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Composite Numerical Integration: Approximating Definite Integrals using Composite Rules, Lecture notes of Calculus

Composite numerical integration, a method for approximating definite integrals by dividing the integration interval into subintervals and applying simple integration rules to each subinterval. Simpson's rule, Trapezoidal rule, and Midpoint rule, providing examples and error analysis.

What you will learn

  • How does composite numerical integration differ from regular numerical integration?
  • How does the error analysis of composite numerical integration differ between the three rules?
  • What is composite numerical integration?
  • What are the advantages of using composite Simpson's rule over composite Trapezoidal rule?
  • What is the role of subintervals in composite numerical integration?

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4.4 Composite Numerical Integration
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4.4 Composite Numerical Integration

Motivation: 1) on large interval, use low order Newton-Cotes formulas

are not accurate.

2) on large interval, interpolation using high degree polynomial is

unsuitable because of oscillatory nature of high degree polynomials.

Main idea: divide integration interval [๐‘Ž๐‘Ž, ๐‘๐‘] into subintervals and use

simple integration rule for each subinterval.

Composite Trapezoidal rule

Let ๐‘“๐‘“ โˆˆ ๐ถ๐ถ

2

[

],

๐‘๐‘โˆ’๐‘Ž๐‘Ž

๐‘›๐‘›

, and ๐‘‘๐‘‘

๐‘—๐‘—

= ๐‘Ž๐‘Ž + ๐‘—๐‘—โ„Ž for ๐‘—๐‘— = 0, โ‹ฏ , ๐‘›๐‘›.

On each subinterval ๏ฟฝ๐‘‘๐‘‘

๐‘—๐‘—โˆ’

๐‘—๐‘—

๏ฟฝ, for for ๐‘—๐‘— = 1, โ‹ฏ , ๐‘›๐‘›, apply Trapezoidal rule:

Figure 1 Composite Trapezoidal Rule

๏ฟฝ ๐‘“๐‘“

( ๐‘‘๐‘‘

) ๐‘‘๐‘‘๐‘‘๐‘‘

๐‘๐‘

๐‘Ž๐‘Ž

= ๏ฟฝ

โ„Ž

2

๏ฟฝ๐‘“๐‘“

( ๐‘‘๐‘‘

0

) + ๐‘“๐‘“

( ๐‘‘๐‘‘

1

) ๏ฟฝ โˆ’

โ„Ž

3

12

๐‘“๐‘“

โ€ฒโ€ฒ

(๐œ‰๐œ‰

1

)๏ฟฝ

๏ฟฝ

โ„Ž

2

๏ฟฝ๐‘“๐‘“

( ๐‘‘๐‘‘

1

) + ๐‘“๐‘“

( ๐‘‘๐‘‘

2

) ๏ฟฝ โˆ’

โ„Ž

3

12

๐‘“๐‘“

โ€ฒโ€ฒ

(๐œ‰๐œ‰

2

)

๏ฟฝ

  • โ‹ฏ

  • ๏ฟฝ

โ„Ž

2

๏ฟฝ๐‘“๐‘“

( ๐‘‘๐‘‘

๐‘›๐‘›โˆ’

) + ๐‘“๐‘“

( ๐‘‘๐‘‘

๐‘›๐‘›

) ๏ฟฝ โˆ’

โ„Ž

3

12

๐‘“๐‘“

โ€ฒโ€ฒ

( ๐œ‰๐œ‰

๐‘›๐‘›

) ๏ฟฝ

=

โ„Ž

2

๏ฟฝ๐‘“๐‘“

( ๐‘Ž๐‘Ž

) + 2 ๏ฟฝ ๐‘“๐‘“๏ฟฝ๐‘‘๐‘‘

๐‘—๐‘—

๏ฟฝ

๐‘›๐‘›โˆ’

๐‘—๐‘—=

  • ๐‘“๐‘“

( ๐‘๐‘

) ๏ฟฝ โˆ’

โ„Ž

3

12

๏ฟฝ ๐‘“๐‘“

โ€ฒโ€ฒ

(๐œ‰๐œ‰

๐‘—๐‘—

)

๐‘›๐‘›

๐‘—๐‘—=

=

โ„Ž

2

๏ฟฝ๐‘“๐‘“

( ๐‘Ž๐‘Ž

) + 2 ๏ฟฝ ๐‘“๐‘“๏ฟฝ๐‘‘๐‘‘

๐‘—๐‘—

๏ฟฝ

๐‘›๐‘›โˆ’

๐‘—๐‘—=

  • ๐‘“๐‘“

( ๐‘๐‘

) ๏ฟฝ โˆ’

๐‘๐‘ โˆ’ ๐‘Ž๐‘Ž

12

โ„Ž

2

๐‘“๐‘“

โ€ฒโ€ฒ

(๐œ‡๐œ‡)

Error, which can be

simplified

Composite Simpsonโ€™s rule

Let ๐‘“๐‘“ โˆˆ ๐ถ๐ถ

2

[

],

๐‘๐‘โˆ’๐‘Ž๐‘Ž

๐‘›๐‘›

, and ๐‘‘๐‘‘

๐‘—๐‘—

for ๐‘—๐‘— = 0, โ‹ฏ , ๐‘›๐‘›.

