First Divided Differences
When writing the general expression of a polynomial, we often think of the most common form. However, other forms are more appropriate for interpolation, for example:
\[ p_n(x) = a_0 + a_1(x-x_0) + a_2(x-x_0)(x-x_1) + a_3(x-x_0)(x-x_1)(x-x_2) + \dots + a_{n-1}(x-x_0)\cdots(x-x_{n-2}) + a_n(x-x_0)\cdots(x-x_{n-1}) \]
The coefficient \(a_n\) contains \(n\) terms of the form \((x-x_i)\), so the polynomial is of degree \(n\). This formula is especially useful when determining the \((n+1)\) coefficients \(a_i\) such that \(p_n(x)\) passes through the \((n+1)\) collocation points \((x_i, f(x_i))\) for \(i=0,1,2,\dots,n\). We must ensure:
\[ p_n(x_i) = f(x_i), \quad i=0,1,2,\dots,n \]
All coefficients vanish at \(x=x_0\) except the first one, giving:
\[ a_0 = f(x_0) \]
Next, for \(x_1\):
\[ a_1 = \frac{f(x_1) - f(x_0)}{x_1 - x_0} \]
🔹 Definition: First Divided Differences
The first divided differences of the function \(f(x)\) are defined as:
\[ f[x_i, x_{i+1}] = \frac{f(x_{i+1}) - f(x_i)}{x_{i+1} - x_i} \]
Thus, the coefficient \(a_1\) can be written as:
\[ a_1 = f[x_0, x_1] \]
🔹 Example: First Divided Difference
Consider the points \( (1, 2) \) and \( (3, 8) \). Compute the first divided difference:
\[ f[x_0, x_1] = \frac{8 - 2}{3 - 1} = 3 \]
Hence, the linear interpolation coefficient is \( a_1 = 3 \), and the first-degree polynomial is:
\[ p_1(x) = f(x_0) + a_1(x - x_0) = 2 + 3(x - 1) \]
🔹 Remark: Second Coefficient
The third coefficient \(a_2\) is determined by:
\[ a_2 = \frac{f(x_2) - f(x_0) - f[x_0,x_1](x_2 - x_0)}{(x_2 - x_0)(x_2 - x_1)} = \frac{f[x_1,x_2] - f[x_0,x_1]}{x_2 - x_0} \]