A family of Padé-type approximants for accelerating the convergence of sequences

A family of Padé-type approximants for accelerating the convergence of sequences

JOURNAL OF COMPUTATIONAL AND APPUED MATHEMATICS ELSEVIER Journal of Computational and Applied Mathematics 102 (1999) 287-302 A family of Pad&type...

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JOURNAL OF COMPUTATIONAL AND APPUED MATHEMATICS

ELSEVIER

Journal of Computational

and Applied Mathematics

102 (1999) 287-302

A family of Pad&type approximants for accelerating the convergence of sequences R. Thukral Pad6 Research Centre, 39 Deanswood Hill, Leeds, West Yorkshire LS17 SJS, UK Received

1 September

1997; received

in revised form 7 September

1998

Abstract We describe a collection of Pad&-type methods for accelerating the convergence of sequence of functions. The construction and connections of Padk’s methods with other similar methods are given. We examine the effectiveness of these new methods, namely integral PadC approximant, modified Pad6 approximant and squared Pad& approximant together with the well-established methods, namely functional Pad6 approximant and classical Pad& approximant, for approximating the characteristic value and corresponding characteristic function. Estimates of characteristic value and characteristic function derived using integral Pad& approximants are found to be substantially more accurate than other similar methods. @ 1999 Elsevier Science B.V. All rights reserved. Keywords: Integral PadC approximant; Classical PadC approximant; Functional Pad& approximant; Modified Pad6 approximant; Squared Pad6 approximant; Integral equation; Neumann series; Convergence acceleration

1. Introduction In this paper, three new methods for accelerating the convergence of sequence of functions are introduced and their effectiveness is examined by determining the characteristic value and the characteristic function of an integral equation. These new methods use functional Pad&type approximants, where we employ the terminology “Pad&type” as introduced by C. Brezinski, which means that the denominator polynomial of the rational approximant is arbitrarily prescribed (on the contrary, in the classical PadC approach the denominator is left free in order to achieve the maximal order of interpolation). We have introduced the appropriate names for the Pad&type approximants as modified Pad& approximant, squared Pad6 approximant and integral Pad& approximant. The modified Pad& approximant method was the first modification of the classical Pad& approximant and this was further improved by the squared Pad& approximant method. Finally, the integral Pad& approximant was developed and this is shown to be a good alternative to functional Pad& approximant. The main reason for developing the appropriate denominators of the new methods was to overcome the essential difficulty encountered by classical Pad6 approximants and functional Pad6 approximants. 0377-0427/99/$-see front matter @ 1999 Elsevier PII: SO377-0427(98)00229-5

Science B.V. All rights reserved.

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The major drawback of these methods is the use of the minimal sensitivity principle [ 1, 21 and the presence of superfluous zeros in the denominator. Hence, we investigated on the basis that these new methods should have a similar order and cofactors arrangement in the determinant of the denominator polynomial as classical PadC approximant and the accuracy of the functional Pad& approximant. Also we use a similar principle of integrating each of the cofactors in the determinant of the denominator polynomial, which was introduced by Graves-Morris [5, 7, II] and applied in functional Pad& approximant method. We describe the fundamentals of the denominator for each of the new Pad&type methods and the numerator is determined naturally. In order to construct these new methods we use a similar procedure as classical Pad& approximant. The construction of the denominator of the modified PadC approximant method is simply obtained by integrating each of the cofactor in the determinant of the denominator polynomial of the classical Pad6 approximant method. This improved method overcomes the use of the minimal sensitivity principle but it lacks the desired precision and therefore we have investigated this method further. We construct the denominator of the squared Pad& approximant method by squaring each of the cofactor in the determinant of the denominator polynomial of the classical Pade approximant method and then we integrate these new cofactors. The precision was good for some cases but we found that this method is not versatile. Hence, we decided to construct the denominator of the integral Pad& approximant method in a different way to those discussed above and in previously published papers [6-8, 111. The construction of the denominator of the integral Pad& approximant method is obtained by combining the coefficients of the generating function as cofactors in the determinant of the denominator polynomial. We found that the integral Pade approximant method is consistent and overcomes all the difficulties encountered in the previous studies [6, 7, II]. We begin with the generating function f(x, 2) of a series of functions given by f(Xp A) =

2

Ci(X)

/I’,

(1)

i=o

in which C,(x) ~L~[a,b] sense. We also suppose (1) converges for values Pad&type approximation

are given and [a,b] is the domain of definition of C,(x) in some natural that J”(x, A) is holomorphic as a function of 2 at the origin 1$= 0. Then of \I( which are small enough. In this paper, we see how the methods of can be used to accelerate the convergence of a series having the form (1).

