1 / 13

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.

kapono
Download Presentation

Introduction to Numerical Analysis I

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related