🔹 Linear System Formulation

Represent the polynomial as:

\( p_n(x) = a_0 + a_1 x + a_2 x^2 + \dots + a_n x^n \)

The coefficients \(a_i\) satisfy the linear system:

\[ \begin{bmatrix} 1 & x_0 & x_0^2 & \dots & x_0^n \\ 1 & x_1 & x_1^2 & \dots & x_1^n \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & x_n & x_n^2 & \dots & x_n^n \end{bmatrix} \begin{bmatrix} a_0 \\ a_1 \\ \vdots \\ a_n \end{bmatrix} = \begin{bmatrix} f(x_0) \\ f(x_1) \\ \vdots \\ f(x_n) \end{bmatrix} \]

This matrix is called the Vandermonde matrix. Its conditioning worsens as n increases. Solving this system directly is rarely used in practice, but it provides an important theoretical framework.

🔹 Example: Degree 3 Polynomial

Find the polynomial passing through the points \( (0,1), (1,2), (2,9), (3,28) \).

Set up the Vandermonde system:

\[ \begin{bmatrix} 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 \\ 1 & 2 & 4 & 8 \\ 1 & 3 & 9 & 27 \end{bmatrix} \begin{bmatrix} a_0 \\ a_1 \\ a_2 \\ a_3 \end{bmatrix} = \begin{bmatrix} 1 \\ 2 \\ 9 \\ 28 \end{bmatrix} \]

Solving (e.g., using LU decomposition) gives:

\([a_0, a_1, a_2, a_3]^T = [1, 0, 0, 1]^T\)

Hence, the interpolating polynomial is:

\( p_3(x) = 1 + x^3 \)

📝 Summary

• The Vandermonde matrix allows systematic construction of interpolation polynomials.
• It represents a linear system relating polynomial coefficients to collocation points.
• Its conditioning deteriorates for large n, so it is mainly of theoretical importance.
• For small n, it can be solved directly to find the unique interpolating polynomial.