Interpolated quadrature formulae containing derivatives for some Cauchy type integrals and for their principal values

INTERPOLATED QUADRATURE FORMULAE CONTAINING DERIVATIVES FOR SOME CAUCHY TYPE INTEGRALS AND FOR THEIR PRINCIPAL VALUES* G. N. PYKHTEEV and I. SHOKAMOLOV Minsk - Dushsnbe (Received

27 May 1969)

IN the solution of problems in the mechanics of continuous media it is often necessary to evaluate integrals of Cauchy type and their principal values of the form ‘)f(z”Y(r)=Y(fla)-

-

Z(4==Z(fl4=

1)

J

dt

f(t)

-i t--r

n

‘)‘(i--2) n

‘

i f(t)

3

’

I(i-P) dt

S+-+y(*_tr) ’

2

E

e 14 il, [-I,

il.

(0.1)

(0.2)

-1

We here understand by \/k? - 1) the branch analytic in the plane with a cut along the segment L-1, 11 of the x-axis, and such that V (2’ - 1) = z + o (l/z). The integral (0.11 is understood as singular in the sense of the Cauchy principal value. The notation Y (flzl or I (fix) emphasizes that these integrals are operators for f(x) and are functions of the points z and x. At the present time various interpolation quadrature formulas containing values of the density f(x) at given points of the segment I-1, 13 have been obtained. Below we consider interpolation quadrature formulas for integrals (0.11, (0.21, containing values of the derivative of the density/(x). Methods of calculating the coefficients of the formulas considered are given, and methods of estimating their remainder terms are indicated. 1. Construction of interpolation quadrature formulas containing derivatives of the density f(x). Method of calculating the coefficients We are given a system of,points x,, x,, . . . , x,_, of the segment l-1, 11, where * Zh. v%hisl.

Mat. mat. Fiz.,

10, 2, 438-444,

208

1970.

Interpolated

209

quadrature formulas

and we construct a Lagrange interpolation polynomial x,&x,,< . ..dx._,, I..,: (f’ Ix), coincident with the derivative f’ (3~)of the density f(x) of the integrals (0.1) and (0.2) at these points. Then ll-i (i.1) &k” (z)P(zr), PC4 = J%n(fl14 c h-0

where n-i

an (4

B*hn(x)= (5 -

x-)0

II

O,(X)= n

‘(Xk)

(x--h).

(1.2)

k-0

Integrating (1.1) and imposing on the function f(x) the condition f(b) = 0, e. E [A, 11, which can obviously always be considered to be satisfied, we obtain the approximate formula

where L Ban (2) =

B.h” (t)dt.

11.4)

s El

In the integrals (O.l), (0.21 we replace f(x) by the polynomial IF (fix) in accordance with (1.31. As a result we obtain the approximate formulas n-l Y(z) = 9, (flz) =

c

&AnW(sr),

(1.5)

A-O

which we will call the interpolation quadrature formulas, and 8,” k) and Akn (x) are their coefficients. The latter have the form &An(z)

=

y(Bh”

Iz),

h”(x)

= z(Bh” IX).

(1.7)

The operators EW~‘(f!Z) = Y(z) --P,(flz), Z(r) -L(flq

EP(j(x) =

(1.8)

(W

210

G. N. Pykhteev

cmd I. Skokamolov

will be called the remainder terms of the interpolation (1.6), and the operators

quadrature formulas t1.5), (1.10) (1.11)

E~(fI4 = f(l) - L”(fi4, E*n(Jlt) = f;(z) -L*“(f’[4

will be called the remainder terms of,the app~xima~ formulas (1.3) and (1.1). Obviously, the operators (1.8)~(1.11) are the errors of the quadrature formulas (1.5), (1.6) and of the approximate formulas (1.3), (1.1) at the points z and X. In studying quadrature formulas for ordinary Riemann integrals most attention is given to estimating the remainder term in different classes [I, 21, since their coefficients are simply calculated. But the coefficients of the quadrature formulas (1.5), (1.6) for a large number of points have a fairly complex form. Hence, before proceeding to the estimation of the remainder terms of the quadrature formulas (1.5), (1.6), we give a method of calculating the coefficients of these formulae. The method proposed is based on known accurate formulae (see, for example, [31) Y(T?nlz) = p(Z).

