**Course Title:** MATLAB Programming: Applications in Engineering, Data Science, and Simulation **Section Title:** Polynomials, Curve Fitting, and Interpolation **Topic:** Curve fitting using polyfit and interpolation techniques (linear, spline, etc.). ### 4.1 Overview of Curve Fitting Curve fitting is a technique used to construct a mathematical model that describes the relationship between a set of data points. In this topic, we will discuss how to use MATLAB's built-in functions, `polyfit` and interpolation techniques, to fit curves to a given dataset. ### 4.2 Using Polyfit for Curve Fitting `polyfit` is a MATLAB function that fits a polynomial curve to a set of data points. The general syntax for `polyfit` is: ```matlab p = polyfit(x, y, n) ``` where `x` and `y` are the data points, and `n` is the degree of the polynomial. **Example 4.1:** Fit a 3rd degree polynomial to the following data points: ```matlab x = [1, 2, 3, 4, 5]; y = [2, 4, 6, 8, 10]; p = polyfit(x, y, 3); ``` In this example, `polyfit` returns the coefficients of the 3rd degree polynomial that best fits the data points. We can then use the `polyval` function to evaluate the polynomial at any point. **Key Concept:** The degree of the polynomial (n) determines the flexibility of the fit. A higher degree polynomial can fit more complex data, but may also result in overfitting. ### 4.3 Interpolation Techniques Interpolation is a technique used to estimate a value between two known data points. MATLAB provides several interpolation functions, including `interp1`, `interp2`, and `interp3`. **Example 4.2:** Use `interp1` to interpolate a value between two data points: ```matlab x = [1, 3, 5]; y = [2, 4, 6]; xi = 2; yi = interp1(x, y, xi) ``` In this example, `interp1` returns the interpolated value at `xi = 2`. **Key Concept:** Interpolation can be used to fill in missing values in a dataset or to estimate a value between two known data points. ### 4.4 Spline Interpolation Spline interpolation is a technique used to fit a piecewise curve to a set of data points. MATLAB provides the `spline` function to perform spline interpolation. **Example 4.3:** Use `spline` to interpolate a value between two data points: ```matlab x = [1, 3, 5]; y = [2, 4, 6]; xi = 2; yi = spline(x, y, xi); ``` In this example, `spline` returns the interpolated value at `xi = 2`. **Key Concept:** Spline interpolation can result in a smoother curve than linear interpolation, but may also result in oscillations. ### 4.5 Practical Takeaways * Use `polyfit` to fit a polynomial curve to a set of data points. * Use interpolation techniques to estimate a value between two known data points. * Choose the degree of the polynomial and the interpolation method based on the complexity and noise of the data. * Be cautious of overfitting and oscillations in the fit. ### Additional Resources For more information on curve fitting and interpolation techniques, please refer to the following resources: * MATLAB Documentation: [Curve Fitting Toolbox](https://www.mathworks.com/help/curvefit/index.html) * MATLAB Documentation: [Interpolation](https://www.mathworks.com/help/matlab/math/interpolation.html) ### Practice Exercises 1. Fit a 2nd degree polynomial to the following data points: `x = [1, 2, 3, 4, 5]`, `y = [2, 4, 6, 8, 10]`. Evaluate the polynomial at `x = 3`. 2. Use `interp1` to interpolate a value between two data points: `x = [1, 3, 5]`, `y = [2, 4, 6]`, `xi = 2`. 3. Use `spline` to interpolate a value between two data points: `x = [1, 3, 5]`, `y = [2, 4, 6]`, `xi = 2`. ### Leave a Comment or Ask for Help If you have any questions or need further clarification on any of the concepts discussed in this topic, please leave a comment below. We will respond to your comment as soon as possible.