Question

Implement Lagrange's interpolation formula. Imagine we have $n+1$ measurements of some quantity $y$ that depends on $x$ : $(x_0,y_0),(x_1,y_1),\dots ,(x_n,y_n)$. We may think of $y$ as a function of $x$ and ask what $y$ is at some arbitrary point $x$ not coinciding with any of the $x_0,\dots ,x_n$. This problem is known as interpolation. One way to solve this problem is to fit a continuous function that goes through all the $n+1$ points and then evaluate this function for any desired $x$. A candidate for such a function is the polynomial of degree $n$ that goes through all the points. This polynomial can be written $$p_L(x)=\sum _{k=0}^n{y}_k{L}_k(x)$$ where $$L_k(x)=\prod _{i=0,i\ne k}^n\frac{x-x_i}{x_k-x_j}$$ The $\mathrm{\Pi }$ notation corresponds to $\sum$, but the terms are multiplied, e.g., $$\prod _{i=0,i\ne k}^n{x}_i=x_0{x}_1\cdots x_{k-1}{x}_{k+1}\cdots x_n$$ The polynomial $p_L(x)$ is known as Lagrange's interpolation formula, and the points $(x_0,y_0),\dots ,(x_n,y_n)$ are called interpolation points. Make a function Lagrange $(x$, points) that evaluates $p_L$ at the point x, given $n+1$ interpolation points as a two-dimensional array points, such that points $[i,0]$ is the $x$ coordinate of point number $i$ and points $[i,1]$ is the corresponding $y$ coordinate. To verify the program, we observe that $L_k\left(x_k\right)=1$ and that $L_k\left(x_i\right)=0$ for $i\ne k$, implying that $p_L\left(x_k\right)=y_k$. Write a function verify (points) that computes $|p_L\left(x_k\right)-y_k|$ at all the interpolation points and checks that the value is approximately zero. Call verify with 5 equally spaced points along the curve $y=\sin (x)$ for $x\in [0,\pi ]$. Then evaluate $p_L(x)$ for an $x$ in the middle of two interpolation points and compare the value of $p_L(x)$ with the exact one: $\sin (x)$. Name of program file: Lagrange_polynomial1.py.

Short answer

Implement Lagrange interpolation and test it using 5 points from the sine function. Ensure the polynomial approximates these points accurately.

Step-by-step solution

Understand Lagrange's Interpolation Formula

Lagrange's interpolation formula is a method used to estimate a polynomial that passes through a given set of points. The formula relies on constructing a polynomial of degree $n$ that fits $n+1$ points $(x_0,y_0),(x_1,y_1),\dots ,(x_n,y_n)$. The polynomial is expressed as $p_L(x)=\sum _{k=0}^n{y}_k{L}_k(x)$, where $L_k(x)$ are Lagrange basis polynomials.

Define Lagrange Basis Polynomials

For each point, the Lagrange basis polynomial $L_k(x)$ is defined as $L_k(x)=\prod _{i=0,ieqk}^n\frac{x-x_i}{x_k-x_i}$. This ensures that $L_k(x_k)=1$ and $L_k(x_i)=0$ for $ieqk$. This property is crucial because it allows $p_L(x)$ to interpolate exactly the given points.

Implement the Lagrange Function

Create a function `Lagrange(x, points)` that takes a target $x$ value and an array of points as input. Iterate over each point to calculate $L_k(x)$ and use it to compute $p_L(x)$. Accumulate these results to get the interpolated value.

Verify the Implementation

Write a function `verify(points)` to ensure the correctness of the Lagrange interpolation. For each point in `points`, compute $|p_L(x_k)-y_k|$ where $x_k$ and $y_k$ are the coordinates of each point. The result should be approximately zero, confirming that the polynomial accurately interpolates the points.

Test with Sine Function Data

Test the implementation using 5 equally spaced points from the sine function over $x\in [0,\pi ]$. Calculate $p_L(x)$ for a value of $x$ located between two interpolation points, then compare this result with $\sin (x)$ to evaluate the accuracy of the interpolation.

Code the Program

Write the `Lagrange_polynomial1.py` file with the implemented functions `Lagrange` and `verify`. Ensure the functions are correctly coded, tested, and debugged to verify that the interpolation and verification work as expected.

Key concepts

Polynomial Interpolation

Polynomial Interpolation is a fundamental technique in numerical analysis used to find a polynomial that fits a given set of data points. This polynomial is used to estimate values at points not explicitly provided in the data set. The idea is to construct a polynomial of degree $n$ that passes through $n+1$ data points. The polynomial created joins the dots and enables predictions between known points.

• Used for estimating unknown values within a range of discrete points.
• Helps in interpolation by constructing a smooth curve.
• Applications can be found in fields like engineering, science, and financial modeling.

The main aim is to achieve a polynomial that not only smoothly passes through all the chosen points but also provides reliable estimates for in-between values.

Lagrange Basis Polynomials

Lagrange Basis Polynomials are at the heart of Lagrange Interpolation. Each basis polynomial $L_k(x)$ is designed to be zero at all data points except one, where it equals one. This specific characteristic ensures that each term in the polynomial contributes exactly the value it should at its corresponding data point.
The formula for the Lagrange basis polynomial $L_k(x)$ is given by: $$L_k(x)=\prod _{i=0,ieqk}^n\frac{x-x_i}{x_k-x_i}$$
This ingenious design:

• Guarantees the interpolating polynomial goes through each given point.
• Allows a unique combination of these basis polynomials to make up the desired polynomial.
• Makes each data point "control" only its corresponding term in the interpolation polynomial.

This process ensures that the final polynomial mimics the behavior of the original function at the provided data points.

Interpolation Verification

Verification is a critical step in ensuring that the interpolation is accurate. For the Lagrange method, the verification involves checking whether the interpolated polynomial correctly matches the original data points.
For each data point $(x_k,y_k)$, compute $|p_L(x_k)-y_k|$. If this difference is approximately zero for all points, the interpolation is deemed successful:

• Confirms that the polynomial passes exactly through each data point.
• Helps identify any potential inaccuracies or errors in implementation.
• Ensures that interpolated values between data points will be more reliable.

By comparing results from the interpolated polynomial with known values, students can gain confidence in their method and calculations.

Numerical Methods

Numerical methods are techniques used for finding approximate solutions to mathematical problems. In the context of polynomial interpolation, these methods are crucial for efficiently computing values that might be too complex or impossible to solve analytically. Lagrange interpolation is a type of numerical method that provides an elegant solution to interpolation problems.
Key advantages include:

• Capability to handle large datasets where analytical solutions are impractical.
• Fast and efficient calculations once set up, given that the Lagrange polynomials are determined.
• Offers a way to deal with real-world inaccuracies by providing estimates rather than exact solutions.

Numerical methods like Lagrange Interpolation expand our ability to work with complex data relationships in practical, computationally friendly ways.

Python Programming

Python is an ideal language for implementing numerical algorithms, due to its ease of syntax and powerful libraries. In Lagrange Interpolation, Python can be used to implement both the construction of the interpolation polynomial and its evaluation at desired points. Here's why Python shines in this context:

• The language's simplicity allows focus on algorithmic logic rather than intricate syntax.
• Libraries such as NumPy provide robust tools for handling numerical data and performing mathematical operations.
• Python scripts can easily verify calculations, automate testing, and debug efficiently, helping ensure the accuracy of implementations.

By employing Python to solve interpolation problems, students harness a modern tool that is both powerful and widely applicable in scientific computing.