6(z) =

2

-T’(zZ-

I(Tml~) ,= U,(x),

1);

u7n(4 = sin m arc cos 5, where the ‘A”,tx) are Chebysehv polynomials of the first kind, and consists the fact that Bkn (x) is calculated in the form n B&“(Z)= emnkTm (x)+ i,‘i~onk c

of

and substituted in (1.7), as a result of which we obtain for the coefficients @; (z) and AL Cd the simple expressions n

Jz$* (2) =

A

c

c,nhfm(z)+

~/2coQk,

VII-i

dk*(X)=

z ?n=I

cm=“Um(t).

This method of calculating the coefficients is convenient because of the fact that for constant cmnkt as a result of the orthogonality of the polynomials T, (xl with density 1/ V(I - 2%) the relations I

s

dz

m = 0, 1, . . . , n.

l~~e~~laied

quadrature

211

formulae

are true. By means of these relations it is possible, for a given system of points x,, x,, . . . , +, to construct a table for determining the cmflk, and then a standard program for calculating the coefficients akn (z) and A,” (x) at any point z and x. 2.

Representation

of the remainder

terms (1.8M1.11)

For the remainder terms (1.9)~(1.11) it is possible to obtain different rep~sentations depending on the class to which f(x) is referred. We here discuss the two classes Waft) and P. Following [Zf, we will refer to the first-class functions defined on the segment [-I, 11, possessing derivatives up to the (r - lkth order inclusive and a piecewise-continuous derivative of the r-th order f(x) satisfying on i-1, 11 the inequality l/(r)(z) 1 Q M, and to the second-class functions continuously differentiable n times on f-1, 11. Let f(s) ez WP+~(I).

Then for its derivative

f’ (x) we have

We replace the derivative f’ (3~)occurring on the right side of (1.11) by this formula, assuming that r,c n. As a result, considering that formula (1.1) is accurate for any polynomial of degree less than or equal to n, after some transformations we obtain for the remainder term (1.11) r = 0,1, . . . , R,

E*” (P/s) = j t-3.1”(5 t)r’+“(t)i%

(2.1)

--f ?%A

1

&r” (s, t) = -

c c L (z-

(r -

1) f

f) -

%I= (+G (a - t) 1

k==O

From this, noticing that &% E”(fls)=

we find a similar representation

s 4

E*n(f’It)dt,

for the remainder term (1.10):

(2.2)

G. N. Pykhteeu

212

and I. Shokamolov

(2.3)

We turn to the remainder term (1.9). We use the following formula for integrating by parts the integral (0.2):

which is obvious for a function f(x) differentiable on k-1, Il. We replace the value of the integral (0.2) by the formula (1.9) just written down, and notice #at &(flt) =Z(L=(f)Is) and (~S/dz)&n(fjs) = L,n(fjt), and then use equation (1.11). As a result we obtain for the remainder term (1.9) the representation

From this, using (2.1), we arrive at another representation (1.9): a,.” (I,

SP(ti.)=j

~)~(r~*)~~)d~,

r =

0,1,.

. *, n,

arn(r,t)=’

--t

Let

f(t)

E

c-1.

of the remainder term

J --t

H(s,t)fL”(;E,Z)dT.

Then, as is known from the theory of the interpolation

(2.5)

of

functions

and consequently, by (2.2) and (2.4) we have for the remainder terms (1.10) and (1.9) the representations %I (~)f(~~i)(~)~~,

E” crjx, = ;i Eo

(2.7)

interpolated

quadmture

213

formulae

We now write down the representation of the modulus of the limiting value of the remainder term (1.8) on the segment i-1, 11 of the real axis. For this we notice that Elpn(flz)=

Y(E”(f)lz)=

(2.9)

WI+&Wz), k-0

and then pass to the limit as z + z E [- 1,1] and use Sokhotskii’s result we obtain the relation (Elgn(flz))* = *En (fls) - iEr” (r]z,,

formula.

As a

= = f--1, 11,

which implies the equation

pP(fiX)l 3.