1.1. The integral Pad6 approximants method (IPA) We define a rational function T(X,1) to be an integral PadC approximant

of type (n, k) for f(x, 2)

if r(x, 2) = N(& n)D(n), where N(x, A), D(n) are polynomials d(N)
a{D}
(2) in 1, N(x, 2) E &[a, b] as a function of x and (3a)

D(0) = 1,

(3b)

N(x, A) - D(A)f(x, 2) = 0(2+‘).

(3c)

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If N(x, A.), D(A) satisfy axiom (3), then there exists a unique Y(X,A) defined by (1). The proofs of existence and uniqueness are similar to that for the classical Pad& approximant [4, 5, 141 and the rate of convergence is similar to functional Pad& approximant. This is evident from this investigation -and in all the-other test examples. For the purpose of this paper, we define the denominator polynomial of an integral Pad& approximant of type (n,k) as h

h

I JlZ I G+~-~(x)G-&)

dx

a

I Cn+k-~(x)Cn(x)dx J”C,+~-,(X)C,-*,*(X)dX ... fkdx)C+~ Wx I

G+,-l(~>G-k,I(X)~

...

JlZ

Ja

S'C,I-,(x)C,~-I+,(x)dx

D(A)=

h

.

a

a

3

. . .

s

b

h

G+~-,(x)G-~(x>dx

.I a

LI

-k

A

s b

G+,-,(x)G(x) dx *k-l

A

...

G+,- I (x)Cn+k-I (x>dx

a

. . .

1 (4)

provided D(0) # 0 and C,(X) are the coefficients of (1). The purpose of integrating the elements in (4) is to make the estimates of the characteristic values independent of the variable x. This is similar to the approach for the functional Pad6 approximant in which we overcome the serious problem, use of [I, 21, with the classical Pad6 approximant [7, 1I]. We take the appropriate roots of denominator polynomial, given by (4), as our estimates of the characteristic value for the integral Pad& approximant. Naturally, the numerator polynomial N(x, A) follows from (3~) as N(x, 2) = KV)f(-% Gl;;,

(5)

where this notation, now and in the sequel, indicates that truncation at degree n in 1 has been effected. If, in the representation (4) D(0) # 0, then r(x, A) defined by (2), (4) and (5) is an integral PadC approximant of type (n, k) for f(x, A). Integral PadC approximants constructed using (3) can be laid out in a table: (090) (190) (2,O)

(0,l) (1,l) (2,l)

(0,2) (1,2) (2,2)

*.. **. ..* *.

(f-5)

and this concept is similar to the classical Pad& approximants [5]. Conjecture 1.1 (Integral PadC approximant).

f(x, 2) = Nx, WW)

Let (7)

be a meromorphic function with precisely k jinite poles. Then, for all n sufJiciently large, there exists a unique rational function r(x, A) of type (n, k) which interpolates to f (x, A). Hence

pir *

n

= f (x, A).

(8)

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Summaries of the advantages of integral PadC approximants are as follows: (9 We do not have to assign a particular value of x in the Neumann series, to obtain an estimate of the characteristic value using classical Pade approximant [7, 111. (ii) Integral Pade approximants produces a substantially more accurate estimates than the classical Pad& approximant, the modified Pad& approximant and the other two methods considered previously, namely Fredholm determinant and Rayleigh-Ritz [7, 111. (iii) The order of the denominator polynomial of an integral Pad& approximant is half the order of the denominator polynomial of the functional Pad& approximant. Therefore, an integral Pad& approximant does not possess superfluous zeros, which was a serious problem with the functional Pad& approximant as noted by the investigators [6, 7, 111. We do not have to construct a further method, known as the hybrid functional Pad& approximant, (iv) to obtain the characteristic function [6, 7, 111. (v) From the last two advantages it is established that the method of the functional Pade approximants needs much more numerical computation than integral Padt approximants. Integral Pad& approximant is applicable to a wider class of generating functions. We found a (vi) major drawback of the squared Pad& approximant is that the numerical performance of this method is not suitable when the generating function possesses an alternating or a negative power series. (vii) Integral Pad& approximant is simpler and more effective method for obtaining the characteristic values and the characteristic functions than other similar methods. 1.2. The modiJied Pad& approximant method (MPA) A modified Pad& approximant of type (n, k) for the given power series (1) is the rational function G, 2) = &,4/&A),

