The main-square error bounds for the gaussian quadrature of analytic functions

(1.1)

THE MEAN-SQUARE ERROR BOUNDS FOR THE GAUSSIAN QUADRATURE

OF ANALYTIC FUNCTIONS Kw

Ko

U

ABSTRACT. Inthis paper we present the L2-estimation for the ker- nelKn of the remaider term for the Gaussian quadrature with re- spect to one of four Chebyshev weight functions and the error bound of the type on the contour

where Z(r) denotes the length of the contour

r.

1.

Introduction

Consider the Gaussian quadrature with respect to the nonnegative weight function w(x) defined on the interval [-1,1] such that the mo- ments J2

^{xkw(x) dx exist for}

= 0,1,2,···. If we apply the residue theorem to the contour integral

1 [

f(z)1r

(x)

21ri Jr (z - x)1r

(z)

where r is the closed contour which contains the simple zeros of or- thogonal polynomial1r

(z), and integrate with respect to the w(x) on [-1,1], we get the formula

[ . f(t)w(t) dt = ~ ^{Akf(n,) + R,. (I),}

Received October24, 1994.

1991 Mathematics Subject Classification: Primary 65D32.

Key words and phrases: Gaussian quadrature, Chebyshev polynomial.

*This work was partially supported by the KOSEF, under grant NO.91-08-00- 01,1993.

where

are the zeros of the nth degree orthogonal polynomial

w) on [-1,1] and Ak are the corresponding Christoffel numbers which are defined by

1

1I"n(t)

Ak

wet) dt.

-1

t -

1I"n

If f is a single-valued analytic function in a domain D which contains [-1,1] and if r is a closed contour in D surrounding [-1,1], then the remainder term Rn (.) can be represented as a contour integral

(1.2) Rn(f) = -2 1

. [ Kn(z)f(z)dz, 11"~ Jr

where the kernel K

is given by

(1.3)

or, alternatively, by

(1.4) K ( )

Pn(z)

n

z 1I"n Z ( ).

Here, 1I"n(z) is the orthogonal polynomial1l"n(' : w) evaluated at z, while Pn(z) is known as the function of the second kind which is defined by

(1.5) 1

1I"n(t)

Pn(Z)

--wet) dt.

-1

z-t See, e.g.,[3].

Gautschi and Varga [5] have shown the error estimates in the following form on the circle and ellipse

(1.6) IRn(f) I ~ ^Z2(r)

^max

^{IKn(z) I max}

^If(z)l,

where Z(r) denotes the length of r. And they [5] have given an explicit

representation for the kernel K

on r, and from these determinl:!d the

maximum points on the circle and ellipse for a Jacobi measure wet) =

(1- t)a(l + ^t)f3 for arbitrary

> -1,{3 > -1. Martin and Stamp [8]

have shown an explicit expression for kernel Kn(z) by the method of the Laurent series expansion. They have given methods for computing the coefficients (in terms of the moments) for the Laurent series of Kn(z). Our goal is to get the other estimate with the following form

(1.7)

that

for a Jacobi measure wet) = (1 - t)a(1 + ^t)f3 to find the L

norm for the kernel and to determine error bound on the ellipse. In section 2 we find the L

norm of the kernel Kn(z) and obtain error bound on the elliptic contour with respect to one of four Chebyshev weight functions. In section 3 we give an example.

2. Main results

In this section we consider contour r as an elliptic contour which is defined by

2.1 Chebyshev measures of the first kind

For the weight function wet) = ^(1-t

)-!, we know that the orthog- anal polynomial is the Chebyshev polynomial T

of the first kind, (2.1.1)

Furthermore, one finds (2.1.2)

1

^{Tn(t) (1}

-1

^{z- J z - l}

⁾ⁿ

-1

z - t

z - cos 0 vz

hence we have the functions of the second kind

(2.1.3)

It follows that

Pn(Z) 411"

(2.1.4) Kn(z) = -(-) = ( _1I"n

U-U- Un Un+U-n ) ^{1) ( , 1} -1)

Z

= 2(u+u .

(2.1.5)

We know that

K ()12 411"2 1

I n

= p2n [a2(p) -cos20][a2n(P) +cos2nO]' where

(2.1.6) aj(p)

2(P' +p-J),

= 1,2,3,···, p> 1.

