Computer Oriented Numerical Methods (BCA) 3rd Sem Previous Year Solved Question Paper 2022

Divided differences and forward differences are related concepts used in polynomial interpolation, specifically in constructing interpolation polynomials using the Newton's divided difference form.

The relationship between divided differences and forward differences can be understood as follows:

1. Divided Differences: Divided differences are used to compute the coefficients of the Newton form of an interpolating polynomial. They are denoted as f[x0​],f[x0​,x1​],f[x0​,x1​,x2​],… and represent the differences in function values at different data points. For example, f[x0​] is simply the function value at the first data point, f[x0​,x1​] is the first forward difference, f[x0​,x1​,x2​] is the second forward difference, and so on.

2. Forward Differences: Forward differences are used to calculate divided differences. The first forward difference is computed as the difference between the function values at adjacent data points, and the second forward difference is the difference between the first forward differences, and so on. For example, the first forward difference is f[x0​,x1​]=f[x1​]−f[x0​], the second forward difference is f[x0​,x1​,x2​]=f[x1​,x2​]−f[x0​,x1​], and so forth.

In summary, divided differences are derived from forward differences, and they play a key role in constructing the Newton form of an interpolating polynomial. The relationship between them is that the coefficients of the divided differences are obtained by computing forward differences, and these coefficients are used to express the interpolating polynomial.