(9)

where A(x, A), B(L) are polynomials in a, A(x, A) E &[a, b] as a function of x and a(A)
=

a(B)


(W

1,

(lob)

A(x, 1) - B(l)f(x,

2) = O(?+' ).

(1Oc)

The construction of the denominator polynomial of the modified PadC approximant of type (n, k) is given as b

s

Cn-k+2

(x>dx

. . .

lbc-k+2 a

(x>

nk

dx

G+l

(x)

dx

s

(x>

dx

a

ll

B(L) =

J b

b

Cn-k+~ (x>dx

b

s”c-k_j a

(x>

dx

. . .

G+2

a

hk-I

...

provided B(0) # 0 and C(X) are the coefficients of (1).

1

(11)

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Naturally, the numerator polynomial A(x, A) follows from (10~) as A(& 2) = [B(l)f(x, a.

(12)

Each approximant of the sequence of (n, k) type modified Pad& approximant has precisely k poles. We take the zeros of (11) as our estimates of the characteristic value for the modified Pad& approximant.

1.3. The squared Pad6 approximant method (SPA) A squared Pad6 approximant of type (n, k) for the given power series (1) is the rational function T(X,2) = G(& n)/H(J),

(13)

where G(x, A), H(i) are polynomials in A, G(x, d) ~Lz[a,b] as a function of x and d(G)
d(H)
(14a)

H(0) = 1,

(14b)

G(x, II) - H(i)f(x,

2) = 0(/Y+‘).

(14c)

The construction of the denominator polynomial of the squared Pad6 approximant of type (n, k) is given as

/bC:-,l,cx, dx /“C;_k+2(x) dx . . . a a a

/“C;+,(x)

H(2) =

/bC~-k+Lx) dx a

/-bC_k+3(x) dx a

...

...

dx

/bC;+Z(x) dx , a

1

(15)

I

provided H(0) # 0 and Ci(x) are coefficients of (1). Naturally, the numerator polynomial G(x, A) follows from (14~) as

G(x,A>= [W~)_f(x,U;;.

(16)

Each approximant of the sequence of (n, k) type Pad& approximant has precisely k poles. We actually take the square root of the zeros formed by (15) as our estimates of the characteristic value. The outline of this paper is as follows. In Sections 2 and 3 we briefly describe two well-known methods, the classical Pad& approximant and the functional Pad& approximant, respectively. Moreover, in Section 4, we demonstrate the similarity of the three methods, namely, the integral Pad& approximant, the squared Pad6 approximant and the functional Pad6 approximant, which produce identical estimates of the characteristic value. In Section 5 we examine the effectiveness of these new methods based on the Pad&type approximants for determining the characteristic values and the characteristic functions of an integral equation. The technique utilised for solving the integral equation is based on successive substitution, which is an iterative procedure, yielding a sequence of

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approximations leading to an infinite power series solution. In the process we make two distinct comparisons of the estimates derived using the integral Pad& approximant. First, we compare estimates formed using the row sequence of an integral Pad& approximant of type (n, 1) with corresponding estimates derived from the modified Pad& approximant of type (n, l), the squared PadC approximant of type (n, l), the functional Padt approximant of type (n, 2) and the classical PadC approximant of type (n, 1). Then we compare estimates based on another row sequence of an integral Pad& approximant of type (n, 2) with corresponding estimates derived from the modified Pade approximant of type (n,2) the squared PadC approximant of type (n, 2), the functional PadC approximant of type (n,4) and the classical PadC approximant of type (n, 2). In Section 6 we illustrate the precision of a particular characteristic function of integral Pad& approximant. The effectiveness of these new methods for accelerating the convergence of a sequence of functions was investigated in the context of the Neumann series of an integral equation. The method of integral Pade approximants proved to be the most effective of the methods considered.

