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Introduction to Numerical Analysis I

Introduction to Numerical Analysis I. Numerical Integration. MATH/CMPSC 455. Numerical Integration. Mathematical Problem:. Example:. Example:. By calculus, find that , then use.

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Introduction to Numerical Analysis I

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  1. Introduction to Numerical Analysis I Numerical Integration MATH/CMPSC 455

  2. Numerical Integration Mathematical Problem: Example: Example:

  3. By calculus, find that , then use Numerical Integration: replace by another function that approximates well and is easily integral, then we have

  4. Newton-Cotes Formulas Idea: 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:

  5. Trapezoid rule Use two nodes: and

  6. Simpson’s rule Use three nodes:

  7. Example: Apply the Trapezoid Rule and Simpson’s Rule to approximate Example: Apply the Trapezoid Rule and Simpson’s Rule to approximate

  8. Error of the trapezoid rule: The trapezoid rule is exact for all polynomial of degree less than or equal to 1.

  9. Error of the Simpson’s rule: The Simpson’s rule is exact for all polynomial of degree less than or equal to 3.

  10. 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

  11. The Composite Simpson’s Rule

  12. Error of Composite Rules Error of the composite trapezoid rule: Error of the composite Simpson’s rule:

  13. Example: Apply the composite Trapezoid Rule and Simpson’s Rule ( 4 subintervals ) to approximate

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