Root finding (root, brentq) in SciPy - Mini Project: Build & Apply

Root Finding with scipy: Using root and brentq

📖 Scenario: Imagine you are an engineer trying to find the temperature at which a machine part reaches a specific stress level. This requires solving an equation where the stress equals zero. You will use Python's scipy library to find the root of this equation.

🎯 Goal: You will create a function representing the stress equation, then use scipy.optimize.root and scipy.optimize.brentq methods to find the temperature where the stress is zero.

📋 What You'll Learn

Create a function stress_equation that takes a variable T and returns T**3 - 6*T**2 + 11*T - 6

Create a variable initial_guess and set it to 2.5

Use scipy.optimize.root with stress_equation and initial_guess to find a root, store result in root_result

Use scipy.optimize.brentq with stress_equation and interval (1.5, 2.5) to find a root, store result in brentq_root

Print both roots with descriptive messages

💡 Why This Matters

🌍 Real World

Engineers and scientists often need to find where a function equals zero to understand critical points like stress, temperature, or pressure thresholds.

💼 Career

Root finding is a fundamental skill in data science and engineering roles, useful for solving equations that model real-world problems.

1

Create the stress equation function

Create a function called stress_equation that takes one argument T and returns the value of T**3 - 6*T**2 + 11*T - 6.

SciPy

# Define the stress_equation function
# Your code here

Hint

This function models the stress depending on temperature T.

2

Set the initial guess for root finding

Create a variable called initial_guess and set it to 2.5.

SciPy

def stress_equation(T):
    return T**3 - 6*T**2 + 11*T - 6

# Set initial_guess to 2.5
# Your code here

Hint

The initial guess helps the root finder start searching near this value.

3

Find roots using root and brentq methods

Import root and brentq from scipy.optimize. Use root with stress_equation and initial_guess, store the result in root_result. Then use brentq with stress_equation and interval (1.5, 2.5), store the result in brentq_root.

SciPy

def stress_equation(T):
    return T**3 - 6*T**2 + 11*T - 6

initial_guess = 2.5

# Import root and brentq from scipy.optimize
# Use root with stress_equation and initial_guess, save to root_result
# Use brentq with stress_equation and interval (1.5, 2.5), save to brentq_root
# Your code here

Hint

Use root to find a root near the initial guess, and brentq to find a root in the interval (1.5, 2.5).

4

Print the roots found

Print the root found by root_result using root_result.x[0] with the message: Root found by root method:. Then print the root found by brentq_root with the message: Root found by brentq method:.

SciPy

from scipy.optimize import root, brentq

def stress_equation(T):
    return T**3 - 6*T**2 + 11*T - 6

initial_guess = 2.5

root_result = root(stress_equation, initial_guess)
brentq_root = brentq(stress_equation, 1.5, 2.5)

# Print the root from root_result.x[0] with message
# Print the root from brentq_root with message
# Your code here

Hint

Use print and f-strings to show the roots clearly.

Practice

1. What is the main purpose of the scipy.optimize.root function in root finding?

easy

A. To calculate the maximum value of a function

B. To integrate a function over an interval

C. To find where a function equals zero using an initial guess

D. To sort an array of numbers

Solution

1. Step 1: Understand the purpose of root finding

   Root finding means locating points where a function's value is zero.

2. Step 2: Identify what scipy.optimize.root does

   This function finds roots starting from an initial guess, aiming to solve f(x) = 0.

3. Final Answer:

   To find where a function equals zero using an initial guess -> Option C

4. Quick Check:

   Root finding = find zero points [OK]

Hint: Root means function value zero, root() finds it from guess [OK]

Common Mistakes:

• Confusing root finding with integration
• Thinking root finds maxima or minima
• Assuming root sorts data

2. Which of the following is the correct way to import the brentq function from scipy.optimize?

easy

A. from scipy.optimize import brentq

B. import brentq from scipy.optimize

C. from scipy import brentq

D. import scipy.optimize.brentq

Solution

1. Step 1: Recall Python import syntax for functions

   To import a function from a module, use: from module import function.

