Easily Generate Equally Spaced Values with linspace

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np.linspace(): Create Evenly or Non-Evenly Spaced Arrays

Table of Contents

Creating Ranges of Numbers With Even Spacing
- Using np.linspace()
- Using range() and List Comprehensions
- Using np.arange()
Customizing the Output From np.linspace()
- The start, stop, and num Parameters
- The dtype Parameter for Changing Output Type
- The endpoint and retstep Parameters
- Nonscalar Values for Higher-Dimensional Arrays
- Summary of Input Parameters and Return Values
- Example: A Food Production Conveyor Belt
Representing Mathematical Functions
- Mathematical Functions With np.linspace()
- Example: Superimposing Traveling Waves
- Two-Dimensional Mathematical Functions
Creating Ranges of Numbers With Uneven Spacing
- Logarithmic Spaces
- Other Nonlinear Ranges
- Example: Simulation of an Orbiting Planet
Conclusion

When working with numerical applications in Python, it is often necessary to create an array of numbers. In many cases, you may want the numbers to be evenly spaced, but there are also instances where non-evenly spaced numbers are needed. One of the key tools for achieving both types of spacing is np.linspace().

In this tutorial, we will explore how to use np.linspace() effectively. This tutorial focuses on providing informative and detailed explanations, including step-by-step sample codes that are executable.

Creating Ranges of Numbers With Even Spacing

There are several ways to create a range of evenly spaced numbers in Python. One of these methods is using np.linspace(), which allows for customization of the range for specific requirements. However, there are alternatives to np.linspace() that can also achieve the same outcome. In the following sections, we will first learn about using np.linspace() and then compare it with other methods of creating ranges of evenly spaced numbers.

Using `np.linspace()`

np.linspace() requires two mandatory parameters: start and stop, which define the beginning and ending of the range:

import numpy as np
np.linspace(1, 10)

The output of this code would be an array of evenly spaced numbers from 1 to 10. By default, np.linspace() generates an array of 50 numbers within the specified range. However, you can also specify the number of numbers you want to generate using the optional num parameter:

np.linspace(1, 10, num=10)

In this case, the output would be an array of 10 evenly spaced numbers from 1 to 10.

Using `range()` and List Comprehensions

Another way to create a range of evenly spaced numbers is by using the range() function along with list comprehensions. For example, you can achieve the same result as the previous example with the following code:

import numpy as np

start = 1
stop = 10
num = 50

arr = [start + (stop - start) / (num - 1) * i for i in range(num)]

This code creates a list comprehension that generates a list of numbers by iterating over the range() function. The formula (stop - start) / (num - 1) calculates the spacing between each number in the range.

Using `np.arange()`

Another alternative is to use np.arange(), which is similar to range(), but returns an array instead of a list. Here is an example:

import numpy as np
np.arange(1, 10, 0.2)

In this case, np.arange() generates an array of numbers starting from 1 and ending at 10, with a step of 0.2 between each number.

Customizing the Output From np.linspace()

np.linspace() provides several optional parameters that allow for customization of the output. These parameters include dtype, endpoint, and retstep, among others:

The `start`, `stop`, and `num` Parameters

As mentioned earlier, the start and stop parameters define the beginning and ending of the range. Additionally, the num parameter allows you to specify the number of numbers you want to generate within the range. Here is an example:

import numpy as np
np.linspace(1, 10, num=10, endpoint=False)

In this example, the endpoint=False parameter excludes the ending value from the generated array.

The `dtype` Parameter for Changing Output Type

By default, np.linspace() returns an array of floating-point numbers. However, you can use the dtype parameter to specify a different data type. For example:

import numpy as np
np.linspace(1, 10, dtype=int)

In this case, the output would be an array of integers instead of floating-point numbers.

The `endpoint` and `retstep` Parameters

The endpoint parameter determines whether the ending value should be included in the generated array. By default, endpoint=True, meaning the ending value is included. You can set endpoint=False to exclude the ending value.

The retstep parameter, when set to True, returns the spacing between each number in the generated array as the second output. Here is an example:

import numpy as np
output, spacing = np.linspace(1, 10, retstep=True)

In this case, output would be the generated array, and spacing would be the spacing between each number.

Nonscalar Values for Higher-Dimensional Arrays

np.linspace() can also generate higher-dimensional arrays by accepting nonscalar values for the start, stop, and num parameters. For example:

import numpy as np
np.linspace([1, 2], [10, 20], num=2)

In this case, the output would be an array with two rows, where the first row consists of the numbers [1, 2] and the second row consists of the numbers [10, 20].

