Introduction

Computing is an essential part of scientific research. Mathematical, engineering, scientific and other technical problems are complex and require computing power and speed. Python provides the SciPy library for solving technical problems computationally.

**This article presents a SciPy tutorial and how to implement the code in Python with examples.**

Prerequisites

- Installed Python 2.7 or Python 3
- A Python environment for running the code.
- SciPy library installed.
- NumPy library installed (Follow our guide: How to Install NumPy).
- Matplotlib library installed for plotting (optional).

**Note:** Python 2 no longer has support as of 2020. Consider switching to Python 3. Install Python 3 by following one of our guides: How to install Python 3 on CentOS 7, How to install Python 3 on CentOS 8, How to install Python 3 on Ubuntu, How to install Python on Windows.

## What is SciPy?

SciPy (**Sci**entific **Py**thon) is an open-source scientific computing module for Python. Based on NumPy, SciPy includes tools to solve scientific problems. Scientists created this library to address their growing needs for solving complex issues.

### SciPy vs NumPy

The NumPy library (**Num**erical **Py**thon) does numerical computation. Scientists use this library for working with arrays since NumPy covers elementary uses in data science, statistics, and mathematics.

SciPy covers advanced calculations and functions. This library adds more data science features, all linear algebra functions, and standard scientific algorithms.

**Note:** Anaconda includes the SciPy, NumPy, and Matplotlib libraries by default. Check out our guides on installing Anaconda: How to Install Anaconda on CentOS 7, How to Install Anaconda on CentOS 8, How to Install Anaconda on Ubuntu 18.04 or 20.04.

## Why Use SciPy?

The SciPy library builds on top of NumPy and operates on arrays. The computational power is fast because NumPy uses C for evaluation.

The Python scientific stack is similar to MATLAB, Octave, Scilab, and Fortran. The main difference is Python is easy to learn and write.

**Note:** Some Python environments are scientific. If you need to choose one, check out our comparison of the Best Python IDEs and Code Editors.

## SciPy Subpackages

The SciPy library has different groups of subpackages. There are two ways to import subpackages from the SciPy module:

`import scipy.<sub package name> as <alias>`

Or alternatively:

`from scipy import <sub package name> as <alias>`

In both importing methods, the alias is optional.

## SciPy Functions

SciPy includes many of the primary array functions available in NumPy and some of the commonly used modules from the SciPy subpackages.

To import a function from a subpackage, use:

`from scipy.<subpackage> import <function>`

**Note:** Some of the NumPy functions available in SciPy show deprecation warnings.

### Basic Functions

To get help and information for any SciPy function, use the ** help()** command:

`help(<name of function>)`

The ** help()** command does not need parameters. After executing without parameters, a prompt appears where you input the function name.

Another quick way to get help with any command in Python is to write the command name, put a question mark at the end, and run the code.

### Special Functions

Special functions in the SciPy module include commonly used computations and algorithms. All special functions accept NumPy arrays as input. The calculations are elementwise.

To import the ** special** subpackage, use:

`import scipy.special as special`

Or alternatively:

`from scipy import special`

To import a specific function from the ** special** subpackage, use:

`from scipy.special import <function name>`

**Factorial**

Evaluate the factorial of any number by running:

`special.factorial(<integer or array>)`

For example, to find the factorial of ten, use:

`special.factorial(10)`

**Permutations and Combinations**

To find the number of permutations, use:

`special.perm(<number of elements>, <number of elements taken>)`

For example, to see the number of permutations of three elements taken two at a time:

`special.perm(6,2)`

Similarly, find the number of combinations with:

`special.comb(<number of elements>, <number of elements taken>, repetition=<True or False>)`

To find the number of combinations of three elements taken two at a time with repetition, enter:

`special.comb(6,2, repetition=True)`

Permutations and combinations are used in computer science sorting algorithms.

**Exponential Functions**

Exponential functions evaluate the exponents for different bases.

Calculate the exponents of base ten with:

`special.exp10(<integer or array>)`

For example:

`special.exp10([0,1,2])`

Computer science often uses exponential functions of base two:

`special.exp2(<integer or array>)`

Calculate the tenth power of base two with:

`special.exp2(10)`

**Logarithmic Sum of Exponentials**

The Logarithmic Sum of Exponentials (LSE or LogSumExp) is an approximation used by machine learning algorithms. Calculate the LSE with:

`special.logsumexp(<integer or array>)`

**Bessel Function**

Bessel functions appear in wave propagation, signal processing, and static potential problems. Find the Bessel function of the first kind with:

`special.jn(<integer order>, <integer or array>)`

Take advantage of the full stack to visualize the Bessel function. To find the second-order Bessel function of the first kind, use:

```
#import stack
import scipy.special as special
import matplotlib.pyplot as plt
import numpy as np
#The X-axis
x = np.linspace(1,50,100)
#Bessel function of the first kind order two
jn1 = special.jn(2,x)
```

Plot the results:

```
#Plotting
plt.title('Bessel function first kind order two')
plt.plot(x, jn1)
```

**Note:** Learn about a crucial mathematical operation in Python, raising a number to a power by reading our article Python Power Operator and Function.

