-joint method (ELLAM) [9, 37], the modified method of characteristics with adjustedadvection (MMOCAA) [20], the characteristic mixed method (CMM) [1, 22], and theEulerian-Lagrangian mixed discontinuous method (ELMDM) [12]. The common fea-ture of this class is that the advective part is handled by a characteristic trackingtechnique (in a Lagrangian framework) and the diffusive part is treated by a spatial(Eulerian) approximation scheme. These characteristic methods can take reasonablylarge time steps and do not numerically diffuse sharp solution fronts, and some of themcan conserve mass. In this paper relationships among these characteristic methods areexamined. In particular, we show that these sometimes diverse methods can be given aunified formulation. We start with ELMDM, from which we recover all other methods.This paper focuses on characteristic finite element methods; similar examination can bepresented for characteristic finite difference methods. Most of the earlier characteristicmethods are based on lowest-order finite elements in their respective setting. In thispaper we extend them to general finite elements.

The outline of this paper is as follows. In the next section, we describe a continuousproblem. In the third section, we state a unified formulation of characteristic methods.In the fourth section, we deduce all the above mentioned methods from this formulation.In the fifth section, we mention some generalizations. We conclude with two remarksin the last section. We mention that we do not consider stability and convergenceproperties of these characteristic methods, which can be found in the cited references.Also, it would be interesting to compare the characteristic methods under considerationcomputationally. This would involve tremendous work and is beyond the scope of thispaper. This paper focuses on the theoretical relationships among these methods.

2. A Continuous Problem

We consider the advection-diffusion equation for u on a bounded domain Ω ⊂ <d,d ≤ 3, with boundary ∂Ω = ΓD ∪ ΓN , ΓD ∩ ΓN = ∅:

(2.1)

∂t(φu) +∇ · (bu− a∇u) = f in Ω× J,

u = gD, on ΓD × J,

(bu− a∇u) · ν = gN , on ΓN × J,

u(x, 0) = u0(x) in Ω,

where

RELATIONSHIPS AMONG CHARACTERISTIC FINITE ELEMENT METHODS 3

• • •e

T1 T2← ν

Figure 1. An illustration for the jump definition.

J = (0, T ] (T > 0), a(x, t) ∈ (L∞(Ω)

)d×d, b(x, t) ∈ (L∞(Ω)

)d, φ(x, t) ∈ L∞(Ω), gD(x, t) ∈L∞(ΓD), gN (x, t) ∈ L∞(ΓN ), f(x, t) ∈ L2(Ω) (for each t ∈ J) and u0(x) ∈ L2(Ω) aregiven functions (the standard Sobolev spaces Hk(Ω) = W k,2(Ω) with the usual normsare used in this paper), and ν is the outer unit normal to ∂Ω.

To introduce a unified formulation, we rewrite this equation as follows:

(2.2)

∂t(φu) +∇ · (bu− σ) = f in Ω× J,

σ = a∇u in Ω× J,


(bu− σ) · ν = gN , on ΓN × J,

u(x, 0) = u0(x) in Ω.

Namely, an auxiliary variable σ is introduced. This variable usually has a physicalmeaning in applications such as the electric field in semiconductor modeling [13, 14] orthe velocity field in petroleum simulation [19, 25]. In the next two sections, we considerthe case where a = (aij) is positive definite:

(2.3) 0 < |ξ|−2d∑

i,j=1

aij(x, t)ξiξj ≤ a∗ < ∞, (x, t) ∈ Ω× J, ξ 6= 0 ∈ <d,

with a∗ being constant. The situation without this assumption will be addressed in thefifth section.

3. A Unified Formulation

For h > 0, let (Th)h be a sequence of finite element partitions of Ω; each subdomainT ∈ Th has a Lipschitz boundary. Let Eo

h denote the set of all interior edges (respectively,faces) e of Th, Eb

h the set of the edges (respectively, faces) e on ∂Ω, and Eh = Eoh ∪ Eb

h.We tacitly assume that Eo

h 6= ∅. Finally, each exterior edge or face has imposed on iteither Dirichlet or Neumann conditions, but not both.

For l ≥ 0, define

H l(Th) =(v ∈ L2(Ω) : v|T ∈ H l(T ), T ∈ Th

).

4 ZHANGXIN CHEN

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tn

tn−1

t

x

xn(x, t)

xn(x, tn−1)

Figure 2. An illustration of characteristics.

With each e ∈ Eh, we associate a unit normal vector ν. For e ∈ Ebh, ν is just the outer

unit normal to ∂Ω. For e ∈ Eoh, with e = T1 ∩ T2 and T1, T2 ∈ Th, ν is the unit normal

exterior to T2 with the corresponding jump definition (see Fig. 1): for v ∈ H l(Th) withl > 1/2, we define the average and jump by

v =12

((v|T1)|e + (v|T2)|e

), [v] = (v|T2)|e − (v|T1)|e.

For e ∈ Ebh, we utilize the convention (from inside Ω)

v = v|e and [v] =

v if e ∈ ΓD,

0 if e ∈ ΓN .

For each positive integer N , let 0 = t0 < t1 < · · · < tN = T be a partition of J intosubintervals Jn = (tn−1, tn], with length ∆tn = tn− tn−1, 1 ≤ n ≤ N . Set vn = v(·, tn)and

∆t = max1≤n≤N

∆tn.

For any x ∈ Ω and two times 0 ≤ tn−1 < tn ≤ T , the hyperbolic part of problem(2.1), φ∂tu + b · ∇u, defines the characteristic xn(x, t) along the interstitial velocityϕ = b/φ:

(3.1)∂txn = ϕ(xn, t), t ∈ Jn,

xn(x, tn) = x.

In general, we cannot follow the characteristic in (3.1) exactly; we can only followit approximately. There are many ways to solve the first order ordinary differentialequation (3.1). Let us consider the Euler method

(3.2) xn(x, t) = x− ϕ(x, tn)(tn − t), t ∈ [t(x), tn],

where t(x) = tn−1 if xn(x, t) does not backtrack to the boundary ∂Ω for t ∈ [tn−1, tn];t(x) ∈ (tn−1, tn] is the time instant when xn(x, t) intersects ∂Ω, i.e., xn(x, t(x)) ∈ ∂Ω,otherwise. See Fig. 2, where, for the purpose of demonstration, the characteristics areshown for constant ϕ in one dimension. If ∆tn is sufficiently small (depending upon


the smoothness of ϕ), the approximate characteristics do not cross each other, whichis assumed here. We denote the inverse of xn(·, t) by xn(·, t). For a function v(x, t), ift ∈ Jn, we define

(3.3) v(x, t) = v(xn(x, t), tn).

Note that v(x, tn−1,+) = vn−1,+(x) follows the characteristics forward from tn−1 to tn

to become vn(x). We shall use this type of functions as test functions below.Let Vh ×Wh be the finite element spaces for the approximation of σ and u, respec-

tively. They are finite dimensional and defined locally on each element T ∈ Th, so letVh(T ) = Vh|T and Wh(T ) = Wh|T . Neither continuity constraint nor boundary valuesare imposed on Vh ×Wh. Examples of Vh ×Wh will be given later. Let (·, ·)S denotethe L2(S) inner product (we omit S if S = Ω).

A unified characteristic scheme for (2.1) is: Find (σh, uh) : t1, . . . , tN → Vh ×Wh

such that(3.4)

(φnunh, v)− (φn−1un−1

h , vn−1,+) +∑

T∈Th

(σnh ,∇v)T ∆tn −

∑

e∈Eh

(σnh · ν, [v])e ∆tn

=∫

Jn

(f, v)−

∑

e∈ΓN

(gN , v)e −∑

e∈ΓD

(gDb · ν, v)e

dt, v ∈ Wh,

∑

T∈Th

((a−1)n

σnh −∇un

h, τ)

T+

∑

e∈Eh

([unh], τ · ν)e =

∑

e∈ΓD

(gnD, τ · ν)e , τ ∈ Vh,

where v is extended by zero outside Ω. The initial approximation u0h can be defined in

any reasonable manner.System (3.4) has been introduced in [12], and stems from a space-time mixed for-

mulation of (2.2). It can be seen [12] that (3.4) has a unique solution and the stiffnessmatrix arising from it is positive definite for any pair of Vh and Wh. All characteristicmethods under consideration will be derived from (3.4).

4. Relationships

In this section we derive various characteristic methods from (3.4) and discuss theirrelationships.

4.1. The Eulerian-Lagrangian mixed discontinuous method (ELMDM). LetTh be a partition into elements, say, simplexes, rectangular parallelepipeds, and/orprisms where edges or faces on ∂Ω may be curved. In ELMDM, Vh(T ) and Wh(T ) canbe any sets of polynomials. For example, they can be chosen as follows:

(4.1) Vh(T ) = (Pr1(T ))d , Wh(T ) = Pr2(T ), r1, r2 ≥ 0,

6 ZHANGXIN CHEN

where Pr(T ) is the set of polynomials of degree at most r on T . Other choices can betaken:

(4.2) Vh(T ) = (Qr1(T ))d , Wh(T ) = Qr2(T ), r1, r2 ≥ 0,

where Qr(T ) is the set of polynomials of degree at most r in each variable on T . Withthese choices, (3.4) is the Eulerian-Lagrangian mixed discontinuous method (ELMDM)introduced in [12]. For its stability and convergence, refer to [12]. Note that in ELMDM,the set Pr(T ) can be used even on rectangular parallelepipeds and prisms. Also, anycombination of Pr1(T ) and Qr2(T ) can be utilized for Vh(T ) and Wh(T ). ELMDMexpresses local conservation of mass along the characteristics [12]. Moreover, it istotally local, and the partition between adjacent elements does not have to match.Thus ELMDM is of high localizability and parallelizability. While it is in mixed form,it can be implemented in nonmixed form without introducing new variables. ELMDMis based on mixed discontinuous finite element methods [4, 11, 13, 14, 17].

4.2. The characteristic mixed method (CMM). Let Th be as in §4.1 and a regularpartition. Associated with the partition Th, let Vh × Wh ⊂ H(div; Ω) × L2(Ω) bethe Raviart-Thomas-Nedelec [31, 36], the Brezzi-Douglas-Fortin-Marini [7], the Brezzi-Douglas-Marini [8] (if d = 2), the Brezzi-Douglas-Duran-Fortin [6] (if d = 3), or theChen-Douglas [15] mixed finite element space, where

H(div; Ω) =(v ∈ (L2(Ω))d : ∇ · v ∈ L2(Ω)

).

Note that Vh ⊂ H(div; Ω) means that the normal components of elements in Vh arecontinuous across interior boundaries. Because of this feature, (3.4) reduces to: Find(σh, uh) : t1, . . . , tN → Vh ×Wh such that

(4.3)


h , vn−1,+) +∑

T∈Th

(σnh ,∇v)T ∆tn −

∑

e∈Eh


=∫

Jn

(f, v)−

∑

e∈ΓN

(gN , v)e −∑

e∈ΓD

(gDb · ν, v)e

dt, v ∈ Wh,

∑

T∈Th

((a−1)n

σnh −∇un

h, τ)

T+

∑

e∈Eh

([unh], τ · ν)e =

∑

e∈ΓD

(gnD, τ · ν)e , τ ∈ Vh.

This is a generalization of CMM introduced in [1, 22], where Vh ×Wh was taken to bethe lowest-order Raviart-Thomas-Nedelec mixed finite element spaces. In particular,Wh is the space of piecewise constants. With this, (4.3) becomes: Find (σh, uh) :


t1, . . . , tN → Vh ×Wh such that

(4.4)


h , vn−1,+)−∑

e∈Eh


=∫

Jn

(f, v)−

∑

e∈ΓN

(gN , v)e −∑

e∈ΓD

(gDb · ν, v)e

dt, v ∈ Wh,

∑

T∈Th

((a−1)n

σnh , τ

)T

+∑

e∈Eh

([unh], τ · ν)e =

∑

e∈ΓD

(gnD, τ · ν)e , τ ∈ Vh,

which is the characteristic mixed method developed in [1, 22]. We remark that apostprocessing procedure similar to that in [38] was used to improve the approximationuh in [1]. This postprocessing procedure is antidiffusive, so a slope limiting process wasexploited to stablize their method. For the stability and convergence analysis of (4.4),see [1]. CMM expresses local conservation of mass along the characteristics as well.Note that with v in place of vn−1,+, backward Euler integration for the three termsin the right-hand side of the first equation in (4.3), and an explicit treatment of theadvection, we can recover the usual mixed finite element method for parabolic problems[39].

4.3. The Eulerian-Lagrangian localized adjoint method (ELLAM). To deriveELLAM, we write (3.4) in Galerkin form (nonmixed form). For this, we introduce thecoefficient-dependent projections Pn

h : L2(Ω) → Vh by

(4.5)((a−1)n(w − Pn

h w), τ)

= 0 ∀τ ∈ Vh,

for w ∈ L2(Ω), and Rnh : H1(Th) → Vh by

(4.6)∑

T∈Th

((a−1)nRn

h(v), τ)T

= −∑

e∈Eh

([v], τ · ν)e +∑

e∈ΓD

(gnD, τ · ν)e ∀τ ∈ Vh,

for v ∈ H1(Ih). We remark that the definition of these two projection operators islocal.

4.3.1. The discontinuous case. Using (4.5) and (4.6), (3.4) can be rewritten as follows[12]: Find uh : t1, . . . , tN → Wh such that(4.7)(φnun

h, v)− (φn−1un−1h , vn−1,+) +

∑

T∈Th

(Pnh (an∇un

h) ,∇v)T ∆tn

−∑

e∈Eh

([unh], Pn

h (an∇v) · ν)e ∆tn −∑

e∈Eh

((Pnh (an∇un

h) + Rnh(un

h)) · ν , [v])e ∆tn

=∫

Jn

(f, v)−

∑

e∈ΓN

(gN , v)e −∑

e∈ΓD

(gDb · ν, v)e

dt

−∑

e∈ΓD

(gnD, Pn

h (an∇v) · ν)e ∆tn ∀v ∈ Wh,

8 ZHANGXIN CHEN

with σh given by

(4.8) σnh = Pn

h (an∇unh) + Rn

h(unh).

Namely, (3.4) is equivalent to (4.7) and (4.8). When a is piecewise constant and thefollowing relation holds:

(4.9) ∇Wh(T ) ⊂ Vh(T ), T ∈ Th,

(4.7) becomes: Find uh : t1, . . . , tN → Wh satisfying(4.10)(φnun

h, v)− (φn−1un−1h , vn−1,+) +

∑

T∈Th

(an∇unh,∇v)T ∆tn −

∑

e∈Eh

([unh], an∇v · ν)e ∆tn

−∑

e∈Eh

(an∇unh · ν , [v])e ∆tn +

∑

T∈Th

((a−1)nRn

h(unh), Rn

h(v))T∆tn

=∫

Jn

(f, v)−

∑

e∈ΓN

(gN , v)e −∑

e∈ΓD

(gDb · ν, v)e

dt

−∑

e∈ΓD

(gnD, (an∇v −Rn

h(unh)) · ν)e ∆tn ∀v ∈ Wh.

While it is derived from (3.4) under the assumption that a is piecewise constant, (4.10)(as it is) is a finite element method regardless of a being variable or constant. When Vh

and Wh are chosen as in §4.1, (4.10) can be thought of as ELLAM with discontinuousfinite elements. In this case, ELLAM also preserves mass locally. For its analysis, referto [12].

4.3.2. The continuous case. We now consider the case where Wh ⊂ H1(Ω). For sim-plicity, let

(4.11) ΓD = (x ∈ ∂Ω : b · ν ≥ 0) , ΓN = (x ∈ ∂Ω : b · ν < 0) .

DefineMh = Wh ∩

(v ∈ H1(Ω) : v

∣∣ΓD

= 0)

.

Note that [v] = 0 on Eh for any v ∈ Mh by continuity and convention. Consequently,(4.10) reduces to: Find un

h ∈ Mh + gnD, n = 1, . . . ,N , such that

(4.12)


h , vn−1,+) + (an∇unh,∇v)∆tn

=∫

Jn

(f, v)−

∑

e∈ΓN

(gN , v)e

dt ∀v ∈ Mh.

This is an extension of ELLAM devised in [9, 37], where piecewise linear polynomialswere used. Also, boundary conditions were treated in an ad hoc manner in [9, 37],while they are incorporated into the weak formulation in a natural way in (4.12).


ELLAM with continuous finite elements expresses global conservation of mass. For itsconvergence with piecewise linear polynomials, refer to [42, 43].

4.4. The modified method of characteristics (MMOC). MMOC has an inherentdifficulty to handle the general boundary boundary in (2.1) [23]. Traditionally, itwas developed for an advection-diffusion transport problem with a periodic boundarycondition [18, 23, 26]. To derive it from (3.4), we shall follow this tradition. Define

(4.13) Mh = Wh ∩H1(Ω).

With a periodic boundary condition and backward Euler integration for the first termin the right-hand side of (4.12), this equation becomes: Find un

h ∈ Mh, n = 1, . . . ,N ,such that

(4.14) (φnunh, v)− (φn−1un−1

h , vn−1,+) + (an∇unh,∇v) ∆tn = (fn, v)∆tn ∀v ∈ Mh.

For each n, let

G(x) ≡ G(x, tn) = x− ϕ(x, tn)∆tn.

We assume that ϕ has bounded first partial derivatives in space. Then, for ∆tn suf-ficiently small, G(·) is a differentiable homeomorphism of Ω into itself. Moreover, theJacobian of this transformation is

J(G(x)

)=

1− ∂x1ϕn1∆tn −∂x2ϕ

n1∆tn −∂x3ϕ

n1∆tn

−∂x1ϕn2∆tn 1− ∂x2ϕ

n2∆tn −∂x3ϕ

n2∆tn

−∂x1ϕn3∆tn −∂x2ϕ

n3∆tn 1− ∂x3ϕ

n3∆tn

,

where ϕ = (ϕ1, ϕ2, ϕ3), and its determinant equals

(4.15)∣∣J(

G(x))∣∣ = 1−∇ · ϕn∆tn + O

((∆tn)2

).

With a change of variable, ∆tn being sufficiently small, and (4.15), the second term inthe left-hand side of (4.14) can be expressed by(4.16)(

φn−1un−1h , vn−1,+

)

=∫

Ωφn−1(x)un−1

h (x)v(xn(x, tn−1)

)dx

=∫

Ωφn−1

(xn(x, tn−1)

)un−1

h

(xn(x, tn−1)

)v(x)

∣∣J(G(x)

)∣∣ dx

=∫

Ωφn−1

(xn(x, tn−1)

)un−1

h

(xn(x, tn−1)

)v(x)

(1−∇ · ϕn∆tn + O

((∆tn)2

))dx.

10 ZHANGXIN CHEN

Consequently, (4.14) can be rewritten as follows: Find unh ∈ Mh, n = 1, . . . ,N , such

that

(4.17)(φnun

h, v)−(φn−1un−1

h , v)

+ (an∇unh,∇v)∆tn

= (fn, v)∆tn +(φn−1un−1

h , v)

O (∆tn) ∀v ∈ Mh,

where un−1h = un−1

h

(xn(x, tn−1)

). Ignoring the last term in the right-hand side of (4.17),

we see that

(4.18) (φnunh, v)−

(φn−1un−1

h , v)

+ (an∇unh,∇v)∆tn = (fn, v)∆tn ∀v ∈ Mh.

Equation (4.18) is an extension of MMOC originally introduced in [23], where φ wasassumed to be independent of t. As mentioned in §4.3.2, (4.12) conserves mass globally.If the coefficients φ and b are constants, it follows from (4.15) that MMOC globallyconserves mass as well. However, in general, a systematic conservation error of sizeO (∆tn) should be expected from MMOC. In the case where ∇ · ϕ = 0, a systematicerror of size O

((∆tn)2

)can occur.

MMOC is considered for continuous finite elements; its analysis can be found in[18, 23]. MMOC can be also developed for discontinuous elements [12], as for ELLAMin (4.10). The transport diffusion method (TDM) [5, 29, 35] and the operator splittingmethod (OSM) [24, 44] are virtually the same as MMOC, although they were presentedin slightly different forms.

4.5. The modified method of characteristics with adjusted advection (MMO-CAA). For MMOC to have a global mass conservation, a scheme different from EL-LAM was developed in [20], i.e., the modified method of characteristics with adjustedadvection (MMOCAA). MMOCAA is defined from MMOC by perturbing the foot ofcharacteristics slightly [20, 21].

With Mh defined in (4.13) and u0h ∈ Mh given, for n ≥ 1 set

Qn−1h =

∫

Ωφn−1(x)un−1

h (x)dx, Qn−1h =

∫

Ωφn−1(x)un−1

h (x)dx.

As mentioned above, Qn−1h 6= Qn−1

h in general. Set

x−n = xn(x, tn−1)− γϕ(x, tn) (∆tn)2 , x+n = xn(x, tn−1) + γϕ(x, tn) (∆tn)2 ,

where γ is a fixed constant, normally chosen to be less than one [20]. Define

un−1h (x) =

max(un−1

h (x−n ) , un−1h (x+

n ))

if Qn−1h < Qn−1

h ,

minun−1

h (x−n ) , un−1h (x+

n ))

if Qn−1h > Qn−1

h ,

andQn−1

h =∫

Ωφn−1(x)un−1

h (x)dx.


If Qn−1h = Qn−1

h , we must accept that mass cannot be conserved; otherwise, find Λn−1 ∈< such that

(4.19) Qn−1h = Λn−1Qn−1

h + (1− Λn−1)Qn−1h .

Define

(4.20) un−1h = Λn−1un−1

h + (1− Λn−1)un−1h ,

and

(4.21) Qn−1h =

∫

Ωφn−1(x)un−1

h (x)dx.

Clearly, Qn−1h = Qn−1

h , so the conservation law is preserved. Now, continue in n withun−1

h in place of un−1h in MMOC (4.18); i.e., find un

h ∈ Mh, n = 1, . . . ,N , such that

(4.22) (φnunh, v)−

(φn−1un−1

h , v)

+ (an∇unh,∇v)∆tn = (fn, v)∆tn ∀v ∈ Mh.

For the analysis of MMOCAA, see [21]. Again, MMOCAA can be considered fordiscontinuous finite elements [12].

4.6. The method of characteristics (MOC). The classical method of characteris-tics is a finite difference method that is based on the forward tracking of particles incells or elements [28, 34, 40]. Here we extend it to the finite element setting. Again, weconsider (2.1) with a periodic boundary condition. With the notation in (3.3) and theabove definition of Mh in (4.13), the explicit finite element method of characteristics isdefined by: Find un

h ∈ Mh, n = 1, . . . ,N , such that

(4.23) (φnunh, v)−

(φn−1un−1

h , v)

+(an−1∇un−1

h ,∇v)

∆tn = (fn, v)∆tn ∀v ∈ Mh.

The Eulerian-Lagrangian method (ELM) developed in [32, 33] is similar to (4.23). Itis known that the forward tracked characteristic method gives rise to the difficultyof distorted grids. Also, this explicit method requires that a Courant-Friedrich-Lewy(CFL) time step constraint be imposed. An implicit forward tracked characteristicmethod, i.e., the finite elements incorporating characteristics (FEIC), was introducedin [41]. FEIC uses space-time elements with edges oriented along characteristics. Again,it distorts grids, particularly in multi-dimensional cases. Further, it has restrictions onthe space-time elements near the boundary of the space domain. In one dimension,for example, at each time step the treatment of boundary conditions in FEIC exploitsone triangular space-time element at the inlet boundary, which effectively limits theCourant number to be of order one.

12 ZHANGXIN CHEN

5. Remarks on a Degenerate Diffusion

So far we have assumed that (2.3) holds. We now consider the case where a = (aij)is symmetric and positive semi-definite:

(5.1) 0 ≤ |ξ|−2d∑

i,j=1

aij(x, t)ξiξj ≤ a∗ < ∞, (x, t) ∈ Ω× J, ξ ∈ <d.

When (5.1) holds, there is a symmetric, positive semi-definite matrix κ such that

(5.2) a = κκ.

Now, (2.2) is of the form

(5.3)

∂t(φu) +∇ · (bu− κσ) = f in Ω× J,

σ = κ∇u in Ω× J,


(bu− κσ) · ν = gN , on ΓN × J,

u(x, 0) = u0(x) in Ω.

Corresponding to (5.3), the counterpart of (3.4) is: Find (σh, uh) : t1, . . . , tN →Vh ×Wh such that(5.4)


h , vn−1,+) +∑

T∈Th

(κnσnh ,∇v)T ∆tn −

∑

e∈Eh

(κnσnh · ν, [v])e ∆tn

=∫

Jn

(f, v)−

∑

e∈ΓN

(gN , v)e −∑

e∈ΓD

(gDb · ν, v)e

dt, v ∈ Wh,

∑

T∈Th

(σnh − κn∇un

h, τ)T +∑

e∈Eh

([unh], κnτ · ν)e =

∑

e∈ΓD

(gnD, κnτ · ν)e , τ ∈ Vh.

All the characteristic methods considered in the previous section except CMM can berecovered from (5.4) in the same fashion. CMM inherently requires that a be positivedefinite. To relax this requirement, we can employ the expanded concept in CMM[2, 10]; we do not pursue this.

6. Concluding Remarks

Relationships among various characteristic methods are examined in this paper. Webegin with a unified scheme, from which we derive ELMDM, CMM, ELLAM, MMOC(TDM, OSM), MMOCAA, and MOC (ELM, FEIC). While ELMDM is in mixed form,it utilizes any finite element spaces, which do not need to satisfy the inf-sup condition.It is also totally local and directly applies to a degenerate diffusion problem. It canbe implemented in nonmixed form without introducing new variables. CMM requires


a nondegenerate diffusion coefficient. Moreover, it uses the property that the normalcomponents of elements in the vector space are continuous across interior boundaries.The former requirement can be relaxed via the expanded concept, while the latter canbe removed by introducing Lagrange multipliers over boundaries [3]. Both ELMDM andCMM conserve mass locally. ELLAM with discontinuous finite elements conserves masslocally, while it with continuous elements conserves mass globally. MMOC has certaindifficulties, especially with regard to mass conservation. MMOCAA (with continuousfinite elements) evolves from MMOC to conserve mass globally, but it still has theinherent difficulty in the treatment of boundary conditions. The classical MOC givesrise to the usual difficulty of distorted Lagrangian grids in over one dimension, whichbacktracking methods avoid.

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14 ZHANGXIN CHEN

[17] B. Cockburn and C.-W. Shu, The local discontinuous Galerkin method for time-dependentconvection-diffusion systems, SIAM J. Numer. Anal. 35 (1998), 2440–2463.

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[38] S. Stenberg, Some new families of finite elements for the Stokes equations, Numer. Math. 56(1990), 827–838.

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[41] E. Varoglu and W. D. L. Finn, Finite elements incorporating characteristics for one-dimensionaldiffusion-convection equations, J. Comp. Phys. 34 (1980), 371–389.

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Department of MathematicsBox 750156, Southern Methodist UniversityDallas, TX 75275-0156, U.S.A.email [email protected]

AN APPROXIMATION OF THE HANKEL TRANSFORM FORABSOLUTELY CONTINUOUS MAPPINGS

N.M. DRAGOMIR, S.S. DRAGOMIR, M. GU, X. GAN, AND R. WHITE

J. KSIAM Vol.6, No.1, 17-31, 2002

Abstract. Using some techniques developed by Dragomir and Wang in the recentpaper [2] in connection to Ostrowski integral inequality, we point out some approxi-mation results for the Henkel’s transform of absolutely continuous mapping.

1. Introduction

Two-dimensional systems may often show circular symmetry, for example opticalsystems are often constructed from components that, in themselves, are circulary sym-metrical.

When circular symmetry exists, that is, when f(x, y) = f(r), r2 = x2 + y2, then thebidimensional Fourier transform can be represented in the following way [8, p. 244 - p.250]

(1.1)∫ ∞

−∞

∫ ∞

−∞f(x, y)e−i2π(xu+yv)dxdy

=∫ ∞

0

∫ 2π

0f(r)e−i2πqr cos(θ−ϕ)rdrdθ =

∫ ∞

0f(r)

[∫ 2π

0e−i2πqr cos(θ−ϕ)dθ

]rdr

= 2π

∫ ∞

0f(r)J0(2πqr)rdr

where x + iy = reiθ, u + iv = qeiϕ, q2 = u2 + v2 and we have used the relation

(1.2) J0(z) =12π

∫ 2π

0e−iz cos βdβ

We refer to G(f)(q) given by

(1.3) G(f)(q) = 2π∫ ∞

0f(r)J0(2πqr)rdr

as the Hankel transform (of zero order) of f(r).The main aim of the present article is to point out some estimates of the Hankel

transform for absolutely continuous mappings defined on an finite interval [a, b] by theuse of some techniques developed by Dragomir and Wang in the recent paper [2] in

17

18 N.M. DRAGOMIR, S.S. DRAGOMIR, M. GU, X. GAN, AND R. WHITE

connection to Ostrowski integral inequality. Some adaptive quadrature formulae whichwill allowe another approach than the classical one will be also derived.