For the detail, we refer to [5].

THEOREM

1. Ifw(t)

(1- t

)-! on (-1,1), then (2.1.7)

Proof Since

= ^!(pe

+ ^p-1e

^{we get Idzl}

^!(p + ^{p-1) dO,}

and the

bound for the kernel K

Setting t

e

we see that dt

itdO and

where C is the unit circle, and applying the residue theorem, we get

where Res(J, x) denotes the residue value of f at x.

To calculate the value 2:;:'1 ^Res(J, _p-1 Wj ), let t2n + ^(;)2n

^(t-

zd(t - Z2)··· (t - Z2n), then we have

1 1

Res(J - -w .)

g(z·) -;---:----:---:-:;----;----;---;- , p

(Zj - Zl) ... (Zj - Zj-1)(Zj - zi+d ... (Zj - Z2n) ,

LEMMA

1. It holds that

IT ^{(Zj - Zi) = 2nz;n-1.}

1<i<2n

i=/;j

Proof Let w;n ^{= -1 and Zj = ;Wj, since}

t2n + ('!)2n

t2n _ ('!w

)2n

t2n _ z~n

J J

=

(t - Zj)(t2n- 1 + t2n- 2zj + ... + z;n-1),

it follows that

(t - Zl)(t - Z2)··· (t - Zj)··. (t - Z2n)

=

(t - Zj)(t2n- 1 + t2n- 2zj + ... + ^z;n-1).

Cancelling (t - Zj) on both sides, one gets

(t - Zl)'" (t - Zj-1)(t - zi+d ... (t - Z2n)

= (t2n- 1 + t2n- 2zj + ... + ^z;n-1),

o

and inserting t into

we have

Therefore we calculate the residue of fat -;Wj in the form

We have the residue by elementary calculation;

1 1 1 1

Res(f, -) = ^{Res(f, --)} = -4 (p2 _

)(p2n

P P

ps +prn-

Now we have to calculate summation;

1 2n z?

2n(p2n - pk) ~ (z; - p2)(z; - ~) ^,

LEMMA

2. It holds that

L

2n ₁ _2nx2n- 1

w3~n =-l.

w· -x - 1+x2n '

j=1 3

Proof. We can write in the form

IT we differentiate both sides with respect to x, we have

o

2nx2n- 1 = ^{(x -}

(x - W2n) + (x - Wl)(X - W3) ... (x - W2n) + ...

+ (x - wJ ... (x - W2n-l).

Dividing by the first equation, we have

2nx2n-1 2n 1

1+x2n =L x - w .·

j=1 3

It follows that

2n 1 2n 1 1 1 -2nx2n- 2

" , , ( ) wJ~n

^-1.

L....J w~

x 2 = ^{L....J 2x}

^{x - w·} + x = ¹ + x 2n '

j=l 3 j=l 3 3

Therefore for

and w;n

-1, we have

2n

2n

" 3 2 " 3

~ (z~ ^- p2)(z~ -~)

^p ~ (w~ ^- p4)(w~ ^-1)

3=1 J J P 3=1 J J

2n b

1

_ 22:

p _

- p

(w~ ^{- p4} + , W~ ^-

a - ^{p4 -}

^p4

3 J

We use Lemma 2, then it follows that

2n Z; ^{np2 (p2n _ pk)}

~ (zJ - p2)(zJ - ~) ^{= -} (p4 - 1)(p2n + pk)'

So we have the residue;

1 1 2n 1 1

Res(j, -) + Res(j, --) + 2: Res(j,

(2 1 )(p2 1 )"

P P

P P -

+

We have proved the £2 bound for the kernel K

p> 1.

o

Now we have to estimate error bound

Since the ellipse r has the length l(r)

4c

E(E), where

E - - - = -

- p+p-1'

we have the error bound for the analytic function on the elliptic contour

(2.1.8)

2.2 Chebyshev measures of the Second Kind

For the Chebyshev measure ofthe second kind, Le., wet) = (1-t

)!,

the nth degree orthogonal polynomial is

(2.2.1)

while

(2.2.2)

We know that

(2.2.3)

where

(2.2.4)

K z

= ~ a2(p) -cos28

I n()1 p2n

a2n+2(P) - cos2(n + ^1)8'

For the detail, we refer to [5].

