# Curve Fitting and Interpolation: Lecture (II)

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Curve Fitting and Interpolation: Lecture (II)
Chapter 5 Curve Fitting and Interpolation: Lecture (II) Dr. Jie Zou PHY 3320

Outline Interpolation: (2) Newton’s Divided-Difference Interpolating Polynomials Linear Interpolation (1st-order polynomial) Quadratic Interpolation (2nd-order polynomial) Nth-order polynomial interpolation (Lecture (III)) Dr. Jie Zou PHY 3320

Newton’s linear interpolation
Case: Two data points are available - {x0, f(x0)} and {x1, f(x1)}. Note: f(x)=the actual function Task: Determine the formula for the linear interpolating function f1(x). Note: f1(x) = the (approximating) interpolation polynomial; “1” indicates the order of the polynomial. Dr. Jie Zou PHY 3320

Linear interpolation formula: Derivation
Consider the similar triangles ADE and ABC: DE/AE=BC/AC, or Solving for f1(x) gives the Linear interpolation formula: Note: a1 represents the finite-divided-difference approximation of the first derivative, df(x)/d(x) or f’(x). Dr. Jie Zou PHY 3320

Example 5.6 Develop a linear interpolation formula for the function e0.5x using the values at x0 = 0 and x1 = 2, and use it to estimate the value of e0.5x at x = 1. Also, find the percent error in the interpolation. Dr. Jie Zou PHY 3320

Case: Three data points are available - {x0, f(x0)}, {x1, f(x1)}, and {x2, f(x2)}. Note: f(x)=the actual function Task: Determine the formula for the quadratic interpolating function, f2(x). Note: f2(x) = the (approximating) interpolation polynomial; “2” indicates the order of the polynomial. Dr. Jie Zou PHY 3320