Journal of Innovative Technology and Education, Vol. 3, 2016, no. 1, 251-264

HIKARI Ltd, www.m-hikari.com

https://doi.org/10.12988/jite.2016.6613

Some Remarks on the Asymptotic

Iteration Method

W. Robin

Engineering Mathematics Group

Edinburgh Napier University

10 Colinton Road, EH10 5DT, UK Copyright © 2016 W. Robin. This article is distributed under the Creative Commons Attribution

License, which permits unrestricted use, distribution, and reproduction in any medium, provided the

original work is properly cited.

Abstract

The asymptotic iteration method is shown to arise naturally from the continued

fraction approach to solving second-order homogeneous linear ordinary differential

equations. This emergence of the asymptotic iteration method from the continued

fraction approach follows when the continued fraction method is (a) conjoined with

the operator factorization method and (b) ‘completed’ by the explicit consideration

of the continued fraction convergents. As well as a specific example being consid-

ered, a general discussion of the emergent methodology is presented.

Mathematics Subject Classification: 30B70, 33C45, 34A05, 34A25, 40A15

Keywords: ordinary differential equations; operator factorization; Riccati equa-

tion; continued fractions; convergents; asymptotic iteration method

1. Introduction

Over the past few years, the asymptotic iteration method (AIM) has proved a pop-

ular technique for the solution of second-order linear homogeneous ordinary differ-

ential equations (ODE) [2, 3, 6, 7 and 18]. The AIM has been applied across a wide

variety of problems arising in the solution of second-order linear ODE and its pro-

ponents can claim a considerable amount of success in such enterprises. Naturally,

the foundations of the AIM have been subject to certain amount of scrutiny and it

has become apparent that the AIM is intimately related to two other well-known

methodologies for solving second-order linear ODE: the factorization method [19]

252 W. Robin

and the continued fraction method (CFM) [4, 10 and 13]. In this paper, we show

that the AIM, for second-order linear ODE, can be deduced from the CFM, but only

by first exhibiting the full formalism of the CFM (see below and [4]); once this is

done, the AIM emerges, then, from the CFM via a basic mathematical induction

argument. This leaves us with an interesting synthesis of the factorization method,

the CFM and the AIM for solving second-order linear homogeneous ODE, a syn-

thesis that effectively ‘completes’ the AIM, when applied to second-order linear

homogeneous ODE, as we discuss below.

The paper is organized as follows. In section 2, we review the CFM in conjunc-

tion with the factorization method and derive, in a compact form, the standard CFM

approach to solving second-order linear homogeneous ODE. As an example of this

approach, we consider the CFM solution of the hypergeometric-type of ODE in

section 3; at this point, we encounter the problem emphasised by Matamala et al

[14] about applying the CFM to the solution of second-order linear ODE – ‘within

CFM a previous laborious backward process is required for the integration of Eq.(6)

[essentially finding the solution of the ODE in question]’.However, as shown in

section 4, this problem can be eliminated, entirely within the theory of continued

fractions (CF), by use of a well-known result from the basic theory of CF [11, 12]

to ‘complete’ the CFM in such a manner that a fully forward iterative solution

scheme can be presented for the CFM (see, in particular [4]). Then, in section 4, it

is shown, also, how the AIM emerges naturally from this ‘complete’ form of the

CFM, through a basic mathematical induction argument. Section 4 is followed, in

section 5, by a general discussion of the ‘pros and cons’ of the AIM, as well as a

consideration of further methodological points in the application of the AIM, and

we then round-off the paper with a summary of our conclusions on the analysis and

discussion presented in the paper.

2. Applying the CFM with the Factorization Method

The problem in question is the solution of the second-order linear ODE [7]

0)()()()()( xyxqxyxpxy (2.1)

for given (sufficiently smooth) functions )(xp and ).(xq Of course, the solution of

(2.1), ),(xy may also depend on initial/boundary conditions. As usual, the dashes

in (2.1) refer to differentiation with respect to the independent variable, .x

To solve (2.1) using the factorization method, we assume that we may rewrite

(2.1) in the factorized form [19]

0)()]()][([ xyxDxD (2.2)

where dxdD / is the usual differential operator, and the (sufficiently smooth)

functions )(x ( )(/)( xyxy from (2.2)) and )(x must be determined such

Some remarks on the asymptotic iteration method 253

that (2.1) and (2.2) are identical. So, evaluating the brackets in (2.2) and compar-

ing the resultant equation with (2.1), we find that )(x and )(x must be solu-

tions of

)()()( xpxx (2.3a)

and

)()()()( xqxxx (2.3b)

Eliminating )(x in (2.3a) in terms of )(x and substituting for said )(x in

(2.3b), we find that )(x must be the solution of the Riccati equation

0)()()()()( 2 xqxxpxx (2.4)

and, given )(x as the solution of (2.4), )(x follows from (2.3a).

