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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
Typology: Lecture notes
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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 [๐๐, ๐๐] into subintervals and use
simple integration rule for each subinterval.
Composite Trapezoidal rule
Let ๐๐ โ ๐ถ๐ถ
2
๐๐โ๐๐
๐๐
, and ๐๐
๐๐
= ๐๐ + ๐๐โ for ๐๐ = 0, โฏ , ๐๐.
On each subinterval ๏ฟฝ๐๐
๐๐โ
๐๐
๏ฟฝ, for for ๐๐ = 1, โฏ , ๐๐, apply Trapezoidal rule:
Figure 1 Composite Trapezoidal Rule
๏ฟฝ ๐๐
( ๐๐
) ๐๐๐๐
๐๐
๐๐
= ๏ฟฝ
โ
2
๏ฟฝ๐๐
( ๐๐
0
) + ๐๐
( ๐๐
1
) ๏ฟฝ โ
โ
3
12
๐๐
โฒโฒ
(๐๐
1
)๏ฟฝ
๏ฟฝ
โ
2
๏ฟฝ๐๐
( ๐๐
1
) + ๐๐
( ๐๐
2
) ๏ฟฝ โ
โ
3
12
๐๐
โฒโฒ
(๐๐
2
)
๏ฟฝ
โฏ
๏ฟฝ
โ
2
๏ฟฝ๐๐
( ๐๐
๐๐โ
) + ๐๐
( ๐๐
๐๐
) ๏ฟฝ โ
โ
3
12
๐๐
โฒโฒ
( ๐๐
๐๐
) ๏ฟฝ
=
โ
2
๏ฟฝ๐๐
( ๐๐
) + 2 ๏ฟฝ ๐๐๏ฟฝ๐๐
๐๐
๏ฟฝ
๐๐โ
๐๐=
( ๐๐
) ๏ฟฝ โ
โ
3
12
๏ฟฝ ๐๐
โฒโฒ
(๐๐
๐๐
)
๐๐
๐๐=
=
โ
2
๏ฟฝ๐๐
( ๐๐
) + 2 ๏ฟฝ ๐๐๏ฟฝ๐๐
๐๐
๏ฟฝ
๐๐โ
๐๐=
( ๐๐
) ๏ฟฝ โ
๐๐ โ ๐๐
12
โ
2
๐๐
โฒโฒ
(๐๐)
Error, which can be
simplified
Composite Simpsonโs rule
Let ๐๐ โ ๐ถ๐ถ
2
๐๐โ๐๐
๐๐
, and ๐๐
๐๐
for ๐๐ = 0, โฏ , ๐๐.
On each consecutive pair of subintervals (for example
0
2
2
4
and ๏ฟฝ๐๐
2๐๐โ
2๐๐
๏ฟฝ) for each ๐๐ = 1, โฏ , ๐๐/2, apply a Simpsonโs rule:
Figure 2 Composite Simpson's rule
๏ฟฝ ๐๐
( ๐๐
) ๐๐๐๐
๐๐
๐๐
= ๏ฟฝ ๏ฟฝ ๐๐
( ๐๐
) ๐๐๐๐
๐ฅ๐ฅ
2๐๐
๐ฅ๐ฅ
2๐๐โ
๐๐/ 2
๐๐=
= ๏ฟฝ
โ
3
๏ฟฝ๐๐๏ฟฝ๐๐
2๐๐โ
๏ฟฝ + 4๐๐๏ฟฝ๐๐
2๐๐โ
๏ฟฝ + ๐๐๏ฟฝ๐๐
2๐๐
๏ฟฝ โ
โ
5
90
๐๐
( 4 )
๏ฟฝ๐๐
๐๐
๏ฟฝ๏ฟฝ
๐๐/ 2
๐๐=
=
โ
3
โ
โ
โ
๐๐
( ๐๐
0
) + 2 ๏ฟฝ ๐๐๏ฟฝ๐๐
2๐๐
๏ฟฝ
๏ฟฝ
๐๐
2
๏ฟฝโ
๐๐=
2๐๐โ
๏ฟฝ
๏ฟฝ
๐๐
2
๏ฟฝ
๐๐=
( ๐๐
๐๐
)
โ
โ
โ
โ
โ
5
90
๏ฟฝ ๐๐
( 4 )
๏ฟฝ๐๐
๐๐
๏ฟฝ
๏ฟฝ
๐๐
2
๏ฟฝ
๐๐=
Error, which can be simplified
Composite Midpoint rule
Theorem 4.6 Let ๐๐ โ ๐ถ๐ถ
2
๐๐โ๐๐
๐๐+
, and ๐๐
๐๐
1)โ for each ๐๐ = โ1, 0, โฏ , ๐๐, ๐๐ + 1. There exists a ๐๐ โ (๐๐, ๐๐) for which
Composite Midpoint rule with its error term is
๐๐
๐๐
2๐๐
(
๐๐
2
)
๐๐=
2
โฒโฒ
Figure 3 Composite Midpoint rule
Exercise 13. Determine the values of ๐๐ and โ required to approximate
1
๐ฅ๐ฅ+
2
0
to within 10
โ
and compute the approximation. Use
a. Composite Trapezoidal rule.
b. Composite Simpsonโs rule.
c. Composite Midpoint rule.