2. The classical Pad& approximant method (CPA) A classical Pad& approximant

of type (n,k)

for the given power series (1) is the rational function

r(x, n> = U(x, A)/V(x, A),

(17)

where U(x, A), V(x, A) are polynomials a(u)


V(0) =

in II, U(x, A) ELZ[U, b] as a function of x and

a{v}
(184

1,

(lgb)

U(x, ;1) - V(X, 2) f(x, ;1) = O(p+k+‘). The construction as

of the denominator

(18~)

polynomial

of classical Pade approximant

C,_,+,(x)

Cn-k+2@)

. *.

G+l(x)

G-,+2(x)

G-k,3(X)

. . .

Cn+2(x>

V(x,A) =

of type (n, k) is given

(19) qk A

1k-l

. . .

1

provided V(x, 0) # 0 and C(x) are coefficients of (1). Naturally, the numerator polynomial U(x, 1) follows from (1%~) as W,

A) = [V(A) j-(x, A)];;.

(20)

Each approximant of the sequence of (n, k)-type PadC approximant has precisely k poles. To determine these zeros, in order to estimate the characteristic value, we must assign a particular value of x in the Neumann series and this is usually done using the principle of minimal sensitivity [ 1, 21. Issues of existence, uniqueness and other related definitions of classical Pad& approximants are treated in [4, 51 and many other texts.

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3. The functional Pad6 approximants method @PA)

We define a rational function r(x, A) to be a functional Pade approximant of type (IZ,2k) for f(x, A) if

4% A>= P(X,q/q(A),

(21)

where p(x, A), q(1) are polynomials in A.,p(x, A) E &[a, b] as a function of x and d(p)


a(q)<2k 41)

- 2a

llbi

p(*.4i2

for CI20,

(2%

dx,

Wb)

q(A) = q*(4,

(22c)

q(0) # 0,

(22d)

P(X, A) - q(A) f(X, 1) = 0 (A”+’ ).

We)

The asterisk in (22~) denotes the functional complex conjugate. If p(x, A), q(1), satisfy (22a)-(e), then r(x, A) defined by (1) is unique; the questions of existence, uniqueness and degeneracy are treated in [9]. The explicit formula for the denominator polynomial is given by 0

MO1

-Mot

0

i

;

q(A)=

-MO,2k-I A2k

* *. 440,2k-I

MO,2k

” ’Ml,2k-I

Ml,2k

i

;

-Ml,2k--1

’. ’

0

22k-'

...

i

;

(23)

M2k--1,2k

1

The elements of (23) are defined by Mj=

'-~'J~~~+~+~-2k+~(~)[~j-~+,,x(x)I* i=O

2k for i=O,l,..., We know that occur in complex the zeros of q(A) The numerator

dx,

(24)

a

andj=i+ l,i+2,..., 2k and taking Cj(X) := 0 if j < 0. the polynomials produced by (23) are strictly positive for 1 E R and their zeros conjugate pairs close to the real axis [6, 7, 111. Ideally, we take the real parts of as our estimates of the characteristic values 1,. polynomial p(x,A) follows from (22e) as

P(X, A) = [q(A) _I%

41;;.

(25)

If, in the representation (23), q(0) # 0, then r(x,L) defined by (21), (23) and (25) is the functional Pad6 approximant of type (n,2k) for f(x,A).

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In addition, the denominator polynomial of a hybrid functional Pad& approximant is defined in terms of the roots of the denominator of the corresponding functional Pad& approximant [6, 7, 111. We actually take the real parts of the roots of the functional Pad& approximant and express the hybrid functional Pad& approximant of type (n,k) as qH(3L> = fi

(A - n” ).

(26)

i=l

The associated numerator polynomial is defined as pH(x, A) = [.0x, A) c?“Gw*

(27)

Since n and k govern the degree of the numerator and denominator, respectively, we express the hybrid functional Pad& approximant of type (n, k) as pH(x,A) = pH(x, Q/qH(Q.

