Understanding Natural Cubic Interpolation: Spline Functions & Cubic Spline, Study Guides, Projects, Research of Numerical Methods in Engineering

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Natural Cubic Interpolation

Jingjing Huang 10/24/

Interpolation

• Construct a function crosses known points

• Predict the value of unknown points

Interpolation

• Polynomial Interpolation

• Same polynomial for all points
• Vandermonde Matrix, ill-conditioned

• Lagrange Form

• Hard to evaluate

• Piecewise Interpolation

• Different polynomials for each interval

Lagrange form

Given k+1 points

Define:

where

Lagrange form

• Example: interpolate f(x) = x^2 , for x = 1,2,

How to represent models

• Specify every point along a model?

• Hard to get precise results
• Too much data, too hard to work with generally

• Specify a model by a small number of “control

points”

• Known as a spline curve or just spline

8

Spline Interpolation Definition

• Given n +1 distinct knots x i such that:

with n +1 knot values yi find a spline function

with each Si ( x ) a polynomial of degree at most n.

Tangent

• The derivative of a curve represents the

tangent vector to the curve at some point

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d x

dt

( ) t

x (^) ( ) t

• Why cubic?
  • Good enough for some cases
  • The degree is not too high to be easily solved

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Natural Cubic Spline Interpolation

• Si(x) = aix^3 + bix^2 + cix + di (Given n points)
  • 4 Coefficients with n-1 subintervals = 4n-4 equations
  • There are 4n-6 conditions
    • Interpolation conditions
    • Continuity conditions
    • Natural Conditions
  • S’’(x 0 ) = 0
  • S’’(xn) = 0
  • O(n^3 )

Thanks