https://r2.erweima.ai/mathbot/image/1726971546439dp21zx11.jpg

The problem asks to find the Hermite polynomial for the given set of data points. Hermite interpolation is used when not only the values \( y_i \) but also the derivatives \( y'_i \) at the points \( x_i \) are known.

Here is the given dataThe problem asks to find the Hermite polynomial for the given set of data points. Hermite interpolation is used when not only the values \( y_i \) but also the derivatives \( y'_i \) at the points \( x_i \) are known.

Here is the given 

\[
\begin{array}{c|c|c}
x_i & y_i & y'_i \\
\hline
-1 & -2 & 2 \\
0 & 2  & 4 \\
1 & 6  & 6 \\
4 & 0  & 8 \\
\end{array}
\]

### Steps to construct the Hermite polynomial:

1. **Double each data point**: Each \( x_i \) appears twice in the divided difference table.
2. **Use divided differences** to incorporate both function values and derivative information.

Let’s proceed to find the Hermite polynomial step by step using divided differences.

#### Step 1: Define the Divided Difference Table

I will now create the divided difference table for this dataset to find the Hermite polynomial. This involves filling the table based on both the values \( y_i \) and their derivatives \( y'_i \).

I will calculate and return the polynomial.The Hermite polynomial that interpolates the given data points and their derivatives is:

\[
H(x) = -0.1372x^7 + 1.1437x^6 - 2.2257x^5 - 1.7873x^4 + 4.3628x^3 + 0.6437x^2 + 2.0x + 2.0
\]

This polynomial approximates the values and derivatives provided in the table.

Do you have any questions or would you like more details on this solution?

### Related Questions:
1. What is the difference between Hermite and Lagrange interpolation?
2. How does the divided difference method work?
3. Can Hermite polynomials be used for non-polynomial functions?
4. How can I apply Hermite interpolation in numerical analysis?
5. What are some real-world applications of Hermite interpolation?

### Tip:
Hermite interpolation is especially useful when both function values and derivatives are known, providing a smoother and more accurate approximation.

Encuentre el polinomio de Hermite para la siguiente tabla: xi, yi, yi'

Learn how to find the Hermite polynomial using numerical methods for the provided dataset. This example demonstrates the use of divided differences and Hermite interpolation to match function values and derivatives. Suitable for university students studying Numerical Analysis.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation