Polynomial Interpolation

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Vandermonde Matrix

Definition

For and , the matrix

is called the Vandermonde matrix

Determinant of the Vandermonde Matrix

Lemma

Let be a Vandermonde Matrix. Then

Corollary

Let be a Vandermonde matrix. is invertible if and only if .

If is invertible, then the equation that is represented by the Vandermonde matrix has a unique solution.

Intuition

If any of are not distinct, then they will be subtracted from the row during Gauss-Jordan Elimination, leaving a row of 0. This means a free variable. Recall, the intuition presented in the theorem the relation between the determinant and matrix invertibility.

Polynomial Data Points

Theorem

Given distinct data points , there exists a unique polynomial of degree

satisfying for each

Intuition

Recall for a line, we had . We have and . So we have two data points for a polynomial of degree 1.

Example Problem: Finding the Polynomial

Find a cubic polynomial whose graph passes through the points .

Let for . For each data point, we evaluate the equation .

This is a linear system, and we can write is as the matrix-vector product or an augmented matrix:

recall that these two are equivalent.

We have 3 options:

  1. Solve the system with Gaussian Elimination or Gauss-Jordan Elimination. The coefficient matrix will be our solution to
  2. Use the matrix inversion algorithm to calculate the inverse of the Vandermonde matrix, then left multiply the matrix to create the identity matrix on the left side, and the vector multiplied by the inverted matrix on the right.
  3. Use the relation between the determinant and matrix invertibility to calculate the matrix inversion, and do the same as above.

Solving the system gives , so

Example Problem: Word Problem

A car manufacturing company uses a wind tunnel to test the force due to air resistance experienced by the car windshield. The following data was collected:

Air velocity (m/s)
Force on windshield (N)

Construct a quadratic polynomial to model this data, and use it to predict the force due to air resistance from a wind speed of 40m/s.

Let where . Using our data points , we obtain the system of equations in matrix notation is:

Now we can solve the system using the three methods specified above.

The result is:

Thus

When , we have

Therefore, when the air velocity is , the windshield experiences approximately of force.