(28)

We find that (28) is identical to the integral Pad& approximant (2) for k = 1. It is obvious that the hybrid functional Pad& approximant is dependent on functional Pade approximant and therefore requires much more numerical computation than the integral PadC approximant. We can easily prove this by comparing the dimensions of the determinant and the degree of the denominator polynomial of functional Pad& approximant and integral Pad& approximant, which are given by (23) and (4), respectively.

4. Equivalence of the estimates Here we shall observe how three methods, namely integral Pad& approximant, the squared PadC approximant and the functional Pad& approximant produce similar estimates of the characteristic value. We shall do this by showing that the limit of the poles of the respective denominator polynomial satisfies identical equations. First we begin by expanding (4), the denominator polynomial of the integral Pad& approximant of type (n, 1):

D(n)=

SbCn(x) C,_,(x) dx a

A.

/-“C.‘(x) dx a 1 ’

which gives b

D(A) =

sa

b

C,(x) C,_,(x)dx

- ;1 s

a

C,‘(x) dx.

(29)

The characteristic value of the integral Pad& approximant of type (n, 1) is calculated by solving (29) and thus we have A=

j-” C,(x)C,,_,(x) (I

dx

s”C,'(x)

I a

dx.

(30)

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Similarly, expanding the denominator polynomial of the functional Pad6 approximant of type (n,2), b

J

0

4(A)= _

a

cl

b

0

C;_,(x)dx

J

a

J Cn-,(x)G(x>dx b

C,2_1(~)dx 2

/I2

1

b

Ja

C,'(x)

dx

1

which gives

q(A) =

J" C,'(x) dx c,‘_,(x) dx J" c,‘_,(x) dx + ;12 a 0

J"

a

Ja

-21

dx [sa (x) 1. b

b

c,‘-I(X)&

1

G(x)C,-,

(31)

Solving (3 1) by a standard quadratic formula and simplifying gives rise to

(32) We know from previous studies [6, 7, 1l] that (32) produces a pair of complex conjugates which are close to the real axis, that is (33) and b

G(x)C-l(X)d.x 1’ - [~bCI(X)dxjlhC~_,(x)dx].

(34)

Therefore, the estimate of the characteristic value of the functional Pad6 approximant of type (n, 2) method is given by 6

A=

G(x)Cn-l(X)

dx

(35)

Alternatively, we can find the estimates of the functional Pad6 approximant by differentiating (3 1) w.r.t. I and with a convenient normalisation we obtain (35) [8, 111. First we begin by expanding (15), the denominator polynomial of the squared Pad& approximant oftype (n,l)

J C,2-,(xWJ~ C,‘(x)dx , b

b

H(;1)

=

a

1

1

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which gives b

H(A) =

sa

b

&(x)dx

- 1

The characteristic we have

sa

(36)

a

value of squared Pade approximant

of type (n, 1) is based on solving (36); thus

b

b A=

C,‘(x) dx. s

c,‘-,w

C,‘(x) dx.

d.x II

If we rearrange

(37)

a

(34), we have (38)

and substituting

the right-hand

side of (38) into (37) we obtain

(39) It has been established in [5, 6, 8, lo] that the zeros, that is characteristic PadC approximant converges as

value, of the functional

lim A,=A2. il-IX This result also applies to the squared PadC approximant. However, we actually take the square root of (37) as our estimates of the characteristic

(40)

value

(41) Therefore, we expect that, for all 12 sufficiently large, (41) is equivalent to (30). It should be noted that this is true mathematically, but we find a serious problem numerically that is when the generating function possesses an alternating or a negative power series. Mathematically, we have demonstrated the fact that these three methods produce a similar accuracy for the estimates of the characteristic value, that is (30), (35) and (41) are similar for the first row sequence. We conjecture that these three methods produce identical estimates for any type of row sequence. However, numerically we find that the integral Pade approximant has better precision than the squared Pad& approximant and the functional Padt approximant. This is empirically evident from Table 2.

