View of Double inequalities of Newton's quadrature rule

Rev. Anal. Num´er. Th´eor. Approx., vol. 35 (2006) no. 2, pp. 141–146 ictp.acad.ro/jnaat

DOUBLE INEQUALITIES OF NEWTON’S QUADRATURE RULE

MARIUS HELJIU^∗

Abstract. In this paper double inequalities of Newton’s quadrature rule are given.

MSC 2000. 65D32.

Keywords. Newton’s inequality, double integral inequalities, numerical inte- gration.

1. INTRODUCTION

In the papers [1], [2], [3], [4], [5], [7] and [8] relatively double inequalities of quadratures rules of the trapezoid and Simpson were given. In this note we will obtain upper and lower error bounds for Newton’s quadrature rule.

Letf : [a, b]→R, f ∈C⁴([a, b]) andx₁, x₂ ∈[a, b] so thatx₁ = ^2a+b₃ , x₂ =

a+2b

3 , then as it is well known [6] the relation is obtained (1.1)

Z b a

f(x)dx= ^b−a₈ [f(a) + 3f(x1) + 3f(x2) +f(b)] +R, where

(1.2) R=

Z b a

ϕ(x)f⁽⁴⁾(x)dx.

The function ϕis given by the relation (h denotes ^b−a₃ ):

(1.3) ϕ(x) =















(x−a)⁴

4! −^3h₈ ^(x−a)_3! ³, x∈[a, x₁]

(x−a)⁴

4! −^3h₈ ^(x−a)_3! ³ − ^9h₈ ^(x−x_3!¹⁾³, x∈[x1, x2]

(x−a)⁴

4! −^3h₈ ^(x−a)_3! ³ − ^9h₈ ^(x−x_3!¹⁾³ −^9h₈ ^(x−x_3!²⁾³, x∈[x2, b].

2. MAIN RESULT

Under the assumptions of the quadrature formula (1.1) we have the next theorem:

∗University of Petro¸sani, Department of Mathematics and Computer Science, Romania, e-mail: [email protected].

Theorem 2.1. Let f ∈C⁴(a, b).Then:

23γ4−15S3

51840 (b−a)⁵ ≤ ^b−a₈ [f(a) + 3f(x1) + 3f(x2) +f(b)]− Z b

a

f(x)dx

≤ ^23Γ₅₁₈₄₀^4−15S3(b−a)⁵, (2.1)

whereγ4, Γ₄ ∈R, γ4 ≤f⁽⁴⁾(x)≤Γ₄,for anyx∈[a, b]andS3 = ^f(3)(b)−f_b−a^(3)(a). If γ₄= min

x∈[a,b]f⁽⁴⁾(x), Γ₄ = max

x∈[a,b]f⁽⁴⁾(x) then inequalities are sharp.

Proof. From (1.2) integrating by parts we get : (2.2)

Z b a

ϕ(x)f⁽⁴⁾(x)dx= Z b

a

f(x)dx−^b−a₈ [f(a) + 3f(x₁) + 3f(x₂) +f(b)]. It is easy to see [6] that we get the equality:

(2.3)

Z b a

ϕ(x)dx=−^(b−a)₆₄₈₀⁵. From (2.2) and (2.3) we get the equalities:

Z b a

hf⁽⁴⁾(x)−γ4

iϕ(x)dx=

= Z b

a

f(x)dx− ^b−a₈ [f(a) + 3f(x1) + 3f(x2) +f(b)] + ₆₄₈₀^γ4 (b−a)⁵ (2.4)

and Z b

a

hΓ₄−f⁽⁴⁾(x)ⁱϕ(x)dx=

=− Z b

a

f(x)dx+^b−a₈ [f(a) + 3f(x₁) + 3f(x₂) +f(b)]− ₆₄₈₀^Γ4 (b−a)⁵. (2.5)

On the other hand:

(2.6)

Z b a

hf⁽⁴⁾(x)−γ₄ⁱϕ(x)dx≤ max

x∈[a,b]|ϕ(x)|

Z b a

f⁽⁴⁾(x)−γ₄dx.

From (1.3) we get:

(2.7) max

x∈[a,b]|ϕ(x)|= ^(b−a)₃₄₅₆⁴. On the other hand the equality follows:

Z b a

f⁽⁴⁾(x)−γ4

dx=

Z b a

(f⁽⁴⁾(x)−γ4)dx

=f⁽³⁾(b)−f⁽³⁾(a)−γ4(b−a) = (S3−γ4)(b−a).

