Root finding (root, brentq) in SciPy - Interactive Code Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to import the root function from scipy.optimize.

SciPy

from scipy.optimize import [1]

result = root(lambda x: x**2 - 4, 1.0)
print(result.x)

Drag options to blanks, or click blank then click option'

Acurve_fit

Bbrentq

Cminimize

Droot

Attempts:

3 left

💡 Hint

Common Mistakes

Importing 'brentq' instead of 'root'.

Using 'minimize' which is for optimization, not root finding.

The root function is imported from scipy.optimize to find roots of equations.

2fill in blank

medium

Complete the code to find the root of the function f(x) = x^3 - 1 using root with initial guess 0.5.

SciPy

from scipy.optimize import root

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

sol = root(f, [1])
print(sol.x)

Drag options to blanks, or click blank then click option'

A0.5

B2.0

C-1.0

D1.0

Attempts:

3 left

💡 Hint

Common Mistakes

Using 1.0 as initial guess instead of 0.5.

Using negative initial guess which is not required here.

The initial guess for the root is 0.5 as given in the instruction.

3fill in blank

hard

Fix the error in the code to correctly find the root of f(x) = cos(x) - x using brentq between 0 and 1.

SciPy

from scipy.optimize import brentq
import numpy as np

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

root_value = brentq(f, [1], 1)
print(root_value)

Drag options to blanks, or click blank then click option'

A0

B1

C-1

D0.5

Attempts:

3 left

💡 Hint

Common Mistakes

Using 1 as the start of the interval which is the end.

Using negative values outside the valid interval.

The brentq method requires an interval where the function changes sign. The interval [0,1] is correct, so the first argument must be 0.

4fill in blank

hard

Fill both blanks to create a dictionary comprehension that maps words to their lengths only if length is greater than 3.

SciPy

words = ['data', 'science', 'ai', 'ml']
lengths = {word: [1] for word in words if [2]

Drag options to blanks, or click blank then click option'

Alen(word)

Blen(word) > 3

Cword

Dword > 3

Attempts:

3 left

💡 Hint

Common Mistakes

Using 'word' instead of 'len(word)' for the value.

Using 'word > 3' which compares string to number.

The dictionary comprehension maps each word to its length using len(word). The condition filters words with length greater than 3 using len(word) > 3.

5fill in blank

hard

Fill all three blanks to create a dictionary comprehension that maps uppercase words to their lengths only if length is greater than 2.

SciPy

words = ['cat', 'dog', 'a', 'elephant']
result = {{ [1]: [2] for w in words if [3] }}

Drag options to blanks, or click blank then click option'

Aw.upper()

Blen(w)

Clen(w) > 2

Dw

Attempts:

3 left

💡 Hint

Common Mistakes

Using 'w' instead of 'w.upper()' for keys.

Not filtering words by length.

The dictionary comprehension uses w.upper() as keys, len(w) as values, and filters words with length greater than 2 using len(w) > 2.

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