Chapter 6 Homework Consider The Definite Integral If

Composite Newton-Cotes Quadrature 1

6.5 Composite Newton-Cotes Quadrature

1. Provide the details of the transformation of the error term associated with the

composite Simpson’s rule from

h5

90

m

X

j=1

f(4)(ξj) to (b−a)h4

180 f(4)(ξ).

Suppose fhas four continuous derivatives. Then the Extreme Value Theorem guar-

It then follows that for each j

Summing over each subinterval [x2j−2, x2j], we ﬁnd that

We can now conclude, by the Intermediate Value Theorem, that there exists ξ∈

2. Derive the Composite Midpoint Rule with error:

Zb

a

f(x)dx = 2h

n

X

j=1

f(xj) + (b−a)h2

6f′′(ξ),

2Section 6.5

Recall that

I(f) =

n

X

j=1 Zxj

xj−1

f(x)dx

The error term needs to be examined more closely. Suppose fhas two continuous

derivatives. Then the Extreme Value Theorem guarantees that there exist two

constants c1, c2∈[a, b]such that

It then follows that for each j

Summing over each subinterval [xj−h, xj+h], we ﬁnd that

Composite Newton-Cotes Quadrature 3

We can now conclude, by the Intermediate Value Theorem, that there exists ξ∈

[a, b]such that f′′(ξ) = 1

nPn

j=1 f′′(ξj). This implies that the error for the com-

posite midpoint rule can be written as

3. (a) Let Qh(f) be an approximation to the deﬁnite integral I(f) obtained using

a generic composite quadrature formula with a subinterval size of h. If

the composite quadrature formula has a theoretical rate of convergence of

O(hk), show that

Qh(f)−Qh/b(f)

Qh/b(f)−Qh/b2(f)≈bk.

(b) What value do we expect from the ratio

Sh(f)−Sh/2(f)

Sh/2(f)−Sh/4(f),

where Sh(f) denotes the composite Simpson’s rule approximation to the

deﬁnite integral I(f) obtained with a subinterval size of h.

(a) Let eh=Qh(f)−I(f);i.e.,ehis the error associated with Qh(f). Since the

for suﬃciently small h.

(b) Because the composite Simpson’s rule has a theoretical rate of convergence of

4. Verify that the composite Simpson’s rule has rate of convergence O(h4) by

approximating the value of R1

0√1 + x3dx.

Consider the deﬁnite integral

approaches 16 as his decreased, thereby providing numerical veriﬁcation that the

rate of convergence is O(h4).

n h Sh(f)Sh(f)−Sh/2(f)

Sh/2(f)−Sh/4(f)

5. (a) Verify that the composite midpoint rule has rate of convergence O(h2) by

approximating the value of R1

0√1 + x3dx.

(b) Repeat part (a) by approximating the value of Rπ

0sin xdx.

(a) Consider the deﬁnite integral

I(f) = Z1

0p1 + x3dx.

Composite Newton-Cotes Quadrature 5

hMh

Mh−Mh/2

Mh/2−Mh/4

1 1.060660 4.606

1/2 1.100103 4.100

(b) Consider the deﬁnite integral

The table below lists composite midpoint rule approximations to I(f)for sev-

hMh|e2h/eh|

π3.141593

In Exercises 6 – 11, verify that the composite trapezoidal rule has rate of con-

vergence O(h2), the composite midpoint rule has rate of convergence O(h2) and

the composite Simpson’s rule has rate of convergence O(h4) by approximating

the value of the indicated deﬁnite integral.

6. R2

1

xdx

Consider the deﬁnite integral

I(f) = Z2

1

xdx.

