Lagrange interpolation constructs a polynomial that passes exactly through all given data points. For each point (xi, yi), it creates a basis polynomial Li(x) that equals 1 when x = xi and 0 at all other xj. The final polynomial is the sum of yi × Li(x).
Linear interpolation is the simplest and most widely used form. When you have exactly two known points, or when you want to estimate a value between two adjacent entries in a table, linear interpolation assumes the function changes at a constant rate between those two points.
When to use linear interpolation: when values in a lookup table change smoothly over small intervals (tax tables, steam property tables, trigonometric tables), when you only have two nearby data points, or when speed matters and the function is approximately linear between points.
When you only have two data points, this calculator automatically produces a linear (degree 1) polynomial, which is identical to the linear interpolation formula. For a dedicated two-point calculator, see the Linear Interpolation Calculator.
Double interpolation (bivariate interpolation) estimates a value from a 2D table where both the row variable and column variable fall between tabulated entries. It is most common in thermodynamics (steam tables), aviation (performance charts), and engineering (property tables).
| Step | Operation | Example (Steam Tables) |
|---|---|---|
| 1 | Find the two rows that bracket your first variable | Find T = 220°C between T = 200°C and T = 250°C rows |
| 2 | Interpolate along the first variable for both columns | Interpolate each property at T = 220°C |
| 3 | Find the two column values that bracket your second variable | Find P = 1.5 MPa between P = 1.0 and P = 2.0 MPa columns |
| 4 | Interpolate the results from step 2 along the second variable | Interpolate the T = 220°C values between the two pressures |
For double interpolation, apply this calculator twice. First, use it with the known column values to interpolate along the row variable. Then use the results as input to interpolate along the column variable. The error in double interpolation is larger than single interpolation because two approximation operations are chained. The Correlation Coefficient Calculator quantifies how strongly two variables are linearly related, which helps determine when linear interpolation between them is reliable.
Excel has limited built-in interpolation support. Understanding what each tool can and cannot do helps you choose the right approach.
| Task | Excel Function | This Calculator |
|---|---|---|
| Linear interpolation (2 points) | =FORECAST.LINEAR(x, y_range, x_range) | Enter 2 points, get y |
| Lookup nearest table value | =VLOOKUP(x, table, col, TRUE) | Not needed - use calc |
| Quadratic interpolation (3 points) | No built-in; manual formula | Enter 3 points directly |
| Cubic interpolation (4 points) | No built-in; complex array formula | Enter 4 points directly |
| Up to 6-point polynomial | No built-in | Enter up to 6 points |
| Batch multiple x values | Drag formula down column | One at a time |
For polynomial interpolation with 3 or more points, this calculator is significantly faster than writing the Lagrange formula manually in Excel. For statistical analysis of what function best describes your data, the Mean Median Mode Calculator provides descriptive statistics for understanding your data distribution before choosing an interpolation strategy.
Given data points (1, 2), (3, 8), and (5, 14), estimate y at x = 2.