scipy

SciPy (pronounced "Sigh Pie") is an open-source Python library used for scientific computing and technical computing. It builds on the NumPy library and provides a vast collection of high-level numerical routines for various scientific and engineering tasks. While NumPy provides the fundamental array object and basic array operations, SciPy offers a wide range of modules for more specialized functionalities.

Key features and modules of SciPy include:

- `scipy.integrate`: For numerical integration routines, including ordinary differential equations (ODEs).
- `scipy.linalg`: Provides advanced linear algebra routines, complementing and extending NumPy's `linalg` module.
- `scipy.optimize`: Offers algorithms for function optimization (minimization, root finding, curve fitting).
- `scipy.interpolate`: For interpolation routines, useful for estimating values between known data points.
- `scipy.stats`: Contains statistical distributions (e.g., normal, Poisson) and functions (e.g., hypothesis testing, descriptive statistics).
- `scipy.fft`: For Fast Fourier Transform (FFT) and other discrete Fourier transforms.
- `scipy.signal`: Provides tools for signal processing, including filtering, convolution, and spectral analysis.
- `scipy.spatial`: For spatial data structures and algorithms, such as k-d trees, distance functions, and Delaunay triangulation.
- `scipy.ndimage`: For N-dimensional image processing, including filtering, morphology, and transformations.
- `scipy.io`: Utilities for reading and writing data in various scientific formats (e.g., MATLAB .mat files, NetCDF).

SciPy is a core component of the scientific Python ecosystem and is widely used in fields like physics, engineering, finance, data science, and machine learning for its efficiency, robustness, and comprehensive collection of algorithms. It leverages compiled C/Fortran code under the hood for many of its functions, making it highly performant.

import numpy as np
from scipy import integrate, optimize, stats

print("--- SciPy Example ---")

 1. Numerical Integration using scipy.integrate.quad
 Define a function to integrate: f(x) = x^2
def f(x):
    return x2

 Integrate f(x) from 0 to 2
result_integral, error_integral = integrate.quad(f, 0, 2)
print(f"\n1. Numerical Integration (f(x) = x^2 from 0 to 2):")
print(f"   Result: {result_integral:.4f} (Expected: 8/3 \u2248 2.6667)")
print(f"   Error estimate: {error_integral:.2e}")

 2. Function Optimization using scipy.optimize.minimize
 Define an objective function to minimize: g(x) = (x - 2)^2 + 1
def g(x):
    return (x - 2)2 + 1

 Initial guess for x
x0 = 0

 Minimize the function
result_optimize = optimize.minimize(g, x0)
print(f"\n2. Function Optimization (minimize g(x) = (x - 2)^2 + 1):")
print(f"   Optimal x: {result_optimize.x[0]:.4f}")
print(f"   Minimum value: {result_optimize.fun:.4f}")

 3. Statistical Functions using scipy.stats
 Calculate the Probability Density Function (PDF) of a standard normal distribution at x=0
mu = 0     Mean
sd = 1     Standard Deviation
pdf_at_0 = stats.norm.pdf(0, loc=mu, scale=sd)
cdf_at_1 = stats.norm.cdf(1, loc=mu, scale=sd)

print(f"\n3. Statistical Functions (Standard Normal Distribution):")
print(f"   PDF at x=0: {pdf_at_0:.4f}")
print(f"   CDF at x=1: {cdf_at_1:.4f}")

print("\n--- End of SciPy Example ---")