### Integration and ODE Functions

SciPy provides a subpackage for calculations with definite integrals. To import the ** integrate** subpackage, use:

`import scipy.integrate as integrate`

Or alternatively:

`from scipy import integrate`

Import a specific function from the subpackage ** integrate** with:

`from scipy.integrate import <function name>`

**General Integration**

Calculate a single variable integral with the ** quad** function from the

**subpackage:**

`integrate`

`integrate.quad(<function>, <lower limit>, <upper limit>)`

The ** function** input is defined using a lambda function.

For example, to calculate the definite integral of the function x+1 between zero and one:

```
from scipy import integrate
f = lambda x: x+1
integrate.quad(f, 0, 1)
```

The output shows two values. The first value is the evaluated integral, and the second is the error of estimation.

### Optimization Functions

SciPy has an optimization subpackage for finding the minimum or maximum of a function. The ** optimize** subpackage includes solvers and algorithms for finding local and global optimal values.

To import the optimize subpackage:

`from scipy import optimize`

Or use:

`import scipy.optimize as optimize`

To import a specific function from the subpackage ** optimize**, run:

`from scipy.optimize import <function name>`

**Minimize a Function**

Finding a minimum of a function is used in machine learning to lower an algorithm’s loss (or error).

For example, you can create a function and find the minimum. To do so, use the ** fmin** function from the

**subpackage in SciPy:**

`optimize`

```
#Import stack
import numpy as np
from scipy import optimize
#Defining inverse sine function
def f(x):
return -np.sin(x)
#X-axis
x = np.linspace(0,5,100)
#Starting point
start = 3
#Simplex algorithm for optimization
optimized = optimize.fmin(f,start)
```

To plot the result, run:

```
import matplotlib.pyplot as plt
plt.plot(x, f(x))
plt.scatter(start,f(start))
plt.scatter(optimized, f(optimized))
plt.legend(['Function -sin(x)', 'Starting point', 'Optimized minimum'])
```

### Fourier Transformation Functions

SciPy includes a subpackage for Fourier transformation functions called ** fftpack**. The transformations are Discrete Fourier Transformations (DFT). All transforms are applied using the Fast Fourier Transformation (FFT) algorithm.

To import the ** fftpack** subpackage, use:

`import scipy.fftpack as fftpack`

Or:

`from scipy import fftpack`

**Fast Fourier Transform**

As an example, create a periodic function as a sum of three sine waves:

```
import numpy as np
freq_samp = 100
#Time
t = np.linspace(0, 1, freq_samp*2, endpoint = False )
#Frequencies
f1, f2, f3 = 1, 5, 20
#Amplitudes
A1, A2, A3 = 3, 2, 1
x1 = A1*np.sin(f1*2*np.pi*t)
x2 = A2*np.sin(f2*2*np.pi*t)
x3 = A3*np.sin(f3*2*np.pi*t)
#Sum of waves
x = x1+x2+x3
```

Plot the waves using ** matplotlib**:

```
import matplotlib.pyplot as plt
plt.subplot(2,1,1)
plt.plot(t,x1,t,x2,t,x3)
plt.subplot(2,1,2)
plt.plot(t,x)
plt.xlabel('Time (s)')
plt.ylabel('Amplitude');
```

Next, apply the ** fft** and

**functions from the**

`fftfreq`

**to do a Fourier transform of the signal.**

`fftpack`

```
from scipy import fftpack
A = fftpack.fft(x)
freq = fftpack.fftfreq(len(x))*freq_samp*2
```

Plot the results to see the frequency domain:

```
plt.stem(freq,np.abs(A)/freq_samp,use_line_collection=True)
plt.xlim(-25,25)
plt.xlabel('Frequency (Hz)')
plt.ylabel('Amplitude')
plt.grid()
```

### Signal Processing Functions

The subpackage ** signal** includes functions used in signal processing. To import

**, run:**

`signal`

`import scipy.signal as signal`

Or alternatively:

`from scipy import signal`

**Convolution**

A common task in signal processing is convolution. The SciPy subpackage ** signal** has the function

**to perform this task. For example, create two signals with different frequencies:**

`convolve`

```
import numpy as np
#Time
t = np.linspace(0,1,100)
#Frequency
f1, f2 = 5, 2
#Two signals of different frequencies
first_signal = np.sin(f1*2*np.pi*t)
second_signal = np.sin(f2*2*np.pi*t)
```

Plot the signals:

```
import matplotlib.pyplot as plt
#Plotting both signals
plt.subplot(2,1,1)
plt.plot(t, first_signal)
plt.subplot(2,1,2)
plt.plot(t, second_signal)
plt.ylabel('Amplitude')
plt.xlabel('Time (s)')
```

Import the ** signal** subpackage from

**. Use the**

`scipy`

**function from the**

`convolve`

**subpackage to convolve the two signals:**

`signal`

```
#Importing the signal subpackage
from scipy import signal
#Convolving two signals
convolution = signal.convolve(first_signal, second_signal, mode='same')
```

Plot the results:

```
#Plotting the result
plt.plot(t, convolution)
plt.xlabel('Time (s)')
plt.ylabel('Amplitude')
```

### Interpolation Functions

Interpolation is used in the numerical analysis field to generalize values between two points. SciPy has the ** interpolate** subpackage with interpolation functions and algorithms.