6Section 6.5

hTh|e2h/eh|Mh|e2h/eh|Sh|e2h/eh|

1/2 0.708333 0.685714 0.69444444

7. R1

0e−xdx

Consider the deﬁnite integral

The table below lists composite trapezoidal rule approximations, composite mid-

point rule approximations and composite Simpson’s rule approximations to I(f)for

hTh|e2h/eh|Mh|e2h/eh|Sh|e2h/eh|

1/2 0.645235 0.625584 0.63233368

1/4 0.635409 3.988 0.630477 3.978 0.63213418 15.652

8. R1

0tan−1xdx

Consider the deﬁnite integral

Composite Newton-Cotes Quadrature 7

The table below lists composite trapezoidal rule approximations, composite mid-

hTh|e2h/eh|Mh|e2h/eh|Sh|e2h/eh|

1/2 0.428173 0.444240 0.43999810

9. R2

1

sin x

xdx

Consider the deﬁnite integral

The table below lists composite trapezoidal rule approximations, composite mid-

point rule approximations and composite Simpson’s rule approximations to I(f)for

approach 4, 4, and 16, respectively as his decreased. This provides numerical

evidence that the composite trapezoidal rule has rate of convergence O(h2), the

8Section 6.5

10. R1

0

1

√1+x4dx

Consider the deﬁnite integral

The table below lists composite trapezoidal rule approximations, composite mid-

point rule approximations and composite Simpson’s rule approximations to I(f)for

approach 4, 4, and 16, respectively as his decreased. This provides numerical

evidence that the composite trapezoidal rule has rate of convergence O(h2), the

hTh

Th−Th/2

Th/2−Th/4Mh

Mh−Mh/2

Mh/2−Mh/4Sh

Sh−Sh/2

Sh/2−Sh/4

1/2 0.911848 4.124 0.934814 4.224 0.93127946 35.694

11. R4

0x√x2+ 9dx

Consider the deﬁnite integral

Composite Newton-Cotes Quadrature 9

hTh|e2h/eh|Mh|e2h/eh|Sh|e2h/eh|

234.422205 31.780399 32.56294014

12. Suppose that there exists a composite quadrature rule, Q(f), with the property

Zb

a

f(x)dx =Q(f)−(b−a)h4

240 f(5)(ξ),

where a < ξ < b and h= (b−a)/n.

(a) What is the rate of convergence associated with this quadrature rule?

What conditions must the integrand satisfy to achieve this rate of conver-

gence? Explain how you would numerically verify the rate of convergence.

(b) What is the degree of precision of this quadrature rule? Explain how to

verify the degree of precision.

(c) What is the smallest value of nneeded to guarantee an approximation to

the value of R2

1

xdx to within 10−5? Justify your response.

(a) Note the error term contains the factor h4f(5)(ξ). Thus, provided the inte-

grand has ﬁve continuous derivatives, the indicated quadrature rule has rate

(b) Because the error term contains the ﬁfth derivative of the integrand, the degree

of precision is equal to 4. To verify this degree of precision, demonstrate that

10 Section 6.5

(c) Since h= (b−a)/n, the error term may be written in the form

13. (a) Determine the smallest value of nwhich guarantees that the composite

midpoint rule approximates the value of R1

0

1

1+x2dx to within 1.25 ×10−5.

(b) Determine the smallest value of nwhich guarantees that the composite

midpoint rule approximates the value of R1

0e−x4dx to within 10−5.

(a) Since h= (b−a)/2n, the error term may be written in the form

so the value of nmust be selected to satisfy the inequality

(b) Since h= (b−a)/2n, the error term may be written in the form

Composite Newton-Cotes Quadrature 11

so the value of nmust be selected to satisfy the inequality

In Exercises 14 – 20, approximate the value of the indicated deﬁnite integral

using the composite trapezoidal rule, the composite midpoint rule and the com-

posite Simpson’s rule. For each method, use the smallest value of nwhich will

guarantee an absolute error not greater than 5 ×10−5.

14. R2

1

xdx

Let f(x) = 1/x. Then

The smallest number of subintervals needed to guarantee an absolute error not

greater than 5×10−5from the composite trapezoidal rule satisﬁes the inequality

respectively. The solutions of these inequalities are

15. R1

0e−xdx

12 Section 6.5

Let f(x) = e−x. Then

For the the composite midpoint rule and the composite Simpson’s rule, the corre-

sponding inequalities are

16. R1

0tan−1xdx

Let f(x) = tan−1x. Then

For the the composite midpoint rule and the composite Simpson’s rule, the corre-

sponding inequalities are