On each consecutive pair of subintervals (for example

[

0

2

]

[

2

4

]

and ๏ฟฝ๐‘‘๐‘‘

2๐‘—๐‘—โˆ’

2๐‘—๐‘—

๏ฟฝ) for each ๐‘—๐‘— = 1, โ‹ฏ , ๐‘›๐‘›/2, apply a Simpsonโ€™s rule:

Figure 2 Composite Simpson's rule

๏ฟฝ ๐‘“๐‘“

( ๐‘‘๐‘‘

) ๐‘‘๐‘‘๐‘‘๐‘‘

๐‘๐‘

๐‘Ž๐‘Ž

= ๏ฟฝ ๏ฟฝ ๐‘“๐‘“

( ๐‘‘๐‘‘

) ๐‘‘๐‘‘๐‘‘๐‘‘

๐‘ฅ๐‘ฅ

2๐‘—๐‘—

๐‘ฅ๐‘ฅ

2๐‘—๐‘—โˆ’

๐‘›๐‘›/ 2

๐‘—๐‘—=

= ๏ฟฝ

โ„Ž

3

๏ฟฝ๐‘“๐‘“๏ฟฝ๐‘‘๐‘‘

2๐‘—๐‘—โˆ’

๏ฟฝ + 4๐‘“๐‘“๏ฟฝ๐‘‘๐‘‘

2๐‘—๐‘—โˆ’

๏ฟฝ + ๐‘“๐‘“๏ฟฝ๐‘‘๐‘‘

2๐‘—๐‘—

๏ฟฝ โˆ’

โ„Ž

5

90

๐‘“๐‘“

( 4 )

๏ฟฝ๐œ‰๐œ‰

๐‘—๐‘—

๏ฟฝ๏ฟฝ

๐‘›๐‘›/ 2

๐‘—๐‘—=

=

โ„Ž

3

โŽ

โŽœ

โŽ›

๐‘“๐‘“

( ๐‘‘๐‘‘

0

) + 2 ๏ฟฝ ๐‘“๐‘“๏ฟฝ๐‘‘๐‘‘

2๐‘—๐‘—

๏ฟฝ

๏ฟฝ

๐‘›๐‘›

2

๏ฟฝโˆ’

๐‘—๐‘—=

  • 4 ๏ฟฝ ๐‘“๐‘“๏ฟฝ๐‘‘๐‘‘

2๐‘—๐‘—โˆ’

๏ฟฝ

๏ฟฝ

๐‘›๐‘›

2

๏ฟฝ

๐‘—๐‘—=

  • ๐‘“๐‘“

( ๐‘‘๐‘‘

๐‘›๐‘›

)

โŽ 

โŽŸ

โŽž

โˆ’

โ„Ž

5

90

๏ฟฝ ๐‘“๐‘“

( 4 )

๏ฟฝ๐œ‰๐œ‰

๐‘—๐‘—

๏ฟฝ

๏ฟฝ

๐‘›๐‘›

2

๏ฟฝ

๐‘—๐‘—=

Error, which can be simplified

Composite Midpoint rule

Theorem 4.6 Let ๐‘“๐‘“ โˆˆ ๐ถ๐ถ

2

[

],

๐‘๐‘โˆ’๐‘Ž๐‘Ž

๐‘›๐‘›+

, and ๐‘‘๐‘‘

๐‘—๐‘—

1)โ„Ž for each ๐‘—๐‘— = โˆ’1, 0, โ‹ฏ , ๐‘›๐‘›, ๐‘›๐‘› + 1. There exists a ๐œ‡๐œ‡ โˆˆ (๐‘Ž๐‘Ž, ๐‘๐‘) for which

Composite Midpoint rule with its error term is

๐‘๐‘

๐‘Ž๐‘Ž

2๐‘—๐‘—

(

๐‘›๐‘›

2

)

๐‘—๐‘—=

2

โ€ฒโ€ฒ

Figure 3 Composite Midpoint rule

Exercise 13. Determine the values of ๐‘›๐‘› and โ„Ž required to approximate

1

๐‘ฅ๐‘ฅ+

2

0

to within 10

โˆ’

and compute the approximation. Use

a. Composite Trapezoidal rule.

b. Composite Simpsonโ€™s rule.

c. Composite Midpoint rule.