In principle, we can now solve (2.1) by integrating (2.2), when we find that

dtduuuccdttxy

x

a

t

a

x

a

)]()([exp)(exp)( 21

(2.5)

with ,xa 1c and 2c constants. The constants 1c and 2c are, of course, the con-

stants of integration and the relation (2.5) is the ’in principle’ general solution of

equation (2.1).

We now consider the basics of the solution of (2.1) via the CFM. The essential

mechanics behind the solution of (2.1), using the CFM, involves the repeated dif-

ferentiation of (2.1) with respect to x [8, 10, 14]. So, differentiating (2.1) k times

with respect to ,x we get the thk )2( - order linear ODE (for integer 0k )

0)()()()()( )()1()2( xyxQxyxPxy kk

kk

k (2.6)

where, for integer 1k

)(

)()()(

1

11

xQ

xQxPxP

k

kkk

(2.7a)

and

)(

)()()()()(

1

1111

xQ

xQxPxPxQxQ

k

kkkkk

(2.7b)

with )()(0 xpxP and ).()(0 xqxQ

254 W. Robin

At this point, instead of requiring a CF solution of (2.1) per se, we realize that

(2.6) can be considered, for every value of ,k as a second-order linear ODE for

),()( xy k that is, we may rewrite (2.6) as ( ,3,2,1,0k )

0)()()()()( )()()(

xyxQxyxPxy kk

kk

k (2.8)

with a corresponding sequence of Riccati equations

0)(),()(),(),( 2 xQkxxPkxkx kk (2.9)

and general solutions along the line of (2.5).

However, from (2.8) the ),( kx are related through the recurrence relation

,,,,, kkxαxP

xQ

xy

xykxα

k

kk

k

3210)1,()(

)(

)(

)(),(

)(

)(

(2.10)

when ),0,()( xαx the solution of (2.4), is given by the CF expansion

)(

)()(

)()(

)()(

)(

)(

)(

)(

)()(

3

32

21

10

0

0

xP

xQxP

xQxP

xQxP

xQ

xP

xQK

xy

xyx

k

k

k (2.11)

The relation (2.10) is a linear fractional transformation and is no surprise, as the

Riccati equation is well-known to be invariant under such transformations, which

are also well-known as generators of CF [11]. Indeed it is relation (2.10) that is

usually applied to obtain CF solutions, (2.11), to the ODE (2.1).

In practice, from (2.11), we are guaranteed a solution of (2.9), and hence (2.8),

if we have (or demand) for some integer nk

0)( xQn (2.12)

The condition (2.12) will terminate the CF expansion (2.11) and yield the actual

form of the solution, ),0,()( xαx to (2.4) and hence (2.1). (If the condition (2.12)

is not guaranteed, or is incapable of being forced, convergence problems will arise:

see below).

As mentioned in the introduction, Matamala et al [14] criticize this approach:

‘within CFM a previous laborious backward process is required for the integration

of Eq.(6) [essentially finding the solution of the ODE in question]’. This is indeed

the case as the resolution of (2.11) as a rational (algebraic) function does require

Some remarks on the asymptotic iteration method 255

(2.11) to be multiplied out (resolved). However, there is an alternative approach to

the CFM and, in section 4, we will derive the AIM from this alternative CFM. First,

we consider an example of the original CF formalism developed above.

3. The Hypergeometric-Type Equation

As an example of the formalism of section 2, we consider an ODE that encompasses

many well-known and important special cases [8, 16, 20], the hypergeometric-type

equation [16]

0)()()()()( xyxyxbxyxa (3.1)

with )(xa (at most) a quadratic function, )(xb a linear function and independent

of .x Apart from its inherent importance, the immediate interest of (3.1), is that

when we apply the methodology of section 2 to (3.1), the technical elements of the

CF solution process appear in closed-form and we achieve an ‘in principle’ com-

plete analytic solution of (3.1), a class of ODE that subsumes a considerable number

of important particular ODE [16].

