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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 .

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Introduction to Numerical Analysis I

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  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:

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