2. An Integral Representation

Let J1 (·) represent the first-order Bessel functions of the first kind, that is,

(2.1) J1 (x) :=12π

∫ π

−πeix sin ϕ−iϕdϕ, x ∈ R.

Define the corresponding Bessel’s mean as:

(2.2) B1 (z, w) =

z if w = zzJ1(z)−wJ1(w)

z−w if w 6= z;w, z ∈ R.

The following representation of Hankel transform holds.

Theorem 1. Let g : [a, b] → K (K = C,R) be an absolutely continuous mapping on[a, b] . Then we have the representation

G (g) (ρ) =1ρB1 (2πbρ, 2πaρ)×

∫ b

ag (s) ds(2.3)

+2π

b− a

∫ b

a

∫ b

ak (r, s) g′ (s) rJ0 (2πrρ) drds

for all ρ ∈ [a, b] , ρ 6= 0, where k (·, ·) : [a, b]2 → R is given by

(2.4) k (u, v) :=

v − a if v ∈ [a, u]v − b if v ∈ (u, b] ; (u, v) ∈ [a, b]2

and J0 (·) is the zeroth-order Bessel function of the first kind, that is,

(2.5) J0 (x) =12π

∫ 2π

0e−ix cos βdβ, x ∈ R.

Proof. Using the integration by parts formula for absolutely continuous mappings on[a, b] , we can write

(2.6)∫ x

a(s− a) g′ (s) ds = (x− a) g (x)−

∫ x

ag (s) ds

and

(2.7)∫ b

x(s− b) g′ (s) ds = (b− x) g (x)−

∫ b

xg (s) ds

for all x ∈ [a, b] .Adding (2.6) and (2.7) , we end up with (see also [2])

(2.8) g (x) =1

b− a

∫ b

ag (s) ds +

1b− a

∫ b

ak (r, s) g′ (s) ds

HANKEL TRANSFORM FOR ABSOLUTELY CONTINUOUS MAPPINGS 19

for all x ∈ [a, b] , which is of importance in itself, too.Now, consider the Hankel transform of g on the interval [a, b] , that is:

(2.9) G (g) (ρ) := 2π∫ b

ag (r) rJ0 (2πrρ) dr

and use the representation (2.8) to get

G (g) (ρ)(2.10)

= 2π

∫ b

a

[1

b− a

∫ b

ag (s) ds +

1b− a

∫ b

ak (r, s) g′ (s) ds

]rJ0 (2πrρ) dr

=1

b− a

∫ b

ag (s) ds · 2π

∫ b

arJ0 (2πrρ) dr

+2π

b− a

∫ b

a

∫ b

ak (r, s) g′ (s) rJ0 (2πrρ) dsdr.

Consider the change of variable r′ = 2πrρ. Then r = r′2πρ , dr = dr′

2πρ and∫ b

arJ0 (2πrρ) dr =

∫ 2πbρ

2πaρ

r′

2πρJ0

(r′

) 12πρ

dr′

=1

(2πρ)2

∫ 2πbρ

2πaρr′J0

(r′

)dr′.

It is a well-known property of Bessel functions that

(2.11)∫ x

0ξJ0 (ξ) dξ = xJ1 (x) , x ∈ R.

Consequently,∫ 2πbρ

2πaρr′J0

(r′

)dr′ =

∫ 2πbρ

0r′J0

(r′

)dr′ −

∫ 2πaρ

0r′J0

(r′

)dr′

= 2πbρJ1 (2πbρ)− 2πaρJ1 (2πaρ)

and

2π

b− a

∫ b

arJ0 (2πrρ) dr =

2π

b− a· 1(2πρ)2

[2πbρJ1 (2πbρ)− 2πaρJ1 (2πaρ)]

=1ρ

2πbρJ1 (2πbρ)− 2πaρJ1 (2πaρ)2πbρ− 2πaρ

=1ρB1 (2πbρ, 2πaρ) .

Using (2.10) , we deduce the desired representation (2.1) .


In practical applications we have a = 0 and b = 1. Consequently, we can state thefollowing corollary.

Corollary 2. Let g : [0, 1] → K (K = C,R) be an absolutely continuous mapping on[0, 1] . Then we have the representation

G (g) (ρ)(2.12)

=J1 (2πρ)

ρ×

∫ 1

0g (s) ds + 2π

∫ 1

0

∫ 1

0k (r, s) g′ (s) rJ0 (2πrρ) drds, ρ ∈ (0, 1]

where k : [0, 1] → R is given by

k (u, v) =

v if v ∈ [0, u]v − 1 if v ∈ (u, 1] .

Now let us define the mapping with real values I (g) : [a, b] → [0,∞) given by

(2.13) I (g) (ρ) := |G (g) (ρ)|2 , ρ ∈ [a, b] .

Using the well known property of complex numbers

(2.14) |x + y|2 = |x|2 + 2 Re (zy) + |y|2 for any x, y ∈ C

we can state the following corollary:

Corollary 3. With the assumptions from Theorem 1, we have

I (g) (ρ) =1ρ2|B1 (2πbρ, 2πaρ)|2

∣∣∣∣∫ b

ag (s) ds

∣∣∣∣2

(2.15)

+4π

ρ (b− a)Re

[B1 (2πbρ, 2πaρ)

∫ b

ag (s) ds

×∫ b

a

∫ b

ak (r, s) rg′ (s)J0 (2πrρ)drds

]

+4π2

(b− a)2

∣∣∣∣∫ b

a

∫ b


∣∣∣∣2


for all ρ ∈ [a, b] , ρ 6= 0.If a = 0, b = 1, then

I (g) (ρ) =1ρ2|J1 (2πρ)|2

∣∣∣∣∫ 1

0g (s) ds

∣∣∣∣2

(2.16)

+4π

ρRe

[J1 (2πρ)

∫ 1

0g (s) ds

×∫ 1

0

∫ 1

0k (r, s) rg′ (s)J0 (2πrρ)drds

]

+4π2

∣∣∣∣∫ 1

0

∫ 1

0k (r, s) g′ (s) rJ0 (2πrρ) drds

∣∣∣∣2

for all ρ ∈ (0, 1] .

3. Integral Inequalities

The main aim of this section is to point out an estimate for the remainder

(3.1) R [g] (ρ) :=2π

b− a

∫ b

a

∫ b


in formula (2.3) .We can state the following integral inequalities.

Theorem 4. Let g be as in Theorem 1. Then∣∣∣∣G (g) (ρ)− 1ρB1 (2πbρ, 2πaρ)×

∫ b

ag (s) ds

∣∣∣∣(3.2)

≤

2πb−a ‖g′‖∞

∫ ba

[(r−a)2+(b−r)2

2

]|r| dr if g′ ∈ L∞ [a, b] ;

2πb−a ‖g′‖p

[∫ ba

[(r−a)q+1+(b−r)q+1

q+1

]|r|q dr

] 1q

if g′ ∈ Lp [a, b] , p > 1, 1p + 1

q = 1;

2π ‖g′‖1

∫ ba |r| dr

for all ρ ∈ [a, b] , ρ 6= 0.

Proof. Using the representation (2.3) , we get∣∣∣∣G (g) (ρ)− 1

ρB1 (2πbρ, 2πaρ)

∫ b

ag (s) ds

∣∣∣∣

≤ 2π

b− a

∫ b

a

∫ b

a|k (r, s)| |r| ∣∣g′ (s)∣∣ |J0 (2πrρ)| drds =: A (ρ)


for all ρ ∈ [a, b] \ 0 .It is obvious that

|J0 (2πrρ)| =∣∣∣∣

12π

∫ 2π

0e−x cos βdβ

∣∣∣∣ ≤12π

∫ 2π

0

∣∣∣e−x cos β∣∣∣ dβ =

12π

∫ 2π

0dβ = 1.

In this way we can state the following inequality

A (ρ) ≤ 2π

b− a

∫ b

a

∫ b

a|k (r, s)| |r| ∣∣g′ (s)∣∣ drds := B (ρ) .

It is obvious that

B (ρ) ≤ ∥∥g′∥∥∞

2π

b− a

∫ b

a

∫ b

a|k (r, s)| |r| drds(3.3)

=∥∥g′

∥∥∞

2π

b− a

∫ b

a

(∫ b

a|k (r, s)| ds

)|r| dr

=∥∥g′

∥∥∞

2π

b− a

∫ b

a

[∫ r

a(s− a) ds +

∫ b

r(b− s) ds

]|r| dr

=∥∥g′

∥∥∞

2π

b− a

∫ b

a

[(r − a)2 + (b− r)2

2

]|r| dr,

and the first inequality in (3.2) is obtained.For the second inequality, we use Holder’s integral inequality for double integrals toget:

B (ρ)(3.4)

≤ 2π

b− a

(∫ b

a

∫ b

a|k (r, s)|q |r|q drds

) 1q(∫ b

a

∫ b

a

∣∣g′ (s)∣∣p dsdr

) 1p

=2π

b− a(b− a)

1p

∥∥g′∥∥

p

(∫ b

a

(∫ r

a(s− a)q ds +

∫ b

r(b− s)q ds

)|r|q dr

) 1q

=2π

b− a(b− a)

1p

∥∥g′∥∥

p

(∫ b

a

[(r − a)q+1 + (b− r)q+1

q + 1

]|r|q dr

) 1q

and the second inequality is also proved.Finally, we observe that

B (ρ) ≤ 2π

b− asup

(r,s)∈[a,b]2|k (r, s)|

∫ b

a|r| dr ·

∫ b

a

∣∣g′ (s)∣∣ ds(3.5)

=2π

b− a(b− a)

∥∥g′∥∥

1

∫ b

a|r| dr = 2π

∥∥g′∥∥

1

∫ b

a|r| dr


and the theorem is proved.

Remark 1. In practical applications a ≥ 0, so the first bound in (3.2) becomes

2π

b− a

∥∥g′∥∥∞

∫ b

a

[(r − a)2 + (b− r)2

2

]rdr

=2π

b− a

∥∥g′∥∥∞

[12

∫ b

a(r − a)2 rdr +

12

∫ b

a(b− r)2 rdr

]

=2π

b− a

∥∥g′∥∥∞

[124

(b− a)3 (3b + a) +124

(b− a)3 (3a + b)]

=2π

b− a

∥∥g′∥∥∞

124

(b− a)3 (4b + 4a)

=π

3

∥∥g′∥∥∞ (b− a)2 (a + b) .

The second term will be:

2π

(b− a) (q + 1)1q

∥∥g′∥∥

p

[∫ b

a(r − a)q+1 rqdr +

∫ b

a(b− r)q+1 rqdr

] 1q

,

and the third term is:π

∥∥g′∥∥

1(b + a) (b− a) .

Consequently, we can state that∣∣∣∣G (g) (ρ)− 1

ρB1 (2πbρ, 2πaρ)

∫ b

ag (s) ds

∣∣∣∣(3.6)

≤

π3 ‖g′‖∞ (a + b) (b− a)2 if g′ ∈ L∞ [a, b] ;

2π

(b−a)(q+1)1q‖g′‖p

[∫ ba (r − a)q+1 rqdr +

∫ ba (b− r)q+1 rqdr

] 1q

if g′ ∈ Lp [a, b] , p > 1, 1p + 1

q = 1;

π ‖g′‖1 (b + a) (b− a)

for all ρ ∈ [a, b] .

Remark 2. If we assume that a = 0 and b > 0 in (3.6) , then we get∣∣∣∣G (g) (ρ)− J1 (2πbρ)

ρ×

∫ b

0g (s) ds

∣∣∣∣(3.7)

≤

π3 ‖g′‖∞ b3 if g′ ∈ L∞ [a, b]π ‖g′‖1 b2 .


The above inequality shows that

G (g) (ρ) ≈ J1 (2πbρ)ρ

×∫ b

0g (s) ds

when b → 0+ and the precision of approximation is 3.

Remark 3. Let us observe that the integral

I =∫ b

a(b− r)2 (r − a)2 dr

can be written in a different form. By using the change of variable r = a (1− t) + bt,t ∈ [0, 1] , we obtain

I = (b− a)∫ 1

0[b− (1− t) a− tb]q [(1− t) a + tb− a]s dt

= (b− a)q+s+1∫ 1

0(1− t)q tsdt = (b− a)q+s+1 β (q + 1, s + 1)

where β (·, ·) is a Beta function, that is,

β (q, s) =∫ 1

0(1− t)q−1 ts−1dt; q, s > 0.

Now, coming back to the second bound in (3.6) for a = 0, we can observe that∫ b

0rq+1rqdr =

∫ b

0r2q+1dr =

b2q+2

2q + 2and ∫ b

0(b− r)q+1 rqdr = b2q+2β (q + 2, q + 1) .

Consequently, we have the inequality∣∣∣∣G (g) (ρ)− J1 (2πbρ)

ρ

∫ b

0g (s) ds

∣∣∣∣(3.8)

≤ 2π

b (q + 1)1q

∥∥g′∥∥

p

[b2q+2

2q + 2+ b2q+2β (q + 2, q + 1)

] 1q

=2πb

1+ 2q

(q + 1)1q

[1

2q + 2+ β (q + 2, q + 1)

] 1q

for all ρ ∈ (0, b] .

In some practical applications the upper limit of integration is b = 1, therefore thefollowing corollary is required.


Corollary 5. Let g : [0, 1] → K (K = C,R) be an absolutely continuous mapping on[0, 1] . Then we have the inequality

∣∣∣∣G (g) (ρ)− J1 (2πρ)ρ

×∫ 1

0g (s) ds

∣∣∣∣(3.9)

≤

π3 ‖g′‖∞ if g′ ∈ L∞ [a, b]

2π

(q+1)1q

[1

2q+2 + β (q + 2, q + 1)] 1

q ‖g′‖p

if g′ ∈ Lp [0, 1] , p > 1, 1p + 1

q = 1;

π ‖g′‖1

for all ρ ∈ (0, 1] .

Using the inequality (3.2) which provides upper bounds for the remainder R [g] (ρ) ,we will point out the following inequality which approximates the mapping I (g) (ρ)(c.f.(2.13)).

Corollary 6. Let g : [a, b] → K (K = C,R) be an absolutely continuous mapping on[a, b] . Then we have the inequality

∣∣∣∣∣I (g) (ρ)− 1ρ2|B1 (2πbρ, 2πaρ)|2

∣∣∣∣∫ b

ag (s) ds

∣∣∣∣2∣∣∣∣∣

≤[2ρ|B1 (2πbρ, 2πaρ)|

∣∣∣∣∫ b

ag (s) ds

∣∣∣∣ + E (g) (ρ)]

E (g) (ρ)

where

E (g) (ρ) :=

2πb−a ‖g′‖∞

∫ ba

[(r−a)2+(b−r)2

2

]|r| dr if g′ ∈ L∞ [a, b] ;

2πb−a ‖g′‖p

[∫ ba

[(r−a)q+1+(b−r)q+1

q+1

]|r|q dr

] 1q

if g′ ∈ Lp [a, b] , p > 1, 1p + 1

q = 1;

2π ‖g′‖1

∫ ba |r| dr

for all ρ ∈ [a, b) , ρ 6= 0.

The proof is obvious by Corollary 3 and Theorem 4.


4. A Quadrature Formula

In this section we point out a quadrature formula for the Hankel’s transform

G (g) (ρ) := 2π∫ 1

0g (s) rJ0 (2πrρ) dr

where g is assumed to be absolutely continuous on [0, 1] .Firstly, let us assume that in formula (3.6) we have 0 ≤ a ≤ b ≤ 1. Then we have

the coarser upper bound∣∣∣∣G (g) (ρ)− 1

ρB1 (2πbρ, 2πaρ)×

∫ b

ag (s) ds

∣∣∣∣(4.1)

≤

2π3 ‖g′‖∞ (b− a)2 if g′ ∈ L∞ [a, b] ;

21+1

q π‖g′‖p

[(q+1)(q+2)]1q

(b− a)2q if g′ ∈ Lp [a, b] , p > 1, 1

p + 1q = 1;

2π ‖g′‖1 (b− a) .

Indeed, from (3.6) we have∣∣∣∣G (g) (ρ)− 1

ρB1 (2πbρ, 2πaρ)×

∫ b

ag (s) ds

∣∣∣∣

≤

π3 ‖g′‖∞ (a + b) (b− a)2 ;

2π

(b−a)(q+1)1q‖g′‖p

[∫ ba (r − a)q+1 rqdr +

∫ ba (b− r)q+1 rqdr

] 1q ;

π ‖g′‖1 (b + a) (b− a) .

As 0 ≤ a ≤ b ≤ 1, then a + b ≤ 2 and hence:π

3

∥∥g′∥∥∞ (a + b) (b− a)2 ≤ 2π

3

∥∥g′∥∥∞ (b− a)2

andπ

∥∥g′∥∥

1(b + a) (b− a) ≤ 2π

∥∥g′∥∥

1(b− a) .

On the other hand,∫ b

a(r − a)q+1 rqdr ≤

∫ b

a(r − a)q+1 dr =

(b− a)q+2

q + 2and ∫ b

a(b− r)q+1 rqdr ≤

∫ b

a(b− r)q+1 dr =

(b− a)q+2

q + 2.


Consequently, we get

2π

(b− a) (q + 1)1q

∥∥g′∥∥

p

[∫ b

a(r − a)q+1 rqdr +

∫ b

a(b− r)q+1 rqdr

] 1q

≤ 2π

(b− a) (q + 1)1q

∥∥g′∥∥

p

[2 (b− a)q+2

q + 2

] 1q

=21+ 1

q π ‖g′‖p (b− a)2q

[(q + 1) (q + 2)]1q

and the second inequality in (4.1) is also proved.Now, consider In : 0 = x0 < x1 < ... < xn−1 < xn = 1 a division of the interval

[0, 1] . Put hi := xi+1 − xi, (i = 0, ..., n− 1) and υ (h) := max hi|i = 0, ..., n− 1 thenorm of the division. Construct the sums

H (g, In, ρ)(4.2)

: =1ρ

n−1∑

i=0

B1 (2πxi+1ρ, 2πxiρ)×∫ xi+1

xi

g (s) ds

=1ρ

n−1∑

i=0

1hi

[xi+1J1 (2πxi+1ρ)− xiJ1 (2πxiρ)]×∫ xi+1

xi

g (s) ds.

We can state the following theorem concerning the approximation of Hankel’s transformin terms of the quadrature formula (4.2) .

Theorem 7. Let g : [0, 1] → K (K = C,R) be an absolutely continuous mapping on[0, 1] . Then we have

(4.3) G (g) (ρ) = H (g, In, ρ) + R (g, In, ρ) , ρ ∈ (0, 1]

where H (g, In, ρ) is as given by the formula (4.2) and the remainder R (g, In, ρ) satisfiesthe estimate

(4.4) |R (g, In, ρ)| ≤

2π3 ‖g′‖∞

∑n−1i=0 h2

i if g′ ∈ L∞ [a, b] ;

21+1

q π

[(q+1)(q+2)]1q‖g′‖p

(∑n−1i=0 h2

i

) 1q

if g′ ∈ Lp [a, b] , p > 1, 1p + 1

q = 1;

2πν (h) ‖g′‖1

for all ρ ∈ (0, 1] .


Proof. Apply formula (4.1) on the subintervals [xi, xi+1] (i = 0, ..., n− 1) to get

∣∣∣∣2π

∫ xi+1

xi

g (r) rJ0 (2πrρ) dr − 1ρB1 (2πxi+1ρ, 2πxiρ)×

∫ xi+1

xi

g (s) ds

∣∣∣∣

≤

2π3 supr∈[xi,xi+1] |g′ (r)|h2

i

21+1

q π

[(q+1)(q+2)]1q

(∫ xi+1

xi|g′ (s)|p ds

) 1p

h2q

i

2π(∫ xi+1

xi|g′ (s)| ds

)hi.

Summing over i from 0 to n− 1 and using the generalized triangle inequality, we get

|R (g, In, ρ)|

≤n−1∑

i=0

∣∣∣∣2π

∫ xi+1

xi

g (r) rJ0 (2πrρ) dr − 1ρB1 (2πxi+1ρ, 2πxiρ)×

∫ xi+1

xi

g (s) ds

∣∣∣∣

≤

2π3

n−1∑i=0

supr∈[xi,xi+1] |g′ (r)|h2i

21+1

q π

[(q+1)(q+2)]1q

n−1∑i=0

(∫ xi+1

xi|g′ (s)|p ds

) 1p

h2q

i

2πn−1∑i=0

(∫ xi+1

xi|g′ (s)| ds

)hi

.

Now, observe that

n−1∑

i=0

supr∈[xi,xi+1]

∣∣g′ (r)∣∣h2i ≤

∥∥g′∥∥∞

n−1∑

i=0

h2i


and then the first inequality in (4.4) is proved.Using Holder’s discrete inequality, we deduce

n−1∑

i=0

(∫ xi+1

xi

∣∣g′ (s)∣∣p ds

) 1p

h2q

i

≤(

n−1∑

i=0

[(∫ xi+1

xi

∣∣g′ (s)∣∣p ds

) 1p

]p) 1p

×[

n−1∑

i=0

(h

2q

i

)q] 1

q

=

(n−1∑

i=0

(∫ xi+1

xi

∣∣g′ (s)∣∣p ds

)) 1p

×(

n−1∑

i=0

h2i

) 1q

=∥∥g′

∥∥p

(n−1∑

i=0

h2i

) 1q

.

Finally, let us observe that

2πn−1∑

i=0

(∫ xi+1

xi

∣∣g′ (s)∣∣ ds

)hi ≤ 2πν (h)

n−1∑

i=0

(∫ xi+1

xi

∣∣g′ (s)∣∣ ds

)

= 2πν (h)∥∥g′

∥∥1

and the theorem is completely proved.

Remark 4. It is obvious thatn−1∑

i=0

h2i ≤ ν (h)

n−1∑

i=0

hi

and then by (4.4) we deduce the coarser upper bound

(4.5) |R (g, In, ρ)| ≤

2π3 ‖g′‖∞ ν (h) ;

21+1

q π

[(q+1)(q+2)]1q‖g′‖p [ν (h)]

1q ;

2πν (h) ‖g′‖1 .

Now, observe that if ν (h) → 0, then every branch on the right hand side of (4.5)goes to 0, which proves that H (g, In, ρ) approximates the Hankel’s transform with anyaccuracy.

In practical problems the interval [0, 1] is devided in equidistant subintervals by thedivision In : xi = i

n , i = 0, ..., n.


If we consider the sum

Hn (g, ρ)(4.6)

: =1ρ

n−1∑

i=0

B1

(2π (i + 1) ρ

n,2πiρ

n

)×

∫ i+1n

in

g (s) ds

=1ρ

n−1∑

i=0

[(i + 1)J1

(2π

i + 1n

ρ

)− iJ1

(2π

i

nρ

)]×

∫ i+1n

in

g (s) ds,

then we can state the following corollary.

Corollary 8. Let g be as in Theorem 7. Then we have

(4.7) G (g) (ρ) = Hn (g, ρ) + Rn (g, ρ) , ρ ∈ (0, 1]

where Hn (g, ρ) is as given in (4.6) and the remainder Rn (g, ρ) satisfies the estimate

(4.8) |Rn (g, ρ)| ≤

2π3n ‖g′‖∞ if g′ ∈ L∞ [0, 1] ;

21+1

q π

[(q+1)(q+2)]1q n

1q‖g′‖p if g′ ∈ Lp [0, 1] , p > 1, 1

p + 1q = 1;

2πn ‖g′‖1 .

Remark 5. Supposing that we know ‖g′‖∞ and would like to approximate G (g) (ρ)with a given error ε > 0, then we have to divide the interval [0, 1] into at least nε ∈ N∗points, where

nε :=[2π

3ε

∥∥g′∥∥∞

]+ 1

where [x] is the integer part of x ∈ R.

Using the tools provided in papers [1], [3]-[7] the authors are going to point outother quadrature formulae for Hankel transform for mappings of bounded variation,monotonic, of r-Holder’s type etc.

For a preprint of this paper containing some numerical experiments, seehttp://rgmia.vu.edu.au/v5(E).html. We omit the details.

References

[1] S.S. Dragomir, On the Ostrowski’s Integral inequality to Lipschitz mappings and applications,Computers and Mathematics with Applications, 38 (1999), 33-37.

[2] S.S. Dragomir and S. Wang, Applications of Ostrowski’s inequality for the estimation of error boundsfor some special means and for some numerical quadrature rules, Applied Mathematics Letters 11(1) (1998), 105 -109.

[3] S.S. Dragomir and S. Wang, A new inequality of Ostrowski’s type in Lp norm, Indian Journal ofMathematics, 40 (3)(1998), 299-304.


[4] S.S. Dragomir, Ostrowski’s inequality for monotonic mapping and applications., J. KSIAM, 3(1)(1999), 127-135.

[5] N.S. Barnett and S.S. Dragomir, An inequality of Ostrowski’s type for cumulative distributionfunctions, Kyungpook Mathematical Journal, 39(2) (1999), 303-311.

[6] S.S. Dragomir, On the Ostrowski’s inequality for mappings of bounded variation, MathematicalInequalities and Applications, 4(1) (2001), 59-66.

[7] S.S. Dragomir, N.S. Barnett and S. Wang, An Ostrowski type inequality for a random variable whoseprobability density function belongs to Lp [a, b] , p > 1, Mathematical Inequalities and Applications,2(4) (1999), 501-508.

[8] R.N. Bracewell, The Fourier Transform and Its Applications, Second Edition, Revised, McGraw-Hill, Inc. , 1986.

School of Communications and InformaticsVictoria University of Technology,PO Box 14428, MC MelbourneCity, Victoria 8001, Australiahttp://matilda.vu.edu.au/˜rgmia/dragomirweb.html

AN APPROXIMATION OF THE FOURIER SINE TRANSFORM VIAGRUSS TYPE INEQUALITIES AND APPLICATIONS FOR

ELECTRICAL CIRCUITS

S.S. DRAGOMIR AND A. KALAM

J. KSIAM Vol.6, No.1, 33-45, 2002

Abstract. An approximation of the Fourier Sine Transform via Gruss, Chebychevand Lupas integral inequalities and application for an electrical curcuit containing aninductance L, a condenser of capacity C and a source of electromotive force E0P (t),where P (t) is an L2−integrable function, are given.

1. Introduction

Consider the electrical oscillation in a circuit containing a resistance R, an inductanceL, a condenser of capacity C, and a source of electromotive force E0P (t), where E0 isa constant and P (t) is a known function of the time t.

If the charge on the plates of the condenser is q, then the potential difference acrossthe plates is

q

c. Similarly, if i is the current flowing through the resistance and the in-

ductance, the differences of potential between their ends are Ri and L(

didt

), respectively.

By the equation of continuity

(1.1) i =dq

dt

so that these potential differences may be written as Rdqdt and Ld2q

dt2respectively. Thus

we obtain the ordinary differential equation [6, p. 93]

(1.2) Ld2q

dt2+ R

dq

dt+

q

c= E0P (t)

for the determination of the charge q which accumulates on the plates of the condenser.If we assume that initially this charge is Q and that a current I is flowing in the

circuit, then we obtain the initial conditions

(1.3)

q (0) = Q,

dq (0)dt

= I.

1991 Mathematics Subject Classification. Primary 26D15; Secondary 41A55.Key words and phrases. Gruss Inequality, Fourier Sine Transform.

33

34 S.S. DRAGOMIR AND A. KALAM

It is well known that if the resistance of the circuit is zero, i.e., R = 0, then thesolution of (1.2) with the initial conditions (1.3) is given by (see for example [6, p. 95])

(1.4) q (t) = Q cos (ωt) +I

ωsin (ωt) +

E0

ωL

∫ t

0P (s) sin [ω (t− s)] ds,

where ω2 = 1LC .

