(0, 1, 2, 4) Interpolation by G -splines

Perio&.a

Mathenmtica

Hungarica

VoZ. 21 (4), (1990),

(0, 1, 2, 4) INTERPOLiiTION A.

SAXENA

pp. 261-271

BY G-SPLINES

(Lucknow)

1. Introduction Saxena and Tripathi [3], [4] h ave introduced spline methods for solving the (0, 1, 3) interpolation problem. They have used spline interpolants aA c L$$\ *t3) of degree six for functions / c C6 to solve the problems ad 8: C fin,6 @(a$

= yp), q = 0, 1, 3, k = O(1) n, $&J

= 9;;

and

are given reals. Their Let / c (Y(I).

convergence

results are as follows:

Then for q = o(l)& /j@’ - fq’ ~~~-~x~,x~+~~ < &h5-’

%@h

II sZq) - fq) lL.-[x~,x~+J < w5-q(%m~ where 14whenk=O 60 when k = l(l)n

-

1,

and I&= Here ti6( -) is the modulus

591 when k = 0 60 when k=l(l)m--2 27 when k = n - 1. of continuity

of f@.

AMS (MOS) Subject du-v~+cutkns (1980/85). Key words ati phmwx. Deficient Chpline.

Primary

4lA.

262

SAXENA:

(0, 1, 2, 4) INTERPOLATION

BY

G-SPLINES

Fawzy [2], in one of his very recent, papers, has solved the (0, 1,3) interpolation problem by constructing a spline method using piecewise polynomials such that for functions f c C4, the method converges faster than the method given in [5] for solving the same interpolation problem, and for functions f c C5, the order of approximation is the same as the best order of approximation using quintic splines. For this purpose he has used special kind of g-splines, to which he refers as lacunary g-splines. Motivated by the method and results of Fawzy [2], we solve here a different interpolation problem, namely, the (0, I, 2,4) interpolation problem, which is formulated as follows: Given a uniform

partition

A: of the interval

q

1 = [0, 1] and real numbers

find 8 in a suitable (1.1)

o=~o<~~<...<~~-~~~~=l

class such that sql$

= yp, q = 0, 1,2,4,

k = O(l)?&.

Our aim here is to introduce a spline method for solving the above problem such that for functions f c C6, the method converges faster than the methods given in [3] and [4]; and a method which, for functions f c C’, gives the same order of approximation as the best order of approximation using splines of degree seven. We find the solution in the class of deficient g-splines S$) for functions f c C6 and in SFfj for f c C7. In Section 2 we define the class Sz$) and construct the spline a* c S$) which is the unique solution of the problem (1.1). Error bound of the spline interpolant is given in Section 3. Same scheme for the class S$$ is carried out in Section 4. Application of these spline interpolants is given in Section 5. We finish the paper by making some remarks in Section 6.

2. The

Class

S:,(g) -

Existence

and

Uniqueness

We define the class of spline functions Sz$) as follows: .T~ c SE(i) if the following conditions are fulfilled: s* c P(1),

(2.1) (5w

aA C n6 in bk

and

zjl+l ] for each k = O(l)n sA

C

x7

in LznMl,4.

-

2, and

Any element

SAXENA:

(0, 1, 2, 4) INTERPOLATION

263

BY G-SPLINES

We construct sA in order that it is a solution of (1.1) for functions For this purpose we set s

sk(x) when xk < x < x~+~, k = O(l)% A - L s~-~(x) when xnmI 5 x < x~.

f c C6( I).

2,

Owing to (1.1) and (2.2), we can write

The coefficients Thus we have

ak,j are determined

in terms of the given data using (2.1).

1

{2.,5)

ak,3

=

-ak

-

+k

d-

$k?

24

(2.6)

h2ak$ = -.&zak

w71

h3tZk,6 = 5ak -

+ 4,%‘ - t yk, 14j3k + 10yk;

and 1 an-l,3 = Ean-l

(2.10)

h3anvl,6=

{2.11)

h4an-l,,

3%-,

-

= --6anml

+ 21&-l

-

21~~-,

+ 7dn-l;

where

hyk = 24 Y;+~ -

h2 y; ---yjf) .

I

;

264

EAXENA:

(0, 1, 2, 4) INTERPOLATION

BY W3PLINF,S

and

THEOREM 2.1. For a uniform unique epline function aA C S$j problem (1.1).

partition /J of the interval I, there exists a which is the solution of the interpolation

3. Error Let f c @(I) and d$) = fd(zk), prove the following

Bounds q = 0, 1, 2,4;

k = O(l)n -

1. We shall

THEOREM 3.1. Let sA c S$i) 3 be the eolution of the problem (1.1). Then for f c @(I), we huve

where q = 0(1)6, k = O(l)n table:

1, and the wn.stants c~,~ are given in the follin&g Table.

ck, @

O
1

-$j-

1ooa

ck,z

ck,8

‘k,4

ck,s

ck,e

11

56 -iii-

-

23 2

19

16

-

-ii-

245

k=n-1

ck, I

1

63

-

3-

For the proof of this theorem

8

739 120

-

-- p’(xk)]

218

5

we shall need the following

LEMMA 3.1. We have for all k = O(l)?a 1ak,j

118

< %,jh6-‘%(h)y

2, j = 3, 5

lemmas:

SAXENA:

(0, 1, 2, 4) INTERPOLATION

BY

265

G-SPLINES

where

If f c @(I),

l?ROOF.

