Interpolation with Unequal Intervals

Definition

Interpolation with unequal intervals is a mathematical technique used to estimate intermediate values of a function when the given data points (x-values) are not spaced at regular or equal intervals. Unlike methods like Newton’s Forward or Backward difference formulas which require a constant step size, methods for unequal intervals allow for the estimation of data even when the independent variable values are arbitrarily distributed.

Main Content

1. Divided Differences

Divided differences are the primary tool used for unequal intervals, representing the rate of change of a function over non-uniform intervals.
The first divided difference for two points $(x_0, y_0)$ and $(x_1, y_1)$ is defined as: $f[x_0, x_1] = \frac{f(x_1) - f(x_0)}{x_1 - x_0}$.

2. Lagrange’s Interpolation Formula

This method provides a direct polynomial that passes through all given points without requiring the calculation of difference tables.
It is highly effective for non-equidistant data as it treats each point as a weight in the resulting polynomial expression.

3. Newton’s Divided Difference Interpolation

This method builds a polynomial step-by-step, making it easy to add new data points to an existing model.
It uses a triangular divided difference table to calculate coefficients for the polynomial.

Visual representation of Data Points:
y |
  |      * (x1, y1)
  |   * (x0, y0)             * (x2, y2)
  |_______________________________________ x
      x0      x1        x2
(Note the horizontal gaps between x0, x1, and x2 are unequal)

Working / Process

1. Constructing the Divided Difference Table

List all $x$ values in the first column and $f(x)$ values in the second column.
Calculate successive divided differences (1st, 2nd, etc.) by dividing the difference of the previous column's values by the difference of the corresponding $x$ values.

2. Formulating the Polynomial

For Newton’s method, the polynomial is written as: $P(x) = f(x_0) + (x-x_0)f[x_0, x_1] + (x-x_0)(x-x_1)f[x_0, x_1, x_2] + \dots$
For Lagrange’s method, use the product of terms: $L(x) = \sum_{i=0}^{n} y_i \prod_{j=0, j \neq i}^{n} \frac{x - x_j}{x_i - x_j}$.

3. Calculating the Unknown Value

Substitute the desired $x$-value (the value for which you want to find $f(x)$) into the resulting polynomial equation.
Simplify the arithmetic to obtain the estimated output $y$.

Advantages / Applications

Versatility: Can be applied to experimental data where sensors or measurements were taken at irregular time intervals.
Computational Efficiency: Lagrange interpolation is useful for simple, one-off calculations because it does not require a table.
Predictive Modeling: Widely used in engineering and physics to fill in "missing" data points in datasets gathered from real-world observations.

Summary

Interpolation with unequal intervals is a numerical approach to estimate values within a dataset where the gaps between independent variables are inconsistent. The most common techniques used are Lagrange’s Interpolation and Newton’s Divided Difference method. These processes allow researchers to construct a polynomial that fits the known points to predict unknown values accurately. Important terms include Divided Differences, Lagrange Polynomials, and Nodes.