5. Application to an integral equation To determine the consistency of these new methods, we actually tested them on a previous investigation [ 1 l] and further examples were taken from Moiseiwitsch [ 121. These findings are

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generalised by illustrating the effectiveness of these new methods for determining the characteristic values and the characteristic function of a familiar linear integral equation in the following example. We investigate the convergence of sequences of these new methods for the Neumann series solution of the linear integral equation

f(x, A)= 1 + /J

J0

’%v)f(y,

2) dy>

where 1+x-y, l+y-x,

k(x, v) =

Odydxdl, O
This integral equation is a Fredholm of the second kind with a nondegenerate kernel and has been previously considered by Graves-Morris [7] and Coope and Graves-Morris [6]. The characteristic functions of this equation can be found by converting it to a second-order ordinary differential equation [3]. The explicit solution of (42) is 2 cash v(x - l/2) f(x, i) =

(43)

2 cash v/2 - 3v sinh v/2 ’

where v = v%. The denominator of (43) is analytic as a function of 1 and has just one simple zero at vc = 1.22290658 . . ., corresponding to a single characteristic value A.,= 0.7477502556.. .. It is familiar that the Neumann series of (42) converges for ]A)< 1, [ 121. The first few terms of this series are f-(x, A) = g

C;(x)2 = 1 + [45 + (_$]A+

[EL+;(x-f):+f(x-t)‘]i:,....

i=o (44) as may be found by iteration of (42). We make two distinct comparisons of the estimates derived using the integral Pade approximants. First, we compare estimates formed using the row sequence of the integral Pad6 approximant of type (n, 1) with corresponding estimates derived from the classical Pad& approximant of type (n, l), the modified Pad& approximant of type (n, 1), the squared Pad& approximant of type (n, 1) and the functional Pad& approximant of type (n,2). The results in Table 1 are the estimates of the characteristic value for each of the five iterative methods described and we find that the estimates from the integral Pad6 approximant gives better approximations that the classical Pad& approximant, the modified Pad& approximant and it is similar to the functional Pade approximant and the squared Pade approximant. The results in Table 2 are the estimates of the characteristic value, but showing results derived from another row sequence of the integral Pad& approximant of type (n,2) and with corresponding estimates derived from the classical Pad6 approximant of type (n,2), the modified Pad& approximant of type (n, 2), the squared PadC approximant of type (n,2) and the functional PadC approximant of type (n,4). We see that the row sequence of the integral PadC approximants gives better approximation than the other methods, including the functional Pad& approximants. In each case, the comparisons with other methods were made using a similar amount of data, which is, using a similar number of terms of (44). The proof that the row sequence of Pad& approximant estimates shown in Table 1 converge geometrically to the characteristic value follows as a consequence of SalI’s [ 141 extension of Montessus’

R. Thukrall Journal of Computational

298 Table 1 Estimates

showing the precision

of the characteristic

and Applied Mathematics

102 (1999)

value 1, derived using the five methods

287-302

described

IPA = FPA

SPA

MPA

CPA

n

X

1,

1,

1,

1 2 3 4 5

0.74766 0.74775011 0.7477502553 0.7477502555635 0.7477502555638444

0.7488 0.747752 0.7477502583 0.747750255568 0.747750255563853

0.75 0.74766 0.747754 0.74775011 0.747750261

0.8 0.745 0.74785 0.747746 0.74775043

aThe exact value of I, = 0.747750255563845043.. Table 2 Estimates

showing the precision

of the characteristic

..

value 1, derived using the five methods

described

IPA II:

FPA

SPA

MPA

CPA

n

n,

1,

2,

1,

1 2 3 4

0.7475011 0.74775025556375 0.747750255563845035 0.7477502555638450433881

0.7453 0.74775025556262 0.74775025556384496 0.7477502555638450433815

0.747755 0.7477502557 0.747750255563858 0.747750255563845045

0.74767 0.7477504 0.7477502545 0.74775025557

0.7486 0.74774 0.7477504 0.74775025

“The exact value of & = 0.74775025556384504338894..