Another reason for considering the CF solution process for (3.1), if one were

needed, is the relation of (3.1) to the more general equation of Nikiforov and Uva-

rov [16], that is

0)()(

)(~)(

)(

)(~

)(2

xzxa

xaxz

xa

xbxz (3.2)

with )(~ xa (at most) a quadratic function and )(~

xb a linear function. Equation (3.1)

is related to equation (3.2) by the substitution ),()()( xyxxz with )(x the so-

lution of a particular ODE (see reference [16] for details).

Moving on, we consider the CF solution of (3.1). So, following the procedure

of section 2, we differentiate (3.1) k times, with ,,,,k 3210 and get

0)()()()()( )()()(

xyxQxyxPxy kk

kk

k (3.3)

where

)(

)()()(

xa

xbxakxPk

and

)()(

xaxQ k

k

(3.4)

and

)()(2

)1(xbkxa

kkk

(3.5)

256 W. Robin

with (3.3) having a corresponding sequence of Riccati equations (2.9), with the

),( kx related through the recurrence relation (2.10), when ),(x the logarithmic

derivative of the solution of (3.1), is given by the CF expansion (2.11).

We expect to have a sequence of polynomial solutions to (3.1) if we apply, to

(2.11), the natural termination condition (2.12) or 0)( xQn exactly; that is, if

,n .,3,2,1,0 n In this case ),(x the solution of the Riccati equation corre-

sponding to (3.1), is

)(

)()()(

1

11

xy

xyxx

n

nn

(3.6)

and, after the CF (2.11) is reduced to lowest form, we expect also that we may pick-

off these polynomial solutions to (3.1) from the denominator of (3.6).

As a particular example, we consider (one form of) the Hermite equation

0)()()( xnyxyxxy (3.7)

which, when differentiated k times, yields

0)()()()( )()()(

xyknxxyxy kkk (3.8)

and we have, in agreement with (3.4) and (3.5)

xxPk )( and nkxQk )( (3.9)

We will work out the first few eigenvalues and eigenvectors of (3.7) using

(2.10)/(2.11). First, from (2.10) and (3.9), we have

,,,,, kkxαx

nk

xy

xykxα

k

k

3210)1,(

)(

)(

)(),(

)(

)(

(3.10)

To get, for example, the first four eigenvalues with corresponding eigenvectors,

from (3.10), we consider (from (2.11))

)4,(

)3(

)2(

)1()(

)()0,(

xαx

nx

nx

nx

n

xy

xyxα

(3.11)

So, first, in (3.11), set 0n and we find that constant,0 y which we choose as

one. Next, in (3.11), set 1n and

Some remarks on the asymptotic iteration method 257

xxy

xyxα

1

)(

)()0,(

1

1

(3.12)

and, to within a constant, x.1 y Now, in (3.11), set 2n and

1)1()(

)()0,(

22

2

x

x

x

nx

n

xy

xyxα (3.13)

And, again to within a constant, .122 xy Finally, in (3.11), set 3n and

xx

x

xx

xxy

xyxα

3

33

1

2

3

)(

)()0,(

3

2

3

3

(3.14)

and ,333 xxy correct to within a constant factor. For further examples of this

type of solution to (3.1), see David [8] (see, also, Sous and Al-Hawari [21]).

In the above example the termination condition 0)( xQn arose quite naturally;

this is not the case for eigenvalue problems in general, when the condition (2.12)

must be imposed and a more technical numerical solution of the problem must be

found (for an example of this, see reference [3]). We consider, next, how the AIM

can be recovered from the (more complete) CFM.

4. The AIM Arising from the CFM

As mentioned above, there is another way of developing a CF solution of (2.1)

directly,

that is, through an entirely ‘forward-moving’ calculation. If we set 0)( xQn in the

CF in (2.11), we get the nth-convergent or approximant of the CF in (2.11) [5, 11.

