Notes on Newton's Forward Formula for Interpolation

Introduction to Interpolation

• Definition: Interpolation is the method of estimating the value of a dependent variable (Y) for an intermediate value of an independent variable (X), given a set of X and Y values.
• Function Representation: Y is represented as a function of X, denoted as Y = f(X).

Example of Interpolation

• Given a table of values for X and corresponding values for Y, we may need to interpolate to find Y when X = 4, which lies between X = 3 and X = 5.
• Goal: Find f(4).

Terminology

• Independent Variable (X): Also known as the argument.
• Dependent Variable (Y): Also known as the entry.

Methods of Interpolation

• Newton's Forward Interpolation
• Newton's Backward Interpolation
• Binomial Method
• Lagrange's Method

Equidistant vs. Unequal Intervals

• Interpolation can be applied to both equidistant and unequal intervals:
  • Equidistant: Use Newton's Forward or Backward Interpolation.
  • Unequal Intervals: Use Lagrange's Interpolation.

Identifying the Method

• Use Newton's Forward Interpolation if the value to be interpolated lies in the forward direction of the table.
• Use Newton's Backward Interpolation if it lies in the backward direction of the table.
• Binomial Method: Used for finding missing terms in the table of X.

Newton's Forward Interpolation Formula

• Formula: The general formula for Newton's Forward Interpolation is:

  [ Y = Y_0 + \frac{U}{1!} \Delta Y_0 + \frac{U(U-1)}{2!} \Delta^2 Y_0 + \frac{U(U-1)(U-2)}{3!} \Delta^3 Y_0 + \ldots ]

Components of the Formula

• U: A value calculated from the formula ( U = \frac{X - X_0}{H} )
  • where ( H ) is the difference between consecutive X values.
  • ( X_0 ): The first value of X in the dataset.

Steps to Use the Formula

1. Determine H: Find the difference between consecutive X values (assumed equal).
2. Calculate U: Substitute the values of X and X0 into the equation.
3. Form a Difference Table: Calculate the differences ( \Delta Y_0, \Delta^2 Y_0, \Delta^3 Y_0, \ldots )
4. Substitute in the Formula: Use the calculated values to find the interpolated Y value.

Application

• The formula is particularly useful for interpolating the value of Y corresponding to X near the beginning of the table.

Next Steps

• In the next video, a problem will be solved using Newton's Forward Interpolation Formula for better understanding.