Estimation

= I’I(@(fl4)*+

(~~“(fl~))21,

of the error of the quadrature (1.6) in the class Wr+l(M)

z=f--l, formulas

11.

(2.10)

(1.9,

In the case where f(x) E W+l(~f), to estimate the error of the quadrature formula (1.6) it is natural to use the representation (2.5) which implies the inequality

IE1n(fl4 I d v WC

r = 0,1, . . . , n,

a,” (5) =

s

]ar”(z,t)]dt.

(34

--1

This inequality determines the error of the quadrature formula (1.6) at a given point z ~(-1, 11. If we use the representation (2.3), we obtain a similar inequality for the remainder term of the approximate formula (1.3):

P(fl4l

=zBr”(Z)K

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

b”(x)

=

J IB,“(z, t) 1dt,

(3.2)

--1

which determines the error of the approximate formula (1.3) at a given point z E I--1,11. The function a,(x) occurring in (3.1). is obviously continuous on l-1, 11, and consequently assumes a maximum value at some point of this segment.

214

G. N.

Therefore,

Pykhteev and I. ~hok~vl~v

for the remainder term of the quadrature formula (1.6) the estimate a,” =

max a,(s). -1CXCI

(3.3)

holds uniformly in X. Just as for the approximate formula (1.3) we have fh*iV,

p(fl~)I.g

I’ =

0, 1, . . . .

R,

&a =

max l%W, -i (X
(3.4)

uniformly in X. By using (3.3) and (3.4), it is possible to obtain an estimate of the remainder term of the quadrature formula (1.5) which is uniform in z. This is made possible by the useful property of the remainder term (1.81, that it is a function of z analytic in the plane with a cut along the segment i-1, 11of the x-axis. This is easily verified by turning to (2.9) and taking into account the definition of the branch d (z2 - 1) given above. This property implies that the modulus of the remainder term (1.8) attains a maximum on the edges of the cut made along the segment f-1, 11 of the x-axis. But the modulus of the ~mainder term (1.8) has the ~~sen~tion (2.10) on the segment l-1, 11, and hence its maximum on the segment El, 11does not exceed the quantity 1 [(a,%)2 + (&m)z] and consequently, the estimate /J&n (112)1 G ‘t’d(a,“j2 + @r’*)21Y,

r = 0, 1, . . . , n,

holds for any point z of the complex plane and for any function

4, Estimation

of the error of the quadrature (1.51, (1.6) in the class Cn+’

If f(s) ECU+’ we

and Mn+* =

max

-l
(3.5)

/(z) E W+‘(M).

formulae

IF n+l)(t) 1, from the representation

obtain the estimate

which defines the error of the quadrature formula (1.6) at a given point In this case the ~p~sen~tion f2.7) implies the estimate

(2.8)

215

valid for the given point z GZt--i, 11. Determining the maximum of the function occurring in (4.11, we obtain a uniform estimate of the error of the quadrature formula (1.6)

valid for any point z E t-1, il. Similarly from (4.2) we obtain

IE~(f12)l~~LW+i, Sk =

mftxikn(2), -i
v-4)

where x is any point of the segment I-1, 11. Now using the estimates (4.3) and (4.4) which are uniform in x, and on the basis of the properties of the modulus of the remainder km of the Quasar formula (1.51, indicated in section 3, and of the ~resen~tion C&10), we arrive at the inequality (4.5)

which holds for any point z of the complex plane, that is, it gives an estimate of the error of the quadrature formula (1.5) which is uniform in 2. 5. Estimation of the error of the quadrature formulae (1.51, (1.6) in terms of the error estimate of the a~p~ximate formula (1.1) The quadrature formulae (1.51, (1.6) were obtained by the approximation of the derivative f” (x1 by formula (1.1). It is natural to pose the question of the dependence of the error of the quadrature formulae (1.51, (1.6) on the error of the approximate formula (1.1). This dependence can be obtained if we begin from the representations (2.4) and (2.2). in fact the following theorem holds. Theorem Let the absolute value of the remainder term of the approximate formula U,l) be over~~ded unifo~ly in x by the quantity E+” > 0, tbat is,

216

G. N. Pykhteeu

and 1. Shokamolov

I.%“(f’If) I G Et”.