12], which, as a rational function, we write as (see also Ince [9] and Camacho and

Movasati [4])

,3,2,1 ,)(

)(

)(

)( 1

01

1

nxP

xQK

xv

xu

k

kn

kn

n (4.1)

In fact, following an induction argument [12], the nth-convergent of the CF in (2.11), can be determined through the well-known recursive relations (again, see

Camacho and Movasati [4])

)()()()()( 21 xuxQxuxPxu kkkkk (4.2a)

)()()()()( 21 xvxQxvxPxv kkkkk (4.2b)

provided [4, 12]

258 W. Robin

0)( ,1)( 12 xuxu (4.3a)

1)( ,0)( 1 2 xvxv (4.3b)

Equations (4.2) and (4.3), along with (2.7), then yield a purely forward iterative

scheme for the solution of the Riccati equation (2.4). When we approximate ),(x

the solution of the Riccati equation (2.4), using the nth-convergent (4.1) as

,3,2,1 ,)(

)()()(

1

11

n

xv

xuxx

n

nn (4.4)

then, from a particular case of (2.5), we get a series of approximations to the solu-

tion of (2.1) of the form ( 1nc constant, ,3,2,1n )

x

a n

nnn dt

tv

tucxyxy

)(

)(exp)()(

1

111 (4.5)

with the hope that, in the absence of a natural termination, ).()( 1 xyLimxy nn

Curiously, Matamala et al [14] actually derive the relationships (2.7) and then

(4.2) and (4.3), via a comparison of (2.8) with the AIM. However, they appear to

have overlooked the primacy of (4.2) and (4.3) in the CFM.

If we substitute (4.4) into the left hand side of (2.4), then we get the residual

equation for the Riccati equation (2.4), that is

)()()(

)()(

)(

)(

)(

)(

1

1

2

1

1

1

1 xRxqxv

xuxp

xv

xu

xv

xun

n

n

n

n

n

n

(4.6)

where the residual, ),(xRn may be expressed as (for example)

2

1

1111111 ][][

n

nnnnnnnn

v

pvuvuvqvuR (4.7)

and is a measure of the error involved in the approximation (4.4). If the solution is

exact, as in the example of section 3, then 0nR [2].

We now show that the AIM emerges from the above CFM via a ‘strong’ induc-

tion proof. With ,,3,2,1 k the AIM may be summarized as the following set of

equations

11)( kkk quuxu (4.8a)

111)( kkkk pvuvxv (4.8b)

with initial conditions

Some remarks on the asymptotic iteration method 259

)()( ),()( 0 0 xpxvxqxu (4.9)

and terminating conditions

0)()()()( 11 xvxuxvxu kkkk (4.10)

We wish to extract the formalism (4.8), (4.9) and (4.10) from the CFM (4.2)

and (4.3), with (2.7) in mind. Apparently, from (4.2) and (4.3), the proposition is

true for .0k We assume, now, that the AIM holds for k and 1k and derive it

for .1k We begin by differentiating (4.2a), to get

)()()()()()()()()( 2121 xuxQxuxPxuxQxuxPxu kkkkkkkkk (4.11)

Next, we eliminate )(xPk and )(xQk in (4.11) using the defining equations (2.7) in

the form

)()]()([)( 1 xQxPxPxQ kkkk (4.12a)

and

)]()()[()()()( 11 xPxPxPxQxQxP kkkkkk (4.12b)

(where we have eliminated )(xQk from (4.12b) using (4.12a)) to get, after collect-

ing like terms

)()()]()()()()[()( 11211 xuxQxuxQxuxPxPxu kkkkkkkk

)}]()()()({)()[( 211 xuxQxuxPxuxP kkkkkk

)]()()[( 12 xuxuxQ kkk

)()()()( 111 xuxQxuxP kkkk )]()()[( 1 xuxuxP kkk

)]()()[( 12 xuxuxQ kkk (4.13)

where we have used (4.2a). Making the induction assumption that (4.8) holds for k

and ,1k and making use of (4.2a) again, we find that (4.13) becomes

)()()()( 1 xuxqxuxu kkk (4.14a)

which was to be proven.

Moving on, we can repeat the above process for )(xvk from (4.8b), when we get

equation (4.13) again, but with )(xvk replacing ).(xuk As before, we make the in-

duction assumption that (4.8) holds for k and ,1k and, allowing for (4.2) yet

again, we find that

)()()()()( 1 xvxpxuxvxv kkkk (4.14b)

260 W. Robin

which was to be proven. The terminating condition (4.10) follows when we set

0)( xQk in (4.2).

Naturally, the equation in section 3 can be handled using either the CFM (4.2)

and (4.3) (with 0)( xQk ), or the AIM (4.8) to (4.10).

