Introduction to Numerical Analysis I
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Transcript of Introduction to Numerical Analysis I
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Introduction to Numerical Analysis I
MATH/CMPSC 455
Numerical Integration
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NUMERICAL INTEGRATION
Mathematical Problem:
Example:
Example:
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By calculus, find that , then use
Numerical Integration: replace by another function that approximates well and is easily integral, then we have
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NEWTON-COTES FORMULASIdea: use polynomial interpolation to find the approximation function
Step 1: Select nodes in [a,b] Step 2: Use Lagrange form of polynomial interpolation to find the approximation function
Step 3:
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TRAPEZOID RULEUse two nodes: and
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SIMPSON’S RULEUse three nodes:
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Example: Apply the Trapezoid Rule and Simpson’s Rule to approximate
Example: Apply the Trapezoid Rule and Simpson’s Rule to approximate
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Error of the trapezoid rule:
The trapezoid rule is exact for all polynomial of degree less than or equal to 1.
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Error of the Simpson’s rule:
The Simpson’s rule is exact for all polynomial of degree less than or equal to 3.
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THE COMPOSITE TRAPEZOID RULE
Why? ? The high order polynomial interpolations are unbounded!
Step 1: Partition the interval into n subintervals by introducing points Step 2: Use the trapezoid rule on each subinterval
Step 3: Sum over all subintervals
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THE COMPOSITE SIMPSON’S RULE
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ERROR OF COMPOSITE RULES
Error of the composite trapezoid rule:
Error of the composite Simpson’s rule:
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Example: Apply the composite Trapezoid Rule and Simpson’s Rule ( 4 subintervals ) to approximate