4.3 numerical integration - university of notre dame open newton-cotes formula see figure 4. let ;...
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4.3 Numerical Integration
Numerical quadrature: Numerical method to compute ∫ ( )
approximately by a sum ∑ ( )
.
The interpolation nodes are given as:
( ( ))
( ( ))
( ( ))
…
( ( ))
Here By Lagrange Interpolation Theorem (Thm 3.3):
( ) ∑ ( ) ( )
( ) ( )
( ) ( )( ( )) (1)
∫ ( )
∫ ∑ ( ) ( )
( ) ∫ ( ) ( ) ( )( ( ))
Quadrature formula: ∫ ( )
∑ ( )
with ∫ ( )
.
Error: ( )
( ) ∫ ( ) ( ) ( )( ( ))
𝑃𝑁(𝑥)
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The Trapezoidal Rule (obtained by first Lagrange interpolating polynomial)
Let ; and (see Figure 1)
∫ ( )
∫ [ ( )
( ) ( )
( )]
∫ ( )( )
( )( ( ))
Thus
∫ ( )
[ ( ) ( )]
( )( )
.
Error term
Note: for Trapezoidal rule.
Figure 1 Trapezoidal Rule
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The Simpson’s (1/3) Rule (error obtained by third Taylor polynomial)
Let
; and
(see Figure 2)
( ) ( ) ( )( ) ( )
( )
( )
( )
( )( )
( )
∫ ( )
∫ ( ( ) ( )( ) ( )
( )
( )
( )
( )( ( ))
( )
)
( )
( )
( )( )
Now approximate ( )
[ ( ) ( ) ( )]
( )( )
Thus
∫ ( )
( ( ) ( ) ( ))
( )( )
Error term
Note:
for Simpson’s rule.
Figure 2 Simpson's Rule
𝑥 𝑎 𝑥 𝑥 𝑏
𝑝(𝑥)
𝑓(𝑥)
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Precision
Definition: The degree of accuracy or precision of a quadrature formula is the largest positive integer such that the formula
is exact for , for each .
∫
; ∫
[ ] Trapezoidal rule is exact for (or ).
∫
=
. ∫
[ ]
. Trapezoidal rule is exact for .
∫
. ∫
[ ]
Trapezoidal rule is NOT exact for .
Remark: The degree of precision of a quadrature formula is if and only if the error is zero for all polynomials of degree
, but is NOT zero for some polynomial of degree .
Closed Newton-Cotes Formulas
Let and
. for .
∫ ( )
∑ ( )
with ∫ ( )
.
Here ( ) is the ith Lagrange base polynomial of degree N.
Trapezoidal rule has degree of accuracy one.
Simpson’s rule has degree of accuracy three.
Figure 3 Closed Newton-Cotes Formulas
𝑃𝑁(𝑥)
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Theorem 4.2 Suppose that ∑ ( ) is the (n+1)-point closed Newton-Cotes formula with and
.
There exists ( ) for which ∫ ( )
∑ ( )
( )( )
( ) ∫ ( ) ( )
,
if is even and [ ], and
∫ ( )
∑ ( )
( )( )
( ) ∫ ( ) ( )
if is odd and [ ].
Remark: is even, degree of precision is is odd, degree of precision is
Examples. N=1: Trapezoidal rule; N=2: Simpson’s rule.
N=3: Simpson’s Three-Eighths rule
∫ ( )
( ( ) ( ) ( ) ( ))
( )( ) where ;
.
a=x-1
xn
f(x)
x0 b=x
n+1
Figure 4 Open Newton-Cotes Formula
𝑃𝑛(𝑥)
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Open Newton-Cotes Formula
See Figure 4. Let
; and for . This implies .
Theorem 4.3 Suppose that ∑ ( ) is the (n+1)-point open Newton-Cotes formula with and
. There exists ( ) for which ∫ ( )
∑ ( )
( )( )
( ) ∫ ( ) ( )
,
if is even and [ ], and
∫ ( )
∑ ( )
( )( )
( ) ∫ ( ) ( )
if is odd and [ ].
Examples of open Newton-Cotes formulas
n=0: Midpoint rule (Figure 5)
∫ ( )
( )
( )( )
where .
a=x
-1
f(x)
x0
0
b=x1
Figure 5 Midpoint rule
𝑝 (𝑥)
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n=1: ∫ ( )
[ ( ) ( )]
( )( ) where .
n=2: ∫ ( )
[ ( ) ( ) ( )]
( )( ) where .