Topic: Python SciPy – From Easy to Top: Part 1 of 6: Introduction and Basics

---

1. What is SciPy?

• SciPy is an open-source Python library used for scientific and technical computing.

• Built on top of NumPy, it provides many user-friendly and efficient numerical routines such as routines for numerical integration, optimization, interpolation, eigenvalue problems, algebraic equations, and others.

---

2. Installing SciPy

If you don’t have SciPy installed yet, use:

pip install scipy

---

3. Importing SciPy Modules

SciPy is organized into sub-packages for different tasks. Example:

import scipy.integrate
import scipy.optimize
import scipy.linalg

---

4. Key SciPy Sub-packages

• scipy.integrate — Numerical integration and ODE solvers.
• scipy.optimize — Optimization and root finding.
• scipy.linalg — Linear algebra routines (more advanced than NumPy’s).
• scipy.signal — Signal processing.
• scipy.fft — Fast Fourier Transforms.
• scipy.stats — Statistical functions.

---

5. Basic Example: Numerical Integration

Calculate the integral of sin(x) from 0 to pi:

import numpy as np
from scipy import integrate

result, error = integrate.quad(np.sin, 0, np.pi)
print("Integral of sin(x) from 0 to pi:", result)

---

6. Basic Example: Root Finding

Find the root of the function f(x) = x^2 - 4:

from scipy import optimize

def f(x):
    return x**2 - 4

root = optimize.root_scalar(f, bracket=[0, 3])
print("Root:", root.root)

---

7. SciPy vs NumPy

• NumPy focuses on basic array operations and linear algebra.

• SciPy extends functionality with advanced scientific algorithms.

---

8. Summary

• SciPy is essential for scientific computing in Python.

• It contains many specialized sub-packages.

• Understanding SciPy’s structure helps solve complex numerical problems easily.

---

Exercise

• Calculate the integral of e^(-x^2) from -infinity to +infinity using scipy.integrate.quad.

• Find the root of cos(x) - x = 0 using scipy.optimize.root_scalar.

---

#Python #SciPy #ScientificComputing #NumericalIntegration #Optimization

https://t.iss.one/DataScienceM

Topic: Python SciPy – From Easy to Top: Part 2 of 6: Numerical Integration and Differentiation

---

1. Numerical Integration Overview

• Numerical integration approximates the area under curves when an exact solution is difficult or impossible.

• SciPy provides several methods like quad, dblquad, and trapz.

---

2. Using `scipy.integrate.quad`

This function computes the definite integral of a function of one variable.

Example: Integrate cos(x) from 0 to pi divided by 2

import numpy as np
from scipy import integrate

result, error = integrate.quad(np.cos, 0, np.pi/2)
print("Integral of cos(x) from 0 to pi/2:", result)

---

3. Double Integration with `dblquad`

Integrate a function of two variables over a rectangular region.

Example: Integrate f(x, y) = x times y over x from 0 to 1, y from 0 to 2

def f(x, y):
    return x * y

result, error = integrate.dblquad(f, 0, 1, lambda x: 0, lambda x: 2)
print("Double integral result:", result)

---

4. Using the Trapezoidal Rule: `trapz`

Useful for integrating discrete data points.

Example:

import numpy as np
from scipy import integrate

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

area = integrate.trapz(y, x)
print("Approximate integral using trapz:", area)

---

5. Numerical Differentiation with `derivative`

SciPy’s derivative function approximates the derivative of a function at a point.

Example: Derivative of sin(x) at x equals pi divided by 4

from scipy.misc import derivative
import numpy as np

def f(x):
    return np.sin(x)

dx = derivative(f, np.pi/4, dx=1e-6)
print("Derivative of sin(x) at pi/4:", dx)

---

6. Limitations of `derivative`

• derivative uses finite difference methods, which can be noisy for non-smooth functions.

• Suitable for simple derivative calculations but not for complex cases.

---

7. Summary

• quad is powerful for one-dimensional definite integrals.

• dblquad handles two-variable integration.

• trapz approximates integration from sampled data.

• derivative provides numerical differentiation.

---

Exercise

• Compute the integral of e to the power of negative x squared from 0 to 1 using quad.

• Calculate the derivative of cos(x) at 0.

• Use trapz to approximate the integral of x squared over \[0, 5] using 50 points.

---

#Python #SciPy #NumericalIntegration #Differentiation #ScientificComputing

https://t.iss.one/DataScienceM