5. General Discussion and Conclusions

We have established the fact that the AIM is actually another way of determining

the CFM formalism, which, with its (the CFM formalism that is) full development

via the factorization method and relations (4.2) and (4.3), appears now in its most

complete and coherent form. However, a big advantage of the AIM representation

is that the AIM defining relations (4.8) and (4.9) deliver the convergent components

)(/)( xvxu nn directly, without actually considering the CF process that they are in-

timately connected with, and this may simplify the construction of solutions or ap-

proximate solutions. On the other hand, the basic factorization/CF approach devel-

ops the theory of the solution of (2.1) in a manner that enables the wholesale incor-

poration of the relevant CF theory into the solution process; in particular, we can

incorporate the theory of the convergence of CF into the factorization/CF method-

ology. For example, for the case of the hypergeometric equation itself, that is [24]

0)()())1(()()1( xyxyxxyxx (5.1)

with , and constants, we find, on differentiating (5.1) k times, with

,,3,2,1,0 k that

)())12(()()1( )()( xyxkkxyxx kk

0)())(( )( xykk k (5.2)

a result which is consistent with the scheme (3.1) to (3.4). Unless otherwise speci-

fied, there is no reason to set 0))(( kk and the CF [(2.11)] would not ter-

minate. In this case it is necessary to discuss the convergence of the CF and it is on

the matter of convergence in problems like this that the AIM has struggled [2].

Recently, following the original work on the convergence of CF solutions of (5.2)

by Ince [9], Camacho and Movasati [4], using a theorem of Poincare’s [17] (see

also Mate and Nevai [15]) discuss the convergence of CF solutions to (2.1) (specif-

ically Fuchsian ODE) and conclude (in the current notation) that the nth-convergent

(4.1) converges to the solution )(/)()( xyxyx of the Riccati ODE (2.4). This

general result seems to cap-off the CFM/AIM techniques for the solution of a wide

class of second-order linear homogeneous ODE.

Given the above, there are still a number of points that require further discus-

sion. First, as Amore et al [2] have shown, the AIM equations are not unique and

their analysis leads to the transformations

Some remarks on the asymptotic iteration method 261

)()()(

xuexu kxp

k

and )()(

)(xvexv k

xpk

(5.3)

which transform the AIM equations (4.8) into a second pair of recurrence relations

for the AIM [2] (compare (4.10) with (5.3))

111 )( kkkk puqvxuu (5.4a)

11 kkk uvv (5.4b)

provided we replace the initial conditions (4.9) with

)()()(

0 xpexuxp

and )()(

)(0 xqexv

xp

(5.5)

It is easy to show that the transformations (5.3) leave the residual analysis basi-

cally unchanged [2].

With this ‘ambiguity’ set aside, another problem that requires attention is the

solution of the AIM recurrence relations, (4.8). Naturally, it is possible to simply

step-through the relations one iteration at a time, but another possibility exists: the

recurrence relations (4.8) may have a solution in series. Cho et al, in a review of the

AIM [6], make use of Taylor series expansions of ku and kv which they substitute

into the (differential) recurrence relations (4.8) to (4.10) and reduce ‘the AIM into

a set of recursion relations [for the Taylor series coefficients] which no longer re-

quire derivative operators.’ However, as noted at the end of section 3, when used in

its more general form in eigenvalue problems [3] it proves necessary to force the

termination of the (CF) expansion and determine the eigenvalues and eigenvectors

numerically.

The third matter that arises for discussion is the necessity, in general, to trans-

form the variables to force the given problem (ODE) into a standard form to which

the AIM may be applied, that is, (2.1). This point was mentioned at the start of

section 3 in relation to the link between the equation (3.2) of Nikiforov and Uvarov

[16] and the hypergeometric-type ODE (3.1). However, more general examples of

this process abound and the technical points of these transformations have a life

independent of any particular solution method; see, for example, references [1],

[22] and [23].

In conclusion, we can say that we have shown that the CFM for second-order

linear ODE can be presented through the factorization method and ‘completed’ by

the well-known relations (4.2)/(4.3) (for example, Camacho and Movasati [4])

when the solution process is expressed as a forward iteration method for the solu-

tion of second-order linear ODE. And, when this ‘completed’ form of the CFM is

considered, it has proven possible to derive the AIM directly from the full CFM

formalism. In addition, we have drawn attention to a recent general theorem on the

convergence of CFM/AIM solutions for a wide class of second-order linear homo-

geneous ODE

262 W. Robin

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Received: June 17, 2016; Published: November 2, 2016