(2.8)

From the relations (2.4), (2.6), (2.7) and (2.8) it follows : Z b

a

f(x)dx−^b−a₈ [f(a) + 3f(x1) + 3f(x2) +f(b)]≤

≤ ^15S₅₁₈₄₀^3−23γ4(b−a)⁵, (2.9)

the first inequality of (2.1).

We also have:

(2.10)

Z b a

h

Γ₄−f⁽⁴⁾(x)ⁱϕ(x)dx≤ max

x∈[a,b]|ϕ(x)|

Z b a

Γ₄−f⁽⁴⁾(x)dx and

(2.11)

Z b a

Γ₄−f⁽⁴⁾(x)dx= Z b

a

(Γ₄−f⁽⁴⁾(x))dx=

= Γ₄(b−a)+f⁽³⁾(a)−f⁽³⁾(b) = (Γ₄−S₃)(b−a).

By analogy from (2.5), (2.7), (2.10) and (2.11) we get:

Z b a

f(x)dx−^b−a₈ [f(a) + 3f(x₁) + 3f(x₂) +f(b)]≥

≥ ^15S₅₁₈₄₀^3−23Γ4(b−a)⁵. (2.12)

The last relation and (2.9) lead us to the inequality (2.1).

To show that inequality (2.1) is sharp we consider the function f given by the relation f(x) = (x−a)⁴.It is easy to see that the equalities f⁽⁴⁾(x) = 24 and γ4 = Γ₄= 24, S3 = 24 are obtained.

Calculating the three members of the inequality (2.1) under the given cir- cumstances, we notice that these have the common value given by the expres- sion ₂₇₀¹ (b−a)⁵.Hence, we deduce that the inequality (2.1) is sharp.

Another relation is given by the next theorem:

Theorem 2.2. Under the assumptions of Theorem 2.1we have:

7γ4−15S₃

51840 (b−a)⁵ ≤ Z b

a

f(x)dx−^b−a₈ [f(a) + 3f(x₁) + 3f(x₂) +f(b)]

≤ ^7Γ₅₁₈₄₀^4−15S3(b−a)⁵. (2.13)

If γ4 = min

x∈[a,b]f⁽⁴⁾(x),Γ₄ = max

x∈[a,b]f⁽⁴⁾(x) then the inequalities (2.13) are sharp.

Proof. From (2.4), (2.6), (2.7)and (2.8) we have:

− Z b

a

f(x)dx+^b−a₈ [f(a) + 3f(x₁) + 3f(x₂) +f(b)]≤

≤ ^15S₅₁₈₄₀^3−7γ4(b−a)⁵. (2.14)

By analogy from (2.5), (2.10), (2.11) and (2.12) we have:

Z b a

f(x)dx−^b−a₈ [f(a) + 3f(x₁) + 3f(x₂) +f(b)]≤

≤ ^7Γ₅₁₈₄₀^4−15S3(b−a)⁵. (2.15)

From (2.14) and (2.15) we will have immediately the inequalities (2.13).

To show that the inequalities are sharp we choose f(x) = (x−a)⁴ and we

follow the steps of the proof for Theorem 2.1.

The next theorem offers us inequalities which do not depend uponS3. Theorem 2.3. Under the assumptions of Theorem 2.1we have:

7γ4−23Γ4

103680 (b−a)⁵ ≤ Z b

a

f(x)dx−^b−a₈ [f(a) + 3f(x1) + 3f(x2) +f(b)]≤

≤ ^7Γ₁₀₃₆₈₀^4−23γ4(b−a)⁵. (2.16)

If γ₄ = min

x∈[a,b]f⁽⁴⁾(x), Γ₄ = max

x∈[a,b]f⁽⁴⁾(x) then the inequalities (2.16) are sharp.

Proof. The inequalities (2.16) are easily deduced by (2.9) and (2.14) , re- spectively (2.12) and (2.15). To show that the inequalities are sharp we follow

the steps of the proof for Theorem 2.1.

