The Logic of Linspace's Endpoint and FFT Frequency Bins

import numpy as np

# Generate 5 evenly spaced points between 0 and 10 (inclusive)
my_array = np.linspace(0, 10, 5)

print(my_array)

[ 0.   2.5  5.   7.5 10. ]

The Logic of the Step

This is the core logic of linspace. To get from a start to a stop with a specific number of points, you need to figure out the number of intervals or gaps between those points. If you have num points, you will always have num - 1 intervals.

For our example:

• num = 5 points
• Number of intervals = 5 - 1 = 4

The formula for the step size is:

Step = (stop - start) / (num - 1)

Plugging in our values: (10 - 0) / (5 - 1) = 10 / 4 = 2.5. This simple but robust calculation is why linspace is so reliable, especially when dealing with floating-point numbers.

The "Why": When to Choose Linspace

So, when is linspace the right tool for the job? It shines whenever your primary concern is the number of points and the exactness of your endpoints.

• Data Visualization: This is the classic use case. To plot a smooth function like y = sin(x), you need a set of x-values. Using linspace ensures your plot covers the exact domain you want, and increasing num gives you a smoother curve.

  import matplotlib.pyplot as plt
  
  x = np.linspace(0, 2 * np.pi, 200) # 200 points for a smooth curve
  y = np.sin(x)
  
  plt.plot(x, y)
  plt.title("A Smooth Sine Wave Thanks to Linspace")
  plt.show()

• Numerical Simulations: When defining a grid for a physics or engineering simulation, you often need a specific number of points across a domain. For example, analyzing temperature across a 1-meter rod with 101 measurement points.
• Machine Learning: Generating a range of values to test a hyperparameter. For instance, testing 20 different regularization strengths between 0.01 and 1.0.
• Digital Signal Processing: Creating a time vector for a signal. If you sample a signal for 2 seconds at 100 Hz, you'll have 200 samples. linspace is perfect for generating the corresponding 200 time points from 0 to 2.

Linspace vs. Arange: A Crucial Distinction

A common point of confusion for newcomers is the difference between np.linspace and np.arange. While both generate number sequences, their fundamental logic is different, and choosing the wrong one can lead to subtle bugs.

arange (short for "array range") works by taking a start, stop, and a step size. You tell it how big each step should be, not how many points you want in total.

Here’s a head-to-head comparison:

Feature	np.linspace	np.arange
Primary Control	The number of points you want (num).	The explicit step size between points.
Endpoint Inclusion	The stop value is included by default.	The stop value is excluded from the output.
Core Question it Answers	"Give me 50 points from 0 to 100."	"Give me points from 0 to 100, in steps of 2."
Floating-Point Behavior	Very reliable. Because it calculates the step based on the endpoints, it's not subject to floating-point accumulation errors.	Can be unpredictable. Using a float for the step (e.g., 0.1) can lead to unexpected results due to how computers store decimals. The last element might be slightly off or omitted entirely.

The floating-point issue is key. If you try np.arange(0, 1, 0.1), you might not get [0.0, 0.1, ..., 0.9] as you'd expect. You might get an extra element, or the last element might be 0.899999.... Because linspace guarantees the endpoints and calculates the step internally, it's the safer and more robust choice for non-integer steps.

Beyond the Basics: Advanced Linspace Tricks

linspace has a few more tricks up its sleeve for specialized situations.

Excluding the Endpoint

What if you don't want to include the stop value? Simply set the optional parameter endpoint=False.

This is perfect for periodic functions, like angles on a circle. A full circle is 0 to 2π radians. But 2π is the same position as 0, so you often want to exclude it to avoid duplication.

# Angles for a full circle, excluding the overlapping endpoint
angles = np.linspace(0, 2 * np.pi, 10, endpoint=False)
print(angles)

Getting the Step Size

Sometimes you need to know the exact step size that linspace calculated. You can get it back by setting retstep=True (return step). This will make the function return a tuple: the array and the calculated step.

points, step_size = np.linspace(0, 10, 5, retstep=True)

print(f"Points: {points}")
print(f"Calculated Step Size: {step_size}")

This is useful when you need to use the step size in other calculations, ensuring perfect consistency.

Key Takeaways: The Linspace Logic

Understanding linspace is a small step that makes a big difference in the quality and reliability of your code. Let's boil it down to the essentials:

• Use linspace when you care about the number of points and the endpoints. It's your go-to for plotting, gridding, and sampling.
• Use arange when you care about the specific step size, and you're typically working with integers.
• The core logic is based on intervals: num points create num - 1 gaps. The step size is the total range divided by these gaps.
• linspace is the safer, more predictable choice for floating-point intervals, as it avoids the common pitfalls of arange.

Next time you're setting up an array of numbers, pause and think: do I know the step size, or do I know how many points I need? If it's the latter, the logic of linspace is your answer.