Numerical Integration Composite Rules CS/SE 4X03
Ned Nedialkov McMaster University February 12, 2021
Outline
Composite trapezoidal rule Composite Simpson & midpoint rules Errors
Composite trapezoidal rule Composite Simpson & midpoint rules Errors
How to increase the accuracy of a rule
• We can increase the degree of the polynomial, but the error might be large
• Apply a basic rule over small subintervals
◦ subdivide [a, b] into r subintervals
◦ h = b−a length of each subinterval r
◦ ti =a+ih,i=0,1,…,r t0 = a, tr = b
b a
f(x)dx =
r t i i=1 ti−1
f(x)dx
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Composite trapezoidal rule Composite Simpson & midpoint rules Errors Composite trapezoidal rule
From the basic rule on [ti−1,ti], i = 1,…,r
ti
ti−1 we derive
f(x)dx ≈
ti − ti−1 h
2 [f(ti−1) + f(ti)] = 2 [f(ti−1) + f(ti)]
b a
r t i
h r
f(x)dx ≈ 2 [f(ti−1) + f(ti)]
f(x)dx = =
i=1 i=1
= h(f(t0)+f(t1)+···+f(tr−1))
2
+h( f(t1)+···+f(tr−1)+f(tr)) 2
r−1
= h [f(a) + f(b)] + hf(ti)
2
i=1
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i=1 ti−1 hrr
2
i=1 f(ti−1) + f(ti)
Composite trapezoidal rule Composite Simpson & midpoint rules Errors Composite Simpson & midpoint rules
Simpson:
b h r/2−1 r/2
a
f(x)dx≈ 3f(a)+2 f(t2i)+4f(t2i−1)+f(b) i=1 i=1
Midpoint:
b r f(x)dx ≈ h
f (a + (i − 1/2)h)
a
i=1
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Composite trapezoidal rule Composite Simpson & midpoint rules
Errors
Errors
• Trapezoidal rule
• Midpoint rule
• Simpson
E(f) = −f′′(η)(b − a)h2 12
E(f) = f′′(ξ)(b − a)h2 24
E(f) = −f(iv)(ζ)(b − a)h4 180
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