THEOREM

2. Ifw(t) = (1-t

)l on (-1,1), then

(2.2.5)

Proof For z

r, we get Idzl

p-l )dO, and the £2 bound for the kernel K

Setting t = ^e

we see that dt = ^{itdO and}

where C is the unit circle, and applying the residue theorem, we get

where Res(j, x) denotes the residue value of f at x.

To calculate the value

Res(f

let t

(t- zI)(t -

(t -

then we have

and

Therefore for

=

^{and wJn+2} = ^{1, we have}

2n+2 1 1 1

L ^{[z; - (p 2 + p2) -} z~] = ^-(2n + 2)(p2 + p2),

j=1 J

since E;:t ² wJ

E;:t ² ~

^O. We have proved the L

bound for

3

the kernel K

o

2.3 Chebyshev Measures of the Type a

-{3 =-!

Let a = -/3 = -!. The orthogonal polynomial in this case is given by

(2.3.1)

Furthermore

1

^Pn(t)

^{.)1 1 - +}

^{dt = Jo. (1r cos(n}

^{+ 1)0 cos +}

^{cosnf) dO}

21r(u+l)

(2.3.2) -(u-u-

)u

+1' z=2(u+u).

We know that (2.3.3)

K 2 47r2 (al(p) + cos"O)2

I n(z)1

p2n+1 [a2(p) - cos20][a2n+1(P) +cos(2n + 1)0]' where

zEr

(2.3.4)

For the detail, we refer to [5].

THEOREM

3. Hw(t)

J~ ^{on (-1,1), then}

(2.3.5)

[IKn(z)1 2

I

dz

[

87r3(p + p-l)2 47r3(p + p-l )]

:::; (p - p-l ) (p4n+2 + 1) - (p4n+2 - 1) , p> 1.

Proof For Z E r, we get Idzl :::; Hp + ^{p-l) dB,} and the L2 bound for the kernel K

Setting

= e

we see that dt = itdB and

where C

the unit circle, and applying the residue theorem, we get

where Res(j, x) denotes the residue value of f at x.

To calculate the value

Res(f

let

2n+

+ ^(1)2n+l

(t - Zl)(t - Z2)'" (t - z2n+d, then we have Res(j,

1

p

1

=

g(Zj) ( Zj -

)

( Zj - Zj-l) Zj - Zj+l ... Zj - Z2n+l ( ) ( )'

o

t2n(t+p)(t+~

) where get) = (t_p)(t_~)(t2n+l+p2n+l)'

Since IT. ^{(Zj - Zi)} = ⁽²ⁿ +

we calculate residue of f at 1<t<2n+1

- ¥j

_!w·_{p J}

in the form

1 1 [(Zj + ^p)(Zj + _~)]

Res(f, --Wj) =

P (2n + l)(p2nH - p2n+l) (Zj - p)(Zj - p) We have the residue by elementary calculation;

1 (p+~)

Res(f, p)

^{(p _} _~)(p2nH + p4n-)'

For Zj

and w;nH = -1, it follows that

2f1 ^{(Zj + p)(Zj + ~)}

2f1 ^{(Wj + p2)(Wj + 1)}

j=1 (Zj - p)(Zj - p) j=1 (Wj - (2)(Wj -1) _ 2L: n

+ 1

[1 a b ] _ 2p2(p2 + ^{1) b _ 2(1} + ^p2)

- + _{W·_p} ,.2+ _w·-l , a - , . 2 _p--l , - _{I -} ,.2'

j=1

Therefore we get

1 2nH (Zj + ^p)(Zj +

(2n + 1)(p2nH - p4n-) ~ (Zj - p)(Zj - ~)

1 (p + ~)

- (p2nH - p2!+l) (p -

+ p2!+1) ,

h .

_(2n+1)x2n 2n+1 1

h

were usmg f-Jj=1

=

,wj = - . ne ave prov the L2 bound for the kernel K

[I ^{K n(z)1 2} I ^dz l

< 21r2(p+p-1) {27r (al(p) + ^{cos 0)2 dO}

- p2nH 10 (a2(p) - cos20)(a2nH(P) + ^cos(2n + ¹⁾⁰⁾

[

81r3(p + p-1)2 41r3(p + p-1 )]

=

(p - p-1 )(p4n+2 + ^{1) - (p4n+2 - 1) .}

REMARK.

