Approximation of Cauchy principal value integrals by piecewise Hermite quartic polynomials by spline

Appl. Math. Lett. Vol. 5, No. 1, pp. 75-78, 1992 Printed in Great Britain. All rights reserved

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APPROXIMATION OF CAUCHY PRINCIPAL VALUE INTEGRALS BY PIECEWISE HERMITE QUARTIC POLYNOMIALS BY SPLINE G.BEHFOROOZ Department

of Mathematics,

1600 Burrstone

Utica College of Syracuse University

Road, Utica, NY 13502-4892, (Received July

U.S.A.

1991)

Abstract-The interpolatory piecewise Hermite quartic polynomials induced by cubic spline has been established by Behforooz and Papamichael [I]. These polynomials can be used to approximate the functions and their derivatives over the entire spline interval and obtain better orders of convergence than use of the cubic spline. In this paper these piecewise polynomials will be used to estimate the Cauchy principal value integrals and their derivatives. This technique yields better results than using the derivatives of the single spline functions which has been considered by Orsi [2].

1. INTRODUCTION For a given data {(x~,~(z~))}~~,, with the equally consider the interpolatory cubic spline S with knots Suppose that S matches with the function y = f(x) where yi = f(xi). If all of k + 1 parameters rni ti-1 5 z 5 xi; i = l(l)I, the value of S(z) can be two point formula,

S(z) = yi + rni(a:- Xi-l)

+

S[zi--1,

xi-l,

xi,

spaced points 2; = a + ih; i = xi on the interval [o, b], with h = at the knots xi, i.e., S(zi) = yi; = .5d1)(zi) are known, then at obtained by use of the following

+ S[Zi_l, G-1, q](L7J- q_1)2 Xi](Z - 2i_1)2(2- Xi),

0(1)/c, we (b - a)/!~. i = O(l)lc, any point Hermite’s

(1.1)

where S[G-1,

S[G-1,

Xi-l,

G-1,

%I = f

%r %I = $

{(Yi -Y&-1) {-2(Yi

- h-l},

- Y&l)

+ qmi

+ m-l)),

are usual notations for divided differences. To determine the Ic + 1 parameters mi, we can use the first derivative consistency relations, which consist of only k - 1 linear equations in L + 1 parameters mi, together with two additional linear equations from end conditions. For a set of proper end conditions, the best orders of convergence that can be achieved by using S and its derivatives are jjy@) -

S@)Ij

= O(h4_‘);

T = 0,

1,

2, 3,

(I-2)

where I] * ]I denotes the uniform norm (L, function norm) on [a, b]. Also, in the literature there are some end conditions such that by using the cubic splines with those end conditions we obtain superconvergence for the first derivative at the knots zi; i = O(l)lc, i.e.,

yi(l) = mi + 0(h4);

i=

0, 1, . . . , k.

(1.3) Typeset by A,&-Tj$

75

76

G. BEHFOROOZ

For the list of some of these types of end conditions, see for example, Behforooz and Papamichael [1,3], or Lucas [4]. By adding one more term to (l.l), the following piecewise Hermite quartic polynomial induced by cubic spline S has been established by Behforooz and Papamichael [l], P(x)

= S(x) + Ai(z - z~-$(x

- xi)‘,

;c E [Q-l

-

i = 1,

xi];

2, . . . , k,

(1.4)

where

Ai = S[xi-l,

xi-19

Ak = S[xk-2,

%k-1,

xi, xi, xi+l];

i=l,

2, . . . . k-l,

and xk-1,

xk,

xk]

=

Ak-1.

These piecewise polynomials are called Hermite quartic polynomials spline S. Now we state the following theorem on the order of convergence For the proof of the Theorem 1.1, see [I]. THEOREM

1.1.

Let S

(?[a, b], at the knots conditions such that

induced by cubic of these polynomials.

be an interpolatory cubic spline which agrees with the function y = f(x) E xi = a + ih; i = 0, 1, . . . , k, with h = (b - a)/k, and satisfies two end Irn.1 - $)I 1

< Ah4;

i=O,l,

. . . . k,

where A is a constant independent of h. Let P be the piecewise Hermite induced by S. Then there exist constants B,; r = 0, 1, 2, 3, 4, such that

[If”’ -

P(‘)II -< B,h5-‘;

(1.5) quartic

polynomials

r = 0, 1, 2, 3, 4.

(1.6)

The results (1.2) and (1.6) show that, under the conditions of the Theorem 1.1, P can be used with very little additional computational effort to produce, at any point x E [a, b], more accurate approximation to the function y = f( x ) and its derivatives than those obtained from the cubic spline S. 2. CAUCHY

PRINCIPAL

VALUE

INTEGRALS

Suppose that F(f; t) d enote the Cauchy principal value integral of the function the interval [a, b] with the weight function W(X) and parameter a < t < b, i.e.,

It is well known that following two limits,

Jbfo

principal

value

the Cauchy

F(f; t) = &,y+ Let F(f(“);

F(f; t> =

integral

F(f@);

value integral t) =

J

b

a

(2.1) is defined

bfo

for the nth derivative

f’“‘(x) I-_t

4x)

dx.

