Question

Make a table for approximations of \(\cos x\). The function \(\cos (x)\) can be approximated by the sum $$ C(x ; n)=\sum_{j=0}^{n} c_{j} $$ where $$ c_{j}=-c_{j-1} \frac{x^{2}}{2 j(2 j-1)}, \quad j=1,2, \ldots, n $$ and \(c_{0}=1\). Make a Python function for computing \(C(x ; n)\). (Hint: Represent \(c_{j}\) by a variable term, make updates term = -term*... inside a for loop, and accumulate the term variable in a variable for the sum.) Also make a function for writing out a table of the errors in the approximation \(C(x ; n)\) of \(\cos (x)\) for some \(x\) and \(n\) values given as arguments to the function. Let the \(x\) values run downward in the rowsand the \(n\) values to the right in the columns. For example, a table for \(x=4 \pi, 6 \pi, 8 \pi, 10 \pi\) and \(n=5,25,50,100,200\) can look like \(\begin{array}{rccccc}\mathrm{x} & 5 & 25 & 50 & 100 & 200 \\ 12.5664 & 1.61 \mathrm{e}+04 & 1.87 \mathrm{e}-11 & 1.74 \mathrm{e}-12 & 1.74 \mathrm{e}-12 & 1.74 \mathrm{e}-12 \\ 18.8496 & 1.22 \mathrm{e}+06 & 2.28 \mathrm{e}-02 & 7.12 \mathrm{e}-11 & 7.12 \mathrm{e}-11 & 7.12 \mathrm{e}-11 \\ 25.1327 & 2.41 \mathrm{e}+07 & 6.58 \mathrm{e}+04 & -4.87 \mathrm{e}-07 & -4.87 \mathrm{e}-07 & -4.87 \mathrm{e}-07 \\ 31.4159 & 2.36 \mathrm{e}+08 & 6.52 \mathrm{e}+09 & 1.65 \mathrm{e}-04 & 1.65 \mathrm{e}-04 & 1.65 \mathrm{e}-04\end{array}\) Observe how the error increases with \(x\) and decreases with \(n .\) Name of program file: cossum.py.

Short answer

Create Python functions to compute and print approximation errors for \( C(x, n) \).

Step-by-step solution

Define the Function for C(x, n)

First, define a Python function named `C(x, n)` to compute the approximation of \( \cos(x) \). Initialize the first term with \( c_0 = 1 \) and store this value in a variable called `term`.

Implement the For Loop

Within the function, create a for loop that iterates from \( j=1 \) to \( n \). In each iteration, update the term using the formula: `term = -term * (x ** 2) / (2 * j * (2 * j - 1))`. Accumulate these terms into a sum variable initialized as 1, to represent \( C(x, n) \).

Return Result

After the loop completes, return the accumulated sum as the result of the function, representing the approximation \( C(x, n) \).

Define Function for Error Table

Create a second function to generate and print a table of errors. This function will take a list of \( x \) values and \( n \) values as input arguments.

Implement Nested Loops for Table

Inside the error table function, use a for loop for each \( x \) value. For every \( x \), compute \( C(x, n) \) for each \( n \) in the list of \( n \) values using your `C(x, n)` function. Calculate the error by subtracting this approximation from `math.cos(x)`.

Print the Error Table

Print the headers for the table. Use formatted strings to align the errors corresponding to different \( n \) values in columns for each \( x \) value.

Implement the Main Function

Finally, implement a main function that uses the error table function. Set specific \( x \) and \( n \) values as variables and pass them to the error table function. This will automatically generate the table as specified.

Key concepts

Cosine Approximation

Numerical methods offer a fascinating approach to approximate complex mathematical functions when exact computation is challenging or impossible. One significant application of these methods is in approximating the cosine function using a series expansion. The cosine function, denoted as \( \cos(x) \), can be approximated by the series: \[ C(x; n) = \sum_{j=0}^{n} c_j \] where the coefficients follow the pattern \( c_0 = 1 \) and for \( j \geq 1 \), \( c_j = -c_{j-1} \frac{x^2}{2j(2j-1)} \).
This expansion allows us to compute an approximate value of \( \cos(x) \) by summing up the first \( n \) terms. The better the approximation, the higher the value of \( n \) needed. However, keep in mind there are computational limits and precision issues to consider. Implementing this method can aid in scenarios where using direct trigonometric functions is not feasible due to constraints in precision or computation speed.
Understanding how a simple series of multiplications and divisions can closely emulate the behavior of the cosine function is key in numerical analysis, showcasing its power and versatility in mathematical computations.

Python Programming

Python is an incredibly powerful language for mathematical computation and numerical methods. Its simplicity and extensive library support make it an excellent choice for implementing functions like the cosine approximation series.
To start computing \( C(x; n) \), define a Python function. You initialize with \( c_0 = 1 \) and store this as a variable `term`. By iterating from \( j=1 \) to \( n \), you update `term` using the recursion formula, accumulating each \( term \) in a variable named `sum`. This loop effectively computes each successive term in the series and culminates in an approximate sum. Finally, the function returns this sum as the approximation of \( \cos(x) \).
This recursive process, specifically updating `term` as ```pythonterm = -term * (x ** 2) / (2 * j * (2 * j - 1))```is a perfect example of utilizing Python's ability to handle complex arithmetic with elegant and simple iterations. Such code not only acts as a practical tool for approximations but also serves as an educational gateway into the world of numerical analysis.

Error Analysis

When approximating mathematical functions, understanding and analyzing error is integral. Error analysis gives insight into the accuracy and reliability of an approximation method. In this cosine approximation, the error is the difference between the actual \( \cos(x) \) calculated using Python's math library and the approximated value \( C(x; n) \).
Computing this error through a Python function helps in visualizing how the approximation performs across different ranges of \( x \) and different values of \( n \). Constructing a table with error values across these variables demonstrates the idea that for larger values of \( n \), the approximation error tends to decrease. However, for large magnitudes of \( x \), precision may degrade due to cumulative errors.

• Increasing \( n \) generally reduces the error.
• Higher \( x \) values might complicate error minimization.

This exercise in error analysis helps not only in refining the approximation process but also in grasping the limitations and practical aspects of numerical computation.

Mathematical Functions

Mathematical functions form the backbone of a wide array of computational tasks. Functions like sine, cosine, and exponential functions are foundational not just in mathematics but in science and engineering applications.
Numerical methods allow for these complex functions to be approximated efficiently. Through this approach, series expansions like the one used for cosine simplifies the process of computing a variety of values through repetitive operations that a computer handles comfortably. This method breaks down the complexity of mathematically intensive calculations into a series of manageable tasks.
Additionally, Python’s `math` module further complements these numerical methods, providing built-in functions for accurate calculations and serving as a benchmark for validating results obtained from approximation techniques. When utilizing functions such as \( \cos(x) \), one can compare the output of numerical methods against Python's library functions to assess performance and accuracy. Thus, understanding and implementing mathematical functions through numerical methods enriches both computational proficiency and theoretical understanding.