Question

Fill lists with function values. A function with many applications in science is defined as $$ h(x)=\frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2} x^{2}} $$ Fig. 5.13 Plot of the dimensionless temperature \(T(z, t)\) in the ground for two different \(t\) values and \(b=2\). Fill lists xlist and hlist with \(x\) and \(h(x)\) values for uniformly spaced \(x\) coordinates in \([-4,4]\). You may adapt the example in Chapter 5.2.1. Name of program file: fill_lists.py.

Short answer

Create a Python function to compute and store values of \(h(x)\) for \(x\) in [-4,4].

Step-by-step solution

Understand the Function

The given function is \( h(x)=\frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2} x^{2}} \). This is a Gaussian function, which is commonly used in statistics and probability theory. We will compute this function for different values of \(x\) in the range \([-4, 4]\).

Define the Range and Step Size

We need to fill the xlist with uniformly spaced values. For example, we can choose 100 values between -4 and 4. Therefore, the step size \( \Delta x \) would be \( \Delta x = \frac{4 - (-4)}{100-1} \).

Initialize the Lists

We initialize two empty lists: xlist for storing \(x\) values, and hlist for storing corresponding \(h(x)\) values. These lists will be populated in the next step.

Calculate Function Values and Fill Lists

Iterate over \(x\) values using the defined step size. For each \(x\), compute \(h(x)\) and append both \(x\) and \(h(x)\) to their respective lists. This can be done in Python using a for loop and the math library for the exponential and square root function.

Write a Python Program

Create a Python file named fill_lists.py. Use a loop to iterate over the \(x\) values, calculate the corresponding \(h(x)\) using the math functions, and append the results to xlist and hlist. This program outputs these lists for further analysis or plotting.

Key concepts

Functions in Python

In Python, functions play a crucial role in organizing and reusing code for various computations. They allow us to create blocks of code that perform specific tasks and can be called upon when needed. Functions can be built-in, like `print()` or `len()`, or user-defined, where you write your custom function using the `def` keyword. By defining a function, you can execute complex operations multiple times without rewriting the logic, enhancing code readability and efficiency. For this exercise, you define a function that calculates the values of a Gaussian function. This function takes an input `x` and returns its corresponding `h(x)`. Here is a simple way to define a function in Python: ```python def gaussian_function(x): import math return (1 / math.sqrt(2 * math.pi)) * math.exp(-0.5 * x**2) ``` By using this function, you can compute `h(x)` for each `x` in the desired range, making your program modular and easy to update or modify in the future.

Numerical Methods

Numerical methods involve techniques to approximate mathematical operations that may not have exact solutions. These methods are used to solve equations, calculate integrals, and perform other operations that can be challenging analytically. In this context, filling lists with function values falls under numerical methods because you are estimating the function values across a range of `x`. This is useful for plots or further analysis where finding a closed-form solution is impractical. To apply numerical methods effectively, you select a range and decide on the number of points, ensuring a balance between computation speed and accuracy. For example, by setting a uniform step size, denoted as \( \Delta x \), decide how many evaluations of the function you need. This choice affects how finely the function will be approximated over the interval \([-4, 4]\). The more points you choose, the finer the approximation, and vice versa.

Data Structures

Data structures in Python are essential for organizing and managing data efficiently. Lists are a basic yet powerful data structure that allows you to store ordered collections of items. You can append, insert, remove, and sort these items. In this example, you use lists to hold your `x` values (xlist) and the computed `h(x)` values (hlist). Lists allow you to store and access function values efficiently, making them ideal for iterative processes where you constantly need to update or retrieve values. Here’s how you might initialize and use lists in Python: ```python xlist = [] # List to hold x values hlist = [] # List to hold h(x) values for x in range_of_x_values: xlist.append(x) hlist.append(gaussian_function(x)) ``` This example shows how lists provide a straightforward way to manage collections of related values, crucial for numerical computations and data analysis.

Mathematical Computations

Mathematical computations in Python are straightforward due to its strong library support, like `math` or `numpy`. These libraries offer a wide range of mathematical functions and constants that are helpful for scientific tasks. For instance, when working with the Gaussian function, you employ exponential and square root calculations. Python's `math` library provides functions like `math.exp()` and `math.sqrt()`, which allow precise calculations: ```python import math h_x = (1 / math.sqrt(2 * math.pi)) * math.exp(-0.5 * x**2) ``` This computation returns the value of the Gaussian function for any `x`, and Python handles the nuances of numerical precision internally. Properly leveraging these libraries ensures that your mathematical computations are not only accurate but also efficient. Embracing these tools aids in tackling various mathematical and engineering challenges with Python.