Consequently, there is a practical need in computing the following Fourier Sine Trans-form:

A (0, t;P, ω, t) :=∫ t

0P (s) sin [ω (t− s)] ds,

which may be a difficult task if P is complicated enough. In this case, a numericalapproach would satisfy the user if the accuracy is good.

The main aim of the present paper is to point out some estimates for the FourierSine Transform by the use of a Gruss type inequality.

2. Inequalities for the Fourier Sine Transform

Let us consider the Fourier Sine transform

(2.1) A (a, b; P, ω, t) :=∫ b

aP (s) sin [ω (t− s)] ds,

where t may be an arbitrary real number.The following result holds.

Theorem 1. If P ∈ L2 [a, b], then we have the inequality:∣∣∣∣A (a, b; P, ω, t) + COS (ω (t− b) , ω (t− a))

∫ b

aP (s) ds

∣∣∣∣(2.2)

≤ (b− a)

[1

b− a

∫ b

aP 2 (s) ds−

(1

b− a

∫ b

aP (s) ds

)2] 1

2

×[12

(1− SIN (2ω (t− b) , 2ω (t− a))− COS2 [ω (t− b) , ω (t− a)]

)],

where

SIN (z, ω) :=sin z − sinω

z − ω, z 6= w,

and

COS (z, ω) :=cos z − cosω

z − ω, z 6= w

are the trigonometric means.

APPROXIMATION OF THE FOURIER SINE TRANSFORM 35

Proof. Using Korkine-Andreief’s integral identity [5, p. 242, 243], i.e., we recall it

1b− a

∫ b

af (t) g (t) dt− 1

b− a

∫ b

af (t) dt · 1

b− a

∫ b

ag (t) dt(2.3)

=1

2 (b− a)2

∫ b

a

∫ b

a(f (t)− f (s)) (g (t)− g (s)) dtds,

we may write:

1b− a

∫ b

aP (s) sin [ω (t− s)] ds− 1

b− a

∫ b

aP (s) ds · 1

b− a

∫ b

asin [ω (t− s)] ds

=1

2 (b− a)2

∫ b

a

∫ b

a(P (u)− P (s)) (sin [ω (t− u)]− sin [ω (t− s)]) duds,

from where we get, via the Cauchy-Schwartz inequality, that∣∣∣∣

1b− a

∫ b

aP (s) sin [ω (t− s)] ds(2.4)

− 1b− a

∫ b

aP (s) ds · 1

b− a

∫ b

asin [ω (t− s)] ds

∣∣∣∣

≤ 12 (b− a)2

∫ b

a

∫ b

a|P (u)− P (s)| |sin [ω (t− u)]− sin [ω (t− s)]| duds

≤ 12 (b− a)2

[∫ b

a

∫ b

a[P (u)− P (s)]2 duds

] 12

×[∫ b

a

∫ b

a(sin [ω (t− u)]− sin [ω (t− s)])2 duds

] 12

=1

2 (b− a)2

[2

((b− a)

∫ b

aP 2 (s) ds−

(∫ b

aP (s) ds

)2)] 1

2

×[2

((b− a)

∫ b

a

(sin2 [ω (t− s)]

)ds−

(∫ b

a(sin [ω (t− s)]) ds

)2)] 1

2

= : K.

However,∫ b

asin [ω (t− s)] ds =

cos [ω (t− b)]− cos [ω (t− a)]ω

= − (b− a)COS (ω (t− b) , ω (t− a)) ,


∫ b

asin2 [ω (t− s)] ds =

∫ b

a

1− cos [2ω (t− s)]2

ds

=12

(b− a)− 12

∫ b

acos [2ω (t− s)] ds

=12

(b− a) +12· sin 2ω (t− b)− sin 2ω (t− a)

2ω

=12

(b− a)− 12

(b− a) SIN (2ω (t− b) , 2ω (t− a))

=12

(b− a) [1− SIN (2ω (t− b) , 2ω (t− a))]

and then

K =1

(b− a)2

[((b− a)

∫ b

aP 2 (s) ds−

(∫ b

aP (s) ds

)2)] 1

2

×[12

(b− a)2 [1− SIN (2ω (t− b) , 2ω (t− a))]

− (b− a)2 COS2 [ω (t− b) , ω (t− a)]] 1

2

=

[1

b− a

∫ b

aP 2 (s) ds−

(1

b− a

∫ b

aP (s) ds

)2] 1

2

×[12

[1− SIN (2ω (t− b) , 2ω (t− a))]− COS2 [ω (t− b) , ω (t− a)]] 1

2

.

Using (2.4), we deduce the desired result (2.2).

Corollary 1. If m ≤ P ≤ M a.e. on [a, b], where m and M are real numbers, then,for any t ∈ R,

∣∣∣∣A (a, b; P, ω, t) + COS (ω (t− b) , ω (t− a))∫ b

aP (s) ds

∣∣∣∣(2.5)

≤ 12

(b− a) (M −m) B (a, b, ω, t) ,

where

B (a, b, ω, t)

: =[12

[1− SIN (2ω (t− b) , 2ω (t− a))]− COS2 [ω (t− b) , ω (t− a)]] 1

2

.


Proof. Using Gruss’ integral inequality [5, p. 296] for the function P , we may write:

(2.6) 0 ≤ 1b− a

∫ b

aP 2 (s) ds−

(1

b− a

∫ b

aP (s) ds

)2

≤ 14

(M −m)2 ,

and then, by (2.2), we deduce (2.5).

Corollary 2. If the function P : [a, b] → R is absolutely continuous on [a, b] andP ′ ∈ L∞ [a, b], then, for any t ∈ R, we have


aP (s) ds

∣∣∣∣(2.7)

≤ 12√

3

∥∥P ′∥∥∞ (b− a)2 B (a, b, ω, t) ,

where ‖P ′‖∞ := ess supt∈[a,b]

|P ′ (t)|.

Proof. Using Chebychev’s integral inequality [5, p. 297] for P , we may write that

(2.8) 0 ≤ 1b− a

∫ b

aP 2 (s) ds−

(1

b− a

∫ b

aP (s) ds

)2

≤ 112

∥∥P ′∥∥2

∞ (b− a)2

and then, by (2.2), we obtain (2.7).

Corollary 3. If the function P : [a, b] → R is absolutely continuous on [a, b] andP ′ ∈ L2 [a, b], then, for any t ∈ R, we have


aP (s) ds

∣∣∣∣(2.9)

≤ (b− a)1+ 1q ‖P ′‖2

πB (a, b, ω, t) .

Proof. Using Lupas’ inequality [5, p. 301], we may write that

(2.10) 0 ≤ 1b− a

∫ b

aP 2 (s) ds−

(1

b− a

∫ b

aP (s) ds

)2

≤ b− a

π2

∥∥P ′∥∥2

2,

and then, by (2.2) we obtain (2.9).

For other Gruss type inequalities which may be applied in a similar fashion, see therecent papers [1] – [4].

We may now state the following result in estimating the solutions of (1.2) with theinitial conditions (1.3).


Theorem 2. Assume that P is absolutely continuous on any compact interval [0, t],t ∈ R+. If q (·) is the solution of equation (1.2) with R = 0 and the initial conditions(1.3), then we have the estimate:

∣∣∣∣q (t)−Q cosωt− I

ωsinωt +

cosωt− 1ωt

· E0

ωL

∫ t

0P (s) ds

∣∣∣∣(2.11)

≤ tE0

2ωLU (ω, t) ·

[1t

∫ t

0P 2 (s) ds−

(1t

∫ t

0P (s) ds

)2] 1

2

≤

E0

2ωLt (Mt −mt) U (ω, t) if mt ≤ P (s) ≤ Mt, s ∈ [0, t] ;

12√

3· E0

ωLt2 ‖P ′‖∞,[0,t] U (ω, t) if P ′ ∈ L∞ [0, t] ;

1π· E0

ωLt

32 ‖P ′‖2,[0,t] U (ω, t) if P ′ ∈ L2 [0, t] ,

where

U (ω, t) :=

[12

(1− sin 2ωt

2ωt

)−

(cosωt− 1

ωt

)2] 1

2

, t > 0.

Proof. If in (2.2), we choose a = 0, b = t, we get∣∣∣∣A (0, t; P, ω, t) +

∫ t

0P (s) ds · cosωt− 1

ωt

∣∣∣∣(2.12)

≤ t

[1t

∫ t

0P 2 (s) ds−

(1t

∫ t

0P (s) ds

)2] 1

2

×[

12

(1− sin 2ωt

2ωt

)−

(cosωt− 1

ωt

)2] 1

2

.

By (1.4), we get

(2.13) A (0, t;P, ω, t) =ωL

E0

[q (t)−Q cosωt− I

ωsinωt

].

Inserting (2.13) in (2.12), we easily deduce the first part of (2.11).The second part follows by Corollaries 1 – 3, and we omit the details.


Remark 1. We observe that U (ω, t) is actually[

1t

∫ t

0sin2 [ω (t− s)] ds−

(1t

∫ t

0sin [ω (t− s)] ds

)2] 1

2

.

If we consider the function f (s) = sin [ω (t− s)], then we may state that;

−1 ≤ f (s) ≤ 1 for any s ∈ [0, t] ,f ′ (s) = −ω cos [ω (t− s)] ,

∥∥f ′∥∥

[0,t],∞ ≤ ω

and

∥∥f ′∥∥

[0,t],2=

(∫ t

0

[f ′ (s)

]2ds

) 12

= ω

(∫ t

0cos2 [ω (t− s)] ds

) 12

= ω

(∫ t

0

1 + cos [2ω (t− s)]2

ds

) 12

= ω

[12t +

12ω

sin (2ωt)] 1

2

.

Consequently, using Gruss’, Chebychev’s and Lupas’s inequalities for f (s) = sin [ω (t− s)],we may state that

U (ω, t) ≤

1,

12√

3ωt,

1π·√

t√2ω

[t +

1ω

sin (2ωt)] 1

2

,

implying that

U (ω, t) ≤ min

1,

12√

3ωt,

1√2π

√tω

[t +

1ω

sin (2ωt)] 1

2

for t ≥ 0.

3. Some Quadrature Formulae

Consider the division of the interval [a, b] given by

In : a = x0 < x1 < x2 < · · · < xn−1 < xn = b,

withhi := xi+1 − xi, i = 0, n− 1 and ν (h) = max

i=0,n−1hi,


and define the quadrature formula:

(3.1) An (In; P, ω, t) := −n−1∑

i=0

COS (ω (t− xi+1) , ω (t− xi))∫ xi+1

xi

P (s) ds.

Then we may state the following result in approximating the Fourier-Sine OperatorA (a, b;P, ω, t) in terms of the sums defined above in (3.1).

Theorem 3. Assume that P : [a, b] → R is integrable on [a, b]. Then we have

(3.2) A (a, b; P, ω, t) = An (In, P, ω, t) + En (In, P, ω, t) ,

where An (In, P, ω, t) are as defined in (3.1) and the error En (In, P, ω, t) satisfies theestimates:

|En (In, P, ω, t)|(3.3)

≤

12

n−1∑i=0

hi (Mi −mi) B (xi, xi+1, ω, t) if mi ≤ P (s) ≤ Mi,

s ∈ [xi, xi+1] , i = 0, n− 11

2√

3· ‖P ′‖[a,b],∞

n−1∑i=0

h2i B (xi, xi+1, ω, t) if P ′ ∈ L∞ [a, b] ;

1π· ‖P ′‖[a,b],2

n−1∑i=0

h32i B (xi, xi+1, ω, t) if P ′ ∈ L2 [a, b] ,

where

B (α, β, ω, t)

: =

[1

β − α

∫ β

αsin2 [ω (t− s)] ds−

(1

β − α

∫ β

αsin [ω (t− s)] ds

)2] 1

2

=[12

(1− SIN (2ω (t− β) , 2ω (t− α))− COS2 [ω (t− β) , ω (t− α)]

)] 12

and α, β ∈ [a, b] , t ∈ R, ω are as given above.

Proof. If we apply Corollary 1 on the intervals [xi, xi+1](i = 0, n− 1

), we may write

that ∣∣∣∣∫ xi+1

xi

P (s) sin [ω (t− s)] ds + COS (ω (t− xi+1) , ω (t− xi))∫ xi+1

xi

P (s) ds

∣∣∣∣

≤ 12hi (Mi −mi) B (xi, xi+1, ω, t) ,


where mi = infs∈[xi,xi+1]

P (s) and Mi = sups∈[xi,xi+1]

P (s).

Summing over i from 0 to n− 1 and using the triangle inequality, we may write:

|A (a, b;P, ω, t)−An (In, P, ω, t)|

≤n−1∑

i=0

∣∣∣∣∫ xi+1

xi

P (s) sin [ω (t− s)] ds

+ COS (ω (t− xi+1) , ω (t− xi))∫ xi+1

xi

P (s) ds

∣∣∣∣

≤ 12

n−1∑

i=0

hi (Mi −mi) B (xi, xi+1, ω, t) ,

and the first part of (3.3) is proved.If we use Corollary 2, on the intervals [xi, xi+1], we may write that:

∣∣∣∣∫ xi+1

xi


xi

P (s) ds

∣∣∣∣

≤ 12√

3

∥∥P ′∥∥∞ h2

i B (xi, xi+1, ω, t) .

Doing in a similar way as above, we get the second part of (3.3).Finally, if we apply Corollary 3 on the intervals [xi, xi+1], we may write that

∣∣∣∣∫ xi+1

xi


xi

P (s) ds

∣∣∣∣

≤ 1π

∥∥P ′∥∥[a,b],2

n−1∑

i=0

h32i B (xi, xi+1, ω, t) , i = 0, n− 1,

from where we get the last part of (3.3).

Remark 2. Since the quantities B (xi, xi+1, ω, t) (i = 0, . . . , n− 1) play an importantrole in evaluating the error estimate in the quadrature formula (3.2), and in practicemay be difficult to compute, we point out here a way of upper bounding them by the useof Chebychev’s inequality (2.8).


Applying (2.8) on the intervals [xi, xi+1], (i = 0, . . . , n− 1), we may write that:

B (xi, xi+1, ω, t)(3.4)

=

[1hi

∫ xi+1

xi

sin2 [ω (t− s)] ds−(

1hi

∫ xi+1

xi

sin [ω (t− s)] ds

)2] 1

2

≤ 12√

3hi

∥∥∥∥d

dssin [ω (t− s)]

∥∥∥∥∞,[xi,xi+1]

≤ 12√

3hiω,

for any i = 0, n− 1.Now, using the evaluation (3.4), we may point out the following corollary which will

contain simpler bounds for the error estimate than Theorem 2.

Corollary 4. With the assumption in Theorem 2, we have (3.2) and the error En (In, P, ω, f)satisfies the estimate

|En (In, P, ω, t)|(3.5)

≤

ω4√

3

n−1∑i=0

(Mi −mi) h2i if mi ≤ P (s) ≤ Mi,

s ∈ [xi, xi+1] ,ω

12· ‖P ′‖[a,b],∞

n−1∑i=0

h3i if P ′ ∈ L∞ [a, b] ;

ω

2π√

3· ‖P ′‖[a,b],q

n−1∑i=0

h52i if P ′ ∈ L2 [a, b] ;

≤

ω

4√

3(M −m) ν (h) (b− a) if m ≤ P (s) ≤ M, s ∈ [a, b] ,

ω

12· ‖P ′‖[a,b],∞ ν2 (h) (b− a) if P ′ ∈ L∞ [a, b] ;

ω

2π√

3· ‖P ′‖[a,b],2 ν

32 (h) (b− a) if P ′ ∈ L2 [a, b] .

Remark 3. We note that all the bounds go to zero if the norm of the division In issmall, i.e., ν (h) → 0. It is obvious that the higher order of accuracy is provided by thesecond bound.

In practical considerations, it is useful to use the equidistant partitioning

En : xi = a +i

n(b− a) , i = 0, . . . , n.


In this partitioning, we consider the quadrature formula:

An (P, ω, t) :=n−1∑

i=0

COS

[ω

(t− a− i + 1

n(b− a)

), ω

(t− a− i

n(b− a)

)](3.6)

×a+ i+1

n(b−a)∫

a+ in

(b−a)

P (s) ds.

Then, we may state the following result in estimating the error of approximating theintegral operator A (a, b, P, ω, t) by the use of quadrature formula An (P, ω, t).

Corollary 5. Assume that P is as in Theorem 2. Then

(3.7) A (a, b, P, ω, t) = An (P, ω, t) + En (P, ω, t) ,

where An (P, ω, t) is given by (3.6) and the error En (P, ω, t) satisfies the bounds:

|En (P, ω, t)|(3.8)

≤

ω (M −m) (b− a)2

4n√

3if m ≤ P (s) ≤ M, s ∈ [a, b] ,

ω ‖P ′‖[a,b],∞ (b− a)3

12n2if P ′ ∈ L∞ [a, b] ;

ω ‖P ′‖[a,b],2 (b− a)52

2πn32

√3

if P ′ ∈ L2 [a, b] .

Now, for a fixed t > 0, let us consider the uniform partitioning of the interval [0, t]:

Un : xi :=i

nt, i = 0, . . . , n.

We can consider the quadrature formula

(3.9) Qn (P, ω, t) := −n−1∑

i=0

COS

(ω · n− i− 1

nt, ω · n− i

nt

)∫ i+1n

t

in

tP (s) ds.

Then we may state the following result in approximating the solution of equation (1.2)with R = 0 and the initial condition (1.3).

Theorem 4. Assume that P is absolutely continuous on any compact interval [0, t],t ∈ R+. If q (·) is the solution of equation (1.2) with R = 0 and the intial conditions(1.3), then we have:

(3.10) q (t) = Q cosωt +I

ωsinωt +

E0

ωLQn (P, ω, t) + Rn (P, ω, t) ,


where the quadrature formula Qn (P, ω, t) is given in (3.9) and the remainder Rn (P, ω, f)in (3.10) satisfies the estimate

|Rn (P, ω, t)|(3.11)

≤

E0

4Ln√

3(Mt −mt) t2 where mt ≤ P (s) ≤ Mt, s ∈ [0, t] ;

E0

12Ln2‖P ′‖[0,t],∞ t3 if P ′ ∈ L∞ [0, t] ;

E0

2Lπn32

√3‖P ′‖[0,t],2 t

32 if P ′ ∈ L2 [0, t] .

Proof. If we apply Corollary 5 for a = 0, b = t, then we get

A (0, t; P, ω, t) : =∫ t

0P (s) sin [ω (t− s)] ds(3.12)

= Q (P, ω, t) + Sn (P, ω, t)

where

(3.13) |Sn (P, ω, t)| ≤

ω (Mt −mt) t2

4n√

3where mt ≤ P (s) ≤ Mt, s ∈ [0, t] ;

ω ‖P ′‖[0,t],∞ t3

12n2if P ′ ∈ L∞ [0, t] ;

ω ‖P ′‖[0,t],2 t32

2πn32

√3

if P ′ ∈ L2 [0, t] .

Using (3.12), (3.13) and (1.4), we deduce (3.10) and (3.11).

Remark 4. If one would like to numerically approximate q (·) on the given interval[0, t] with a theoretical accuracy better than a given ε > 0 (ε-small), then the minimalnumber of nodes n0 to reach that accuracy as given by the second bound in (3.11)) willbe

n0 :=

⌈(E0

12Lε

∥∥P ′∥∥[0,t],∞ t3

) 12

⌉+ 1,

where dae denotes the integer part of a ∈ R.

References

[1] S.S. DRAGOMIR and I. FEDOTOV, An inequality of Gruss type for Riemann-Stieltjes integraland applications for special means, Tamkang J. Math., 29(4) (1998), 287-292.


[2] S.S. DRAGOMIR, A generalisation of Gruss’ inequality in inner product spaces and applications,J. Mathematical Analysis and Applications, 237 (1999), 74-82.

[3] S.S. DRAGOMIR, A Gruss’ type integral inequality for mappings of r−Holder’s type and applica-tions for trapezoid formula, Tamkang Journal of Mathematics, 31(1) (2000), 43-47.

[4] S.S. DRAGOMIR, Some integral inequalities of Gruss type, Indian J. Pure Appl. Math., 31(4)(2000), 397-415.

[5] D.S. MITRINOVIC, J.E. PECARIC and A.M. FINK, Classical and New Inequalities in Analysis,Kluwer Academic, Dordrecht, 1993.

[6] I.N. SNEDDON, Fourier Transforms, McGraw-Hill Book Company, New York, Toronto, London,1987.

School of Communications and InformaticsVictoria University of TechnologyPO Box 14428Melbourne City MCVictoria 8001, Australiaemail [email protected] [email protected]://rgmia.vu.edu.au/SSDragomirWeb.html

J. KSIAM Vol.6, No.1, 47-52, 2002

DECODING OF LEXICODES S10,4

D. G. Kim

Abstract. In this paper we propose a simple decoding algorithm for the 4-ary lexico-graphic codes (or lexicodes) of length 10 with minimum distance 4, write S10,4. It isbased on the syndrome decoding method. That is, using a syndrome vector we detect anerror and it will be corrected an error from the four parity check equations.

1. Introduction

In this paper, we shall introduce the surprising arithemetical operations which areused in the Game of Nim. Under these operations, the lexicodes are linear over somefinite field. Their definition is derived from a greedy algorithm, that is, each codewordis chosen as the first word not prohibitively near to previous codewords.

The main aim of this paper is to find an decoding algorithm of the 4-ary [10, 6, 4]lexicodes, write S10,4. Using a syndrome vector and the four parity check equations,we correct one error in received vector.

This paper is arranged as follows. The nim operation is introduced in section 2,the lexicodes with base 22a

are discussed in section 3. In particular we obtain the sixbasis of the 4-ary lexicodes S10,4. Section 4 gives a decoding algorithm and decodingexamples for this code.

2. Nim operation

First, we define the two operations which are called the nim-addition ⊕ and nim-multiplication ⊗ in that game.

Research partially supported by Chungwoon University Grant.1991 Mathematics Subject Classification. 94B35.Key words and phrases. nim-operations, minimum distance, lexicographic codes, parity check ma-

trix, syndrome.

Typeset by AMS-TEX

47

48 D. G. Kim

Definition 1. Let x′ be a variable that ranges over all elements strictly less than x andmex the least non-negative integer not of the form. Then we define the two operations:(1) a⊕ b = mexa′ ⊕ b, a⊕ b′(2) a⊗ b = mex(a′ ⊗ b)⊕ (a⊗ b′)⊕ (a′ ⊗ b′)

Two operations, ⊕ and⊗, convert the numbers 0, 1, 2, · · · into a field of characteristic2. Also, for a ≥ 0, the numbers less than 22a

form a subfield and isomorphic to theGalois field GF(22a

).

Theorem 2 ([2]). The nim-operations turn the set of non-negative integers into afield of characteristic 2.

Using the field laws, we shall fill out the first 4 by 4 corner of the addition andmultiplication tables in nim. Consider the nim-addition of any two numbers from0, 1, 2, 3.

Theorem 3 ([1]). We have x⊕ 0 = 0⊕ x = x, for every number x.

Since 0, 1, 2, 3 is a field of characteristic 2, we have x⊕x = 0 for all x ∈ 0, 1, 2, 3.By Theorem 3, 1⊕ 2 can not be one of 0, 1, 2 and so must be 3. Since 1⊕ 3 6= 0, 1, 3, itmust be 2. In the same way, we have 2⊕ 3 = 1. Therefore the sum of any two distinctnumbers from 1, 2, 3 is the third.

⊕ 0 1 2 30 0 1 2 31 1 0 3 22 2 3 0 13 3 2 1 0

There is a nim-multiplication ⊗ which together with nim-addition ⊕ converts theintegers into a field [2]. With nim-multiplication, we know that 0⊗ x must be 0 whichis the zero of the field. Also 1⊗x must be x. Since the elements other than 0, 1 satisfyx2 = x ⊕ 1 (here x2 means x ⊗ x) in the field GF(4), we have 2 ⊗ 2 = 2 ⊕ 1 = 3 and3⊗ 3 = 3⊕ 1 = 2. Next 2⊗ 3 can not be one of 0, 2, 3 and so must be 1.

⊗ 0 1 2 30 0 0 0 01 0 1 2 32 0 2 3 13 0 3 1 2

The following is a rule enabling us to perform nim-additions. In its statement, theterm 2-power means a power of 2, such as 1, 2, 4, 8, · · · , in the ordinary sense:

DECODING OF LEXICODES S10,4 49

(i) If x is a 2-powers and y < x, then x⊕ y = x + y.(ii) x⊕ x = 0 for any x.

For example, 15⊕ 5 = (8⊕ 4⊕ 2⊕ 1) ⊕(4⊕ 1) = 8⊕ 2 = 10, since both 4’s and 1’sare cancelled.

For finite numbers, the nim-multiplication follows from the following rules, similarto those for nim-addition. In the following statement, the term Fermat 2-power meansthe number 22n

, such as 2, 4, 16, 256 · · · , in the ordinary sense:

(i) If x is a Fermat 2-powers and y < x, then x⊗ y = x× y.(ii) x⊗ x = 3

2 × x for any Fermat 2-power x.

For example 16⊗2 = 32, since 16 = 222. By an equation (ii), we have 22 = 2× 3

2 = 3,

42 = 4× 32 = 6, 162 = 16× 3

2 = 24, · · · .Using the associative and distributive laws, 19 ⊗ 11=(16 ⊕ 2 ⊕ 1) ⊗ (8 ⊕ 2 ⊕ 1) =(16⊗ 8)⊕ (16⊗ 2)⊕ (16⊗ 1) ⊕ (2⊗ 8)⊕ (2⊗ 2)⊕ (2⊗ 1) ⊕ (8⊕ 2⊕ 1) =128⊕ 32⊕ 16⊕(2⊗8)⊕2⊕8 = 128⊕32⊕16⊕4⊕2 =182, since 2⊗8 = 2⊗(4⊗2) = 4⊗22 = 4⊗3 = 8⊕4.Next, we compute the inverse value 15−1 satisfying 15 ⊗ 15−1 = 1. 15 ⊗ 4= (8 ⊕4 ⊕ 2 ⊕ 1) ⊗ 4=(8 ⊗ 4) ⊕ (4 ⊗ 4) ⊕ (2 ⊗ 4) ⊕ (1 ⊗ 4) = (2 ⊗ 4 ⊗ 4) ⊕ 6 ⊕ 8 ⊕ 4=(2⊗6)⊕ (4⊕2)⊕8⊕4=(2⊗ (4⊕2))⊕2⊕8 = 8⊕3⊕2⊕8 = 3⊕2=1. Hence 15−1 = 4.

3. Lexicodes

Consider the lexicodes with base B = 22a

. A word of this codes is a sequencex = · · ·x3x2x1, xi ∈ 0, 1, · · · , 22a −1. For a convenience, we omit leading zeros ( i.e.,012 = 12). The set of words is ordered lexicographically, i.e., the word x = · · ·x3x2x1

is smaller than the word y = · · · y3y2 y1, written x < y, if for some n we have xn < yn,but xN = yN for all N > n. For example, 123 < 132, 312 < 1032.

Lexicodes are defined by saying a word is in the code if it does not conflict with anyearlier codewords. That is, the lexicode with minimum distance d is defined by sayingthat two words do not conflict if the Hamming distance between them is not less thand. We write Sn,d for the lexicode consisting of the codewords with base 4, length n orless and minimum distance d.

Example 1. Applying the greedy algorithm, then the lexicode S4,3 contains thecodewords, 0, 111, 222, 333, 1012, 1103, 1230, 1321, 2023, 2132, 2201, 2310, 3031,3120, 3213, 3302.

In [3], Conway and Sloane show that the lexicode with base B = 2a is closed undercoordinatewise nim-addition, and if B = 22a

, the lexicode is closed under coordinate-wise nim-multiplication by scalars k, k ∈ 0, 1, · · · , 22a − 1. As a result we providethe following Lexicode Theorem.

50 D. G. Kim

Theorem 4 ([3]). If B is of the form 22a

, then the lexicode is a linear code over theGalois field GF(B).

Now we consider the lexicodes S10,4. Let ei be the basis of lexicode S10,4. It is easilychecked that we have the first 3 bases e1 = 1111, e2 = 10123 and e3 = 100132. SinceS10,4 is a 6-dimensional vector space, this code has 6 bases. So we need to find thebasis e4, e5 and e6 of this code.

Theorem 5. For each i (3 ≤ i ≤ 5), if ei+1 is the smallest codeword with more digitsthan ei, then e4 = 11000011. Moreover we have e5 = 101000023 and e6 = 1001000032.