In use the next theorem is important:

Theorem 2.4. Under the assumptions of Theorem 2.1we have:

23γ4−15S3

51840n⁴ (b−a)⁵≤

≤ ^b−a_8n

"

f(a) +f(b) + 2

n−1

X

i=1

f(xi) + 3

n

X

i=1

f(x⁰_i) + 3

n

X

i=1

f(x⁰⁰_i)

#

− Z b

a

f(x)dx

≤ ^23Γ_51840n^4−15S4³(b−a)⁵, (2.17)

where xi = a+ih, h = ^b−a_n , i = 0,1, ..., n and x⁰_i, x⁰⁰_i divide every interval [x_i, x_i+1]in three equal parts.

Proof. We divide each interval [x_i, x_i+1] in the equal parts by points x⁰_i, x⁰⁰_i, then we use Theorem 2.1 on the interval [xi, xi+1] :

23γ4−15S₃ⁱ

51840 (xi+1−xi)⁵ ≤

≤ ^xi+1₈^−xif(xi) + 3f(x⁰_i) + 3f(x⁰⁰_i) +f(xi+1)− Z xi+1

xi

f(x)dx

≤ ^23Γ₅₁₈₄₀^4−15S3i(x_i+1−x_i)⁵, (2.18)

whereS₃ⁱ = ^{f(3)(xi+1)−f}_h ^(3)(xi), i= 0,1, ..., n−1.By adding the formula we have got so far for i= 0,1, ..., n−1 and by noticing that

n−1

X

i=0

S₃ⁱ = ^f(3)(b)−f_h ^(3)(a) we

get the relation we wanted.

From Theorem 2.2, following the steps done, we get:

Theorem 2.5. Under the assumptions of Theorem 2.1we have:

7γ4−15S₃

51840n⁴ (b−a)⁵ ≤

≤ Z b

a

f(x)dx−^b−a_8n

"

f(a) +f(b) + 2

n−1

X

i=1

f(x_i) + 3

n

X

i=1

f(x⁰_i) + 3

n

X

i=1

f(x⁰⁰_i)

#

≤ ^7Γ_51840n^4−15S₄³(b−a)⁵. (2.19)

Proof. Proving this theorem is similarly to proving Theorem 2.4. Then we

use Theorem 2.2.

Also Theorem 2.3 leads us to:

Theorem 2.6. Under the assumptions of Theorem 2.1we have:

7γ4−23Γ4

103680n⁴(b−a)⁵≤

≤ Z b

a

f(x)dx−^b−a_8n

"

f(a) +f(b) + 2

n−1

X

i=1

f(xi) + 3

n

X

i=1

f(x⁰_i) + 3

n

X

i=1

f(x⁰⁰_i)

#

≤ ^7Γ_103680n^4−23γ4⁴(b−a)⁵. (2.20)

Proof. Proving this theorem is similarly to proving Theorem 2.4. Then we

use Theorem 2.3.

REFERENCES

[1] Cerone, P.,Three points rules in numerical integration, Non. Anal. Theor. Meth. Appl., 47(4), pp. 2341–2352, 2001.

[2] Cerone, P. and Dragomir, S. S., Midpoint-type Rules from an Inequalities Point of View, Handbook of Analytic-Computational Methods in Applied Mathematics, Editor:

G. Anastassiou, CRC Press, New York, pp. 135–200, 2000.

[3] Cerone, P. andDragomir, S. S.,Trapezoidal-type Rules from an Inequalities Point of View, Handbook of Analytic-Computational Methods in Applied Mathematics, Editor:

G. Anastassiou, CRC Press, New York, pp. 65–134, 2000.

[4] Dragomir, S. S.,Agarwal, R. P. andCerone, P.,On Simpson’s inequality and appli- cations, J. Inequal. Appl.,5, pp. 533–579, 2000.

[5] Dragomir, S. S.,Pecari´c, J. andWang, S.,The unified treatment of trapezoid, Simpson and Ostrowski type inequalities for monotonic mappings and applications, Math. Comput.

Modelling,31, pp. 61–70, 2000.

[6] Ionescu, D. V.,Cuadraturi numerice, Editura Tehnic˘a, Bucure¸sti, 1957.

[7] Ujevi´c, N.,Some double integral inequalities and applications, Acta Math. Univ. Come- nianae,71(2), pp. 187, 2002.

[8] Ujevi´c, N.,Double integral inequalities of simpson type and applications, J. Appl. Math.

& Computing,14, (1–2), pp. 213–223, 2004.

Received by the editors: January 12, 2006.