Summary of Input Parameters and Return Values

In summary, here is a table that summarizes the input parameters and return values of np.linspace():

Parameter	Description
start	Starting value of the range
stop	Ending value of the range
num	Number of numbers to generate within the range
dtype	Data type of the output array
endpoint	Whether to include the ending value in the generated array
retstep	Whether to return the spacing between each number as the
	second output

Example: A Food Production Conveyor Belt

To illustrate the use of np.linspace(), let’s consider an example of a food production conveyor belt. Suppose the belt consists of five sections, and each section has a different processing time. We can represent the processing times as an array using np.linspace():

import numpy as np
processing_times = np.linspace(5, 15, num=5)

In this case, processing_times would be an array containing the processing times of each section of the conveyor belt.

Representing Mathematical Functions

np.linspace() can also be used to represent mathematical functions in a discrete form. By defining the range of inputs and specifying the number of points to generate within that range, you can obtain the discrete representation of the function. Here is an example:

import numpy as np

x = np.linspace(0, 2 * np.pi, num=100)
y = np.sin(x)

In this example, x represents the inputs to the sine function, and y represents the corresponding outputs. By plotting x against y, you can visualize the sine function.

Example: Superimposing Traveling Waves

Using np.linspace(), you can also superimpose multiple traveling waves to create complex wave patterns. Here is an example that illustrates this concept:

import numpy as np
import matplotlib.pyplot as plt

x = np.linspace(0, 2 * np.pi, num=500)
y1 = np.sin(4 * x)
y2 = np.sin(8 * x)

y = y1 + y2

plt.plot(x, y)
plt.show()

In this example, x represents the inputs to the sine functions, and y1 and y2 represent the outputs of two sine functions with different frequencies. By adding y1 and y2 together, we obtain the composite wave y, which can be visualized by plotting x against y.

Two-Dimensional Mathematical Functions

np.linspace() can also be used to create two-dimensional arrays that represent mathematical functions over a grid of input values. Here is an example:

import numpy as np
x = np.linspace(-5, 5, num=100)
y = np.linspace(-5, 5, num=100)
X, Y = np.meshgrid(x, y)
Z = np.sin(X) * np.cos(Y)

In this example, X and Y represent the inputs to the function Z, which is a mathematical function involving the sine and cosine functions. By evaluating Z over a grid of input values, we obtain a two-dimensional array that represents the function.

Creating Ranges of Numbers With Uneven Spacing

In addition to evenly spaced numbers, np.linspace() can also be used to generate ranges of numbers with uneven spacing.

Logarithmic Spaces

One way to create unevenly spaced numbers is by using logarithmic spacing. By specifying a logarithmic scale, you can generate numbers that increase or decrease exponentially. Here is an example:

import numpy as np
np.logspace(1, 10, num=10, base=10)

In this example, np.logspace() generates an array of numbers starting from 10^1 and ending at 10^10, with a total of 10 numbers. The base parameter specifies the base of the logarithmic scale.

Other Nonlinear Ranges

np.linspace() can also generate ranges of numbers with nonlinear spacing. This can be achieved by providing nonscalar values for the start, stop, and num parameters. Here is an example:

import numpy as np
np.linspace([1, 2], [10, 20], num=7)

In this case, the output would be an array with seven rows, where each row represents a different nonlinearly spaced range of numbers.

Example: Simulation of an Orbiting Planet

To illustrate the use of np.linspace() for nonlinear ranges, let’s consider an example of simulating the motion of an orbiting planet. By specifying the initial position, velocity, and acceleration of the planet, you can generate a range of positions over time using np.linspace(). Here is an example:

import numpy as np

# Constants
G = 6.67430e-11  # Gravitational constant
M = 5.972e24  # Mass of the Earth
R = 6.371e6  # Radius of the Earth

# Initial conditions
r0 = np.array([R + 1e6, 0])  # Initial position (meters)
v0 = np.array([0, 1e3])  # Initial velocity (meters per second)
a0 = -G * M / np.linalg.norm(r0)**3 * r0  # Initial acceleration (meters per second squared)

# Time range
t = np.linspace(0, 10000, num=100)

# Position over time
r = r0 + v0 * t + 0.5 * a0 * t**2

In this example, t represents a range of time values, r0 represents the initial position of the planet, v0 represents the initial velocity, and a0 represents the initial acceleration. By evaluating the position of the planet over time using the equations of motion, we obtain a range of positions that can be visualized or used for further analysis.

Conclusion

In this tutorial, we explored the various aspects of using np.linspace() to create evenly or non-evenly spaced arrays of numbers. We learned about the input parameters and how they can be used to customize the output. We also saw examples illustrating the use of np.linspace() in representing mathematical functions and creating ranges with uneven spacing.

By understanding the capabilities of np.linspace(), you can enhance your numerical programming skills and effectively handle complex numerical applications in Python.