Question

(a) Evaluate the function $$ f(t)=.05 \sin (1000 t)+.5 \cos (\pi t)-.4 \sin (10 t) $$ at the 101 points given by \(0: 01: 1\). Plot the resulting broken line interpolant. (b) In order to study the slow scale trend of this function, we wish to find a low degree polynomial (degree at most 6 ) that best approximates \(f\) in the least squares norm at the above 101 data points. By studying the figure from part (a) find out the smallest \(n\) that would offer a good fit in this sense. (Try to do this without further computing.) (c) Find the best approximating polynomial \(v\) of degree \(n\) and plot it together with \(f\). What are your observations?

Short answer

Answer: The smallest degree n for a good fit can be gauged by analyzing the number of distinct peaks or troughs in the graph, plus or minus 1. This requires some knowledge and experience in curve analysis, and the estimated degree n might vary depending on the specific function and interpolant plotted.

Step-by-step solution

1. Evaluate the function at 101 points.

First, we need to evaluate the function \(f(t) = .05 \sin (1000 t)+.5 \cos (\pi t)-.4 \sin (10 t)\) at 101 points given by the interval \(0: 0.01: 1\). We can achieve this by using a programming language or software, like MATLAB or Python, to extract the values of the function at these points.

2. Plot the broken line interpolant.

Using the obtained function values at the 101 points, plot the broken line interpolant to visualize the trend of the function. This can also be done using computational software for graphical and numerical data, such as MATLAB or Python.

3. Find the smallest n.

Based on the graph of the broken line interpolant observed, one can gauge the smallest degree n that would offer a good fit for the function without further calculations. This requires some knowledge and experience in curve analysis but generally, you should look for the number of distinct peaks or troughs in the graph, plus or minus 1, as a rough estimate of the degree n.

4. Find the best approximating polynomial of degree n.

Utilizing the least squares method, the best approximating polynomial of degree n can be derived. One approach is to set up a system of linear equations and obtain the coefficients of the polynomial using computational tools. The method involves minimizing the sum of the squared differences between the values of the function and the approximating polynomial.

5. Plot the approximating polynomial together with the original function.

Finally, plot the derived polynomial together with the original function \(f(t)\) to compare the approximations and observe how well the polynomial has fitted the original data. Analyze how well the approximating polynomial represents the original function and any discrepancies or overshooting the polynomial encounters. This can again be achieved using programming languages or software like Python or MATLAB.

Key concepts

Least Squares Approximation

Least Squares Approximation is a statistical method used to determine the best-fitting curve or line to a given set of data by minimizing the sum of the squares of the vertical distances (residuals) of the points from the curve. This technique is overwhelmingly used when the data is imperfect, which is typically the case in real-world scenarios.

When we talk about fitting a polynomial to a function based on specific data points, Least Squares helps us find the polynomial coefficients that result in the smallest possible difference between the values of the polynomial and the function at these points. This process transforms an overdetermined system (more equations than unknowns) into a solvable one.

To break it down, imagine plotting all the data points on a graph and then drawing a line or curve that passes as closely as possible to these points. This curve is your least squares approximation. For the polynomial case, this approach results in a set of linear equations (known as normal equations) that can be solved for the polynomial coefficients. The result is an equation for a line, curve, or surface that represents the trend of the data in the simplest form possible.

Polynomial Approximation

Polynomial Approximation involves finding a polynomial of a certain degree that approximates a function or set of data points with a desired degree of accuracy. Depending on the context, different methods can be utilized to obtain this polynomial, with one of the most common being the Least Squares Approximation.

The choice of the degree of the polynomial is crucial in this process. A polynomial that is too simple (of too low degree) may not capture all the tendencies and features of the data. On the other hand, a polynomial that is too complex (of too high degree) may lead to overfitting, where the polynomial passes too closely to the points but performs poorly when predicting or approximating new data points (phenomenon often referred to as the Runge's phenomenon).

The goal is to strike a balance, finding the lowest degree polynomial that still provides a good fit. This doesn't just mean visually on a graph, but also minimizing a specific error measure, such as the least squares error in least squares approximation.

Evaluating Functions

Evaluating functions is a fundamental concept in mathematics and numerical methods. It involves computing the output value of a function for a particular input value or set of values. In practical terms, when we have a formula or function like \(f(t)\), evaluating this function at various points -- say, 101 equidistant points between 0 and 1, as in our example -- allows us to understand the behavior of the function over that interval.

When dealing with complex functions, it often requires more than just plugging in numbers. Sophisticated software tools or programming languages are essential for efficiently evaluating functions at multiple points, especially when the functions involve operations like trigonometry or exponentials. This step is the basis for further applications like creating graphs, interpolating data, and setting up problems for numerical optimization, such as finding the best least squares fit.

Plotting Interpolants

Plotting interpolants involves graphically representing a curve or surface that passes through a set of data points. Interpolation, the method used to construct such a curve, helps to fill in the gaps between known data points, giving us a visual insight into the structure and trends that might not be evident just by looking at the raw data.

Broken line interpolants, or piecewise linear interpolants, are one type of graphical representation where straight lines connect the dots (data points). This kind of plot provides a simple, visual way to interpret the data. A more sophisticated approach uses higher-degree polynomials to smooth out the curves between the points, which can often result in a better understanding of the underlying function being studied.

Software tools make plotting these interpolants straightforward, allowing students and researchers to quickly visualize and assess different approximation strategies or the behavior of the function across the range of data, which is critical when making decisions based on the data analysis.