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Numerical integration continued --- Simpson’s rules - We can add more segments OR - We can use a higher order polynomial
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Simpson’s 1/3 rule use a second order interpolating polynomial If we use Lagrange form
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Integrate and do some algebra
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If we use a=x0 and b=x2, and x1=(b+a)/2 widthAverage height Error for Simpson’s 1/3 rule
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As with Trapezoidal rule, can use multiple applications of Simpson’s 1/3 rule need even number of segments, odd number of points 9 points, 4 segments
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As in multiple trapzoid, break integral up Substitute Simpson’s 1/3 rule for each integral and collect terms
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Example: Numerically integrate from 0 to 1 using 1) single trapezoid, 2) multiple trapezoid, 3) single Simpson’s 1/3 and 4) multiple Simpson’s 1/3
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True, analytic value of I is 0.4749
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Really quite bad 1) Single trapezoidal rule
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2) Multiple trapezoidal rule
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3) Single Simpson’s 1/3 rule
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4) Multiple Simpson’s 1/3 rule
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Simpson’s 1/3 rule is limited to applications with equally-spaced data even number of segments odd number of points Simpson’s 3/8 rule used when there are odd number of segments even number of points
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Simpson’s 3/8 rule uses a third order Lagrange polynomial Four equally spaced points, separated by or
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Can do multiple segment application of Simpson’s 3/8 rule. Can also mix and match Simpson’s 1/3 and 3/8 to fill up segments
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Example: 12 points, 11 segments Each 3/8 rule application takes 3 segments Each 1/3 rule application takes 2 segments
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Neither 2 nor 3 go into 11 But 3 3’s and a 2 do. 1/3 rule 3/8 rule
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Higher order Newton-Cotes closed formulas Simpson’s 1/3 - 2nd order Lagrange Simpson’s 3/8 - 3rd order Lagrange we can keep going but don’t usually - Simpson is accurate enough when applied in multiple segments
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Integration with unequal segments If all unequal, stuck with multiple trapezoid rule application If you can find some sets of equal segments, use Simpson’s rules
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