2. Step 2: Apply to brentq from scipy.optimize

   The correct syntax is: from scipy.optimize import brentq.

3. Final Answer:

   from scipy.optimize import brentq -> Option A

4. Quick Check:

   Correct import syntax = from module import function [OK]

Hint: Use 'from module import function' to import specific functions [OK]

Common Mistakes:

• Using 'import function from module' which is invalid
• Trying to import directly from scipy without optimize
• Using dot notation in import statement incorrectly

3. What will be the output of this code snippet?

from scipy.optimize import brentq

def f(x):
    return x**3 - 2*x - 5

root = brentq(f, 2, 3)
print(round(root, 3))

medium

A. 2.095

B. 2.0946

C. 2.0945

D. 2.0946 (rounded to 4 decimals)

Solution

1. Step 1: Understand the function and interval

   The function is f(x) = x³ - 2x - 5. We look for root between 2 and 3.

2. Step 2: Use brentq to find root and round output

   Using brentq finds root near 2.0945514815. Rounded to 3 decimals is 2.095.

3. Step 3: Check rounding precision in options

   Options B (2.0946), C (2.0945), D (notes 4 decimals) are near unrounded root, but print(round(root,3)) outputs 2.095.

4. Final Answer:

   2.095 -> Option A

5. Quick Check:

   Root near 2.095 rounded 3 decimals = 2.095 [OK]

Hint: brentq finds root in interval; round final answer carefully [OK]

Common Mistakes:

• Confusing rounding digits
• Using wrong interval causing error
• Misunderstanding brentq output as exact integer

4. Identify the error in this code snippet and choose the correct fix:

from scipy.optimize import brentq

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

root = brentq(g, 3, 5)
print(root)

medium

A. No error; code runs fine

B. Interval must include a sign change; use brentq(g, 1, 3)

C. Function g(x) is incorrect; change to g(x) = x**2 + 4

D. Import statement is wrong; use import brentq from scipy.optimize

Solution

1. Step 1: Check function values at interval ends

   g(3) = 9 - 4 = 5 (positive), g(5) = 25 - 4 = 21 (positive). No sign change.

2. Step 2: Understand brentq requires sign change

   brentq needs f(a) and f(b) to have opposite signs to find root.

3. Step 3: Fix interval to include root

   Use interval (1, 3) because g(1) = -3 (negative), g(3) = 5 (positive), sign change exists.

4. Final Answer:

   Interval must include a sign change; use brentq(g, 1, 3) -> Option B

5. Quick Check:

   brentq needs sign change in interval [OK]

Hint: Check function signs at interval ends before brentq [OK]

Common Mistakes:

• Ignoring sign change requirement
• Changing function unnecessarily
• Incorrect import syntax

5. You want to find the root of h(x) = cos(x) - x using scipy.optimize.root. Which initial guess is most appropriate and why?

from scipy.optimize import root
import numpy as np

def h(x):
    return np.cos(x) - x

sol = root(h, ?)
print(sol.x)

hard

A. -1, because negative guesses always work better

B. 2, because cos(2) is near zero

C. 5, because root is always positive and large

D. 0.5, because root is near where cos(x) crosses x close to 0.5

Solution

1. Step 1: Understand the function behavior

   h(x) = cos(x) - x crosses zero near 0.7 (between 0 and 1).

2. Step 2: Choose initial guess close to root

   Initial guess 0.5 is close to root, helping root() converge quickly.

3. Step 3: Evaluate other options

   2 and 5 are far from root; -1 is negative and root is positive, so less suitable.

4. Final Answer:

   0.5, because root is near where cos(x) crosses x close to 0.5 -> Option D

5. Quick Check:

   Good initial guess near root speeds root() convergence [OK]

Hint: Pick initial guess near expected root for root() [OK]

Common Mistakes:

• Choosing initial guess far from root
• Assuming root is always positive and large
• Believing negative guesses always work better