W(X) be such that

w(j)(,) = d)(b) =

0,

forallj=O,

by the sum

W(Z) dx. lim c-o+ J t+( 2 - t

x-t

over

W)

W(X) dx.

t-cf&9_ W(X) dx + J (I

t) be the C auchy principal

Also, if the weight function

* x-t

y = j(x)

1, 2, . . . . n-l,

of the

(2.2)

of f; i.e.,

(2.3)

Cauchy principal value integrals

then the integrals

77

of the form

bfow(x)dx.,

diF(f; t, _ ” dtj

J

dtj

(I x-t

a
can be written as a linear combination of the integrals of the form (2.1), because, by using integration by parts we can obtain the following identity dj

z

I

*

W(X) dx =

x-t

(I

in this case,

[f(x)w(x)l(i) da:

x-t

Ja

In general there are two basic approaches to approximate the Cauchy principal value integral (2.1). The first types are called global methods and second types are local methods. In the global methods the obtained formulas are called product rules and for some of them uniform convergence theorems have been established, see for example, Paget and Elliot [5]. The local methods are based on piecewise polynomials, and the obtained formulas are called the quadrature rules. Some of these methods were used as early as 1950, see Gerasoulis [6] and Stewart [7]. 3. IMPROVED Recently Orsi [2] h as considered some end conditions to approximate F(S(“);

ORDERS

OF APPROXIMATION

the interpolatory spline S(x) of odd degree m > n + 1 with the Cauchy principal value integral F(f(“); t) by the integral

6so(z)

w(x)

t) =

/ (1

dx;

x-t

(3.1)

a
The following theorem shows the uniform convergence of the above quadrature rule based cubic spline with not-a-knot end conditions. For the proof of the Theorem 3.1, see [2].

on

cubic spline with not-a-knot end THEOREM 3.1. Suppose that S(x) is the unique interpolatory conditions which matches the function y = f(x) at the knots xi = a + ih; i = 0, 1, . . . , R, where f E C3[a, b], and f4 is bounded on the interval [a, b]. Let us consider the quadrature rule (3.1) to approximate the Cauchy principal value integral (2.3), then lim F(S(“);

b f”(x) I (I x-t

t) =

h-0

Furthermore,

w(x)

dx;

a
(3.2)

if W(x) E L’[a, b] rl Hp(a, b), wivith 0 < p 5 1, then for all t E [a,b], the quantities

Jlb 4x1 Idx exist and are bounded

and the convergence

(3.3)

L-_t

(I

is uniform

in the whole closed interval

[a, b].

Now by using the piecewise Hermite polynomials induced by cubic spline, we can extend the idea of Orsi [2] to obtain a better and improved approximation. Here first we establish the unique interpolatory cubic spline S(x) with some specific end conditions such that (1.3) holds. Then we P(n) in the use S(x) in (1.4) to approximate the derivatives fcn)by Pen). Finally, we substitute integral (2.3) to present the following new quadrature rule to evaluate, numerically, the Cauchy principal value integral and its derivatives F(@‘);

t)

=

’ $!$ J (I

This is our procedure to obtain our quadrature of convergence of this quadrature rule. AML 5:1-F

w(x) dx;

a
rule, and the following

theorem

is on the order

G. BEHFOROOZ

78

THEOREM 3.2. With the assumptions quantities (3.3) exist and are bounded. the Cauchy

principal

value integral

F(p);

of the Theorem 1.1, suppose Let us consider the quadrature

(2.3),

then for all t E [a, b],

b l+)(x)

t) -

that for all t E [u,b], the rule (3.4) to approximate

z--t

w(x) dx =

0(h5-“).

(3.5)

Ja

PROOF. From (2.3) and (3.4) we have

IF(p); t) - F(P@); t)l

b 4x1 If’“‘(x) - P(n)(x)ldz.

5

z-t

Jla Then the result (3.5) result (1.6).

can be obtained

I

easily by considering

the bounded

quantities

(3.3)

and the

REFERENCES 1.

G. Behforoozand splines, BIT

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3.

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T.R.

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5. 6. 7.

(to

orders of approximation

derived from interpolatory

cubic

for Cauchy principal

value integrals,

J. of Compufational

and

appear).

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End conditions

for cubic spline interpolation,

J. Inst. Mafh.

Appl.

23,

(1979).

Lucas,

Error bounds for interpolating

11, 569-584

Comput.

19, 199-211

J. Numer.

for the numerical evaluation of certain Cauchy principal value

integrals,

(1972).

A. GerasouLis, Piecewise polynomial Stewart,

SIAM

cubic splines under various end conditions,

(1974).

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Improved

Orsi, Spline approximation

Mathematics

355-366

N. Papamichael,

19, 19-26 (1979).

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(4), 891-902

23 (4),

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