Proof. In [3, Table IV], 7 digit codewords are not possible. So we find the smallesteight digit codeword. For k, a ∈ 0, 1, 2, 3, 10aaaaaa is impossible for the samereasons that 01aaaaaa is impossible. Also 11000000, 1100000a and 110000a0 conflictwith 00000000. So e4 may be 110000aa. Assume e4 = 11000011. If c is a linear sum ofany two bases of e1, e2 and e3, then d(e4, k⊗ ei) = d(e4, c) ≥ 4, i = 1, 2, 3, from thelast 2 places of e4, and at least 2 places of k ⊗ ei and c.

If c is a linear sum of e1, e2 and e3, then d(e4, c) ≥ 5 from the last 2 places of e4

and at least 3 places of c.By the similar way, we can obtain the bases e5 and e6. ¤

4. Decoding Method

In this section, we shall obtain a 4 by 10 parity check matrix H using the 6 bases ofS10,4. For a given received vector r, this matrix H gives a syndrome vector s = r⊗HT ,where HT is a transpose of H. If the syndrome is nonzero, this implies that an erroroccurred in the received vector.

Let r be a received vector, r = r10 r9 r8 r7 r6 r5 r4 r3 r2 r1. If ri (i = 1, 2, 3, 7)is incorrect, these equations yield three 0s and one nonzero, and respectively threenonzeros and one 0 if ri (i = 4, 5, 6, 8, 9, 10) is incorrect. In other cases, we concludethat more than one error has been made. In particular, if the syndrome s is a multipleof the ith column vector of H, then ri is not correct. Using the syndrome vector, wecan detect an errored coordinate in the received vector.

Now, all the arithmetic operations are in the nim-sense (nim-additon and nim-multiplication). So we write x + y for x⊕ y, and xy for x⊗ y.

Note : Let c be a codeword, c = c10 c9 c8 c7 c6 c5 c4 c3 c2 c1, c =∑6

i=1 xiei,xi ∈ 0, 1, 2, 3 Then we have c1 = x1 + 3x2 + 2x3 + x4 + 3x5 + 2x6, c2 = x1 + 2x2 +3x3 + x4 + 2x5 + 3x6, c3 = x1 + x2 + x3, ci+3 = xi (i = 1, 2, 3), c7 = x4 + x5 + x6

and ci+4 = xi (i = 4, 5, 6). If r has no error, then the four parity check equations yield0, 0, 0, 0 as the following these :

DECODING OF LEXICODES S10,4 51

r10 + r9 + r8 + r7 = 0(1)

r6 + r5 + r4 + r3 = 0(2)

3r10 + 2r9 + r8 + 3r6 + 2r5 + r4 + r2 = 0(3)

2r10 + 3r9 + r8 + 2r6 + 3r5 + r4 + r1 = 0(4)

From the four parity check equations and a property of x ⊕ x = 0, we obtain acoordinate ci, where c1 = r4+3r5+2r6+r8+3r9+2r10, c2 = r4+2r5+3r6+r8+2r9+3r10,c3 = r4 +r5 +r6, c4 = r3 +r5 +r6, c5 = r3 +r4 +r6, c6 = r3 +r4 +r5, c7 = r8 +r9 +r10,c8 = r7 + r9 + r10, c9 = r7 + r8 + r10, c10 = r7 + r8 + r9. Therefore we can obtain adesired codeword.

These the four equations give a parity check matrix H as the following this :

H =

1 1 1 1 0 0 0 0 0 00 0 0 0 1 1 1 1 0 03 2 1 0 3 2 1 0 1 02 3 1 0 2 3 1 0 0 1

.

For example, let r = 3012221020 be a received vector. Then by nim-multiplicationof matrix, we have the syndrome rHT = (0, 1, 2, 3). Since this vector is the 5th columnvector of H, the 5th coordinate of r is not correct. Therefore we obtan c5 = r3+r4+r6 =0 + 1 + 2 = 3 and then have a desired codeword c = 3012231020.

Now, we give a decoding algorithm of S10,4.

Algorithm

Step 1 : First, we compute the syndrome vector s. If s is a multiple of the ithcolumn of H, we go to step 2.

Step 2 : Since ri is not correct, ri is replaced by ci.

Example 2. Let r = 1232012331. Since s = (2, 0, 0, 0) is a multiple of the 7th columnvector of H, then r7 is not correct. Hence c7 = r8 + r9 + r10 = 1 + 2 + 3 = 0, and sowe get the desired codeword c = 1230012331.

Example 3. Let r = 2131112202. Since s = (1, 0, 3, 2) is a multiple of the 10th columnvector of H, then r10 is not correct. So we have c10 = 3r1 +3r4 +2r5 +r6 +3r8 +2r9 =1 + 1 + 2 + 1 + 2 + 2 = 3. Hence we get c = 3131112202.

52 D. G. Kim

Example 4. Let r = 3012221020. Since s = (0, 1, 2, 3) is a multiple of the 5th columnvector of H, then r5 is not correct, and so c5 = 3r1+2r2+r4+r8+r9 = 0+3+1+1+0 = 3.Therefore we get c = 3012231020.

Example 5. Let r = 213313011. Since s = (0, 1, 0, 0) is a multiple of the 3th columnvector of H, then r3 is not correct. Hence we obtain c3 = r4 + r5 + r6 = 3 + 1 + 3 = 1.Therefore we get c = 213313111.

References

[1] J.H. Conway, Integral Lexicographic Codes, Discrete Mathematics 83(1990) 219-235.[2] J.H. Conway, On Numbers and Games, Academic Press, New York, 1976.[3] J.H.Conway and N.J.A. Sloane, Lexicographic Codes: Error-Correcting Codes from Game Theory,

IEEE Trans. Inform. Theory IT-32(3) (1986) 337-348.

Liberal Arts and ScienceChungwoon University, HongsungChungnam 350-701, Koreae-mail [email protected]

J. KSIAM Vol.6, No.1, 53-57, 2002

BRANCHED SINGULARITIES OF HARMONIC MAPS

Heayong Shin

Abstract. In this paper we give an example of energy minimizing harmonic maps forwhich the set of singular points are two or more lines intersecting at a point.

1. Introduction

It is well known that a harmonic map between Riemannian manifolds are not neces-sarily continuous. In general, even energy minimizing harmonic maps can have pointsof discontinuity which we will call the singularity. The Hausdorff dimension of thesingularity of a minimizing harmonic map is known to be less than or equal to m− 3where m is the dimension of the domain [10].

The shape of the singularity and the behavior of a minimizing harmonic map nearthe singularity have been studied by many mathematicians. For example, when thedimension of the domain m = 3, the singularity of minimizing harmonic map is consistof isolated points [10]. But when m ≥ 4 the singular set can have more complicatedstructure. For the structure of singularity, it has been proved by Leon Simon that thesingular set of a minimizing harmonic map into a real analytic manifold is a union ofa pairwise disjoint locally (n − 3) rectifiable locally compact subsets [9]. But it isnot known whether there is a branch points in singular sets even for simplest case asmaps from B4 to S2. In fact, we do have only few explicit examples of minimizingharmonic maps with singularity.

In this note, we introduce a minimizing harmonic map for which the singular setis made of finite lines crossing at a point. So far this is the only known example ofbranched singular set of minimizing harmonic maps.

AMS subject classification : 53C43 58E20

Key Words : harmonic map, singularity

This paper is supported by KOSEF Grant No. 96-0701-02-01-3.

Typeset by AMS-TEX

53

54 HEAYONG SHIN

2. Minimizing harmonic map with branched singularity

On a C∞ manifold N, we define a degenerate metric h to be a positive semi-definite (0, 2)-tensor field which is not necessarily continuous. So, for any vector fieldV on N, h(V, V ) is a nonnegative function on N.

Now consider a weakly differentiable map u : M → N from a Riemannian manifoldM into a manifold N, and let h be a degenerate metric on N. We define theenergy Eh(u) of u with respect to the degenerate metric h as

Eh(u) =∫

x∈M

trMh(∇u,∇u) (x) dV (x),

where trMh(∇u,∇u) =∑

gij(x)h(u∗( ∂∂xi

), u∗( ∂∂xj

)) and gij = (gij)−1 is a Riemann-ian metric on M . Note that trMh(∇u,∇u) is a nonnegative function on M. Alsonote that if h is the Riemannian metric on N, then Eh(u) is same as the usualenergy E2(u). We say a map u0 is an Eh-minimizing map if Eh(u0) has minimumenergy among the weakly differentiable maps with v|∂M = u0|∂M .

The next lemma states that when we accumulate degenerate metrics on the targetmanifold, the energy minimizing property of a given map is preserved.

Lemma 1. Let hi|i = 1, · · · , k is a family of degenerate metric on a C∞-manifoldN, and let M be a Riemannian manifold. If u0 : M → N is Ehi-minimizing foreach i, then u0 is an Eh-minimizing map where the degenerate metric h is givenby

h(V,W ) =∑

i

hi(V, W ) for any V, W ∈ TxN for all x ∈ N

proof. For any weakly differentiable map v, trMhi(∇v,∇v) is non-negative. So,

Eh(v) =∫

x∈M

∑

i

trMhi(∇v,∇v)dV (x)

=∑

i

∫

x∈M

∫

M

trMhi(∇v,∇v)dV (x) =∑

i

Ehi(v).

Since u0 is an Ehi-minimizing map for each i,

Eh(v) =∑

i

Ehi(v) ≥

∑

i

Ehi(u0) = Eh(u0)

for any v such that v|∂M = u0|∂M . ¤

BRANCHED SINGULARITIES OF HARMONIC MAPS 55

Using the above lemma, we may construct examples of minimizing harmonic maps byaccumulating simpler degenerate metrics on target manifold for which we know that thegiven map is an energy minimizer. By this method we construct a minimizing harmonicmap from B4 into (R6, g) which is not Euclidean with branched singularities whereg is some Riemannian metric.

Theorem 2. Let B4 be the unit ball in the Euclidean 4-space. There exists a Rie-mannian metric on R6 such that the map u0 : B4 → R6 given by

u0(x1, x2, x3, x4) = ((x1, x2, x3)|(x1, x2, x3)| ,

(x2, x3, x4)|(x2, x3, x4)| ) .

is an energy minimizing harmonic map with singularities along x1-axis and x4-axis.

proof. First, we construct a Riemannian structure on R3 so that u(X) = X|X| : B3 →

R3 becomes a minimizing map. Let N+a = (X, x) ∈ R3×R | |X|2 + x2

a2 = 1, x ≥ 0be the upper half of a 3-ellipsoid with Riemannian metric induced from the Euclideanmetric on R4. When a is sufficiently large, u(X) = ( X

|X| , 0) : B3 → N+a is a

minimizing map. In fact, Helein[5] showed that u is minimizing when a > 8.For such an a, let P : R3 → N+

a be a Lipschitz map defined as follows.

P (X) =

(√

2|X|2−|X|4|X| X, a(1− |X|2) ) if |X| ≤ 1

( X|X| , 0 ) if |X| > 1 .

Then we may construct a smooth metric h on R3 so that , h = P ∗(ds2N+

a) for |X| ≤ 1,

and h− P ∗(ds2N+

a) is positive definite if |X| > 1.

For any weakly differentiable map v : B3 → (R3, h) such that v|∂B3 = u|∂B3 ,we have

Eg(v) =∫

B3trB3g(∇v,∇v)dV ol

≥ E2(P∗v) ( by (i) and (ii) )

≥ E2(u) (Since u is an energy minimizing map)

= Eg(u)

So, u(X) = X|X| : B3 → R3 is an Eh-minimizing map.

Now let h1 and h2 be the two degenerate metrics on R6 given by hi = Π∗i (h),i = 1, 2, where Π1, Π2 : R3×R3 → R3 are the projections such that Π1(X, Y ) = X ∈

56 HEAYONG SHIN

R3, Π2(X, Y ) = Y ∈ R3. Then from the above computation, it follows immediatelythat u0 : B4 → R6 is a Ehi

-minimizing map for each i. By applying Lemma 1,u0 : B4 → (R6, g) is an energy minimizing map where the Riemannian structure onthe target manifold is g = h1 + h2. The singular set of this energy minimizing map isSu0 = B4 ∩ x1 = x2 = x3 = 0 or x2 = x3 = x4 = 0 which is a union of two linesegments crossing at the origin. ¤

For the above minimizing map u0, we can notice that the image lies in the productof spheres T = S2 × S2 ⊂ S5√

2⊂ R6, and the induced metric on the torus T is

the standard product metric. Of course, the map u0 is also a energy minimizingmap considered as a map into the target manifold T or as a map into S5√

2with the

induced metric.In the above proof we can notice that by introducing more and more complicated

structure on the target manifold one can produce minimizing tangent map in Bm

whose singular set is any finite union of linear subspaces with codimension bigger than3. Hence, the singularity of minimizing harmonic map can cross finite times with anypositive angle at the branch point.

References

[1] M. Avellaneda and F.-H. Lin, Fonctions quasi affines et minimisation de∫ |∇u|p,

C. R. Acad. Sci. Paris 306, Serie I (1988), 355-358.[2] H. Brezis, J.-M. Coron and E. H. Lieb, Harmonic maps with defects, Comm. Math.

Phys. 107 (1986), 649-705.[3] J. M. Coron and R. Gulliver, Minimizing p-harmonic maps into spheres, J. reine

angew. Math. 401 (1989), 82-100.[4] J.-M. Coron and F. Helein, Harmonic diffeomorphisms, minimizing harmornic maps

and rotational symmetry, Compositio Math. 69 (1989), 175-228.[5] F. Helein, Regularity and Uniqueness of harmonic maps into an ellipsoid, Manuscripta

Math. 60 (1988), 235-257.[6] W. Jager and H. Kaul, Rotationally symmetric harmonic maps from a ball into

a sphere and the regularity problem for weak solution of elliptic system, J. reineangew. Math. 343 (1983), 146-161.

[7] F.-H. Lin, A remark on the map x|x| , C. R. Acad. Sci. Paris 305 (1987), 529-532.

[8] H. Shin, Degree 1 singularities of energy minimizing maps to a bumpy sphere, Math.Ann. 293 (1992), 509-521.

[9] L. Simon, Rectifiability of the singular set of energy minimizing maps, Calc. Var.Partial Differential Equations 3 (1995), 1-65.

[10] R. Schoen, and K. Uhlenbeck, A regularity theory for harmonic maps, J. Diff.Geom. 17 (1982), 307-335.

BRANCHED SINGULARITIES OF HARMONIC MAPS 57

Department of MathematicsChung-Ang University, Seoul, Korea [email protected]

J. KSIAM Vol.6, No.1, 59-65, 2002

A NOTE ON ENERGY MINIMIZING MAP

ON MANIFOLD WITH ISOLATED PEAKS

Heayong Shin

Abstract. In this paper, we consider some homogeneous maps from a cone over 2-spheres and determines whether they become energy minimizing maps or not. In fact,any homogeneous map from a standard cone over 2-sphere of radius smaller than 1 cannot be a minimizing harmonic map.

1. Introduction

There has been some efforts to extend the theory of harmonic maps to singularspaces by several mathematicians. Korevaar & Schoen has developed sobolev theoryof maps from Riemannian domains into general complete metric spaces and have provedthe regularity of harmonic maps into nonpositively curved metric spaces. The similarregularity result has also been proved by Jost independently. But when the domain isnot a Riemannian manifold, the harmonic map theory becomes more ambiguous. Josthas defined the energy density of such a map with respect to a given measure on thedomain which satisfy certain conditions. But as far as the author knows, the minimumstructure of the domain where the theory of harmonic map can be properly consideredis not known.

When the domain is a smooth manifold with C∞ Riemannian metric given excepton some isolated points and the tangent cone at the singular points can be given,the regularity of the minimizing harmonic map at those points can be examined byconsidering the existence of homogeneous minimizing harmonic map on the targetcone. In this paper, we consider 3-dimensional cone over 2-spheres and will study someconditions for which the homogeneous harmonic map is or is not minimizing.

AMS subject classification : 53C43 58E20Key Words : harmonic map, singularityThis paper is supported by KOSEF Grant No. 96-0701-02-01-3.

Typeset by AMS-TEX

59

60 HEAYONG SHIN

2. Homogeneous Minimizing Map in a Cone

Let S be a 2-dimensional topological sphere with a given metric ds2, and letKS = O ∪ ((0, 1]×S) be a cone over S, with the metric given by dρ2 + ρ2ds2. For amap h : S → N with finite Dirichlet energy we consider the homogeneous extensionh : KS → N given by h(r, p) = h(p). When S is the standard unit sphere S2, KS

is the unit ball in R3 and the map h is given by h(x) = h(

x|x|

). We will examine

some cases where h can or can not be an energy minimizer.

Theorem 1. When S is the Euclidean sphere of radius k, k < 1, and N is aRiemannian manifold diffeomorphic to a sphere, h(r, p) = h(p) : KS → N is not anenergy minimizing map unless h is a constant.

proof. If h is an energy minimizing map, then h : S → N must be harmonic.Therefore, we may assume that h is a conformal map from S into N. Consider Sas the unit sphere S2 with metric k2ds2, where ds2 is the standard metric in S2.Then KS can be identified as the unit ball B3

1(0) with the metric.

Ψ = dr2 + k2r2dθ2 + k2r2 sin2 θdσ2,

for standard spherical coordinate r, θ, σ centered at the origin and z-axis as thepole. In this coordinate, the Dirichlet energy E(h) of h is given by

E(h) =∫

KS

|∇h|2 =∫

S

|∇h|2 = 2∫ 2π

0

dσ

∫ π

0

dψ

∣∣∣∣∂h

∂ψ

∣∣∣∣2

N

.

Let 0 < a < 1 and A = (0, 0, a). We introduce another polar coordinate ρ, ϕ, σcentered at A with the z-axis as the pole, so that ρ(x) = |x−A|, ψ(x) is the polarangle and σ(x) is the azimuthal angle of x from A. Then between ρ, ϕ, σ andr, θ, σ we have the following relations

ρ sin ϕ = r sin θ, ρ cos ϕ = r cos θ − a .

Let R(ϕ) denote the maximum allowed radius in B3 for given angle ϕ (i,e,(R(ϕ), ϕ, σ) ∈ S2 ), and let ψ(ϕ) be the standard polar angle centered at the originof the point (R(ϕ), ϕ, σ).

We define ua(ρ, ϕ, σ) = h(ψ(ϕ), σ) with respect to the coordinate (ρ, ϕ, σ) in Kc

and the standard polar coordinate (ψ, σ) on S = S2.In local coordinate (ρ, ϕ, σ), the energy density of ua can be computed as

A NOTE ON ENERGY MINIMIZING MAP 61

|∇ua|2N =∣∣∣∣∂ua

∂r

∣∣∣∣2

N

+1

k2r2

∣∣∣∣∂ua

∂θ

∣∣∣∣2

N

+1

k2r2 sin2 θ

∣∣∣∣∂ua

∂σ

∣∣∣∣2

N

=(∣∣∣∣

∂ϕ

∂r

∣∣∣∣2

+1

k2r2

∣∣∣∣∂ϕ

∂θ

∣∣∣∣2)∣∣∣∣

∂ua

∂ϕ

∣∣∣∣2

N

+1

k2r2 sin2 θ

∣∣∣∣∂ua

∂σ

∣∣∣∣2

N

and the volume element is

dV = k2ρ2 sin ϕ dρ dϕ dσ .

Here, | ∗ |N is the norm as vectors in N.Since

∂ϕ

∂r=

a sin ϕ

ρ√

ρ2 + 2aρ cosϕ + a2and

∂ϕ

∂θ=

ρ + a cosϕ

ρ2,

the energy of ua is

E(ua) =∫ 2π

0

dσ

∫ π

0

dϕ

∫ R(ϕ)

0

dρ

(k2a2 sin2 ϕ + (ρ + a cos ϕ)2

ρ2 + 2aρ cos ϕ + a2

)sin ϕ

∣∣∣∣∂ua

∂ϕ

∣∣∣∣2

N

+1

sin ϕ

∣∣∣∣∂ua

∂σ

∣∣∣∣2

N

=∫ 2π

0

dσ

∫ π

0

dϕ

∫ R(ϕ)

0

dρ

sin ϕ

∣∣∣∣∂ua

∂ϕ

∣∣∣∣2

N

+1

sin ϕ

∣∣∣∣∂ua

∂σ

∣∣∣∣2

N

+ (k2 − 1)∫ 2π

0

dσ

∫ π

0

dϕ

∫ R(ϕ)

0

dρa2 sin2 ϕ

ρ2 + 2aρ cos ϕ + a2sin ϕ

∣∣∣∣∂ua

∂ϕ

∣∣∣∣2

N

= E1(ua) + E2(ua)

For E2(ua),

E2(ua) = (k2 − 1)∫ 2π

0

dσ

∫ π

0

dϕ

∫ R

0

dρa2 sin2 ϕ

ρ2 + 2aρ cosϕ + a2sin ϕ

∣∣∣∣∂ua

∂ϕ

∣∣∣∣2

N

= (k2 − 1)∫ 2π

0

dσ

∫ π

0

dϕ

∫ R

0

dρ

a2 sin2 ϕ sin ϕ

ρ2 + 2aρ cos ϕ + a2

∣∣∣∣∂ψ(ϕ)

∂ϕ

∣∣∣∣2∣∣∣∣

∂h

∂ψ

∣∣∣∣2

N

= (k2 − 1)∫ 2π

0

dσ

∫ π

0

dϕ a sin2 ϕ

∣∣∣∣∂ψ

∂ϕ

∣∣∣∣2∣∣∣∣

∂h

∂ψ

∣∣∣∣2

N

tan−1

(R + a cos ϕ

a sin ϕ

)

− tan−1

(cos ϕ

sinϕ

).

Now, we change the parameter of the above integral by ψ.

62 HEAYONG SHIN

Since, R(ϕ) sin ϕ = sin ψ and R(ϕ) cos ϕ = cos ψ − a,

R2 = 1− 2a cosψ + a2 ,∂ψ

∂ϕ=

1− 2a cos ψ + a2

1− a cos ψ.

Hence,

E2(ua) = (k2 − 1)∫ 2π

0

dσ

∫ π

0

dψa sin2 ψ

(1− a cos ψ)

∣∣∣∣∂h

∂ψ

∣∣∣∣2

N

·

tan−1

(1− a cos ψ

a sin ψ

)− tan−1

(cos ψ − a

sin ψ

).

Therefore, E2(u0) = 0, and unless h is a constant,

d

da

∣∣∣∣a=0+

E2(ua) = (k2 − 1)∫ 2π

0

dσ

∫ π

0

dψ

∣∣∣∣∂h

∂ψ

∣∣∣∣2

N

sin2 ψ

(π

2− tan−1

(cos ψ

sin ψ

))

< 0

By the same computation,

E1(ua) =∫ 2π

0

dσ

∫ π

0

dψ

∣∣∣∣∂h

∂ψ

∣∣∣∣2

N

sinψ

((1− a cos ψ) +

1− 2a cos ψ + a2

1− a cosψ

)

So, E1(u0) = E(h) and by taking z-axis in opposite direction if necessary we mayassume that

d

da

∣∣∣∣a=0

E1(ua) =∫ 2π

0

dϕ

∫ π

0

dψ

∣∣∣∣∂h

∂ψ

∣∣∣∣2

N

sin ψ(−2 cos ψ) ≤ 0 .

Therefore, for sufficiently small a , we have E(ua) = E1(ua) + E2(ua) < E(h). Thisproves that h is not an energy minimizer. ¤

Theorem 2. Let S be the Euclidean sphere with radius k ≥ 1, and N be aRiemannian manifold diffeomorphic to a sphere. Considering S = (S2, k2ds2) andN = (S2, gds2) where S2 is the unit sphere and g2ds2 is the metric on N conformalto that of S2, the map u(r, x) = x : KS → N is an energy minimizer if g satisfies

∫

s∈S2(s · E) g2(s) dAS2(s) = 0


for any unit vector E in Rn where s · E is the inner product in R3.

proof. As in Theorem 1, we may assume that KS = (B31 , ψ) where

ψ = dr2 + k2r2dθ2 + k2r2 sin2 θdσ and u(x) =x

|x| ,

Let A be the set of mappings v ∈ W 1,2(B31 , S2) such that v = id on ∂B3

1 in thesense of trace, and

A = v ∈ W 1,2(B1, S1)| v is smooth except finite points, v|∂B1 = id .

Then, A is a dense subset of A. [3] So we only need to show that E(u) ≤ E(v) for allv ∈ A. For any v ∈ A, let S = p1, p2, . . . , pk, n1, n2, . . . , nk−1 be the singularitiesof v where the singular point of positive (negative) degree d are listed |d| times inp1, · · · , pk(n1, · · · , nk−1). Then, for any s ∈ S2, v−1(s) contains a union of curvesjoining ni to pσ(i) and s to pσ(k) for some permutation σ of 1, 2, . . . , k. Then,the 1-dimensional Hausdorff measure H1(v−1(s)) of the set v−1(s) ⊂ KS is

H1(v−1(s)) ≥ Σk−1i=1 d(ni, pσ(i)) + d(s, pσ(k)).

Therefore,

H1(v−1(s)) ≥ minσ

k−1∑

i=1

d(ni, pσ(i)) + d(s, pσ(k))

.

where the distance d is the distance in KS .¿From the coarea formula,

∫

KS

|∇v|2NdV ≥ 2∫

KS\S∪0J(v)dV = 2

∫

N

H1(v−1(s))dAN (s) .

where J(v) is the absolute value of the determinant of dv restricted to the spaceorthogonal to v−1(s) and dV, dA are volume elements of KS , N , respectively.Hence,

∫

KS

|∇v|2NdV ≥ 2∫

N

minσ

k−1∑

i=1


dAN

= 2∫

S2min

σ

k−1∑

i=1


dµ(s) .

64 HEAYONG SHIN

where dµ = dAN is a positive measure on the unit sphere S2 in R3.Now we will show that, for some X0 ∈ B3

1 ,

∫

S2min

σ

k−1∑

i=1


dµ(s) ≥ minx∈B3

∫

S2d(s,X0)dµ(s) .

By approximation, we may assume that µ =∑q

i=1 αiδbiwhere αi ≥ 0,

∑αi = 1,

and δbiis the Dirac measure at bi ∈ S2. Then, the left side of the above inequality

becomesq∑

l=1

αl

k−1∑

i=1

d(ni, pσl(i)) + αld(bl, pσl(k))

for some permutations σl of 1, 2, . . . , n. Using the Birkhoff’s Theorem inductively(in the exactly same way as in [2] Lemma 7.7 or [11] Lemma 5), we can show that theabove summation is bigger than or equal to

minX0∈B3

q∑

l=1

αld(X0, bl) .

for some X0 ∈ N . Therefore,

E(v) ≥ 2 minX0∈B3

∫

S2d(s,X0)dµ(s) .

Now we consider∫

S2 d(s,X0)dµ(s) =∫

S2 d(s,X0)g2(s)dAS2(S) for some X0 ∈ B3.By rotation, we may assume that X0 = (0, 0, l), −1 ≤ l ≤ 1, then from the construc-tion of metric d of KS , we have

d(s,X0) =√

sin2(kθ) + (cos kθ − l)2 , if θ ≤ π/k

and d(s,X0) = 1 + l if θ ≥ π/k , where θ is the polar angle of s. Since k ≥ 1,d(s,X0) ≥ |s−X0| where | ∗ | is the standard norm in R3. Therefore,

∫

KS

|∇v|2dV ≥ 2 minX0∈B3

∫

S2d(s,X0)g2(s)dAS2 ≥ 2

∫

S2|s−X0|g2(s)dAS2

and from the condition that∫

S2

(s · X0

|X0|)

g2(s)dAS2(s) = 0 we have

2∫

S2|s−X0|g2(s)dAS2(s) ≥ 2

∫

S2g2(s)dAS2(s) = E(u) .

This implies the minimizing property of the map u . ¤


By the above theorems, we may have some informations about the existence ofminimizing tangent map at an isolated singularity (peak) of the domain when thetangent cone at the peak is a cone over a standard sphere. When the tangent cone is acone over a sphere of radius smaller than 1, there doesn’t exist nonconstant minimizingtangent map at the peak. On the contrary, where the tangent cone is a cone over asphere of radius ≥ 1, we may have singularity of minimizing harmonic map at thepeak. When the tangent cone is a cone over a general topological 2-sphere, it is notknown yet when a nonconstant minimizing tangent map can exist at such a point. Theanswer to the above question would give crucial information about the regularity ofminimizing harmonic maps from singular spaces such as Alexandrove spaces.

