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

2. 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

3. 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

4. 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

5. 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

6. Newton’s quadratic interpolation
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

7. Quadratic interpolation formula: Derivation
Assume the quadratic interpolation formula of the form:
f2(x) = a0 + a1(x–x0) + a2(x–x0)(x-x1)
Determine the coefficients a0, a1, and a2 using the data points:
Since f2(x0)= f(x0) = a0, a0 = f (x0)
Since f2(x1) = f(x1)= a0 + a1(x1 - x0),
Since f2(x2) = f(x2) = a0 + a1(x2 - x0) + a2(x2 - x0)(x2 - x1),
Finite-divided-difference approx. of f’’(x).

8. Example 5.7
Develop a quadratic interpolation formula for the function e0.5x using the values at x0, x1 = 2, and x2 = 4 and use it to estimate the value of e0.5x at x = 1. Also, check the percent error in the interpolation. Compare with the linear interpolation in Example 5.6.
Dr. Jie Zou PHY 3320

9. Comparison
Quadratic (2nd-order) interpolation
Linear (1st-order) interpolation
Note: The error usually decreases with an increase in the order of the polynomial, n.
Dr. Jie Zou PHY 3320