1 / 9

Introduction to Numerical Analysis I

Introduction to Numerical Analysis I. Splines. MATH/CMPSC 455. Spline. Suppose that n+1 points has been specified and satisfy . A spline of degree k is a function such that: . On each subinterval , is a polynomial of degree .

sine
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 Splines MATH/CMPSC 455

  2. Spline Suppose that n+1 points has been specified and satisfy . A spline of degree k is a function such that: • On each subinterval , is a polynomial of degree • has a continuous (k-1)-th derivate on Spline is a piecewise polynomial of degree at most k, and has continuous derivatives of all order up to k-1.

  3. Example: Spline of degree 0 Example: Spline of degree 1

  4. Cubic Spline A cubic spline is a piecewise cubic polynomial • is cubic polynomial (piecewise polynomial) • (Interpolation) • , (Continuity)

  5. Question: Can we uniquely determine the cubic spline? Unknowns (coefficients): • Conditions: • Interpolation: • Continuity of 1st order derivative: • Continuity of 2nd order derivative: • Total: We have two degrees of freedom!

  6. Derive the Cubic Spline • Step 1: 2nd order derivative is piecewise linear; • (use the continuity of 2nd order derivative) • Step 2: Take integration twice, get the cubic spline with undetermined coefficient; • Step 3: Determine the coefficient of the low order terms; (use the interpolation property) • Step 4: Determine the remaining coefficient by solving a symmetric, tri-diagonal system; • (use the continuity of 1st order derivative)

  7. Where: Nature Cubic Spline:

  8. Clamped Cubic Spline:

More Related