theorem [13]. Likewise, the proof that the functional Pad& approximant estimates shown in the Table 1 converge geometrically to the characteristic value, follows as a consequence of the row convergence theorem of Graves-Morris and Saff [lo]. All these mathematical results are empirically evident from Tables 1 and 2. It is also clear that the integral PadC approximant method converges much faster than the other methods. This is due to the standard theory of Fredholm integral equations of the second kind [12, 151, that is expressed by (45) it is known that when g(x) fkY)=

?? L~[a,b]

and k(x, y) is an L2 kernel,

m, a) Q(q

(46)

3

where (i) P(x, A) is holomorphic as a function (ii) P(x, A) E L2[a, b] for a fixed I, (iii) Q(n) is holomorphic function of 1,

of 2,

(iv) QW # 0. We have noted that the more serious practical difficulty with the classical Pad& approximant method is that the estimates of the characteristic values of 1 depend on the value of x selected contrary to the exact result (46). If we use the integral Pad& approximants for accelerating convergence of the Neumann series, the estimates of the characteristic values are independent of the variable x, as expressed in (46). This observation goes some way towards explaining the remarkable precision

R Thukrall Journal of Computational and Applied Mathematics 102 (1999) 287-302

299

of estimates of the characteristic values derived from the integral PadC approximants as shown in Tables 1 and 2.

6. Precision of the approximate solution In Fig. 1 we display the exact (analytic) solution and its approximations obtained using the integral Pad& approximant of type (2, 1 ), the modified Pad& approximant of type (2, 1), the squared Pad& approximant of type (2, 1) and the functional Padt approximant of type (2,2). Also in Fig. 1 we see a remarkable precision of the integral Pad& approximant, where graphically there is no significant difference between the exact and the integral Pade approximant. Accordingly, in Table 3, we show the errors incurred by the integral Pad& approximant, the modified PadC approximant, the squared Pad& approximant and the functional Pad& approximant methods for x = 0(0.1)0.5 in the solution of (42). We list the appropriate rational functions displayed. Solution of the integral equation (42) using the integral Pad& approximant of type (2, 1) is

N(x, 2) = 1+(X*-X+~)~+(~x4-fX3+~X*+$&jx-&J~*

r(x,;l) = -

(47)

1-72337A

D(n)

54090

Fig. 1. The analytic solution (exact) of (42) for 1= 1. The curves IPA and MPA are indistinguishable curve. Whereas the curve FPA is perceptibly different and SPA is beyond the exact curve.

from the exact

Table 3 Errors occurring in the solution of (42) using the integral Pade approximant, the modified Pad& approximant, the squared Pad& approximant and the functional Pade approximant method

X

0

0.1 0.2 0.3 0.4 0.5

IPA a=1

MPA a=1

SPA a=1

FPA a=1

-0.00088 -0.00066 -0.0002 1 0.0003 1 0.00070 0.00085

-0.0036 -0.0032 -0.0026 -0.0020 -0.0016 -0.0014

-3.39 -3.19 -3.04 -2.93 -2.86 -2.84

0.069 0.050 0.013 -0.026 -0.053 -0.063

300

R. ThukrallJournal

of Computational and Applied Mathematics 102 (1999) 287-302

Solution of (42) based on the squared Pad& approximant

W, 4 = 1 + (x’ - x + fi)

r(x,ll) = ~ H(l)

p(x,/l) ___ 4(l)

A + (ix” - fx’ - fix’ 1-723371

=

1 + (x’ -x

PadC approximant

&x,/q

B(A)

- =)A” (48)

=

of type (2, 2) is

- %)A. + (ix” - ix’ - $$x’ - $&x

- $$$)A’ (49)

1-$$/A+~1*2

Solution of (42) based on the modified Pad& approximant t-(x,2) = ___

+ Ex

40446

Solution of (42) based on the functional y(x,A)=

of type (2, 1) is

1+

(x2 -

x +

of type (2, 1) is

6)” + (ix” - ix’ + $x2+ &x - $)A’. l-En

(50)

The rational functions above were suitably normalised. It has been established in previous studies [6, 7, 1 l] that the estimates of characteristic function based on the functional Pad& approximants are inferior to those from the hybrid functional Pad& approximants. The reason for the poor performance of the functional Pad& approximants for the estimation of the characteristic function is that the denominator polynomials (11) possess superfluous zeros, and thus the hybrid functional Pad& approximant method was introduced. We do not use the hybrid functional Pad& approximant, not because it produces identical results to the integral PadC approximant, but because of much more numerical computation involved in deriving it. Furthermore, we have found that the precision of the characteristic function for the squared Pad& approximant method is considerably smaller than the integral Pad& approximant. Finally, although the approximate solution for the characteristic function for the modified PadC approximant is satisfactory, it lacks the precision of the integral Pad& approximant.