Import the ** interpolate** subpackage with:

`import scipy.interpolate as interpolate`

Or:

`from scipy import interpolate`

**One Dimensional Interpolation**

The SciPy ** interpolate** subpackage has the

**function for one dimensional interpolation of data. As an example, create**

`interp1d`

*toy*data using

**:**

`numpy`

```
import numpy as np
#Create toy data
x = np.arange(0,10,0.5)
y = np.sin(x)
```

Interpolate the data with ** interp1d** from the

**subpackage:**

`interpolate`

```
from scipy import interpolate
#Interpolate
f = interpolate.interp1d(x, y)
#Create interpolation function
x_i = np.arange(0,10,3)
y_i = f(x_i)
```

Plot the results:

```
#Plot results
plt.scatter(x,y)
plt.plot(x_i, y_i)
plt.legend(['Interpolation', 'Data points'])
```

## Linear Algebra

SciPy has a fully-featured linear algebra subpackage. The SciPy linear algebra subpackage is optimized with the ATLAS LAPACK and BLAS libraries for faster computation.

To import the linear algebra package from SciPy, run:

`import scipy.linalg as linalg`

Or use:

`from scipy import linalg`

All the linear algebra functions expect a NumPy array for input.

### Determinant

Calculate the determinant of a matrix with ** det** from the

**subpackage:**

`linalg`

`linalg.det(<numpy array>)`

For example:

```
import numpy as np
#Generate a 2D array
A = np.array([[1,2],[3, 4]])
from scipy import linalg
#Calculate the determinant
linalg.det(A)
```

### Inverse Matrix

Determine the inverse matrix by using ** inv**:

`linalg.inv(<numpy array>)`

For example:

```
import numpy as np
#Generate a 2D array
A = np.array([[1,2],[3,4]])
from scipy import linalg
#Calculate the inverse matrix
linalg.inv(A)
```

### Eigenvectors and Eigenvalues

Eigenvectors and eigenvalues are a matrix decomposition method. The eigenvalue-eigenvector problem is a commonly implemented linear algebra problem.

The ** eig** function finds the eigenvalues and eigenvectors of a matrix:

`linalg.eig(<numpy array>)`

The output returns two arrays. The first contains eigenvalues, and the second has eigenvectors for the given matrix. For example:

```
import numpy as np
#Generate a 2D array
A = np.array([[1,2],[3, 4]])
from scipy import linalg
#Calculate the eigenvalues and eigenvectors
linalg.eig(A)
```

## Spatial Data Structures and Algorithms

Spatial data structures are objects made of points, lines, and surfaces. SciPy has algorithms for spatial data structures since they apply to many scientific disciplines.

Import the ** spatial** subpackage from SciPy with:

`import scipy.spatial as spatial`

Or:

`from scipy import spatial`

A notable example of a spatial algorithm is the Voronoi diagram. For a given set of points, Voronoi maps divide a plane into regions. If a new point falls into a region, the point in the region is the nearest neighbor.

**Note:** Voronoi diagrams relate to the k-Nearest Neighbor algorithm in machine learning.

As an example, create a Voronoi diagram from twenty random points:

```
from scipy.spatial import Voronoi
import numpy as np
points = np.random.rand(20,2)
voronoi = Voronoi(points)
from scipy.spatial import voronoi_plot_2d
fig = voronoi_plot_2d(voronoi,show_vertices=False)
```

## Image Processing

SciPy has a subpackage for various n-dimensional image processing. To import the ** ndimage** subpackage, run:

`import scipy.ndimage as ndimage`

Or use:

`from scipy import ndimage`

The SciPy ** misc** subpackage contains a sample image for demonstration purposes. To import the

**subpackage and show the image:**

`misc`

```
from scipy import misc
from matplotlib import pyplot as plt
raccoon = misc.face()
#show image
plt.imshow(raccoon)
plt.show()
```

Import the ** ndimage** subpackage and apply a

**to the image. Show the image to see the results:**

`uniform_filter`

```
from scipy import ndimage
filtered = ndimage.uniform_filter(raccoon)
plt.imshow(filtered)
```

## File IO (File Input / Output Package)

SciPy has a file input and output subpackage called ** io**. The

**subpackage is used for reading and writing data formats from different scientific computing programs and languages, such as Fortran, MATLAB, IDL, etc.**

`io`

Import the ** io** subpackage from SciPy with:

`import scipy.io as sio`

Or use:

`from scipy import io as sio`

Conclusion

This tutorial provided the necessary ScyPy examples to get started. Python is easy to learn for beginners and scripts are simple to write and test. Combining SciPy with other Python libraries, such as NumPy and Matplotlib, Python becomes a powerful scientific tool. The SciPy subpackages are well documented and developed continuously.

For further reading, check out our tutorial on the Pandas library: Introduction to Python Pandas.