References

[1] M. Avellaneda and F.-H. Lin, Fonctions quasi affines et minimisation deR |∇u|p, C. R. Acad.

Sci. Paris 306, Serie I (1988), 355-358.[2] H. Brezis, J.-M. Coron and E. H. Lieb, Harmonic maps with defects, Comm. Math. Phys. 107

(1986), 649-705.[3] J. M. Coron and R. Gulliver, Minimizing p-harmonic maps into spheres, J. reine angew. Math.

401 (1989), 82-100.[4] J. Eells and L. Lemaire, A report on harmonic maps, Bull. London Math. Soc. 10 (1978), 1-68.[5] R. Gulliver and B. White, The rate of convergence of a harmonic map at a singular point, Math.

Ann. 283 (1989), 539-549.[6] W. Jager and H. Kaul, Rotationally symmetric harmonic maps from a ball into a sphere and the

regularity problem for weak solution of elliptic system, J. reine angew. Math. 343 (1983), 146-161.[7] J.Jost, Equilidrium maps between metric spaces.[8] N.Korevaar and R.Scheon, Sobolev spaces and harmonic maps for metric space targets, Comm.

Anal. Geom. 1 (1993), 561-659.[9] F.-H. Lin, A remark on the map x

|x| , C. R. Acad. Sci. Paris 305 (1987), 529-532.

[10] Y. Shi, A partial regularity result of harmonic maps from manifolds with bounded measurableRiemannian metrics, Comm. Anal. Geom. 4 (1996), 121-128.

[11] H. Shin, Degree 1 singularities of energy minimizing maps to a bumpy sphere, Math. Ann. 293(1992), 509-521.

[12] H. Shin, Nonexistance of singularities with symmetry for minimizing harmonic maps, preprints.[13] R. Schoen, and K. Uhlenbeck, A regularity theory for harmonic maps, J. Diff. Geom. 17 (1982),

307-335.

Department of MathematicsChung-Ang University, Seoul, Korea [email protected]

J. KSIAM Vol.6, No.1, 67-80, 2002

SYMMETRIC DUALITY FOR A CLASS OF NONDIFFERENTIABLE

VARIATIONAL PROBLEMS WITH INVEXITY

WON JUNG LEE

Abstract. We formulate a pair of nondifferentiable symmetric dual variational problems with a

square root term. Under invexity assumptions, we establish weak, strong, converse and self duality theorems

for our variational problems by using the generalized Schwarz inequality. Also, we give the static case of our

nondifferentiable symmetric duality results.

1. Introduction

Symmetric duality in nonlinear programming was introduced in Dorn [4] by defininga symmetric dual program for quadratic proqrams. Subsequently Dantzig, Eisenbergand Cottle [3] first formulated a pair of symmetric dual nonlinear programs in whichthe dual of the dual equals the prime and established weak and strong duality forthese problems concerning convex and concave functions. Later on, Mond and Hanson[9] extended the symmetric duality results to variational problems, giving continuousanalogues of the results of [3]. Since the invexity conditions on functions were firstdefined by Hanson [5] as a generalization of convexity ones, many authors ([6], [10],[13]) have extended the concepts of invexity and generalized invexity to continuousfunctions. Smart and Mond [13] extended the symmetric duality results to variationalproblems by using the continuous version of invexity.

On the other hand, Mond [8] and Mehndiratta [7] gave symmetric dual theoremsfor certain nondifferentiable programs which involve square roots of quadratic forms inthe objective functions ([1], [2], [11]).

Mond and Smart [11] extended the duality theorems for a class of static nondif-ferentiable problems with Wolfe type and Mond-Weir type duals under invexity, andfurther extended these for the continuous analogues.

2000 Mathematics Subject Classification: Primary 90C26, secondary 90C46.

Key words and phrases: Symmetric duality, nondifferentiable, variational problems, invexity.

Typeset by AMS-TEX

67

68 WON JUNG LEE

We are motivated to consider symmetric duality for nondifferentiable variationalproblems involving weaker convexity assumptions [1].

In this paper, we formulate a pair of nondifferentiable symmetric dual variationalproblems with a square root term. Under invexity assumptions, we establish weak,strong, converse and self duality theorems for our variational problems by using thegeneralized Schwarz inequality. Also, we give the static case of our nondifferentiablesymmetric duality results.

2. Notations and Statement of the Problems

The following conventions for vectors x = (x1, x2, · · · , xn)T and y = (y1, y2, · · · , yn)T

in Rn will be used:

x < y ⇔ xi < yi, i = 1, 2, · · · , n;

x 5 y ⇔ xi 5 yi, i = 1, 2, · · · , n;

x ≤ y ⇔ xi 5 yi, i = 1, 2, · · · , n but x 6= y;

x y is the negation of x ≤ y.

Let [a, b] be a real interval and f : [a, b] × Rn × Rn × Rm × Rm → R. Considerthe real scalar function f(t, x, x′, y, y′), where t ∈ [a, b], x and y are functions of twith x(t) ∈ Rn and y(t) ∈ Rm, and x′ and y′ denote the derivatives of x and y,respectively, with respect to t. Assume that f has continuous fourth-order partialderivatives with respect to x, x′, y and y′. fx and fx′ denote the gradient vectors off(t, x, x′, y, y′) at (t, x, x′, y, y′) with respect to x and x′, respectively. Similarly fy andfy′ denote the gradient vectors of f(t, x, x′, y, y′) at (t, x, x′, y, y′) with respect to y andy′, respectively. Subsequently fyy, fy′y′ , fy′y, fyx, fyx′ , fy′x and fy′x′ will denote the(m×m), (m×m), (m×m), (n×m), (n×m), (n×m) and (n×m) matrices of secondorder partial derivatives, respectively. Similarly, fxx, fx′x′ , fx′x, fxy, fxy′ , fx′y andfx′y′ will denote the (n× n), (n× n), (n× n), (m× n), (m× n), (m× n) and (m× n)matrices of second order partial derivatives, respectively.

Let C([a, b], Rn) be the space of piecewise smooth functions x : [a, b] → Rn withnorm given by ‖x‖ = ‖x‖∞ + ‖Dx‖∞, where D is the differentiation operator definedby

u = Dx ⇔ x(t) = α +∫ t

a

u(s)ds,

where α is a given boundary value. So D = ddt except at discontinuities.

We consider the problem of finding functions x : [a, b] → Rn and y : [a, b] → Rm,with (x′(t), y′(t)) piecewise smooth on [a, b], to solve the following pair of symmetricdual variational problems with a sequare root term:

NONDIFFERENTIABLE VARIATIONAL PROBLEMS 69

Primal (P)

Minimize F (x, y, w) =∫ b

a

f(t, x, x′, y, y′)

− y(t)T

[fy(t, x, x′, y, y′)− d

dtfy′(t, x, x′, y, y′)

]+

(x(t)T B(t)x(t)

) 12

dt

subject to x(a) = x0, x(b) = x1, y(a) = y0, y(b) = y1,[fy(t, x, x′, y, y′)− d


]− C(t)w(t) 5 0, (1)

w(t)T C(t)w(t) 5 1, (2)

Dual (D)

Maximize G(u, v, z) =∫ b

a

f(t, u, u′, v, v′)

− u(t)T

[fx(t, u, u′, v, v′)− d

dtfx′(t, u, u′, v, v′)

]− (v(t)T C(t)v(t))

12

dt

subject to u(a) = x0, u(b) = x1, v(a) = y0, v(b) = y1,[fx(t, u, u′, v, v′)− d


]+ B(t)z(t) = 0, (3)

z(t)T B(t)z(t) 5 1, (4)

where (1) and (3) may fail to hold at corners of (x′(t), y′(t)) and (u′(t), v′(t)) respec-tively, but must be satisfied for unique right-hand and left-hand limits.

3. Symmetric Duality

We consider the following nondifferentiable variational problem:

Minimize∫ b

a

f(t, x(t), x′(t)) + (x(t)T B(t)x(t))

12

dt(P0)

subject to x(a) = α, x(b) = β,

g(t, x(t), x′(t)) 5 0, t ∈ [a, b],

where f : [a, b] × Rn × Rn → R, g : [a, b] × Rn × Rn → Rm are assumed to becontinuously differentiable functions,B(t) is an n×n positive semidefinite (symmetric)matrix, with B(·) continuous on [a, b].

Let K be the set of feasible solutions for (P0) given by

K = x ∈ C ([a, b], Rn) | x(a) = α, x(b) = β, g (t, x(t), x′(t)) 5 0, t ∈ [a, b].

Now we define the invexity as follows.

70 WON JUNG LEE

Definition 3.1. The functional∫ b

a

f(t, ·, ·, v, v′) + (·)T B(t)z(t)

dt is invex in x and

x′ if for each y : [a, b] → Rm, with y′ piecewise smooth, there exists a function η :[a, b]×Rn ×Rn ×Rn ×Rn → Rn such that

∫ b

a

(f(t, x, x′, v, v′) + x(t)T B(t)z(t)

)− (f(t, u, u′, v, v′) + u(t)T B(t)z(t)

)dt

=∫ b

a

η(t, x, x′, u, u′)T

[fx(t, u, u′, v, v′)− d

dtfx′(t, u, u′, v, v′) + B(t)z(t)

]dt

for all x : [a, b] → Rn, u : [a, b] → Rn with (x′(t), u′(t)) piecewise smooth on [a, b].

Definition 3.2. The functional − ∫ b

a

f(t, x, x′, ·, ·)− (·)T C(t)w(t)

dt is invex in y

and y′ if for each x : [a, b] → Rn, with x′ piecewise smooth, there exists a functionξ : [a, b]×Rm ×Rm ×Rm ×Rm → Rm such that

−∫ b

a

(f(t, x, x′, v, v′)− v(t)T C(t)w(t)

)− (f(t, x, x′, y, y′)− y(t)T C(t)w(t)

) dt

= −∫ b

a

ξ(t, v, v′, y, y′)T

[fy(t, x, x′, y, y′)− d

dtfy′(t, x, x′, y, y′)− C(t)w(t)

]dt

for all v : [a, b] → Rm, y : [a, b] → Rm with (v′(t), y′(t)) piecewise smooth on [a, b].

In the sequel, we will write η(x, u) for η(t, x, x′, u, u′) and ξ(v, y) for ξ(t, v, v′, y, y′).

Theorem 3.1 (Weak Duality). Let (x, y, w) be feasible for (P) and (u, v, z) be fea-sible for (D). Assume that for all t ∈ [a, b],

∫ b

af(t, ·, ·, y, y′)+ (·)T B(t)z(t)

dt is

invex in x and x′, and − ∫ b

a

f(t, x, x′, ·, ·)− (·)T C(t)w(t)

dt is invex in y and y′, with

η(x, u) + u(t) = 0 and ξ(v, y) + y(t) = 0 (except perhaps at corners of (x′(t), y′(t)) or(u′(t), v′(t))). Then we have

inf(P ) = sup(D).

Proof. Let (x, y, w) be feasible for (P) and (u, v, z) be feasible for (D). Then

∫ b

a

f(t, x, x′, y, y′)− y(t)T

[fy(t, x, x′, y, y′)− d


]

+(x(t)T B(t)x(t)

) 12

dt


−∫ b

a

f(t, u, u′, v, v′)− u(t)T

[fx(t, u, u′, v, v′)− d


]

− (v(t)T C(t)v(t)

) 12

dt

=∫ b

a

f(t, x, x′, y, y′)− y(t)T

[fy(t, x, x′, y, y′)− d


]

+(x(t)T B(t)x(t)

) 12

(z(t)T B(t)z(t)

) 12

dt

−∫ b

a

f(t, u, u′, v, v′)− u(t)T

[fx(t, u, u′, v, v′)− d


]

− (v(t)T C(t)v(t)

) 12

(w(t)T C(t)w(t)

) 12

dt

(From (4) and (2))

=∫ b

a

f(t, x, x′, y, y′)− y(t)T

[fy(t, x, x′, y, y′)− d


]

+x(t)T B(t)x(t)

dt

−∫ b

a

f(t, u, u′, v, v′)− u(t)T

[fx(t, u, u′, v, v′)− d


]

−v(t)T C(t)w(t)

dt

(Using the generalized Schwarz inequality)

=∫ b

a

[η(x, u) + u(t)]T

[fx(t, u, u′, v, v′)− d

dtfx′(t, u, u′, v, v′) + B(t)z(t)

]

− [ξ(v, y) + y(t)]T[fy(t, x, x′, y, y′)− d

dtfy′(t, x, x′, y, y′)− C(t)w(t)

]dt

(since∫ b

a

f(t, ·, ·, v, v′) + (·)T B(t)z(t)

dt is invex in x and x′,

and −∫ b

a

f(t, x, x′, ·, ·) + (·)T C(t)w(t)

dt is invex in y and y′)

= 0

(from (3) and (1) together with η(x, u) + u(t) = 0 and ξ(v, y) + y(t) = 0) .

Hence the result of Theorem 3.1 holds.

Before presenting the Wolfe type symmetric dual to (P), we state the following FritzJohn necessary optimality conditions [Lemma 3.1] for the nondifferentiable problem

72 WON JUNG LEE

(P0) and it can be easily derived by invoking the result of Valentine [14] or those ofZhang and Mond [15].

Lemma 3.1. If x∗ is an optimal solution of (P0), then there exist γ ∈ R, piecewisesmooth ρ : [a, b] → Rm and z : [a, b] → Rn such that for t ∈ [a, b]

γ(fx(t, x∗, x∗

′) + B(t)z(t)

)= ρT gx(t, x∗, x∗

′),

ρT g(t, x∗, x∗′) = 0,

z(t)T B(t)z(t) 5 1,(x∗(t)T B(t)x∗(t)

) 12 = x∗(t)T B(t)z(t),

(γ, ρ) = 0,

(γ, ρ) 6= 0.

The following the generalized Schwarz inequality [12, p.262] is required in the sequel:

xT Bw 5 (xT Bx)12 (wT Bw)

12 ,

for all x, w ∈ Rn and equality holds if Bx = γBw for some γ = 0.

In the following theorems and proofs, f∗ represents f(t, x∗, x∗′, y∗, y∗′) and partialderivatives are similarly denoted.

Theorem 3.2 (Strong Duality). Let (x∗, y∗, w∗) be an optimal solution for (P).Suppose that the system

[p(t)T

(f∗yy −

d

dtf∗y′y

)+

d

dt

(p(t)T d

dtf∗y′y′

)

+d2

dt2(− p(t)T f∗y′y′

)]p(t) = 0 (5)

only has the solution p(t) = 0 for all t ∈ [a, b]. Assume that for all t ∈ [a, b]

∫ b

a

f(t, ·, ·, y, y′) + (·)T B(t)z(t)

dt is invex in x and x′,

and

−∫ b

a

f(t, x, x′, ·, ·) + (·)T C(t)w(t)

dt is invex in y and y′


(except perhaps at corners of (x′(t), y′(t)) or (u′(t), v′(t))). Then (x∗, y∗, w∗) is anoptimal solution for (D), and the objective values of (P) and (D) are equal.

Proof. since (x∗, y∗, w∗) is an optimal solution of the primal problem (P), by Lemma3.1, there exist α ∈ R, β : [a, b] → Rm and γ ∈ R such that

H∗ ≡α

f∗ − y∗T [f∗y −

d

dtf∗y′ ] + (x∗T Bx∗)

12

+ βT

(f∗y −

d

dtf∗y′ − Cw∗

)+ γ(w∗T Cw∗ − 1)

satisfies

H∗x −

d

dtH∗

x′ +d2

dt2H∗

x′′ = 0, (6)

H∗y −

d

dtH∗

y′ +d2

dt2H∗

y′′ = 0, (7)

−βT C + 2γCw∗ = 0, (8)

βT (f∗y −d

dtf∗y′ − Cw∗) = 0, (9)

γ(w∗T Cw∗ − 1) = 0, (10)

z∗T Bz∗ 5 1, (11)(x∗T Bx∗

) 12 = x∗T Bz∗, (12)(α, β, γ) = 0, (13)(α, β, γ) 6= 0, (14)

throughout [a, b] (except at corners of (x∗′(t), y∗′(t)) where (6) and (7) hold for uniqueright-and left-hand limits). α, β(t) and γ cannot be simultaneously zero at any t ∈ [a, b],and β is continuous except perhaps at corners of (x∗′(t), y∗′(t)).

Equation (6) now becomes

α(f∗x −d

dtf∗x′) + (β − αy∗)T

(f∗yx −

d

dtf∗y′x

)+ αBx∗

(x∗T Bx∗

)− 12

− d

dt

(β − αy∗)T

(f∗yx′ − f∗y′x −

d

dtf∗y′x′

)+

d2

dt2

− (β − αy∗)T

f∗y′x′

= 0. (15)

Equation (7) gives

74 WON JUNG LEE

(β − αy∗)T

(f∗yy −

d

dtf∗y′y

)

+d

dt

(β − αy∗)T d

dtf∗y′y′

+

d2

dt2

− (β − αy∗)T

f∗y′y′

= 0. (16)

Multiplying (16) by β − αy∗ yields[(β − αy∗)T (f∗yy −

d

dtf∗y′y) +

d

dt

(β − αy∗)T d

dtf∗y′y′

+d2

dt2− (β − αy∗)T f∗y′y′

](β − αy∗) = 0.

Thus by the assumption (5),

β = αy∗. (17)

Now multiplying (8) by w∗(t)T = w∗T , we get

w∗T Cβ = 2γw∗T Cw∗. (18)

This gives α 6= 0, since if α = 0, then by (17), (18), and (10), β = γ = 0, everywhere,contradicting the necessary condition (14). Hence α > 0 from (13). Now equation (18)with the aid of (17) and the fact α > 0 gives,

y∗T C =(2γ

α

)Cw∗. (19)

Thus

y∗Cw∗ =(y∗T Cy∗

) 12

(w∗T Cw∗

) 12 . (20)

If γ > 0, then (10) gives w∗T Cw∗ = 1, and so (20) yields

y∗Cw∗ =(y∗T Cy∗

) 12 .

If γ = 0, then (19) gives y∗C = 0. So we will get y∗T Cw∗ =(y∗T Cy∗

) 12 . Therefore in

either case, we obtain

y∗T Cw∗ =(y∗T Cy∗

) 12 . (21)

Equation (15) with (17) and (12) together with α > 0 now becomes


fx∗ − d

dtf∗x′ + Bz∗ = 0. (22)

By (22) and (11), (x∗, y∗, z∗) is feasible for (D).Multiplying (15) by x∗(t), and using (17) and α > 0 in succession, we get

−x∗T(

fx∗ − d

dtf∗x′

)− x∗T Bz∗ = 0. (23)

Hence

∫ b

a

f(t, x∗, x∗′, y∗, y∗′)

−y∗(t)T

[fy(t, x∗, x∗′, y∗, y∗′)− d

dtfy′(t, x∗, x∗

′, y∗, y∗′)]

+(x∗(t)T B(t)x∗(t)

) 12

dt

=∫ b

a

f(t, x∗, x∗′, y∗, y∗′)− y∗(t)T C(t)w∗(t) + x∗(t)T B(t)z∗(t)

dt

(using (9) and (17) with α > 0, and then (12))

=∫ b

a

f(t, x∗, x∗′, y∗, y∗′)− (

y∗T Cy∗) 1

2 − x∗T(

f∗x −d

dtf∗x′

)dt

(using (21) and (23))

=∫ b

a

f(t, x∗, x∗′, y∗, y∗′)

−x∗(t)T

[fx(t, x∗, x∗′, y∗, y∗′)− d

dtfx′(t, x∗, x∗

′, y∗, y∗′)]− (

y∗(t)T C(t)y∗(t)) 1

2

dt.

If the invexity conditions of Theorem 3.1 are satisfied, then, by weak duality,(x∗, y∗, z∗) is optimal for (D), and the extreme values of (P) and (D) are equal.

A converse duality theorem may be stated; the proof would be analogous to that ofTheorem 3.2.

Theorem 3.3 (Converse Duality). Let (x∗, y∗, z∗) be an optimal solution for (D).Suppose that the system

[p(t)T

(f∗xx −

d

dtf∗x′x

)+

d

dt

(p(t)T d

dtf∗x′x′

)

76 WON JUNG LEE

+d2

dt2(− p(t)T f∗x′x′

)]p(t) = 0

only has the solution p(t) = 0 for all t ∈ [a, b]. Assume that for all t ∈ [a, b]

∫ b

a

[f(t, ·, ·, y, y′) + (·)T B(t)z(t)

]dt is invex in x and x′,

and

−∫ b

a

[f(t, x, x′, ·, ·)− (·)T C(t)w(t)

]dt is invex in y and y′

(except perhaps at corners of (x′(t), y′(t)) or (u′(t), v′(t))). Then (x∗, y∗, w∗) is anoptimal solution for (P), and the objective values of (P) and (D) are equal.

Now we establish the self duality of (P).

Assume that m = n, C = B, z = w and f(t, x, x′, y, y′) = −f(t, y, y′, x, x′) (i.e., f isskew-symmetric) for all (x(t), y(t)), t ∈ [a, b] such that (x′(t), y′(t)) is piecewise smoothon [a, b] and that x0 = y0, x1 = y1.

It follows that (D) may be rewritten as a minimization problem:(D′)

Minimize∫ b

a

f(t, y, y′, x, x′)

−x(t)T

[fy(t, y, y′, x, x′)− d

dtfy′(t, y, y′, x, x′)

]+ (y(t)T B(t)y(t))

12

dt

subject to x(a) = x0, x(b) = x1, y(a) = x0, y(b) = x1,

fy(t, y, y′, x, x′)− d

dtfy′(t, y, y′, x, x′)− C(t)w(t) 5 0,

w(t)T C(t)w(t) 5 1.

(D′) is formally identical to (P); that is, the objective and constraint functions andinitial conditions of (P) and (D′) are identical. This problem is said to be self dual.

It is easily seen that whenever (x, y, z) is feasible for (P), then (y, x, z) is feasible for(D), and vice versa.


Theorem 3.4 (Self Duality). Assume that (P) is self dual and that the invexityconditions of Theorem 3.1 are satisfied. If (x∗, y∗, z∗) is an optimal solution for (P),and the system (5) only has a zero solution, then (y∗, x∗, z∗) is an optimal solution forboth (P) and (D), and the common optimal value is 0.

Proof. By Theorem 3.2, (x∗, y∗, z∗) is an optimal solution for (D), and the optimalvalues of (P) and (D) are equal to F (x∗, y∗, z∗).

From self duality, (y∗, x∗, z∗) is feasible for both (P) and (D), so Theorems 3.1 and3.2 give optimality in both problems, and thus objective values of F (y∗, x∗, z∗).

Now it remains to show that F (x∗, y∗, z∗) = 0.Since f is skew-symmetric, we have

∫ b

a

f(t, y∗, y∗′, x∗, x∗′)dt = −∫ b

a

f(t, x∗, x∗′, y∗, y∗′)dt.

So

F (x∗, y∗, z∗)

=∫ b

a

f(t, x∗, x∗′, y∗, y∗′)− y∗(t)T

[fy(t, x∗, x∗′, y∗, y∗′)− d

dtfy′(t, x∗, x∗

′, y∗, y∗′)]

+(x∗(t)T B(t)x∗(t)

) 12

dt

=∫ b

a

f(t, x∗, x∗′, y∗, y∗′)− y(t)∗T

C(t)w∗(t) +(x∗(t)T B(t)x∗(t)

) 12

dt

=∫ b

a

f(t, x∗, x∗′, y∗, y∗′)− (y(t)∗T

C(t)y∗(t))12 +

(x∗(t)T B(t)x∗(t)

) 12

dt

andF (x∗, y∗, z∗) = F (y∗, x∗, z∗).

Hence

F (x∗, y∗, z∗)

= F (y∗, x∗, z∗)

=∫ b

a

f(t, y∗, y∗′, x∗, x∗′)− (

x∗(t)T B(t)x∗(t)) 1

2 +(y∗(t)T B(t)y∗(t)

) 12

dt

=∫ b

a

−f(t, x∗, x∗′, y∗, y∗′)− (

x∗(t)T B(t)x∗(t)) 1

2 +(y∗(t)T C(t)y∗(t)

) 12

dt

= −∫ b

a

f(t, x∗, x∗′, y∗, y∗′) +

(x∗(t)T B(t)x∗(t)

) 12 − (y∗(t)T C(t)y∗(t))

dt

= −F (x∗, y∗, z∗).

78 WON JUNG LEE

Thus

F (x∗, y∗, z∗) = −F (x∗, y∗, z∗).

Therefore

F (x∗, y∗, z∗) = 0.

4. The Static Case of NondifferentiableSymmetric Duality

In the time dependency of programs (P) and (D) is removed and f is considered tohave domain Rn ×Rm , we obtain the nondifferentiable symmetric dual pair given by

(SP) Minimize f(x, y)− yT fy(x, y) +(xT Bx

) 12

subject to fy(x, y)− Cw 5 0,

wT Cw 5 1,

(SD) Maximize f(u, v)− uT fx(u, v) +(vT Cv

) 12

subject to fx(u, v) + Bz = 0,

zT Bz 5 1.

The following duality theorems can be proved along the lines of Theorems 3.1, 3.2and 3.3.

Theorem 4.1 (Weak Duality). Let (x, y, w) be feasible for (SP) and (u, v, z) befeasible for (SD). Assume that f(·, y)+(·)T Cw is invex in x and −

f(x, ·)− (·)T Cw

is invex in y with η(x, u) + u = 0 and ξ(v, y) + y = 0. Then we have

inf(SP ) = sup(SD).

Theorem 4.2 (Strong Duality). Let (x∗, y∗, w∗) be an optimal solution for (SP).Suppose that f∗yy is positive or negative definite.

If, in addition, the invexity conditions of Theorem 4.1 are satisfied, then (x∗, y∗, z∗)is an optimal solution for (SD), and the objective values of (SP) and (SD) are equal.


Theorem 4.3 (Converse Duality). Let (x∗, y∗, z∗) be an optimal solution for (SD).Suppose that f∗xx is positive or negative definite.

If, in addition, the invexity conditions of Theorem 4.1 are satisfied, then (x∗, y∗, w∗)is an optimal solution for (SP), and the objective values of (SP) and (SD) are equal.

The pair (SP) and (SD) will be self dual when m = n, C = B, z = w and f(x, y) =−f(y, x) (i.e. f is skew-symmetric for all x, y ∈ Rn ).

We state without proof a static version of Theorem 3.4.

Theorem 4.4 (Self Duality). Assume that (SP) is self dual and that the invexityconditions of Theorem 4.1 are satisfied. If (x∗, y∗, z∗) is an optimal solution for (SP),and f∗yy is positive or negative definite, then (y∗, x∗, z∗) is an optimal solution for both(SP) and (SD), and the common optimal value is 0.

References

[1] S. Chandra, B. D. Craven and B. Mond, Generalized concavity and duality with asquare root term, Optimization, 16 (1985), 653-662.

[2] S. Chandra and I. Husain, Symmetric dual nondifferentiable programs, Bull. Austral.Math. Soc., 24 (1981), 259-307.

[3] G. B. Dantzig, E. Eisenberg, and R. W. Cottle, Symmetric dual nonlinear programs,Pacific J. Math., 15 (1965), 809-812.

[4] W. S. Dorn, A symmetric dual theorem for quadratic programs, J. Oper. Res. Soc.Japan, 2 (1960), 93-97.

[5] M. A. Hanson, On sufficiency of the Kuhn-Tucker conditions, J. Math. Anal. Appl.,80 (1981), 545-550.

[6] D. S. Kim, G. M. Lee, J. Y. Park and K. H. Son, Control problems with generalizedinvexity, Mathematica Japonica, 38 (1993), No. 2, 263-269.

[7] S. L. Mehndiratta, Symmetric and self duality in nonlinear programming, Numer.Math., 10 (1967), 103-109.

[8] B. Mond, A class of nondifferentiable mathematical programming problems, J. Math.Anal. Appl., 46 (1974), 167-174.

[9] B. Mond and M. A. Hanson, Symmetric duality for variational problems, J. Math.Anal. Appl., 23 (1968), 161-172.

[10] B. Mond and I. Husain, Sufficient optimality criteria and duality for variationalproblems with generalized invexity, J. Austral. Math. Soc. Ser. B, 31 (1989),108-121.

[11] B. Mond and I. Smart, Duality with invexity for a class of nondifferentiable staticand continuous programming problems, J. Math. Anal. Appl., 141 (1989), 373-388.

