Polynomial interpolation

Objective. Construct and assess interpolating polynomials.

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2. Worked example
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Learn: the key idea

Polynomial interpolation constructs the unique polynomial of degree n−1 passing exactly through n given data points, commonly built using Lagrange's formula P(x) = Σ yᵢLᵢ(x), where each Lᵢ(x) is 1 at xᵢ and 0 at the other nodes. For points (0,1) and (2,5), the linear interpolant is P(x) = 1·((x−2)/(0−2)) + 5·((x−0)/(2−0)) = 1 + 2x. Higher-degree interpolation through many equally spaced points can oscillate wildly near the edges, known as Runge's phenomenon, so fitting many points with one high-degree polynomial is often unwise.

Worked example

Find the linear interpolation polynomial through the points (1,3) and (4,9) for polynomial interpolation.

1 $Slope: (9-3)/(4-1) = 2$
2 $Point-slope form: P(x) = 3 + 2(x-1)$
3 $Expand: P(x) = 2x + 1$
4 $Check at x=4: 2(4)+1=9 ✓$

Answer: P(x) = 2x + 1

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Common mistake. Believing a higher-degree interpolating polynomial is always a better fit; with many equally spaced nodes it can oscillate badly between points (Runge's phenomenon), making the approximation worse, not better.

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