7. Remarks and conclusion Three new methods for producing a sequence of rational approximations to the solution of a linear integral equation have been described and their effectiveness has been investigated in many examples. These new methods are essentially for accelerating the convergence of a sequence of functions. In the context of a familiar linear integral equation, the method of integral Pad& approximants is shown to be much more efficient in calculating the characteristic value and substantially more accurate for calculating the approximate solution than other similar techniques. The purpose of demonstrating the integral Pad& approximants method for two types of row sequence is to illustrate the accuracy of the approximate solution, the stability of the convergence and the consistency of the method. Furthermore, we have demonstrated that this new method is much more efficient and does not have the drawbacks of the functional Padt approximants and the classical Pad& approximants. From this illustrated example and in all other test examples, it is clear that the hybrid functional Pad& approximants method is inferior to the integral Pad& approximants for k >, 2, because of the accuracy of the characteristic values of the functional Pade approximants is less than the integral Pad& approximants method. Finally, an analytical investigation of the integral PadC approximant is a subject of further research.

R. ThukrallJournal

301

of Computational and Applied Mathematics 102 (1999) 287-302

Acknowledgements I am greatly indebted to an anonymous on this paper.

referee and Professor

D.B. Ingham for helpful comments

Appendix. Application to another integral equation We illustrate the effectiveness of new methods by taking another well known example. We investigate the convergence of sequences of integral Pade approximants for the Neumann series solution of the linear integral equation f(x,A)=

1 +A

’ 4x, vV(v) s -I

dx

(A.11

where @,

Y>=

8-71

+ y)(l

88’(1 +x)(1

-x),

-1
- y),

-1 <~dy
1,

This integral equation is a Fredholm of the secod kind with a nondegenerate kernel previously considered by Graves-Morris and Thukral [l I]. The characteristic functions of this equation can be found by converting it to a second-order ordinary differential equation [3]. The explicit solution of (1) is f(x, &) = sin[22’s7r( 1 +x)1, Table 4 Estimates

showing the precision

n

IPA= n;

1 2 3 4

10 9.871 9.86962 9.8696046

s E N,

of the characteristic

FPA

a The exact value of 11 = r? = 9.8696044010.. Table 5 Estimates

showing

the precision

IPA

value 11 derived using the five methods

described

SPA 11

MPA

CPA

II

II

10.95 9.877 9.86969 9.8696054

12 10 9.88 9.871

8 9.6 9.84 9.866

.

of the characteristic

value Ii derived using the five methods

described

n

1;

FPA 11

SPA /21

MPA II

CPA 3II

1 2 3 4 5

9.871 9.8696046 9.8696044013 9.8696044010896 9.8696044010893593

9.24 9.869609 9.8696044025 9.869604401091 9.869604401089361

10.03 9.86964 9.86960444 9.86960440115 9.86960440108946

10.25 9.8751 9.86975 9.869609 9.8696046

10.14 9.878 9.8699 9.86962 9.869605 1

a The exact value of /21= n2 = 9.8696044010893586188..

R. Thukrall Journal of Computational and Applied Mathematics 102 (1999) 287-302

302 Table 6 Estimates

showing the precision

of the characteristic

value 13 derived using the three methods

described

IPA

FPA

SPA

MPA

CPA

n

4

A3

A3

13

13

1

-

-

-

-

2 3 4 5

90.1 88.97 88.845 88.8289

104.5 90.26 88.999 88.848

170.1 102.1 92.6 90.1

64.79 78.31 84.52 87.15

a The exact value of 13 = 9n2 = 88.8264396..

with corresponding As = (s7#,

characteristic

126.1 90.99 89.07 88.858

.

value

SEN.

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