[12] F. Riesz and B. Sz-Nagy, Functional Analysis, Ungar, New York, 1955.[13] I. Smart and B. Mond, Symmetric duality with invexity in variational problems, J.

Math. Anal. Appl., 152 (1990), 536-545.

80 WON JUNG LEE

[14] F. A. Valentine, The problem of Lagrange with differential inequalities as added sideconditions, in Contributions to the Calculus of Variations, Univ. of Chicago Press,Chicago, 1937, 407-448.

[15] J. Zhang and B. Mond, Duality for a nondifferentiable programming problem, Bull.Austral. Math. Soc., 55 (1997), 29-44.

Department of Applied MathematicsPukyong National UniversityPusan 608-737Koreae-mail: [email protected]

J. KSIAM Vol.6, No.1, 81-89, 2002

EXISTENCE OF SOLUTIONS OF FUZZY DELAY DIFFERENTIALEQUATIONS WITH NONLOCAL CONDITION

K.BALACHANDRAN AND P.PRAKASH

Abstract. In this paper we prove the existence of solutions of fuzzy delay differentialequations with nonlocal condition. The results are obtained by using the fixed pointprinciples.

1. Introduction

The theory of fuzzy differential equations has been studied by many authors [2-5,9,10] by using the H-differentiability for the fuzzy valued mappings of a real variablewhose values are normal, convex, upper semicontinuous and compactly supported fuzzysets in Rn. Seikkala [8] defined the fuzzy derivative which is generalization of theHukuhara derivative in [6]. The local existence theorems are given in [9], and theexistence theorems under compactness-type conditions are investigated in [10], for theCauchy problem x′ = f(t, x), x(t0) = x0 when the fuzzy valued mapping f satisfies thegeneralized Lipschitz condition. Park et al [5] studied the fuzzy differential equationwith nonlocal condition. Nieto [4] proved an existence theorem for fuzzy differentialequations on the metric space (En, D).

In this paper we prove the existence of solutions of fuzzy delay differential equationswith nonlocal condition of the form

x′(t) = f(t, x(σ1(t)), x(σ2(t)), · · · , x(σn(t))), t ∈ J = [0, a] (1)x(0) − g(t1, t2, · · · , tp, x(·)) = x0,

where σi : J → J, i = 1, 2, · · · , n are continuous functions and f : J × En2 → En islevelwise continuous function and σi(t) ≤ t for all t ∈ J, g : Jp × En → En satisfiesthe Lipschitz condition. The symbol g(t1, t2, · · · tp, x(·)) is used in the sense that in theplace of ′·′, we can substitute only elements of the set t1, t2, · · · , tp. For example,g(t1, t2, · · · , tp, x(·)) can be defined by the formula

g(t1, t2, · · · , tp, x(·)) = c1x(t1) + c2x(t2) + · · ·+ cpx(tp),

2000 AMS Subject Classification : 45G10Key words: Fuzzy delay differential equations, Nonlocal condition

81

82 K.BALACHANDRAN AND P.PRAKASH

where ci(i = 1, 2, · · · , p) are given constants.

2. Preliminaries

Let PK(Rn) denote the family of all nonempty, compact, convex subsets of Rn.Addition and scalar multiplication in PK(Rn) are defined as usual. Let A and B betwo nonempty bounded subsets of Rn. The distance between A and B is defined by theHausdorff metric

d(A, B) = max

supa∈A

infb∈B

||a− b||, supb∈B

infa∈A

||a− b||

,

where || · || denote the usual Euclidean norm in Rn. Then it is clear that (PK(Rn), d)becomes a metric space. Let I = [t0, t0 + a] ⊂ R (a > 0) be a compact interval and letEn be the set of all u : Rn → [0, 1] such that u satisfies the following conditions:

: (i) u is normal, that is, there exists an x0 ∈ Rn such that u(x0) = 1,: (ii) u is fuzzy convex, that is, u(λx + (1 − λ)y) ≥ minu(x), u(y), for any

x, y ∈ Rn and 0 ≤ λ ≤ 1,: (iii) u is upper semicontinuous,: (iv) [u]0 = clx ∈ Rn : u(x) > 0 is compact.

If u ∈ En, then u is called a fuzzy number, and En is said to be a fuzzy numberspace. For 0 < α ≤ 1, denote [u]α = x ∈ Rn : u(x) ≥ 0. Then from (i)-(iv), it followsthat the α-level set [u]α ∈ PK(Rn) for all 0 ≤ α ≤ 1.

If g : Rn × Rn → Rn is a function, then using Zadeh’s extension principle we canextend g to En ×En → En by the equation

g(u, v)(z) = supz=g(x,y)

minu(x), v(y).

It is well known that [g(u, v)]α = g([u]α, [v]α) for all u, v ∈ En, 0 ≤ α ≤ 1 and contin-uous function g. Further, we have [u + v]α = [u]α + [v]α, [ku]α = k[u]α, where k ∈ R.Define D : En × En → [0,∞) by the relation D(u, v) = sup

0≤α≤1d([u]α, [v]α), where d is

the Hausdorff metric defined in PK(Rn). Then D is a metric in En.

Further we know that [7]: (i) (En, D) is a complete metric space,: (ii) D(u + w, v + w) = D(u, v) for all u, v, w ∈ En,: (iii) D(λu, λv) = |λ|D(u, v) for all u, v ∈ En and λ ∈ R.

It can be proved that D(u + v, w + z) ≤ D(u, w) + D(v, z) for u, v, w and z ∈ En

Definition 2.1.[2] A mapping F : I → En is strongly measurable if for all α ∈ [0, 1] theset-valued map Fα : I → PK(Rn) defined by Fα(t) = [F (t)]α is Lebesgue measurablewhen PK(Rn) has the topology induced by the Hausdorff metric d.

FUZZY DELAY DIFFERENTIAL EQUATIONS 83

Definition 2.2.[2] A mapping F : I → En is said to be integrably bounded if there isan integrable function h(t) such that ‖x(t)‖ ≤ h(t) for every x(t) ∈ F0(t).

Definition 2.3. The integral of a fuzzy mapping F : I → En is defined levelwiseby [

∫I F (t)dt]α =

∫I Fα(t)dt = The set of all

∫I f(t)dt such that f : I → Rn is a mea-

surable selection for Fα for all α ∈ [0, 1].

Definition 2.4.[1] A strongly measurable and integrably bounded mapping F : I → En

is said to be integrable over I if∫I F (t)dt ∈ En.

Note that if F : I → En is strongly measurable and integrably bounded, then F isintegrable. Further if F : I → En is continuous, then it is integrable.

Proposition 2.1. Let F, G : I → En be integrable and c ∈ I, λ ∈ R. Then

: (i)∫ t0+a

t0F (t)dt =

∫ c

t0F (t)dt +

∫ t0+a

cF (t)dt;

: (ii)∫

I(F (t) + G(t))dt =

∫

IF (t)dt +

∫

IG(t)dt,

: (iii)∫

IλF (t)dt = λ

∫

IF (t)dt,

: (iv) D(F,G) is integrable,

: (v) D

(∫

IF (t)dt,

∫

IG(t)dt

)≤

∫

ID(F (t), G(t))dt.

Definition 2.5 A mapping F : I → En is Hukuhara differentiable at t0 ∈ I if for someh0 > 0 the Hukuhara differences

F (t0 + ∆t)−h F (t0), F (t0)−h F (t0 −∆t)

exist in En for all 0 < ∆t < h0 and there exists an F ′(t0) ∈ En such that

lim∆t→0+

D((F (t0 + ∆t)−h F (t0))/∆t, F ′(t0)) = 0

andlim

∆t→0+D((F (t0)−h F (t0 −∆t)/∆t, F ′(t0)) = 0.

Here F ′(t) is called the Hukuhara derivative of F at t0.

Definition 2.6. A mapping F : I → En is called differentiable at a t0 ∈ I if, forany α ∈ [0, 1], the set-valued mapping Fα(t) = [F (t)]α is Hukuhara differentiable atpoint t0 with DFα(t0) and the family DFα(t0) : α ∈ [0, 1] define a fuzzy numberF (t0) ∈ En.

If F : I → En is differentiable at t0 ∈ I, then we say that F ′(t0) is the fuzzy deriva-tive of F (t) at the point t0.


Theorem 2.1. Let F : I → En be differentiable. Denote Fα(t) = [fα(t), gα(t)].Then fα and gα are differentiable and [F ′(t)]α = [f ′α(t), g′α(t)].

Theorem 2.2. Let F : I → En be differentiable and assume that the derivativeF ′ is integrable over I. Then, for each s ∈ I, we have

F (s) = F (a) +∫ s

aF ′(t)dt.

Definition 2.7. A mapping f : I ×En → En is called levelwise continuous at a point(t0, x0) ∈ I × En provided, for any fixed α ∈ [0, 1] and arbitrary ε > 0, there exists aδ(ε, α) > 0 such that

d([f(t, x)]α, [f(t0, x0)]α) < ε

whenever |t− t0| < δ(ε, α) and d([x]α, [x0]α) < δ(ε, α) for all t ∈ I, x ∈ En.

Corollary 2.1 [2] Suppose that F : I → En is continuous. Then the function

G(t) =∫ t

aF (s)ds, t ∈ I

is differentiable and G′(t) = F (t).Now, if F is continuously differentiable on I, then we have the following mean valuetheorem

D(F (b), F (a)) ≤ (b− a) · supD(F ′(t), 0), t ∈ I.As a consequence, we have that

D(G(b), G(a)) ≤ (b− a) · supD(F (t), 0), t ∈ I.Theorem 2.3. Let X be a compact metric space and Y any metric space. A subsetΩ of the space C(X, Y ) of continuous mappings of X into Y is totally bounded in themetric of uniform convergence if and only if Ω is equicontinuous on X, and Ω(x) =φ(x) : φ ∈ Ω is a totally bounded subset of Y for each x ∈ X.

3. Main Results

Definition 3.1. A mapping x : J → En is a solution to the problem (1) if and onlyif it is levelwise continuous and satisfies the integral equation

x(t) = x0 + g(t1, t2, · · · , tp, x(·)) +∫ t

0f(s, x(σ1(s)), x(σ2(s)), · · · , x(σn(s)))ds (2)

for all t ∈ J .Let Y = ξ ∈ En : H(ξ, x0) ≤ b be the space of continuous functions with

H(ξ, ψ) = sup0≤t≤γ

D(ξ(t), ψ(t)) and b is a positive number.

Theorem 3.1. Assume that:


: (i) The mapping f : J × Y → En is levelwise continuous in t on J and thereexists a constant G0 such that

D(f(t, x1, x2, · · · , xn), f(t, y1, y2, · · · , yn)) ≤ G0

n∑

i=1

D(xi, yi)

: (ii) There exists a constant G1 such that for all x, y ∈ Y and σi : J → J, i =1, 2, · · · , n

D(x(σi(t)), y(σi(t))) ≤ G1D(x(t), y(t))

: (iii) g : Jp× Y → En is a function and there exists a constant G2 > 0 such that

D(g(t1, t2, · · · , tp, x(·)), g(t1, t2, · · · , tp, y(·))) ≤ G2D(x, y).

Then there exists a unique solution x(t) of (1) defined on the interval [0, γ] where

γ = mina, (b−N)/M, (1−G2)/G0G1,M = maxD(f(t, x(σ1(t)), x(σ2(t)), · · · , x(σn(t))), 0)) and

N = D(g(t1, t2, · · · , tp, x(·)), 0), 0 ∈ En.

Proof: Define an operator Φ : Y → Y by

Φx(t) = x0 + g(t1, t2, · · · , tp, x(·)) +∫ t

0f(s, x(σ1(s)), x(σ2(s)), · · · , x(σn(s)))ds.(3)

First, we show that Φ : Y → Y is continuous whenever ξ ∈ Y and that H(Φξ, x0) ≤ b.Since f is levelwise continuous and σ is continuous, we take

M = maxD(f(t, x(σ1(t)), x(σ2(t)), · · · , x(σn(t))), 0)

D(Φξ(t + h), Φξ(t))

= D

(x0 + g(t1, t2, · · · , tp, ξ(·)) +

∫ t+h

0f(s, ξ(σ1(s)), ξ(σ2(s)), · · · , ξ(σn(s)))ds,

x0 + g(t1, t2, · · · , tp, ξ(·)) +∫ t

0f(s, ξ(σ1(s)), ξ(σ2(s)), · · · , ξ(σn(s)))ds

)

≤ D

(∫ t+h

0f(s, ξ(σ1(s)), ξ(σ2(s)), · · · , ξ(σn(s)))ds,

∫ t

0f(s, ξ(σ1(s)), ξ(σ2(s)), · · · , ξ(σn(s)))ds

)

≤∫ t+h

tD(f(s, ξ(σ1(s)), ξ(σ2(s)), · · · , ξ(σn(s))), 0)ds

= hM → 0 as h → 0.


That is, the map Φ is continuous. Now

D(Φξ(t), x0)

= D

(x0 + g(t1, t2, · · · , tp, ξ(·)) +

∫ t

0f(s, ξ(σ1(s)), ξ(σ2(s)), · · · , ξ(σn(s)))ds, x0

)

≤ D(g(t1, t2, · · · , tp, ξ(·)), 0) +∫ t

0D(f(s, ξ(σ1(s)), ξ(σ2(s)), · · · , ξ(σn(s))), 0)ds)

= N + Mt

and so

H(Φξ, x0) = sup0≤t≤γ

D(Φξ(t), x0) ≤ N + Mγ ≤ b.

Thus Φ is a mapping from Y into Y . Since C([0, γ], En) is a complete metric spacewith the metric H, we only show that Y is a closed subset of C([0, γ], En). Let ψnbe a sequence in Y such that ψn → ψ ∈ C([0, γ], En) as n →∞. Then

D(ψ(t), x0) ≤ D(ψ(t), ψn(t)) + D(ψn(t), x0),

that is,

H(ψ, x0) = sup0≤t≤γ

D(ψ(t), x0) ≤ H(ψ, ψn) + H(ψn, x0)

≤ ε + b

for sufficiently large n and arbitrary ε > 0. So ψ ∈ Y . This implies that Y is closedsubset of C([0, γ], En). Therefore Y is a complete metric space.

By using Proposition 2.1 and assumptions (i),(ii) and (iii), we will show that Φ is acontraction mapping. For ξ, ψ ∈ Y ,

D(Φξ(t), Φψ(t))

= D

(x0 + g(t1, t2, · · · , tp, ξ(·)) +

∫ t

0f(s, ξ(σ1(s)), ξ(σ2(s)), · · · , ξ(σn(s)))ds,

x0 + g(t1, t2, · · · , tp, ψ(·)) +∫ t

0f(s, ψ(σ1(s)), ψ(σ2(s)), · · · , ψ(σn(s)))ds

)

≤ D(g(t1, t2, · · · , tp, ξ(·)), g(t1, t2, · · · , tp, ψ(·)))+

∫ t

0D(f(s, ξ(σ1(s)), ξ(σ2(s)), · · · , ξ(σn(s))),

f(s, ψ(σ1(s)), ψ(σ2(s)), · · · , ψ(σn(s))))ds

≤ G2D(ξ(·), ψ(·)) +∫ t

0G0G1D(ξ(s), ψ(s))ds


Then we obtain

H(Φξ,Φψ) ≤ supt∈γ

G2D(ξ(·), ψ(·)) +

∫ t

0G0G1D(ξ(s), ψ(s))ds

≤ G2D(ξ(·), ψ(·)) + γG0G1D(ξ(t), ψ(t))≤ (G2 + G0G1γ)H(ξ, ψ).

Since γG0G1 + G2 < 1, Φ is a contraction map. Therefore Φ has a unique fixed pointx ∈ C([0, γ], En) such that Φx = x, that is,

x(t) = x0 + g(t1, t2, · · · , tp, x(·)) +∫ t

0f(s, x(σ1(s)), x(σ2(s)), · · · , x(σn(s)))ds.

Theorem 3.2. Let f, σ and g be as in Theorem 3.1. Denote by x(t, x0), y(t, y0)the solutions of equation (1) corresponding to x0, y0, respectively. Then there existsconstant q > 0 such that

H(x(·, x0), y(·, y0)) ≤ qD(x0, y0)

for any x0, y0 ∈ En and q = 1/(1−G2 − γG0G1).

Proof: Let x(t, x0), y(t, y0) be solutions of equations (1) corresponding to x0, y0, re-spectively. Then

D(x(t, x0), y(t, y0))

= D

(x0 + g(t1, t2, · · · , tp, x(·)) +

∫ t

0f(s, x(σ1(s)), x(σ2(s)), · · · , x(σn(s)))ds,

y0 + g(t1, t2, · · · , tp, y(·)) +∫ t

0f(s, y(σ1(s)), y(σ2(s)), · · · , y(σn(s)))ds

)

≤ D(x0, y0) + D(g(t1, t2, · · · , tp, x(·)), g(t1, t2, · · · , tp, y(·)))+

∫ t

0D(f(s, x(σ1(s)), x(σ2(s)), · · · , x(σn(s))),

f(s, y(σ1(s)), y(σ2(s)), · · · , y(σn(s))))ds

≤ D(x0, y0) + G2D(x(·), y(·)) +∫ t

0G0G1D(x(s), y(s))ds

Thus, H(x(·, x0), y(·, y0)) ≤ D(x0, y0) + (G2 + γG0G1)H(x(·, x0), y(·, y0)),that is, H(x(·, x0), y(·, y0)) ≤ 1/(1−G2 − γG0G1)D(x0, y0).

This completes the proof of the theorem.

Next we generalize the above theorem for the fuzzy delay differential equation (1)with nonlocal condition.

Theorem 3.3. Suppose that f : J ×En2 → En is level wise continuous and bounded,σi : J → J (i = 1 · · ·n) are continuous and g : Jp × En → En is continuous. Then the


initial value problem (1) possesses at least one solution on the interval J .

Proof: Since f is continuous and bounded and g is a continuous function there existsr ≥ 0 such that

D(f(t, x(σ1(t)), x(σ2(t)), · · · , x(σn(t))), 0) ≤ r, t ∈ J, x ∈ En.

Let B be a bounded set in C(J,En). The set ΦB = Φx : x ∈ B is totally boundedif and only if it is equicontinuous and for every t ∈ J , the set ΦB(t) = Φx(t) : t ∈ Jis a totally bounded subset of En. For t0, t1 ∈ J with t0 ≤ t1, and x ∈ B we have that

D(Φx(t0), Φx(t1)) =

D

(x0 + g(t1, t2, · · · , tp, x(·)) +

∫ t0

0f(s, x(σ1(s)), x(σ2(s)), · · · , x(σn(s)))ds,

x0 + g(t1, t2, · · · , tp, x(·)) +∫ t1

0f(s, x(σ1(s)), x(σ2(s)), · · · , x(σn(s)))ds

)

≤ D

(∫ t0

0f(s, x(σ1(s)), x(σ2(s)), · · · , x(σn(s)))ds,

∫ t1

0f(s, x(σ1(s)), x(σ2(s)), · · · , x(σn(s)))ds

)

≤∫ t1

t0D(f(s, x(σ1(s)), x(σ2(s)), · · · , x(σn(s))), 0)ds

≤ |t1 − t0| · supD(f(t, x(σ1(t)), x(σ2(t)), · · · , x(σn(t))), 0) t ∈ J, ≤ |t1 − t0| · r.

This shows that ΦB is equicontinuous. Now, for t ∈ J fixed. we have

D(Φx(t), Φx(t′)) ≤ |t− t′| · r, for every t′ ∈ J, x ∈ B.

Consequently, the set Φx(t) : x ∈ B is totally bounded in En. By Ascoli’s theoremwe conclude that ΦB is a relatively compact subset of C(J,En). Then Φ is compact,that is, Φ transforms bounded sets into relatively compact sets.

We know that x ∈ C(J,En) is a solution of (1) if and only if x is a fixed point of theoperator Φ defined by (3).

Now, in the metric space (C(J,En),H), consider the ball

B = ξ ∈ C(J,En), H(ξ, 0) ≤ m, m = a · r.


Thus, ΦB ⊂ B. Indeed, for x ∈ C(J,En),

D(Φx(t), Φx(0)) = D (x0 + g(t1, t2, · · · , tp, x(·))+

∫ t

0f(s, x(σ1(s)), x(σ2(s)), · · · , x(σn(s)))ds,

x0 + g(t1, t2, · · · , tp, x(·)))≤

∫ t

0D(f(s, x(σ1(s)), x(σ2(s)), · · · , x(σn(s))), 0)ds

≤ |t| · r ≤ a · r.Therefore, defining 0 : J → En, 0(t) = 0, t ∈ J we have

H(Φx,Φ0) = supD(Φx(t), Φ0(t)) : t ∈ J.Therefore Φ is compact and, in consequence, it has a fixed point x ∈ B. This fixed

point is a solution of the initial value problem (1).

References

[1] R.J.Aumann, Integrals of set-valued functions, Journal of Mathematical Analysis and Applications,12 (1965), 1-12.

[2] O.Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems, 24 (1987), 301-317.[3] P.E.Kloeden, Remarks on Peano-like theorems for fuzzy differential equations, Fuzzy Sets and

Systems, 44 (1991), 161-163.[4] J.J.Nieto, The Cauchy problem for continuous fuzzy differential equations, Fuzzy Sets and Systems,

102 (1999), 259-262.[5] J.Y.Park, H.K.Han and K.H.Son, Fuzzy differential equation with nonlocal condition, Fuzzy Sets

and Systems, 115 (2000), 365-369.[6] M.L.Puri and D.A.Ralescu, Differentials of fuzzy functions, Journal of Mathematical Analysis and

Applications, 91 (1983), 552-558.[7] M.L.Puri and D.A.Ralescu, Fuzzy random variables, Journal of Mathematical Analysis and Ap-

plications, 114 (1986), 409-422.[8] S.Seikkala, On the fuzzy initial value problem, Fuzzy Sets and Systems, 24 (1987), 319-330.[9] C.Wu, S.J.Song and E.Lee, Approximate solutions, existence and uniqueness of the Cauchy prob-

lem of fuzzy differential equations, Journal of Mathematical Analysis and Applications, 202 (1996),629-644.

[10] C.Wu and S.J.Song, Existence theorem to the Cauchy problem of fuzzy differential equations undercompactness type conditions, Journal of Informations Sciences, 108 (1998), 123-134.

Department of MathematicsBharathiar UniversityCoimbatore - 641 046INDIAbalachandran [email protected]

J. KSIAM Vol.6, No.1, 91-107, 2002

SCALING METHODS FOR QUASI-NEWTON METHODS

ISSAM A.R. MOGHRABI

Abstract. This paper presents two new self-scaling variable-metric algorithms. Thefirst is based on a known two-parameter family of rank-two updating formulae, thesecond employs an initial scaling of the estimated inverse Hessian which modifiesthe first self-scaling algorithm. The algorithms are compared with similar publishedalgorithms, notably those due to Oren, Shanno and Phua, Biggs and with BFGS (thebest known quasi-Newton method). The best of these new and published algorithmsare also modified to employ inexact line searches with marginal effect. The newalgorithms are superior, especially as the problem dimension increases.

1. Introduction

The theoretical and practical merits of the quasi-Newton family of methods for un-constrained optimization have been systematically explored since the classic 1963 paperof Fletcher and Powell [1]. In the 1970’s the self-scaling variable-metric algorithms wereintroduced, showing significant improvement in efficiency over earlier methods. In par-ticular, in a series of papers, Oren [2,3], Oren and Luenberger [4], Oren and Spedicato[5], Shanno and Phua [6] developed these algorithms for minimizing an unconstrainednonlinear function f(x), where x ∈ Rn, f ∈ R and the gradient g(x) is available for anygiven x. Variable-metric algorithms begin with an estimate x1 to the minimiser xmin

and a numerical estimate H1 of the inverse Hessian matrix, G−1(x). A sequence ofpoints xk is then defined by:

xk+1 = xk − λkHkgk, (1)

where gk =∇ f(xk ) and k is a scalar chosen so as to reduce the value of f at eachiteration. H is updated by:

Hk+1 = [HkHkykyTk Hk/yT

k Hkyk + φkwkwTk ] + ρksks

Tk /sT

k yk (2)

Key Words: Unconstrained optimization, variable metric, self-scaling

91

92 ISSAM A.R. MOGHRABI

withsk = xk+1− xk, yk = gk+1− gk,

wk = (yTk Hkyk)1/2[sk/sT

k ykHkyk/yTk Hkyk] (3)

where φk and ρk are scalars.

The updating is performed so that:

Hk+1yk = ρksk. (4)

This condition is commonly satisfied with ρk = 1, and is then called ”the Secant equa-tion”. With this restriction on (2) we have the so-called ’Broyden family’ of algorithms[7]. For a quadratic function, G−1 is constant and satisfies sk = G−1yk for any corre-sponding yk and sk; clearly the objective of such updating formulae is that Hk tends (insome sense) to the inverse Hessian, G−1(xk ), for a general function. It is well-knownthat if f is a quadratic, and exact line searches are curried out, then after n iterations,Hn+1 = G−1. However, perhaps the strongest result concerning the convergence of theH-matrices towards G−1 for quadratic functions is that of Oren and Luenberger [4].This is derived from the following result for variable metric methods using exact linesearch:

Theorem (I): If f, a quadratic objective function, is minimized by the sequence xkdefined by the iteration xk+1 = xk - λHkgk, where x1 is a given starting point and H1

is any positive definite matrix, and where k minimizes f(xk- Hkgk), then

f(xk+1)− f(xmin)[(K(Rk)− 1)/(K(Rk) + 1)]2(f(xk)− f(xmin)) (5)

whereRk = G1/2 Hk G1/2 , and K(.) denotes the condition number of any matrix. For

proof see [4].The quantity [(K(Rk)-1)/(K(Rk) + 1)]2, referred to by Oren and Luenberger as the”single-step convergence rate”, is increased or decreased according as K(Rk) is increasedor decreased. Thus, the rate of convergence is maximized if K(Rk) approaches 1.Unfortunately the Broyden family cannot guarantee to reduce K(Rk) at each iteration.Hence Oren and Luenberger introduced a preliminary step to the updating processwhereby Hk is first scaled by a factor k. They are then able to prove:-

Property 1: Provided k ∈[0,1] and provided

SCALING METHODS FOR QUASI-NEWTON METHODS 93

µk = βsTk H−1

k sk/sTk yk + (l − β)sT

k yk/yTk Hkyk (6)

for some [0,1], then K(Rk) is monotonically decreasing and the eigenvalues of Rktend monotonically to unity as k increases. For proof see [2].

lf an estimate of the inverse Hessian is maintained (rather than an estimate of the Hes-sian itself which is sometimes preferred) then there is a strong motivation for choosingβ= 0 in (6). This gives:

µk = sTk yk/yT

k Hkyk (7)

The most successful member of the Broyden family is the BFGS, which corresponds toa choice ρk = 1 in (2). We can now summarize the scaled BFGS algorithm due to Orenand Luenberger:-

Algorithm 1: (Oren)Start with any initial point x1.Step 1: Set k=1 and choose H1 to be any positive definite matrix (usually H1 = I).Step 2: Determine the step-size k to minimize f(xk + λdk) where dk = - Hkgk , and

obtain xk+1 = xk + λk dk.Step 3: Set

Hk+1 = (HkHkykyTk Hk/yT

k Hkyk + wkwTk )µk + sks

Tk /sT

k yk (8)

where wk is defined in (3) and µk in (7).Step 4: If not converged increase k by 1 and return to step 2.

Algorithm I possesses the following properties for a quadratic function:-(a) If k minimizes f(xk - Hkdk) for all k, then the vectors dk are mutually conjugate

(with respect to G) and hence the solution is obtained in at most n iterations.(b) The condition number of the matrix Rk is strictly monotonically decreasing.(c) lf k = 1 for all k, then the algorithm converges ”two-step superlinearly”,i.e.

limitk→∞||xk+1 − xmin||/||xk − xmin|| = 0 (9)

The proofs of these properties can be found in [8].We also note that the value of k in (7) is chosen so that


µkHkyk = sk (10)

Thus the Secant condition is satisfied before and after updating. The principaltheoretical reservation to algorithm I is the failure in the quadratic case of the H-matrices to converge to G−1 in a finite number of steps. Experimental evidence inthis paper supports the view of McCormick and Ritter [9] that the finite terminationproperty for quadratics is desirable when minimizing general functions.

