📚 An Introduction to Optimization (2023)

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

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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.

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2. Installing SciPy

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

pip install scipy

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3. Importing SciPy Modules

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

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

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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.

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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)

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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)

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7. SciPy vs NumPy

• NumPy focuses on basic array operations and linear algebra.

• SciPy extends functionality with advanced scientific algorithms.

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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.

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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.

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#Python #SciPy #ScientificComputing #NumericalIntegration #Optimization

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Topic: Python SciPy – From Easy to Top: Part 3 of 6: Optimization Basics

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1. What is Optimization?

• Optimization is the process of finding the minimum or maximum of a function.

• SciPy provides tools to solve these problems efficiently.

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2. Using `scipy.optimize.minimize`

This function minimizes a scalar function of one or more variables.

Example: Minimize the function f(x) = (x - 3)^2

from scipy import optimize

def f(x):
    return (x - 3)**2

result = optimize.minimize(f, x0=0)
print("Minimum value:", result.fun)
print("At x =", result.x)

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**3. Minimizing Multivariable Functions**

Example: Minimize f(x, y) = (x - 2)^2 + (y + 3)^2

def f(vars):
    x, y = vars
    return (x - 2)**2 + (y + 3)**2

result = optimize.minimize(f, x0=[0, 0])
print("Minimum value:", result.fun)
print("At x, y =", result.x)

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**4. Using Bounds and Constraints**

You can restrict the variables within bounds or constraints.

Example: Minimize f(x) = (x - 3)^2 with x between 0 and 5

result = optimize.minimize(f, x0=0, bounds=[(0, 5)])
print("Minimum with bounds:", result.fun)
print("At x =", result.x)

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5. Root Finding with `optimize.root_scalar`

Find a root of a scalar function.

Example: Find root of f(x) = x^3 - 1 between 0 and 2

def f(x):
    return x**3 - 1

root = optimize.root_scalar(f, bracket=[0, 2])
print("Root:", root.root)

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6. Summary

• SciPy’s optimization tools help find minima, maxima, and roots.

• Supports single and multivariable problems with constraints.

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Exercise

• Minimize the function f(x) = x^4 - 3x^3 + 2 over the range \[-2, 3].

• Find the root of f(x) = cos(x) - x near x=1.

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#Python #SciPy #Optimization #RootFinding #ScientificComputing

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