(5.1)

Then the errors of the quadrature formulae (1.6) and (1.5) are estimated by the inequalities

and *.

]&P(~]z) 1 G III1 + a2(50)lEtn,

a(b) = mWl--

50, I+

Eo),

(5.3)

where z E l--1,1], and z is any point of the complex plane. The estimate (5.2) is easily obtained from the representation the inequalities

(2.4) if we use

1 H(z,t)>Q

s

H(x,

t)dt < 1,

-1

the first of which is obvious, and the derivation of the second will not be discussed here. The estimate (5.3) follows from the inequality (5.2), the which is obtained from the representation (2.2), inequality ID(flx) 1 < o(go)Etn, and the representation (2.10), if we use the properties of the modulus of the remainder term (1.8) indicated in section 3. The estimates (5.2) and (5.3) are interesting because they make it possible to reduce the determination of the error of the quadrature formulae (1.6) and (1.5) to the investigation of the remainder term of the approximate formula (l.l), which is a simpler problem than the determination of the error of these formulas by means of the estimates obtained in section 3 and 4. Consequently, the estimates (5.2) and (5.3) make it possible to obtain inequalities estimating the error of the quadrature formulae (1.6) and (1.5), which though cruder, are also simpler, than the inequalities (3.3), (4.3), (3.5) and (4.5). We obtain these inequalities in two cases: when f(x) E W+‘(M) and when f(x) E cn+i. If f(x) E FVr+l(M), (2.1) implies the inequality

I& n (PI x) I s IL” (x)M,

r = 0, 1, . . . , n,

e..qx,=J

Ip.r”(z,t)ldt, --1

Intorpoluted

quadraturt!

217

formulae

from which it is obvious that in the case where f(x) E Wr+f(M), we can take as Ejn the quantity

and hence by (5.2), (5.3), we have the estimates

z E [--1, 11, and z is any point of the complex plane. In the case where f(Z) E Cn+‘, the representation (2.6) is valid, which implies that in this case it is possible to take as Ein the quantity

where

;nfn+1, After substituting

where

sn

=

max 10, (2) I. _-i=Gr
this quantity in (5.2) and (5.3) we obtain the estimates

I E I--1, II, and z is any point of the complex plane.

We notice that in the last inequality the least value of a (t,) will be a (0) = 1. Hence in the case where the choice of E, is not restricted by any condition, we should put t, = 0 in this inequality. We also notice that the quantity on occurring in the last two inequalities, depends on the choice of the interpolation point. It is known (see, for example, [41), that on will have its least value when the zeros of the Chebyshev polyare taken as the interpolation points. nomial Tn(s) = (l/2”-*)cosnarccos~. In this case Wn = I/ 2”-’ and consequently, the moduli of the remainder terms of the quadrature formulae (1.6) and (1.5) are overbounded by the quantities (1 / n!L!“-‘)Mn+’ and (I / n!2n-f)j’[l + aZ(Eo)]Nn+*. respectively. Therefore, the last two inequalities give the best error estimates for the given number n of interpolation points for interpolation quadrature formulae, having as interpolation points the zeros of the Chebyshev polynomial T, (5) = (I/ 2n-‘) cos n BPC cos X. Translated

by J. Berry

218

G. N. Pykhtcev

and I. Shokamolou

REFERENCES

1.

KRYLOV. V. I. Approximate Evaluation of integrals integralov). Fizmatgiz, Moscow, 1959.

2.

NIKOL’SKII,

S. M. Quadrature

Formulae

(Priblizhennoe

(Kvadraturnye

formuly),

vychislenie

Fizmatgiz,

Moscow,

1958.

3.

4.

PYKHTEEV, G. N. Accurate open contour. App. mat. GONCHAROV, V. L. Theory (Teoriya interpolirovaniya

methods of evaluating 1965.

Cauchy type integrals

along an

10,4. 351-373, of Interpolation i priblizheniya

and Approximation funktsii)

of Functions

Gostekhizdat,

Moscow,

1954.