2. Shanno & Phua’s Initial Self-Scaling Method

It would seem a desirable property that the sequence Hk (defined in (2)) should beinvariant under scaling of f(x) by a constant. This is indeed achieved by the self-scalingmethods just described, but as Shanno and Phua [6] point out, it is only necessary toscale H at the first iteration to achieve this. They consider two possible scalings of H1

and they show that in addition to providing the desired invariance, they also greatlyenhance the numerical stability of members of Broyden’s family.

In both of their scalings it is assumed that H1 = I may be used initially to determinex2 where 1 is chosen according to some steplength or linear search criterion to ensuresufficient reduction in the function f . Once x2 has been determined, but before H2 iscalculated, they scale H1 to H+

1 and use H+1 instead of H1 in the formula to define H2.

In their first alternative,

H+1 = λ1H1 (Algorithm I) (11)

and in their second alternative

H+1 = µ1H1 (Algorithm II) (12)

with µ1 given by (7). In either case, the Broyden family (BFGS in particular) be-comes ’self-scaling’ in this invariance sense without the need to apply k at subsequentiterations.Shanno and Phua’s computational results show that for large problems theupdate (12) is preferable.

3. Biggs’ Method

A consequence of the QN-condition is that

yTk Hk+1yk = yT

k sk (13)


Thus yTk Hk+1yk is matched to yT

k sk, a ’first-difference’ estimate of the true directionalcurvature sT

k G(xk+l)sk of f at xk+l. The estimate is exact if f is quadratic.Biggs [10,11] observed that a more accurate estimate of this curvature can be obtained.Four independent pieces of information are available along sk, namely the functionvalues and directional derivatives at xk and xk+l. Thus a cubic model of f along sk canbe constructed, fielding an estimate ρkyT

k sk of the curvature, where

ρk = sTk yk/[4sT

k gk+1 + 2sTk gk − 6(fk+1 − fk)] (Algorithm III) (14)

Biggs therefore suggests the use of this value in the formula(2). Note that for a quadratic function, k = I and hence algorithm III is identical tothe Broyden family.

4. A new self-scaling family of Methods

In this section we consider the Broyden family with φk+1 = 1, i.e. the BFGS update,and it will suffice to omit subscripts k, k+1 and simply use ’*’ to denote new values inthe rest of this paper.The standard BFGS update can be separated into two components, H(1) and H(2), sothat H∗BFGS = H(1) + H(2), where

H(1) = H −HyyT H/yT Hy + wwT (15)

H(2) = ssT /sT y (16)

where w is the vector defined in (3), with subscripts omitted.Oren and Biggs’ modifications of the BFGS formula can then be written as

H∗Oren = µH(1) + H(2) (17)

H∗Biggs = H(1) + ρH(2) (18)

This suggests a more general family of the form

H∗ = αH(1) + γH(2) (19)

which will satisfy


H∗y = γs (20)

This relaxation of the QN-condition is of particular interest in deriving algorithms fornon-quadratic objective functions.

Several choices of α and γ have been investigated but the most effective one (in numeri-cal comparisons), presented here, is readily interpreted in terms of the earlier algorithmsand their properties. We define

H∗new = H(1) + σH(2) (Algorithm IV) (21)

where σ= 1/µ.

The method is similar to Biggs in that only H(2) is scaled, but we now have

yT H∗newy = σsT y = yT Hy (22)

Unfortunately the nice property in Biggs’ method of reverting to the Broyden family inthe quadratic case is lost. However, it can also be viewed in terms of the Oren methodas follows.

H∗new = 1/µH∗

Oren (23)

i.e. H is scaled by µ, the BFGS update is applied and then the resulting matrix isscaled by 1/ µ.Both BFGS and Oren’s update generate identical conjugate gradient search directionsprovided that the function is quadratic and exact line searches are used. To prove thatthe new update IV will also satisfy this property, consider first:Property 1 : Let f be given by

f(x) =12xT Gx + bT x, (24)

where G is symmetric positive definite. Choose an initial approximation H1 = H, whereH is any symmetric positive definite matrix of appropriate order. Obtain H∗new from Hwhere d = - Hg is the search direction and assuming exact line searches then

Hi+1g∗ = Hg∗, for 0 ≤ i < k ≤ n (25)


Proof: Apply induction on i. Let H1 = H, on the above assumptions so that

H1g∗ = Hg∗ (26)

Using the formula (IV) we have

Hi+1g∗ = Hg∗yT Hg∗/yT Hy.Hy + wT g∗w + σsT g∗/sT ys (27)

Now assuming that the property is true for i, namely, that

Hig∗ = Hg∗ (28)

we prove it is true for i + 1 by using the following two standard properties (not provedhere) which are known to be true for quadratic functions and exact line searches:

(a) sTj g∗ = 0 for j=1,2,...,k

(b) gTj Hjg∗ = 0 for j=1,2,...,k

(the orthogonality property satisfies when f is quadratic and exact line search is used).Substituting in (27) we have

Hi+1g∗ = Hg∗ (29)

since yTi Hig∗ = 0 for i < k, and also

wTi g∗ = (yT

i Hiyi)1/2[sTi g∗/sT

i yi(Hiyi)T g∗/yTi Hiyi = 0 (30)

Since the property already holds for i = 1 the normal inductive proof is established forany i. We now prove the following theorem:Theorem (II): Assume that f(x) be the quadratic function defined in (24) and that theline searches are exact: if H is any symmetric positive definite matrix (or appropriateorder) and we define an updating

H∗new = HyyT H/yT Hy + wwT + σssT /yT y (31)

where σ= yT Hy /yT s, then the search direction


d∗new = H∗g∗ (32)

is identical to the conjugate-gradient direction [12] d∗CG defined by

d∗CG = −g, for k=0 and d∗CG = [g∗ + yT g∗/yT d]d, for k=1 (33)

Proof: The update (31) can be written as:

H∗new = H − syT H/sT yHysT /sT y + (σ + yT Hy/sT y)ssT /sT y, (34)

Now

d∗new = Hg∗ + (yT Hg∗/sT y)s + sT g∗.Hy/sT y − 2yT Hy.sTg∗/(yT s)2s (35)

= −Hg∗ + (yT Hg∗/yT s)s (36)

using the property sT g∗ = 0, quoted earlier which holds for exact line searches.

The vector g∗ can be substituted for Hg∗ by using property 2. Therefore

d∗new = g∗ + (yT g∗/yT snew)snew (37)

We also know that d∗BFGS and d∗CG are identical (see Nazareth [13] ), and d∗new isidentical to d∗BFGSwith exact line searches. Hence equation (38) becomes

d∗new = g∗ + (yT g∗/yT dCG)dCG = d∗CG, (38)

and the proof follows.

5. A New Single Scaling : an Initial Scaling of H1

Single scalings are normally applied at the start of the optimization procedure, andare based on the fact that choosing I (or other arbitrary matrix) as an estimate of G−1

is committing the algorithm to a sequence of poor estimates H of G−1. We consider asimple initial scaling which is identical to the new scaling algorithm IV in the case of aquadratic function; for the general function comparative testing shows it can improvethe performance of the new scaling algorithm IV.


We assume here that H1 =I and that H1 is used to determine λ1 such that x2= x1

+ λ1d1 where λ1 minimizes f(x1-λ1H1g1). Shanno’s algorithm II applied to the initialstep only can be written as

H∗Shanno = λH(1) + H(2) (39)

whereλ = l, for k≥ 2.

Multiplying equation (40) by the scalar, σ, yieldsH ∗

Snewh = λσ H(1) + σH(2) (Algorithm V)where

λ σ= 1 for k≥ 2.Theorem (III): Let f be a quadratic function defined as in (24). Assume that H1 =I and exact line search is used; then we have λ1 σ1 = 1 and the algorithms IV and Vare identical.Proof:For the QN-method

d1 = −H1g1 =⇒ s1 = −λ1g1. (40)

Since H1 = I, it follows that sT1 s1 = - λ1gT

1 s1.Now

σ1 = yT1 H1y1/yT

1 s1= yT1 y1/yT

1 s1.since H1 =I,

λ1σ1 = (sT1 s1/gT

1 s1)(yT1 y1/yT

1 s1). (41)

Now substitute (41) in (42) to get:λ1 σ1 = (gT

1 g1/gT1 g1)(yT

1 y1/yT1 g1) = yT

1 y1/yT1 y1= 1.0

(since exact line search is used).

6. A Second Self-Scaling algorithm using Non-Exact Searches

In this section the Brodlie analysis [14], which he applied to the Broyden family(including BFGS), is applied directly to the self-scaling BFGS algorithms, with theone-parameter family of correction formulae. We define the updating

H∗ = (H −HyyT H/yT Hy + wwT ) + ssT /yT y (42)


where w is a vector defined in (3), and y, H, s as defined earlier.Proceeding exactly as in [14] we can show that the new search direction d∗ = -H∗g∗ forsuch algorithms can be expressed as a combination of two vectors p1 and p2. Now thevector H∗g∗ can be written as:

H∗g∗ = [Hg∗yT Hg∗/yT Hy.Hy+(yT Hy)sT g∗/sT yyT Hg∗/yT Hys/sT y −Hy/yT Hy]θ + ρ(sT g∗/sT y)s (43)

If we let

[sT g∗/sT y]− yT Hg∗/yT Hy = τ (44)

and express

Hy = Hg∗ −Hg = Hg∗ + s/λ (45)

then the expression (44) gives:H∗g∗ = [Hg∗ − (yT Hg∗/yT Hy).Hg∗ − τHg∗ − yT Hg∗/(λyT Hy)s

+τ(yT Hy/yT s− 1/λ)s] + (sT g∗/sT y)s (46)

But again using (46) we can show that

[yT Hy/yT s]1/λ = (λyT Hy − sT y)/λsT y = yT Hg∗/sT y (47)

so that, by substitution,H∗g∗ = [(1− yT Hg∗/yT Hy − τ)ϑ)]Hg∗+

(ρsT g∗/sT y − yT Hg∗/λyT Hyϑ + τϑyT Hg∗/sT y)s. (48)

Again, substituting for 1/λ we have

yT Hg∗/λyT Hy = −(yT Hg∗/yT Hy)(yT Hg/sT y) = [yT Hg∗/sT y](1− yT Hg∗/yT Hy)(49)

Hence (49) becomes:


H∗g∗ = θ(1− yT Hg∗/yT Hy − τ)[Hg∗ − (yT Hg∗/sT y)s] + ρ(sT g∗/sT y)s(50)

Let ψ= θ(1 - yT Hg∗/yT Hy - τ)then

H∗g∗ = ψ[Hg∗ − (yT Hg∗/sT y)s] + ρ(sT g∗/sT y)s

= ψ[1− syT /sT y]Hg∗ + (ρsT g∗/sT y)s (51)

Now

d∗ = −H∗g∗ = −ρp1 − ψp2 (52)

where

p1 = (sT g∗/sT y)s, and p2 = [I − syT /sT y]Hg∗. (53)

Equation (54) therefore gives a family of self-scaled BFGS algorithms.In particular we concentrate on the specific new self-scaling algorithm defined by

ρ = σ and θ = 1, (where σ is defined earlier) (Algorithm VI ). (54)

However, Oren’s algorithm, modified to incorporate inexact searches, is defined by

ρ = 1 and θ = µ, (AlgorithmVII) (55)

where is the scalar defined in (7).

Again, for Shanno and Phua’s algorithm we have

ρ = 1 and θ = µ1 or λ1, for k = 1 and ρ = 1, θ = 1, for k ≥ 2. (56)

and finally for our new initial self-scaling algorithm

ρ = σ and θ = λ1σ1 (for k=1) (57)


and all set to one for k≥ 2.In (54) with exact line searches, sT g∗ = 0, and hence we have p1 null. In general,however, p1 is not null and the quantity p1 may be regarded as a correction term forthe inexact search. From the family of self-scaled BFGS algorithms we select thosedefined by equations (56) and (57) for our comparative numerical tests.

7. Computational Results

The comparative tests involve twenty six well-known test functions (given explicitlyin the Appendix). The comparative performances of the algorithms are evaluated byconsidering both the total number of iterations and the number of function evaluations.We define ’iteration’ to mean the step carrying a point x along the direction d = −Hgto a new point x∗, and the number of function calls quoted is that required to reducethe value of f(x) below 1.0−10. The cubic interpolation technique, fully described in[15], is used as the linear search subprogram unless stated otherwise.

Five algorithms are tested and compared: (i) the original BFGS algorithm (ii) Oren’salgorithm I (iii) Shanno’s algorithm II with 1 as an initial sealing factor (iv) algorithmIV and (v) algorithm V were compared over the twenty six test functions. For conve-nience, our numerical results for exact line searches are presented in two tables: table1contains the results for dimensionality n < 10, while table 2 contains the results forproblems in the range 10≤n≤100.

For the first group (table 1) all the established algorithms perform about the sameexcept for poor results with Shanno’s method on Oren and Spedicato’s power function(already reported by Shanno [6] ). The new self-scaling algorithms save about 30% inNOF with negligible additional overheads.For the second group (table 2) all the self-scaling methods perform much better thanBFGS, but the new algorithms again save about 30% in NOF on the best establishedself scaling algorithm.

For inexact line searches our numerical results, presented in table 3, show that the newalgorithm VI beats the modified Oren’s algorithm VII by about 30% in the numberof function calls although the two algorithms perform about the same when comparedby the total number of iterations. For completeness, the best new self-scaling algo-rithm in its exact line search form ( IV) is compared with the latest NAG self-scalingalgorithm OPVM (Biggs’ algorithm), released March 1989, with numerical results inTable 4. Since the line searches are different a measure of total work is included in thecomparison: on this measure the new algorithm serves about 67% on TWORK overall,but this saving is entirely at higher dimensions since Biggs’is more efficient at low di-mensionality. The theoretical observation, e.g. in [6] Shanno and Phua that for someextended test functions (like Rosenbrock’s or Powell’s) the number of iterations and


function evaluations must remain constant with increasing n with perfect arithmeticconfirmed in our numerical computation.

Finally, the computational results presented here show that while self-scaling generallyimproves computational efficiency on small problems (say within < 4) it can some-times decrease efficiency : however, it normally improves the performance of the BFGSmethod on large problems, and the relative improvement increases monotonically withdimensionality n. The effect of changing to inexact line searches is marginal. Oren’s al-gorithm requires slightly more NOF, whereas the new algorithm IV slightly less. Thereare no significant variations with particular test functions. In conclusion, for this par-ticular set of test functions and at the required accuracy levels, the new algorithmsperform very much better than the several well-known methods.


All algorithms terminate when | f - fmin | < 1 x 10−10

TABLE : (1)BFGS OREN SHANNO NEWH SNEWH

TESTFunction N ALG(I) ALG(II) ALG(IV) ALG(V)

NOI(NOF) NOI(NOF) NOI(NOF) NOI(NOF) NOI(NOF)ROSEN 2 22(72) 21(89) 21(72) 21(59) 21(59)CUBE 2 15(60) 22(85) 22(77) 24(65) 21(59)BEALE 2 9(26) 9(33) 9(26) 8(20) 8(20)BOX 2 8(38) 8(43) 8(39) 8(43) 8(44)FREUD 2 6(21) 7(34) 6(25) 6(18) 6(18)RECIPE 3 5(17) 6(21) 5(18) 6(19) 6(19)PIGS 3 11(39) 13(47) 10(36) 11(31) 10(27)HPLICAL 3 19(59) 19(72) 19(61) 18(41) 21(49)POWELL 4 18(76) 29(99) 26(110) 30(66) 30(66)WOOD 4 57(162) 18(66) 19(73) 18(42) 17(40)MIELE 4 29(112) 30(103) 29(120) 31(100) 30(93)TIXON 10 29(73) 18(56) 17(46) 18(40) 18(40)OREN 10 23(122) 12(55) 48(294) 12(64) 12(64)TOTAL NOI 253 212 239 211 208

(NOF) (877) (803) (997) (608) (598)

TABLE : (2)TEST FUNCTION N BFGS OREN

ALG.(I)SHANNOALG.(II)

NEWHALG.(IV)

SNEWHALG.(V)

NOI(NOF) NOI(NOF) NOI(NOF) NOI(NOF) NOI(NOF)NON-DIGN 20 40(102) 22(91) 20(73) 20(61) 20(61)

OREN 30 71(374) 21(85) 125(747) 21(95) 21(93)TRI-DIGN 30 28(57) 28(85) 28(85) 28(57) 28(57)

FULL 40 39(79) 39(123) 39(123) 39(79) 39(79)SHALLOW 40 8(25) 8(30) 8(30) 8(18) 8(18)EX-POWEL 60 32(101) 27(115) 27(115) 36(79) 35(77)EX-WOOD 60 167(452) 19(73) 19(73) 18(42) 17(40)EX-ROSEN 60 169(485) 22(75) 22(75) 22(61) 22(61)

WOLFE 80 63(127) 37(114) 37(114) 37(75) 37(75)EX-POWELL 80 35(105) 27(115) 27(115) 39(85) 39(85)NON-DIGN 90 46(112) 20(74) 20(74) 21(61) 21(61)EX-WOOD 100 230(641) 20(73) 19(73) 18(42) 17(40)EX-ROSEN 100 249(712) 22(75) 22(75) 22(61) 22(61)

TOTAL NOI 1177 330 413 329 326(NOF) (3372) (1772) (1772) (816) (808)


TABLE : (3)

TEST ALGORITHM(6) ALGORITHM(7)FUNCTIONS N NOI(NOF) NOI(NOF)ROSEN 2 22(54) 25(97)CUBE 2 22(57) 24(92)BEALE 2 8(20) 9(33)BOX 2 9(41) 9(44)FREUD 2 6(18) 7(34)BIGGS 3 11(31) 13(47)HELICAL 3 18(39) 18(73)RECIPE 3 6(19) 6(21)MIELE 4 30(83) 30(103)POWILL 4 31(67) 29(99)WOOD 4 19(42) 18(66)DIXON 10 17(37) 18(56)OREN 10 12(64) 12(55)NON-DIGN 20 20(46) 22(84)TRI-DIGN 30 28(57) 28(83)OREN 30 21(95) 21(85)SHALLOW 40 6(18) 6(30)FULL 40 39(79) 39(121)EX-ROSEN 60 23(57) 26(101)EX-POWELL 60 40(83) 35(119)EX-WOOD 60 60(42) 18(66)EX-POWELL 80 43(88) 39(139)WOLFE 80 80(75) 37(112)NON-DIGN 90 23(53) 22(95)EX-WOOD 100 19(42) 18(66)EX-ROSEN 100 23(57) 26(101)TOTAL NOI 551 555

(NOF) (1364) (2022)

References

[1] R. Fletcher and M.J.D. Powell, ”A rapidly convergent descent method for minimization”, Com-puter Journal 6 (1963), 163-168.


[2] S.S. Oren, ”Self-scaling variable metric algorithm, Part II”, Management Science 20 (1974), 863-874.

[3] S.S. Oren, ”On the selection of parameters in self-scaling variable metric algorithms”, Mathemat-ical Programming 3 (1974) 351-367.

[4] S.S. Oren and D.G. Luenberger, ”Self-scaling variable metric algorithm, Part I”, ManagementScience 20 (1974) 845-862.

[5] S.S. Oren and E. Spedicato, ”Optimal conditioning of self-scaling variable metric algorithms”,Mathematical Programming l0 (1976) 70-90.

[6] D.F. Shnnno and K.H. Phua, ”Matrix conditioning and nonlinear optimization”, MathematicalProgramming14 (1978) 149-160.

[7] C.G. Broyden, ”The convergence of a class of double rank minimization algorithms II. The newalgorithm”, Journal of the Institute of Mathematics and its Applications 6 (1970) 221-231.

[8] S.S. Oren, ”Self-scaling variable metric algorithm without line search for unconstrained minimiza-tion”, Mathematics of Computation 27 (1973), 873-885.

[9] G.P. McCormick and K.. Ritter, ”Methods of conjugate directions versus quasi-Newton methods”,Mathematical Programming 3 (1972) 101-l16.

[10] M.C. Biggs, ”Minimization algorithms making use of non-quadratic properties of the objectivefunction”, Journal of the Institute of Mathematics and its Applications 8 (1971) 315-327.

[11] M.C. Biggs, ”A note on minimization algorithms which make use of non-quadratic properties ofthe objective function”, Journal of Institute of Mathematics and its Applications12 (1973) 337-338.

[12] H.R. Hestenes and E. Stiefel, ”Methods of conjugate gradients for solving linear systems”, Journalof Research of the National Bureau of Standards, 49 (1952), 409-436.

[13] L. Nazareth, ”A relationship between BFGS and conjugate-gradient algorithms and its implemen-tations for new algorithms”, SIAM Journal on Numerical Analysis,16 (1979), 794-800.

[14] K.W. Brodlie, ”Some topics in unconstrained minimization”, Ph.D. thesis, University of Dundee,(1973).

[15] B. Bunday, ”A Basic Optimization Methods”, Edward Arnold, Bedford Square, London, (1984).

AppendixAll the test functions used in this paper are from general literature

1. Rosenbrock banana function, n=2,f=100 (x2-x2

1)2+(1-x1)2 , x0 =(-1.2,1.0)T .

2. Cube function, n=2,f =100 (x2-x3

1)2+(1-x1)2, x0=(-1.2,1.0)T .

3. Scale function, n=2,f=(1.5-x1 (1-x2))2+ (2.25 - x1 (1 - x2

2))2+ (2.625 - x1(1 - x3

2))2, x0 = (0,0)T .

4. Box function, n = 2,f =

∑ni=1( e−x1zi - e -x2zi - e−zi + e-10zi) 2 ,where zi = (0.1)i and x0 = (5,0)T , i=1,...

5. Frudenstein and Roth function, n=2,f = [-13 + x1 + ((5-x2)x2-2)x2]2 + [-29 + x1 + ((1+x2)x2-14)x2]2, x0 = (30,3)T .6. Recipe function, n = 3,f = (x1-5)2 + x2

2 + x23/(x1-x2)2, x0 = (2,5,1)T .

7. Biggs function, n=3,f =

∑ni=1(e

−x1zi - x3 e−x2zi- e−zi + 5e−10zi) 2 , where zi = (0.1) i and x0= (1,2,1)T ,i=1...8. Helical Valley function, n=3,f = 100 [x3-1.0 ]2 + [r-1]2 + x2

3,where r=1/2 arctan (x2/xl ), for xl > 0


and r = 1/2 + 1/2 arctan (x2/xl ) for xl < 0, x0 = (-1,0,0)T .9. Miele and Cornwell function, n = 4,f = (ex1 - 1)2 + tan 4 (x3 - x4) + 100 (x2- x3)2 + 8x1+ (x4- l)2, x0= (1, 2, 2, 2)T .10. Dixon function, n = 10,f = (1- xl )2 + (1- x10)2 +

∑ni=1(x

2i - xi+1)2 , x0 = (-1;.. )T , i=2,...

11. Oren and Spedicato power function, n = 10,30,f =

∑ni=1(i - x2

i )2 , x0 = (1,.....)T .12. Non diagonal variant of Rosenbrock function, n = 20, 90,f =

∑ni=1[100 (xl - x2

i )2 + ( 1 - xi )2 ] , x0 = (-1, ..) T , i=1,..

13. Tri-diagonal function, n = 30,f = [

∑ni=2(2xi - xi−1 )2 ] , x0 = (1;....)T .

14. Full set of distinct eigenvalues problem, n = 40,f = (xl-1)2 +

∑ni=2(2xi - xi−1)2 , x0 = (1;..)T .

15. Shallow function (Generalized form), n = 40,f =

∑n/2i=1(x

22i−1 - x2i )2 + (1 - x2i−1 )2 , x0 = (-2;. )T .

16. Powell function (Generalized form), n = 60, 80,f =

∑n/4i=1[( x4i−3 + l0 x 4i−2)2 + 5 (x 4i−1 - x4i )2 + (x4i−2-2 x4i−1)4 + 10 (x4i−3 -

x4i)4],x0 = (3,-1,0,1;...)T .17. Wood function(Generalized form), n = 60,100,∑n/4

i=1f = [100 (x4i−2 - x24i−3)

2 + (1- x4i−3)2+ 90 (x4i - x24i−1)

2 + (1 - x4i−1)2

+ 10.1 (x 4i−2 - 1)2 + (x4i- 1)2 + 19.8 (x4i−2 - 1)(x4i−l), x0 = (-3,-1;-3,-1,...)T .

Math. Dept., Faculty of ScienceBeirut Arab UniversityP.O. Box 11-5020, Beirut, LebanonEmail: [email protected]

J. KSIAM Vol.6, No.1, 109-119, 2002

THE ORPHAN STRUCTURE OF BCH(3,m) CODE

Geum-sug Hwang

AbstractIf C is a code, an orphan is a coset without any parent. We investigate the structure

of orphans of the code BCH(3, m). All weight 5 cosets and all weight 3 reduced cosetsare orphans, and all weight 1,2 and 4 are not orphans. We conjecture that all weight3 unreduced cosets are not orphans. We prove this conjecture for m = 4, 5.

1. Introduction

An [n, k] code C over Fq is a k-dimensional subspace of the n-tuple space GF (qn).An [n, k] code C can be specified by k linearly independent vectors in C. A k by nmatrix G over Fq whose rows forms a basis of C is called generator matrix of C andC = x = uG | u = (u1, u2, · · · , uk), ui ∈ Fq (∗1). Also C can be specified byn− k linearly independent homogeneous equations. A n− k by n matrix H such thatC = (x1, x2, · · · , xn) | Hxt = 0, xi ∈ Fq (∗2) is called parity check matrix for C.(∗1) and (∗2) together imply that G and H are related by GHt = 0 and HGt = 0. Acoset of a code C is the set a+C = a+x | x ∈ C for any vector a. Each vector b is insome coset and each coset contains qk vectors. For a vector b, s = Hbt is the syndromeof b where s is a column vector of length n− k. Two vectors are in same coset if andonly if Hat = Hbt. Hence there are one to one correspondence between syndromesand cosets. A minimum weight vector in a coset is called a coset leader and the cosetweight is the weight of a coset leader. The cosets of C are partially ordered by definingfor two cosets C ′ and C ′′ of C, C ′ ≤ C ′′ provided there is a coset leader x′ of C ′ anda coset leader x′′ of C ′′ such that x′ ≤ x′′. Here for the vectors x′ = (x′1, x

′2, · · · , x′n)

and x′′ = (x′′1 , x′′2 , · · · , x′′n), x′ ≤ x′′ means that x′′i 6= 0 whenever x′i 6= 0. The coset C ′

is a child of C ′′, and C ′′ is a parent of C ′, provided C ′ ≤ C ′′ and there is no coset Dwith C ′ < D < C ′′. An orphan is a coset without any parent.

Key words and phrases. orphan, reduced coset, unreduced coset.This research was supported by PUFS under grant PUFS-2000

Typeset by AMS-TEX

109

110 Geum-sug Hwang

Let BCH(t,m) denote the binary Bose-Chaudhuri-Hocquenghem code of primitivelength n = 2m− 1 and design distance δ = 2t+1. We investigate the orphan structureof the code BCH(3,m) code. The BCH(3,m) code, m ≥ 4, is the null space of the 3by n matrix H over GF (2m) given by

H =

1 α α2 · · · αn−1

1 α3 α6 · · · α3(n−1)

1 α5 α10 · · · α5(n−1)

where α is a primitive element of GF (2m). The syndrome s of a received word r =(r0, r1, · · · , rn−1) is s = Hrt = (S1, S3, S5), Sj ∈ GF (2m). The cosets are the setC(s) = r : Hrt = s. Given an arbitrary binary n-tuple a = (a0, a1, · · · , an−1) ofweight ω, the locator polynomial of a is the polynomial of degree ω defined by

σ(X) =∏

i:ai 6=0(X + αi) = Xω + σ1X

ω−1 + · · ·+ σω.

The roots of the locator polynomial of a indicate the coordinate positions which are1 in a. There is a one to one correspondence between binary n-tuples and locatorpolynomials. A locator polynomial σ(X) =

∏ωi=1(X + Ai) of degree ω is called an

error locator polynomial with syndrome s provided it is the locator polynomial of acoset leader of a coset C(s), s = (S1, S3, S5) of weight ω. This implies that Sj =∑ω

i=1 Aji , j = 1, 3, 5. We give the relation between the coefficients σi of the locator

polynomial σ(X) and the components Sj of its syndrome, namely S1 = σ1, S3 =σ1S

21 + σ2S1 + σ3 and S5 = σ1S

41 + σ2S3 + σ3S

21 + σ4S1 + σ5.