The well-known identity for Jacobi polynomials, 1r$f,a) (z)

= (_l)n1r~a,.B)(z), implies IK,<!,a)(z)1 = IK~a,.B)( -z)1 = IK~a,.B)( -z)l.

Thus for the case a

-f3

!, we have the same £2 bound (2.3.5) for the kernel K

3. Example

In this section we apply the results in section 2 to obtain the £2 error estimate for Gaussian quadrature.

EXAMPLE.

1)1

¹

_V ^[1=t _l+t

_t, ^II) 1

^vT=t2 _2-t

I) We consider wet)

J~+: as the Jacobi weight with parameter a = - f3 = !. ^{To bound} f on the elliptic contour r, note that

1 1 1

If(z)1

12 - zl

2 - Izl

2 - Hp + p-1)"

Therefore we get the error bound in the form

(3.1)

(p _ p-1 )(p4n+2 + 1)

1r(p + p-1) (p4n+2 -1)

where

p+~-l' ¹ <

< 2 + y'3.

II) We consider w( t)

(1-t

)! as the Jacobi weight with parameter a = f3 = l and we get the error bound in the form

(3.2) IRn(J) I

.jc

E(E)

h -

were

1 <

< 2 + y'3.

Table 3.1 (On the ellipse) f(x)

2~x' wet)

(1- t)! (1 + t)!

n bound rho True error supremum norm

5 5.2100( -5) 3.3984 1.4906( -6) 5.4156( -5) 10 1.9254(-10) 3.5558 2.8436(-12)

15 5.4507(-16) 3.6123 -1.1102(-16) 5.6433(-16)

Table 3.2 (On the ellipse) f(x) = 2~x' wet) ^{= (1-} t)!(l + ^t)-!

n bound rho True error supremum norm

5 1.2187( -4) 3.4066 1.8543( -6) 1.8257( -4) 10 4.4492(-10) 3.5579 3.5374(-12) 6.5903(-10) 15 1.2544(-15) 3.6133 -6.6613(-16) 1.8507(-15)

We have expressed the error bound as a parameter of

Number in parentheses of Table 3.1 and Table 3.2 indicate decimal exponents.

As n incresses, the number

approaches to 2 + J3. This

due to the nature of weak singularity of denominator factor. And we have compared with the supremum norm(Gautschi and Varga [5]).

References

[1] P. J. Davis, Interpolation and Approximation, Blasidell, New York, 1963.

[2] P. J. Davis and P. Rabinowitz, Methods of numerical integration, Academic Press, New York, 1975.

[3] W. Gautschi, A survey of Gauss-Christoffel quadrature formulae, in E. B.

Christoffel, The Influence of his Work on Mathematics and Physical Sciences, P. L., Butzer and F. Feher, eds., Birkhauser, Basel, 1975, p. 72-147.

[4] W. Gautschi andE.Tychopoulos, A note on the contour integral representation of the remainderterm for a Gauss-Chebyshev quadrature, SIAM. J. Numer.

Anal. 27 (1990), 219-224.

[5] W. Gautschi and R. S. Varga, Error bounds for Gaussian quadrature of analytic functions, SIAM. J. Numer. Anal.20 (1983), 1170-1186.

[6] W. Gautschi and Shikang Li, The remainder term for analytic functions of Gauss-Radau and Gauss-Lobatto quadrature rules with multiple end points, Journal of Computation and Applied Mathematics 33 (1990),315-329.

[7J - - , Gauss-Radau and Gauss-Lobatto quadratures with double end points, Journal of Computation and Applied Mathematics 34 (1991), 343-360.

[8] C. Martin and M. Stamp, A Note on the Error in Gaussian Quardature, Ap- plied Mathematics And Computation42 (1992), 25-35.

[9] G. Szego, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Pub!., Amer.

Math. Soc., Providence, R. 1., 1975.

Kwan Pyo Ko

Department of Computer Engineering Dongseo University

Pusan 617-716, Korea

E-mail: [email protected] U Jin Choi

Department of Mathematics

Korea Advanced Institute of Science and Technology 373-1 Kusungdong, Yusonggu

Taejon 305-701, Korea