= O(l)?& -

then

using

Taylor’s

formula

we have

for

2, cxk= 12Oy;’

+ 6h2yf) + h3f@)(lk),

= 6Oy;’

+ 5hzyf)

+ hsf@+j,J,

yk = 24yi”

+ 4h2yf)

+ h3f@)(zYk);

bk

and where xk

SubstituCng LEMMA

<

lkp

qky

‘k

<

xk+l*

these values in (2.5)-(2.7),

3.2. The foh?owing 1 f&l,j

-

e&mat@

fG’(XnpJ]

1*&,6

-

f@‘td~

we at once get the lemma.

are valid.

< Cn-l,jh6-‘C06(h), < 7hV4

%-l

j =

3, 5,

< x < x,,,

and

where cn-

PROOF.

For

f c@(I)

17 -120

and q-l,3

= 11.

we have by Taylor’s

formula

I,3

=

anMl = 840~;:~

+ 42h2y$il

+ 7h?f@)(&-J,

&-I

= 3W&‘Ll

A- 30h2y$yl

+ 6h3f@+jn-J,

yn-l

= 12oy;:1

+ 20h2yfll

+ 5h3f@)(Ql),

and dnml = 6h2yzLl Substituting PROOF

+ 3h3fc6)(A,&

these values in (2.8)-(2.11) of Theorem

3.1. Let first be

we get the lemma.

k E

266

SAXENA:

From (2.3) and writing

(0, 1, 2, 4) INTERPOLATION

finite Taylor

-t- ‘xt~~~~q

BY

G-SPLINES

sums for j(z) and its derivatives,

(G,~ - f6~(i%,q)) when

we have

q = 0, 1, 2, 3,

when q = 4,5, a k,6 -

f@)(x) when q = 6;

where xk

Now

using the estimates

<

tk,q

c

‘k+l-

given in Lemma

3.1 we obtain

the theorem

for

k=O(l)n-2.

The other part of the theorem, the same way using Lemma 3.2.

4 . The

when xn-r < x < x~, can be. obtained

Class

L5$$ denotes the class of functions (4.1) (4.2)

in

Surf’

GA(x) such that

a* c Q2Uh GA C 7z7 in [xk, xk+J,

for each k = O(l)n

-

1,

and (4.3)

GF(xk + 0) = Gz)(xk - 0), k = O(l)n

We construct GA such that it is the solution f c V(I). For this purpose we set GA = Gk(x), k = O(1) n Owing to conditions

(1.1) and (4.2), we can write

-

1.

of the problem 1.

(1.1)

for

SAXENA:

(0, 1, 2, 4) INTERPOLATIOX

BY G-SPLIKES

267

If we apply the continuity requirement (4.1) and the condition (4.3), we obtain the coefficients bk,j, (j = 3, 5,6, 7) in terms of t,he given data as follows:

(4.6) (4.7)

h3bk,C = 3ak - T,fIk

+ 10~~ -f&,

and (4.8)

h4bk,7 = -6q.

+ 21& -

21yk + 7&;

where

and dk = 6h(yfi

I -

yj,?)).

It is now obvious that GA is a unique element of Se!) which solution of! the interpolation problem (1 .l). The following convergence theorem can easily be established.

is t,he

THEOREM 4.1. Let GA c S,,, c2d be the unique solution of the interpolation problem (1.1). Then for f c V(I), we have for aU q = O(l)? and k = O(l)n - 1,

where the co&ants

c& are given in the following Table

2

table:

SAXENA:

(0, 1, 2, 4) INTERPOLATION

5. Application

BY G-SPLINES

to Differential

Equation

In this section we develope a global method for approximating solution Y(X) of the Gauchy’s initial value problem (5-l)

9” = fb

the exact

92 Y’L !/@I = !/II, Y’o4 = z.4;

over the entire interval [0, 11. Here we wurne that

and that it satisfies the Lipschitz (5.2)

lfqT%

3% 99

-

f@%%

condition

Y27 !A)1

<

q

(1.2). !Jl

-

Y2l

+ I 34 - !I; II?

(q = O(l)?-) for all x c [0, 1] and all reals yi, y2, &, &. These conditions ensure the existence of unique solution of problem (5.1). Our method consists in converting a discrete variable method into a global method through spline interpolation. For this purpose we use Fawzy’s definition given in [l] to obtain the numerical value of

which are approximations

to the exact values

With these values we construct a spline function ZA c AS$ (respectively aA c E$$‘)) which will be an approximation to the exact solution of (5.1). The set F@J)is defined as:

Xk

=-,

’ n

k=O(l)a,

h=L n

SAXENA:

The error qualities (5.3)

jygl

(0, 1, 2, 4) INTERPOLATION

of the approximate

- g&j

values $&I

< cjcor+2(h)h~+~,

BY

269

G-SPLINES

are estimated

AT= O(l)% -

by the ine-

I, j = O(l)?. + 2.