We define the syndrome (T1, T3, T5) to be reduced provided that T1 = 0. A cosetwith reduced syndorme is called a reduced coset and a coset with T1 6= 0 is called anunreduced coset. The transform of C(s), s = (S1, S3, S5) is the reduced coset C(t) withsyndrome t, t = (T1, T3, T5) = (0, S3 +S3

1 , S5 +S51). Note that two different cosets can

have the same transform. Any coset C(s) of weight 1 has syndrome s = (S1, S31 , S5

1),and so its transform is the code C(0). Hence if t 6= 0 then C(s) has weight > 1. Thecovering radius of a code is the largest weight of orphan. The existence of orphans ofweight less than covering radius complicates the determination of the covering radiusof a code. Let’s start with the following characterization of orphan given by R. A.Brualdi and V. S. Pless.

Theorem 1.1 Let C ′ be a coset of C with weight ω. Then C ′ is an orphan if andonly if the vectors of C ′ with weights ω and ω + 1 cover all coordinate positions.

Proof We first note that each parent of C ′ is of the form ei +C ′ for some unit vectorei, 1 ≤ i ≤ n. If the vectors of weight w and w +1 of C ′ cover all coordinate positions,then the weight of ei + C ′ is either w − 1 or w and hence ei + C ′ cannot be a parent

THE ORPHAN STRUCTURE OF BCH(3, m) CODE 111

of C ′. Now suppose that C ′ is an orphan. If there is a coordinate position j which isnot coverd by any vector of weight w or w + 1 of C ′, then ej + C ′ contains a vectorof weight w + 1 but contains no vectors of weight w, and it follows that ej + C ′ is aparent of C ′.

Let a coset C ′ of a code C of distance d have weight ω, ω < b(d − 1)/2c. If thereare two vectors u, v in C ′ of weight ω or ω + 1, then the vector u + v is a codewordand its weight is less than d, contradicting the distance of C is d. Hence such a cosetC ′ cannot be an orphan by theorem 1.1.

Since the distance of BCH(3,m) is 7, all cosets of weight 1 and 2 are not orphans.Since the maximal coset weight of BCH(3,m) is 5, it is trivial that all cosets of weight5 are orphans. Hence it remains only to investigate cosets of weight 3 and 4. We notethat a coset of weight 3 has a unique coset leader. We now use the notation σk(X) todenote a locator polynomial of degree k.

Lemma 1.2 Let σ2k−1(X) =∏2k−1

i=1 (X + Ai), k ≥ 1, be the locator polynomialof a vector of a reduced coset C(t). If Lσ2k−1(L) 6= 0 for some L ∈ GF (2m), then(X +L)σ2k−1(X +L) is a locator polynomial of degree 2k with syndrome t. Conversely,if σ2k(X) is any even degree locator polynomial with syndrome t and L is one of itsroots, then σ2k(X + L)/X is a locator polynomial of degree 2k − 1 with syndrome t.

Proof Since σ2k−1(X) is a locator polynomial, its roots Ai, i = 1, · · · , 2k − 1 aredistinct nonzero elements of GF (2m). It follows from the condition Lσ2k−1(L) 6= 0 thatL and Ai + L are also distinct and nonzero so that σ2k(X) = (X + L)σ2k−1(X + L)is also locator polynomial. To show that σ2k(X) has syndrome t it suffices to showthat Lj +

∑2k−1i=1 (Ai + L)j =

∑2k−1i=1 Aj

i , j = 1, 3, 5. Since∑2k−1

i=1 Ai = 0, we also have∑A2

i = 0 and∑

A4i = 0. Hence L +

∑(Ai + L) =

∑Ai = 0 and Lj +

∑(Ai + L)j =∑

Aji by expanding (Ai + L)j , j = 3, 5.

Conversely suppose that L is one of the roots of an even degree locator polynomialσ2k(X). Let A1, · · · , A2k be the roots of σ2k(X) and assume L = A1. Since allAi, i = 1, · · · , 2k are nonzero and distinct, L + Ai = A1 + Ai are also distinct andnonzero. Hence σ2k(X + L)/X is a locator polynomial of degree 2k − 1. Since Lj +∑2k

i=2(Ai)j = 0, j = 1, 2, 4

2k∑

i=2

(Ai + L)j =∑

(Ai)j + L(∑

(Ai)j−1) + Lj−1(∑

Ai) + Lj

=∑

(Ai)j + LLj−1 + Lj−1L

=∑

(Ai)j , j = 1, 3, 5.

Thus σ2k(X + L)/X also has syndrome t.

112 Geum-sug Hwang

Henceforth we denote a binary n- tuple A of weight ω with 1’s in positions i1, i2, · · · , iωby A = A1, A2, · · · , Aω = αi1 , αi2 , · · · , αiω.

Corollary 1.3 Any weight 4 vector of weight 3 reduced coset C(t) has the formL,A1 + L,A2 + L,A3 + L for some L ∈ GF (2m), L 6= 0, Ai, i = 1, 2, 3 whereA1, A2, A3 is the unique coset leader of C(t).

Proof Let σ3(X) =∏3

i=1(X + Ai) be the error locator polynomial of C(t). For anynonzero L ∈ GF (2m), if L 6= Ai, i = 1, 2, 3 then Lσ3(L) 6= 0. Hence (X +L)σ3(X +L)is locator polynomial of degree 4 with syndrome t by Lemma 1.2. This implies thatL,A1+L,A2+L,A3+L is a weight 4 vector of C(t). Since the distance of BCH(3,m)is 7, any two distinct locator polynomials of degree 4 with syndrome t have no commonroot. From the converse part of Lemma 1.2 and uniqueness of the coset leader of C(t),any weight 4 vector of C(t) has this form.

Theorem 1.4 The weight ω of a reduced coset C(t) is either zero or an odd integer≥ 3.

Proof Because any coset of weight 1 has syndrome s = (S1, S31 , S5

1), S1 6= 0, wcannot be one. Assume that w is positive and even, say w = 2k. Let σ2k(X) bean error locator polynomial with syndrome t, and let L be a root of σ2k(X). Defineσ2k−1(X) = σ2k(X + L)/X. Then σ2k−1(X) is a locator polynomial with syndrome tby Lemma 1.2, contradicting w is the weight of C(t).

We get the relation between error locator polynomial of coset C(s) and that of itstransform C(t) from the next theorem which is in [2]T. Berger and V. A. Van DerHorst. Henceforth we denote a binary n-tuple A of weight ω with 1’s in positionsi1, i2, · · · , iω by A = A1, A2, · · · , Aω = αi1 , αi2 , · · · , αiω. Two vectors are disjointprovided their locator polynomials have no common roots.

Theorem 1.5 Let C(s), s = (S1, S3, S5) be a coset of weight ω > 1. Then anerror locator polynomial σ(X) with syndrome s can be obtained from an error locatorpolynomial σ(X) of its transform by

σ(X) =

σ(X), if S1 = 0σ(X)/(X + S1), if S1 6= 0, ω evenσ(X + S1), if S1 6= 0, ω odd.

Proof If S1 = 0, then t = s and σ(X) = σ(X), so we need only consider S1 6= 0.Case 1 : ω is even. By Theorem 1.4, ω equals either ω − 1 or ω + 1. Assume that

ω = ω − 1. Then σ(S1) cannot equal zero because that implies σ(X)/(X + S1) is alocator polynomial of degree ω− 1 = ω− 2 with syndrome s, thereby contracting C(s)


has weight w. Thus σ(X + S1) has distinct nonzero roots and is a locator polynomial.Therefore σ(X + S1) has weight w = w − 1 with syndrome s because we have

w∑

i=1

(Ai + S1)j =∑

(Ai)j + S1(∑

(Ai)j−1) + Sj−11 (

∑Ai) +

∑Sj

1

= Tj + Sj1, j = 1, 3, 5

since (∑

Ai)j−1 =∑

Ai = 0 where Ai, i = 1, · · · , w are roots of σ(X). This contra-dicts that C(s) has weight w, so w = w + 1. It follows that σ(S1) 6= 0. Otherwise,σ(X)/(X +S1) is a locator polynomial with syndrome t and degree w−1, which wouldcontradict that C(t) has weight w = w + 1. Since we now know that σ(S1) 6= 0 andw = w + 1, σ(X) = (X + S1)σ(X) is an locator polynomial with syndrome t, orσ(X) = σ(X)/(X + S1).

Case 2 : w is odd. By Theorem 1.4, w = w and σ(X + S1) is a locator polynomialwith syndrome s and the degree w of σ(X + S1) equals w. Thus σ(X + S1) is an errorlocator polynomial with syndrome s.

Corollary 1.6 No orphan has weight 4.Proof Let σ(X) be an error locator polynomial of weight 4 coset C(s) with coset

leader A1, A2, A3, A4 with syndrome s = (S1, S3, S5), S1 6= 0. By Theorem 1.5, anerror locator polynomial σ(X) of the transform C(t) of C(s) is σ(X) = σ(X)(X +S1).This means that S1, A1, A2, A3, A4 is a coset leader of C(t), and C(t) is a parent ofC(s). Thus a coset of weight 4 is not orphan.

Theorem 1.7 All reduced cosets of weight 3 are orphans. Furthermore, such cosetshave exactly (n− 3)/4 weight 4 vectors.

Proof Let C(t) be a reduced coset of weight 3 with coset leader A = A1, A2, A3.For any nonzero L ∈ GF (2m), L 6= Ai, i = 1, 2, 3, L = L,L+A1, L+A2, L+A3 is aweight 4 vector in C(t). Since distance is 7, A and L are disjoint. Hence A and weight4 vectors of C(t) cover all coordinate positions. Therefor, any two distinct weight 4vectors are also disjoint, so there are exactly (n− 3)/4 weight 4 vectors of C(t).

We define the trace mapping from GF (2m) to GF (2) by

Tr(A) = A + A2 + · · ·+ A2m−1, A ∈ GF (2m).

The following lemma shows the properties of trace mappings which can be found in[8]F. J. MacWilliams and N. J. A. Solane.

Lemma 1.8 The followings hold:

114 Geum-sug Hwang

(i) Exactly half of the elements A in GF (2m) have Tr(A) = 0 and exactly half haveTr(A) = 1.

(ii) Tr(A + B) = Tr(A) + Tr(B), A, B ∈ GF (2m).(iii) Tr(A2i

) = Tr(A), i = 1, · · · , m− 1.

We next obtain sufficient conditions for a weight 3 coset not to be an orphan by usingthe trace mapping. [4]E. R. Berlekamp, H. Rumssey and G. Solomon characterizedquadratic equations over fields of characteristic two which have roots and we recordtheir result in the next lemma.

Lemma 1.9 The quadratic equation, X2 + AX + B = 0, A,B ∈ GF (2m), A 6= 0,has solutions in GF (2m) if and only if Tr(B/A2) = 0.

Lemma 1.10 Any reduced coset with syndrome (0, 0, T5), T5 6= 0 has weight 5.Proof Let C(t) has syndrome t, t = (0, 0, T5). Since T1 = 0, C(t) has weight 3 or 5

by Theorem 1.4. Assume that C(t) has weight 3 and let σ(X) = X3+σ1X2+σ2X +σ3

be the error locator polynomial of C(t). we have σ1 = T1 = 0 and σ3 = T3 = 0. Thenσ(X) has zero as its root which contradicts that σ(X) is a locator polynomial. HenceC(t) has weight 5.

Lemma 1.11 Assume m is odd. Any reduced coset C(t) with syndrome t =(0, T3, 0), T3 6= 0 has weight 5.

Proof Since T1 = 0, C(t) has weight 3 or 5 by Theorem 1.4. Assume that C(t)has weight 3 with coset leader A1, A2, A3. Since A1 + A2 + A3 = 0, 0 = T5 =T3(A1A2 + A2A3 + A1A3) = T3(A2

1 + A22 + A1A2). Since T3 6= 0, A2

1A22 + A1A2 = 0

and so A2 is a root of X2 + A1X + A21 = 0. By Lemma 1.8, Tr(A2

1/A22) = Tr(1) = 0,

contradicting to that m is odd. Hence C(t) has weight 5.

2. Main Theorems

Theorem 2.1 Let C(t), t = (0, T3, T5) be a reduced coset of weight 3 and C(s), s =(S1, S3, S5) be a unreduced coset whose transform is C(t). If Tr(T3/S3

1) = 0, then C(s)is not an orphan.

Proof Let A = A1, A2, A3 be the coset leader of C(s). Then A1+S1, A2+A1, A3+S1 is the coset leader of C(t) by Theorem 1.5. Note that C(s) is not an orphan ifand only if there exists a nonzero L ∈ GF (2m) such that A′ = A1, A2, A3, L is acoset leader of weight 4 coset. Since, by Lemma 1.9 Tr(T3/S3

1) = 0 if and only ifX2 + S1X + T3/S1 = 0 has a solution, there exists a L ∈ GF (2m) such that LS1(L +S1) + T3 = 0. If L = 0 then T3 = 0, and so C(t) has weight 5 by Lemma 1.10.Hence L 6= 0. We now show that L 6= Ai, i = 1, 2, 3. Assume that L = A1. ThenA1S1(A1 + S1) = T3 = (A1 + S1)3 + (A2 + S1)3 + (A3 + S1)3 = (A1 + S1)(A2 +S1)(A3 + S1) = (A1S1 + A1A2)(A1 + S1) implies A2A3 = 0, contradicting A has


weight 3. Thus A′ = A1, A2, A3, L is a weight 4 vector of some coset C(s′), wheres′ = (S1 + L, S3 + L3, S5 + L5). Then the transform C(t′) of C(s′), has syndrome(0, T3 +LS1(L+S1), T5 +LS1(L3 +S3

1)). Since T3 +LS1(L+S1) = 0, the coset weightof C(t′), t′ = (0, 0, T5 + LS1(L3 + S3

1)) is 5 by Lemma 1.10. Hence C(s′) has weight4 by Theorem 1.5. Thus the weight 4 vector A′ is a coset leader of C(s′), and henceC(s′) is a parent of C(s). Therefore C(s) is not an orphan.

Theorem 2.2 Assume m is odd. Let C(t), t = (0, T3, T5) be a reduced coset ofweight 3. There exists (n − 7)/2 weight 3 unreduced cosets whose transform is C(t),and they are not orphans. Furthermore, there are at least n(n− 1)(n− 7)/12 weight 3cosets which are not orphans.

Proof Let A = A1, A2, A3 be the coset leader of C(t). By Theorem 1.5 and theuniqueness of the coset leader of C(t), for any nonzero L ∈ GF (2m) with L 6= Ai, thecoset C(l), l = (L, T3+L3, T5+L5) has weight 3 with coset leader A1+L, A2+L,A3+L and C(t) is a transform of C(l). Hence we want to count L such that Tr(T3/L3) = 0,L 6= 0, A1, A2, A3. Since m is odd, n = 2m − 1 is not divisible by 3. This means α3 isa primitive element whenever α is a primitive element of GF (2m). Thus, for the givenT3, T3/L3 | L ∈ GF (2m), L 6= 0 is the set of all nonzero elements of GF (2m). By (i)in Lemma 1.8, there are exactly (n− 1)/2 nonzero L such that Tr(T3/L3) = 0. But,

Tr(T3/L3) = Tr((A31 + A3

2 + A33)/A

31) = Tr(A1A2A3/A

31)

= Tr((A23 + A1A3)/A2

1) = Tr((A3/A1)2 + Tr(A3/A1)

= Tr(A3/A1) + Tr(A3/A1) = 0,

using A1 + A2 + A3 = 0 and (ii), (iii) in Lemma 1.8. We conclude that if L =Ai, i = 1, 2, 3 then Tr(T3/L3) = 0, but coset C(l), l = (L, T3 + L3, T5 + L5) does nothave weight 3. Therefore there are (n + 1)/2 − 4 = (n − 7)/2 weitght 3 unreducedcosets whose transform is C(t) and they are not orphans by Theorem 2.1. We nowcount the number of weight 3 reduced cosets with syndrome (0, T3, ∗) for some fixedT3 ∈ GF (2m) and arbitrary ∗ ∈ GF (2m). This is equivalent to counting the numberof coset leaders of these cosets since each coset has only one coset leader. Let C(t)be a weight 3 reduced coset with syndrome (0, T3, ∗) and let A1, A2, A3 be the cosetleader of C(t). Then, by Lemma 1.9, T3 = A3

1 + A32 + A3

3 = A31 + A3

2 + (A1 + A2)3 =A1A2(A1 + A2) (or A2A3(A2 + A3)). Therefore

A1, A2, A3 is the coset leader of a coset C(t), t = (0, T3, ∗)if and only if Ai is a root of X2 + AjX + T3/Aj = 0, i 6= j i, j = 1, 2, 3

if and only if Tr(T3/A31) = Tr(T3/A

32) = Tr(T3/A

33) = 0.

We have already noted that there are (n − 1)/2 nonzero L ∈ GF (2m) such thatTr(T3/L3) = 0, so there are 1/3((n − 1)/2) weight 3 reduced cosets with syndrome

116 Geum-sug Hwang

(0, T3, ∗) for each nonzero T3 ∈ GF (2m). Therefore we have at least n(1/3((n −1)/2)((n − 7)/2) = n(n − 1)(n − 7)/12 weight 3 unreduced cosets which are not or-phans.

Theorem 2.3 Assume m is an even. There are at least nβ(β − 1) + (n/8)(n −2β)(n−2β−5) weight 3 unreduced cosets which are not orphans where β is the numberof nonzero elements αj ∈ GF (2m) such that the trace of αj is zero and j ≡ 0 (mod 3).

Proof Let C(t), t = (0, T3, T5) be a weight 3 reduced coset with coset leader A =A1, A2, A3. Since m is even, n = 2m − 1 is divisible by 3. So T3/L3 | L ∈GF (2m), L 6= 0 is not the set of all nonzero elements of GF (2m). To count thenumber of nonzero L such that Tr(T3/L3) = 0, define β to be the cardinality of Ψwhere Ψ = αj ∈ GF (2m) | αj 6= 0, T r(αj) = 0, j ≡ 0 (mod 3). Let T3 = αj forsome j. We separate the remainder of the proof into two cases according to whether jis divisible by 3 or not.

Case 1 : Let T3 = α3k, for some k. Then if T3/L3 = R for some R ∈ Ψ, thenT3/(Lαn/3)3 = T3/(Lα2n/3)3 = R and L ∈ GF (2m). Hence, by the same argument inTheorem 2.2, there exist 3β nonzero L such that Tr(T3/L3) = 0, and we have 3β − 3weight 3 unreduced cosets whose transform is C(t) and by Theorem 2.1 they are notorphans. Also we have β weight 3 reduced cosets with syndrome (0, T3, ∗) for somefixed T3 ∈ GF (2m), and there are n/3 nonzero elements of GF (2m), T3 = α3k for somek. This means that there are at least (n/3)(β)(3β − 3) = nβ(β − 1) weight 3 cosetswhich are not orphans.

Case 2 : Let T3 = αk, k = 1, 2 (mod 3). Exactly half of the elements in GF (2m)have trace zero, so we have (n − 1)/2 − β = 1/2(n − 1 − 2β) nonzero R = αj suchthat Tr(R) = 0, j is not divisible by 3. Note if j ≡ 1 (mod 3), then 2j ≡ 2 (mod 3).Thus, there are (n− 1− 2β)/4R such that Tr(R) = 0, j ≡ 1 or 2 (mod 3) respectively.Since there exists L ∈ GF (2m) such that T3/L3 = R ∈ Ψ if and only if k ≡ j (mod 3),there are weight 3 unreduced cosets whose transform is C(t) and ((n− 1− 2β)/4)− 3weight 3 reduced cosets with syndrome (0, T3, ∗) for some fixed nonzero T3 ∈ GF (2m).Therefore we have at least 2[(n/3)((n−1−2β)/4)((3(n−1−2β)−12)/4)] = (n/8)(n−1− 2β)(n− 5− 2β) weight 3 unreduced cosets which are not orphans.

From Case 1 and Case 2, there are at least nβ(β−1)+(n/8)(n−2β−1)(n−2β−5)weight 3 unreduced cosets which are not orphans.

Theorem 2.4 Assume that m is odd. Let C(t), t = (0, T3, T5) be a reduced coset ofweight 3 and C(s), s = (S1, S3, S5) be an unreduced coset whose transform is C(t). IfTr(T5/S5

1) = 0, then C(s) is not an orphan.Proof Let A1, A2, A3 be the coset leader of C(t). Then A1+S1, A2+S1, A3+S1 is

the coset leader of C(s). Since Tr(T5/S51) = 0, by Lemma 1.9, X2 + S2

1X + T5/S1 = 0has roots P, Q ∈ GF (2m) such that P + Q = S2

1 and PQ = T5/S1. ThereforeX4 +S3

1X +T5/S1 = (X2 +S1X +P )(X2 +S1X +Q) (∗3) for P, Q ∈ GF (2m). Since


P + Q+ = S21 , T r(P/S2

1) + Tr(Q/S21) = Tr(1) = 1. Thus only one of Tr(P/S2

1) andTr(Q/S2

1), say Tr(P/S21), equals to zero. By Lemma 1.9, there exists L ∈ GF (2m) such

that L is a root of X2 +S1X +P = 0. From (*3), L is a root of X4 +S31X +T5/S1 = 0,

and so S1L4 + S4

1L = S51 + L5 + (S1 + L)5 = T5. Hence S1, L, S1 + L is coset leader

of weight 3 reduced coset C(p) with syndrome (0, P, T5) where P = S1L(S1 + L). ByLemma 1.10, a coset C(p′) = C(t) + C(p) with syndrome (0, T3 + P, 0) has weight 5.Now A = A1, A2, A3, S1, L, S1 + L is a vector of C(p′) has a vector of weight lessthan 5, contradicting to that C(p′) has weight 5. This A is a vector in C(p′) of weight6. Thus A1 + S1, A2 + S1, A3 + S1, L, L + S1 is a weight 5 vector in C(p′) and is acoset leader. Since any descendent of coset leader is also coset leader of some coset,A1 +S1, A2 +S1, A3 +S1, L is a coset leader of some coset which is a parent of C(s).Therefore C(s) is not an orphan.

We have shown that many weight 3 unreduced cosets are not orphans. We conjecturethat all weight 3 unreduced cosets are not orphans. We prove that this conjecture form = 4 and 5.

Lemma 2.5 Let C(s), s = (S1, S3, S5) be a weight 3 unreduced coset. For eachweight 4 vector A = A1, A2, A3, A4 of C(s) with Ai 6= S1, i = 1, · · · , 4, we haveA = A1 + S1, A2 + S1, A3 + S1, A4 + S1 is also a weight 4 vector of C(s).

Proof Since the Ai are distinct nonzero elements different from S1, the elements Ai+S1 are nonzero and distinct. We calculate

∑4i=1(Ai+S1)j =

∑4i=1 Aj

i +S1(∑

(Ai)j−1)+Sj−1

1 (∑

Ai) +∑

Sj1 =

∑Aj

i , since∑

Aj−1i = Sj−1

1 , j = 1, 3, 5.

Corollary 2.6 Any weight 4 coset C(s) has at least two coset leaders.

Lemma 2.7 A locator polynomial of a weight 4 vector of the weight 3 unreducedcoset C(s) and a locator polynomial of weight 4 vector of the transform C(t) of C(s)have at most one common root.

Proof Let Q = Q1, Q2, Q3, Q4 and P = P1, P2, P3, P4 be weight 4 vectors inC(s) and C(t) respectively. Without loss of generality, assume that Q1 = P1 andQ2 = P2. We claim that Q3, Q4, P3, P4 ∈ C(s′), s′ = (S1, S

31 , S5

1), a coset of weight1. This follows since Qj

3 + Qj4 = Sj + Qj

1 + Qj2 = Sj + P j

1 + P j2 = Sj + Tj + P j

3 + P j4 =

Sj1 + P j

3 + P j4 , j = 1, 3, 5. Therefore S1, Q3, Q4, P3P4 is a codeword, contradicting

the fact that the minimum distance of BCH(3,m) is 7.

Lemma 2.8 Suppose that the locator polynomial σ(X) of weight 4 vector of weight 3unreduced coset C(s) has one common root with the locator polynomial σ(X) of weight4 vector of its transform C(t). If S1 is neither a root of σ(X) nor σ(X), then σ(X)cannot have a common root with σ(X+S1), where σ(X+S1) is also a locator polynomialof weight 4 vector of C(s).

118 Geum-sug Hwang

Proof Let A = A1, A2, A3 be a coset leader of C(t), and let P = P1, P2, P3, P4and Q = Q1, Q2, Q3, Q4 be weight 4 vectors of C(t) abd C(s) respectively. Sup-pose that the locator polynomial σ(X) of P has one common root with the locatorpolynomial σ(X) of Q, say P1 = Q1. We can say that P is of the form Pi+1 =P1 + Ai, i = 1, 2, 3 since P1, P1 + A1, P1 + A2, P3 + A3 is a weight 4 vector of C(t)and any two distinct weight 4 vectors are disjoint. By Lemma 2.5 and Qi 6= 0, Q =Q1 + S1, Q2 + S1, Q3 + S1, Q4 + S1 is a weight 4 vector of C(s) and σ(X + S1) is thelocator polynomial of Q. So suppose that P and Q have a common nonzero position.If P1 = Qi + S1 for some i, then Q1 + Qi = S1, since P1 = Q1. This contradicts thefact that the weight of Q is 4. Without loss of generality, assume that P2 = Q2 + S1.Then Q2 + S1 = P2 = P1 + A1 = Q1 + A1, Q1 + Q2 + S1 = Q3 + Q4 = A1. So we haveQ3 = Q4 +A1. Hence Q4, Q4 +A1, Q4 +A2, Q4 +A3 = Q4, Q3, Q4 +A2, Q4 +A3 isweight 4 vector in C(t) which has two common nonzero positions with Q, contradictingLemma 2.7. Hence P cannot have a common nonzero position with Q.

Theorem 2.9 No weight 3 unreduced coset is an orphan for m = 4 and 5.Proof Let C(s) be weight 3 coset and let C(t) be its transform with coset leader

A = A1, A2, A3. Then S1, A1, A2, A3 is a weight 4 vector of C(s) since S1 6= 0, Ai.We claim that this is the only weight 4 vector of C(s). To get a contradiction, assumethat Q = Q1, Q2, Q3, Q4, Qi 6= S1, Aj i = 1, · · · , 4; j = 1, 2, 3 is another weight4 vector of C(s). Then Q = Q1 + S1, Q2 + S1, Q3 + S1, Q4 + S1 is also weight 4vector of C(s) by Lemma 2.5. Define P (i) = Qi, Qi + A1, Qi + A2, Qi + A3 andP (i) = Qi + S1, Qi + S1 + A1, Qi + S1 + A2, Qi + S1 + A3 for i = 1, · · · , 4. ThenP (i) and P (i) are weight 4 vectors of C(t). It is sufficient to show that these 8 weight4 vectors are distinct since C(t) has only (n − 3)/4 < 8, (m = 4, 5) weight 4 vectorsby Theorem 1.7. If P (i) = P (j), i 6= j then we have Qi = Qj + Ak for some k, sothe locator polynomial of P (j) has two common roots with locator polynomial of Q,contradicting Lemma 2.7. Thus we have P (i) 6= P (j), and P (i) 6= P (j) for i 6= j.If P (i) = P (i) then Ai = S1, contradicting C(s) has weight 3. Now assume thatP (i) = P (j), i 6= j, say i = 1, j = 2. Then Q1 = Q2 + S1 + Ak for some k. Thisimplies Q3 = Q4 + Ak, so the locator polynomial of P (4) has two common roots withQ contradicting Lemma 2.7. Thus all these weight 4 vectors are distinct, contradictingTheorem 1.7. Hence C(s) has only one weight 4 vector and so is not an orphan byTheorem 1.1.

References

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[3] E. R. Berkamp1968], Algebrac coding Theory, McGraw-Hill, New York.[4] E. R. Berkamp, H. Rumsy and G. Solomon1967], On the solution of algebrac equa-

tions over finite fields, information and Control, 10, 553-564.[5] T. Helleseth[1973], All binary 3-error correcting BCH codes oh length - 1 have

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Division of Information and Management Science,College of Information and Science,Pusan University of Foreign Studies,55-1, Uam-Dong, Nam-Gu, Busanemail: gshwang@ taejo.pufs.ac.krClassification number C020705

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