Interpolation estimates unknown function values from known data points by building a curve that passes through those points. This cheat sheet compares three core methods: Lagrange interpolation, Newton divided differences, and cubic spline interpolation. College students need these tools for numerical analysis, data modeling, engineering computation, and scientific programming.
The goal is to choose a method that is accurate, efficient, and stable for the data at hand.
Lagrange interpolation gives one polynomial written directly from the data points, while Newton interpolation builds the same polynomial in a nested form that is easier to update. Divided differences measure changes across nodes and become the coefficients in the Newton polynomial. Cubic splines use piecewise cubic functions to avoid the oscillations that high-degree polynomials can produce.
Error depends on node placement, function smoothness, spacing, and the degree or spline conditions used.
Key Facts
- The Lagrange interpolating polynomial through n + 1 points is P_n(x) = sum from i = 0 to n of y_i L_i(x), where L_i(x) = product over j not equal to i of (x - x_j)/(x_i - x_j).
- Each Lagrange basis polynomial satisfies L_i(x_k) = 1 if i = k and L_i(x_k) = 0 if i is not equal to k.
- The Newton interpolating polynomial is P_n(x) = f[x_0] + f[x_0,x_1](x - x_0) + ... + f[x_0,...,x_n](x - x_0)(x - x_1)...(x - x_(n-1)).
- A divided difference is defined recursively by f[x_i,...,x_(i+k)] = (f[x_(i+1),...,x_(i+k)] - f[x_i,...,x_(i+k-1)])/(x_(i+k) - x_i).
- The polynomial interpolation error has the form f(x) - P_n(x) = f^(n+1)(c)/(n + 1)! times product from i = 0 to n of (x - x_i), for some c in the interval.
- A cubic spline uses a different cubic polynomial S_i(x) on each interval [x_i, x_(i+1)] and requires continuity of S, S', and S'' at interior knots.
- A natural cubic spline uses endpoint conditions S''(x_0) = 0 and S''(x_n) = 0, while a clamped spline specifies endpoint slopes S'(x_0) and S'(x_n).
- High-degree interpolation with equally spaced nodes can show Runge oscillation, so Chebyshev nodes or splines are often more stable choices.
Vocabulary
- Interpolation
- Interpolation is the process of estimating a function value between known data points using a curve that matches those points.
- Node
- A node is an input value x_i where the function value y_i or f(x_i) is known.
- Lagrange basis polynomial
- A Lagrange basis polynomial L_i(x) is a polynomial that equals 1 at node x_i and 0 at all other interpolation nodes.
- Divided difference
- A divided difference is a recursive slope-like quantity used as a coefficient in the Newton interpolation formula.
- Knot
- A knot is a node in spline interpolation where adjacent polynomial pieces meet.
- Natural cubic spline
- A natural cubic spline is a cubic spline whose second derivative is set to zero at both endpoints.
Common Mistakes to Avoid
- Using repeated x-values in an interpolation table is wrong because standard Lagrange and Newton formulas require distinct nodes to avoid division by zero.
- Extrapolating far outside the given nodes is risky because interpolation formulas are designed for values inside the data interval and may grow inaccurately outside it.
- Assuming a higher-degree polynomial is always more accurate is wrong because high-degree interpolation can create large oscillations, especially with equally spaced nodes.
- Forgetting continuity conditions in cubic splines is wrong because spline pieces must match in value, first derivative, and second derivative at interior knots.
- Mixing the order of nodes when using Newton divided differences can cause coefficient errors because the nested polynomial must use the same node order as the divided difference table.
Practice Questions
- 1 Given points (1, 2), (2, 5), and (4, 17), construct the Lagrange interpolating polynomial and estimate P(3).
- 2 For data f(0) = 1, f(1) = 3, and f(2) = 9, compute the divided differences f[0,1], f[1,2], and f[0,1,2].
- 3 A natural cubic spline is built on nodes x_0 = 0, x_1 = 1, x_2 = 3, and x_3 = 4. State the two endpoint second derivative conditions and the continuity conditions required at x_1 and x_2.
- 4 Explain why cubic spline interpolation may be preferred over a single high-degree polynomial when modeling many smooth data points.
Understanding Interpolation Methods Reference
A useful first distinction is between interpolation and fitting. Interpolation treats every listed value as exact, so the constructed curve must pass through each point. That is appropriate for values produced by a trusted formula or a high precision simulation.
Measured data are different because they contain noise. Forcing a curve through noisy readings can make it chase random errors rather than reveal the underlying trend.
In that setting, regression or smoothing may be better. Students should always ask whether their data points are exact constraints or imperfect observations before choosing an interpolation method.
The main practical difference between polynomial forms appears when data change. A Lagrange expression is easy to derive because each basis function selects one data value at its own node. However, adding one new point usually means rebuilding much of the expression.
Newton form stores information in a divided difference table. When a new node is added, the earlier coefficients remain useful and only a new part is computed.
Its nested structure is efficient for evaluating the polynomial on a computer. For repeated evaluation with fixed nodes, a barycentric form of Lagrange interpolation is often preferred in numerical software because it reduces repeated work and can be more stable than directly multiplying many factors.
Spline conditions have a physical meaning. Continuity of the function prevents jumps in position. Continuity of the first derivative prevents sudden changes in slope, which would look like a sharp corner.
Continuity of the second derivative prevents abrupt changes in curvature. These properties matter when a curve represents motion, a road profile, a machine path, or a smoothly changing temperature field. The endpoint choice matters most near the ends of the data range.
A natural spline assumes zero curvature at each end. That can be reasonable when the graph should flatten there, but it can be wrong for a function that is still bending. A clamped spline uses known endpoint slopes, such as measured velocity at the beginning and end of a motion experiment.
Error formulas explain why interpolation is not equally reliable everywhere. The product of distances from the evaluation point to the nodes becomes small near a node, so estimates there are often better. Outside the node interval, those distances can grow rapidly.
This is why extrapolation is much riskier than interpolation, even when the curve fits every known point perfectly. High derivatives matter too. A function with rapid bending or oscillation is harder for one polynomial to follow.
Closely spaced nodes can help in a region where the function changes quickly, though extremely close nodes may magnify rounding errors in computer arithmetic. Chebyshev nodes place more points near interval ends, where ordinary evenly spaced polynomial interpolation is most vulnerable. When checking work, students should plot the interpolant, inspect endpoint behavior, compare against any unused data values, and keep enough decimal precision during divided difference calculations.