For the values fjI and gi, even sharper estimates (5.4)

IYl - ?&I < %%+*wr+49

are valid (see [1] Lemma 2.2.1 and IZ++~(~) is the modulus of For the construction of Here we replace yk’a by &‘a in Gk,]. We prove the following

IY; - !xj < w%+2v4~r+3

and 2.2.3). Here cj’s denote different constants continuity of y@+‘)(z). g*(z) we use the spline method of Section 2. the definition of S*(Z) and the numbers u@,~by

5.1. Let g(x) be the exact .soWion of (5.1). Then

THEOREM

1y(q+-$ - $,$f(x)j < Aqh6-qm6(h), q = 0(1)6.

(5.5)

The proof of this theorem

is an easy consequence of the following

LEMMA

is the spline function

5.1. If Q(X)

c fl;$’

wnstvucted with the exact

values yi?, then (5.6)

1s?(x) - gcq)(x)/ < Kqh6-qco6(h),

q = 0(1)6.

PROOF. Owing to (2.3) we have

k = O(l)n - 2. using (1.3) for Y = 4 we have from (2.5)-(2.7) 1 ak,3

-

‘k,3

J <

K,,$3w4~~~

and 1 ‘k,6

-

‘k,6

1<

K6,3%th)s

270

SAXENA:(O,l,2,4)INTERPOLATION

BY G-SPLINES

This proves the lemma for q = 0. For other values of q, the lemma proved by successive differentiation.

Now using Theorem

3.1 and Lemma

1y@)(x)

- #l)(x)/

proves the theorem. The following theorem (5.1) as n + co.

<

can be

5.1, we obtain ~qh6-qu6(h),

q = 0(1)6;

which

Therefore

owing to Lipschitz I w4

shows that iA

condition

- f(G q&L +mt4

which, on using Theorem

q&9)

- &Jxll

satisfies the differential

equation

(5.2) we have 1 < 1T;(x) - !f(x)j + j!/t4

+

- gJx)lJ

5.1 for q = 0, 1, 2 at once gives (5.7).

Using the method of Section 3 and adopting the same procedure as above, we construct the spline function GA(z) c iS’$$ which will be an approximation to the solution when j 6 C’(I). We omit all details and only state the convergence theorems. THEOREM

5.3.

If

y(x)

THEOREM

5.4. We have

i8 the exact aohtion

oj (5.1), then

SAZiENA:

(0, 1, 2, 4) INTERPOLATION

BY G-SPLINES

271

6. Conclwion

Looking at the results of this’paper and those of [2], one finds the soluproblem in the class S$) for f c 0?(I) and in the tion of (0, 2) interpolation , denotes the class of functions gA(z) c C(1) class S$$ for f c C3(I). Here S z(i) which is quadratic in [x~, x,,+~1, lc = 0(1)~& - 2 and cubic in [xnWI, xJ; and Se$ is the class of functions GA(x) c C(1) which is a, cubic in [x~, Q+J, k = O(l).T&- 1. Regarding the efficacy of approximation of these interpolants, we have 11ii2) - icqJjjm < ~~h’-~q(h),

if / c (F(I),

~~@jf - fq) ]I= < c~~~-~u@),

if / c C3(I).

and

It will be int,eresting tion for T 2 5.

to generalize the result to (0, 1, . . . , Y -

2, T) interpola-

REFERENCES [1]

T.

[2]

T.

[3] [4] [6]

FAWZY, Spline functions and the Cauchy’s problem II, A&z A4d. Aad. Sci. Hungur., vol. 29 (3-4), (1977), pp. 269-271. MR 80~: 66148 FAWZY, (0, 1, 3) Lacunwy int,erpoMion by G-splinea, Ann&es Univ. Sci. Budapest,

(1986), pp. 63-67. ZbZ 664, 41006 (0, 2, 3) and (0, 1, 3) int,erpol&,ion through splines, Acta M&h. Acad. Sci. Hungar., vol. 60 (l-2) (1987), pp. 63-69. R. B. SAXENA and H. C. !CRIPATHI, (0, 2, 3) and (0, l., 3) interpolation by six degree splines, J. of Comptutional ad Applied Mathem&cs, vol. 18 (1987), pp. 396-401. VARMA, A. K., Lacunary int,erpol&ion by splines II, Acta M&h. Acad. Sci. HungaT.. vol. 31 (3-4), (1978), pp. 193-203. Section Mathematics, R. B. SAXENA and H.

vol.

XXIX

C. TRIPATFII,

(Received ANJULA SAXENA DEPARTMENT OF IvlATHE&fATICZ & ASTRONOMY LUCKNOW UNIVERSITY, LUCEi?OW INDIA

October 5, 1988)