1 / 9

Introduction to Numerical Analysis I

Introduction to Numerical Analysis I. Root Finding without Derivatives . MATH/CMPSC 455. Newton’s Method. Give . No. Converge?. Yes. Out put . Quadratic convergence of Newton’s method. Definition: The iteration is quadratically convergent if .

nicole
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 Root Finding without Derivatives MATH/CMPSC 455

  2. Newton’s Method

  3. Give No Converge? Yes Out put

  4. Quadratic convergence of Newton’s method Definition: The iteration is quadratically convergent if Theorem: Let be twice continuously differentiable. If , then Newton’s method is locally and quadratically convergent to the root.

  5. Example: Use Newton’s method to find root of Theorem: Assume that the (m+1)-times continuously differentiable function has a multiplicity m root. Then Newton’s method is locally and linearlyconvergent to the root. Modified Newton’s Method:

  6. Example: Apply Newton’s Method to the following function with starting guess 1 -1

  7. Secant Method

  8. Give No Converge? Yes Out put

  9. Other Methods without Derivatives • Method of False Position: • Similar to Bisection method, but where the midpoint is replaced by a Secant Method-like approximation. • Muller’s Method: • Muller’s Method uses three previous points to draw a parabola through them, and intersect the parabola with the x-axis. • Inverse Quadratic Interpolation: • IQI is similar with Muller’s Method. However, the parabola is of the form • Hybrid Method

More Related