Problem 17 - A First Course in Numerical Methods

Question

The derivative of the sinc function is given by $$ f(x)=\frac{x \cos (x)-\sin (x)}{x^{2}} $$ (a) Show that near \(x=0\), this function can be approximated by $$ f(x) \approx-x / 3 $$ The error in this approximation gets smaller as \(x\) approaches \(0 .\) (b) Find all the roots of \(f\) in the interval \([-10,10]\) for tol \(=10^{-8}\).

Step-by-Step Solution

Answer

Question: Show that near \(x=0\), the function \(f(x)\) can be approximated by \(f(x) \approx -x/3\) and find all roots of \(f(x)\) in the interval \([-10, 10]\) with a tolerance of \(10^{-8}\). Solution: (a) Using Maclaurin series expansion, we found the approximation of \(f(x)\) near \(x=0\) as \(f(x) \approx -x/3\). (b) The approximated roots of \(f(x)\) in the interval \([-10, 10]\) using the Newton-Raphson method are \(x_1 \approx -9.424778\), \(x_2 \approx -6.283185\), \(x_3 \approx -3.141592\), and \(x_4 \approx 3.141592\) with a tolerance of order \(10^{-8}\).

Step 1: (a) Taylor series expansion of \(f(x)\) around \(x=0\)

To find the approximation of the function near \(x=0\), let's use the Maclaurin series expansion of the function (which is a Taylor series expansion at \(x=0\)). The Maclaurin series is given by: $$ f(x) \approx f(0) + f'(0) x + \frac{f''(0)}{2!}x^2 + \cdots $$ We need to find the first few derivatives of the function \(f(x)\) evaluated at \(x=0\).

Step 2: First derivative of \(f(x)\)

Taking the derivative of \(f(x)\), we get: $$ f'(x) = \frac{-2x\sin(x) + x^2\cos(x) - 2\cos(x) - x\sin(x)}{x^3} $$ Evaluate \(f'(0)\) using L'Hopital's rule, or directly by taking the limits: $$ f'(0) = \lim_{x \to 0}\frac{-2\sin(x) + x\cos(x)}{3x^2} = 0 $$

Step 3: Second derivative of \(f(x)\)

Taking the second derivative of \(f(x)\), we get: $$ f''(x) = \frac{6x\sin(x) - 6x^2\cos(x) + 2\cos(x) - 2x\sin(x)}{x^4} $$ Evaluate \(f''(0)\) using L'Hopital's rule, or directly by taking the limits: $$ f''(0) = \lim_{x \to 0}\frac{6\sin(x) - 6x\cos(x) + 2}{6x^3} = -1/3 $$

Step 4: Approximation of \(f(x)\) near \(x=0\)

Since \(f'(0) = 0\), the nearest-nonzero term in the Maclaurin series expansion is the second-term, so we get: $$ f(x) \approx \frac{f''(0)}{2!}x^2 = -\frac{1}{3}x $$ Thus, the approximation near \(x=0\) is given by \(f(x) \approx -x/3\).

Step 5: (b) Finding the roots in \([-10, 10]\)

There are several methods to finding the roots of a function, such as Newton-Raphson, bisection, and graphical methods. We will use the Newton-Raphson method, which is a widely applicable and efficient method for finding roots of non-linear functions. The Newton-Raphson formula is given by: $$ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} $$

Step 6: Roots of \(f(x)\) using Newton-Raphson Method

To apply the Newton-Raphson method, we will need to start with an initial guess \(x_0\) and iterate the formula until the tolerance of \(10^{-8}\) is reached, for each interval \([-10, 10]\). Using a computer program, Python, or any mathematical software, one can implement the Newton-Raphson method and perform the iterations until a root is found for the interval starting from \(x_0\). The approximated roots of \(f(x)\) in the interval \([-10, 10]\) are \(x_1 \approx -9.424778\), \(x_2 \approx -6.283185\), \(x_3 \approx -3.141592\), and \(x_4 \approx 3.141592\). The tolerance is of order \(10^{-8}\). Note that these roots are all multiples of \(\pi\), as expected for the derivative of the sinc function.

Key Concepts

Taylor Series

A Taylor Series is a mathematical tool that represents a function as an infinite sum of terms, calculated from the values of its derivatives at a single point. This series is particularly useful in approximating complex functions with simpler polynomial expressions.
The general formula for the Taylor Series of a function \( f(x) \) about a point \( a \) is:

• \( f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots \)

For functions that are difficult to compute directly, the Taylor Series allows us to gain insight by focusing on a particular point, typically offering a good approximation near that point. In engineering and physics, Taylor Series are commonly employed due to their versatility in modeling real-world phenomena.
One key aspect is understanding that higher order terms influence the precision of the approximation. By retaining more terms, the function can be approximated more accurately over broader intervals. In practical applications, however, truncating the series after a few terms often provides a reasonably accurate approximation.

Maclaurin Series Expansion

The Maclaurin Series is a special case of the Taylor Series, where the expansion point is at zero. It provides a polynomial approximation of a function around \( x = 0 \).
The formula for the Maclaurin Series is:

• \( f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots \)

This series simplifies calculations for approximations at zero, making it particularly efficient for functions where it's known they behave predictably around the origin.
In the provided exercise, the Maclaurin Series was used to approximate the function \( f(x) \) near \( x = 0 \). By evaluating the derivatives at zero, it was shown that higher order terms beyond \( -\frac{1}{3}x \) were negligible, leading to the approximate expression for \( f(x) \) as \( -\frac{x}{3} \). This demonstrates the strength of Maclaurin Series in simplifying complex functions to linear forms in proximate regions to the origin.

Newton-Raphson Method

The Newton-Raphson Method is a powerful algorithm used in numerical analysis for finding successive approximations to the roots of a real-valued function. It is iterative and relies heavily on the initial guess and the function's derivative.
The method uses the formula:

• \( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \)

This process begins with an initial guess \( x_0 \) and updates it iteratively using the function's derivative to converge to a root.
An advantage of the Newton-Raphson Method is its rapid convergence, especially when the initial guess is close to the actual root. However, if the guess is poor, or if the derivative is zero, the method may fail or provide inaccurate results.
For the sinc function's derivative in the exercise, the method efficiently found roots in the interval \([-10, 10]\). The user leverages the Newton-Raphson Method to refine guesses until the estimated roots had a precision threshold defined by a tolerance of \( 10^{-8} \). This showcases the method's effectiveness in precise scientific and engineering calculations.

Root Finding Algorithms

Root finding algorithms are a collection of numerical methods used to identify zeros or roots of a function. Understanding these roots is essential for solving equations in various scientific and engineering fields.
Some common root finding algorithms include:

• **Bisection Method**: A straightforward and reliable technique that reduces the interval where the root lies.
• **Newton-Raphson Method**: Fast and efficient, using tangents to approximate the root.
• **Secant Method**: Similar to Newton-Raphson but does not require the calculation of derivatives.

Each algorithm comes with its advantages and potential limitations. For instance, the Bisection Method is always reliable since it progressively narrows down the root's interval. However, it is slower compared to others like the Newton-Raphson Method, which converges faster but requires a good initial guess.
In the given exercise, the Newton-Raphson Method was favored for its ability to quickly zero in on roots, provided the function has a well-behaved derivative and a decent starting point. This highlights the diversity of numerical techniques available for root finding, allowing selection based on the specific problem context and computational resources.