Root finding (root, brentq) in SciPy - Step-by-Step Execution

Concept Flow - Root finding (root, brentq)

Define function f(x)

↓

Choose interval or initial guess

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Call root finding method

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Iterative search for root

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Check if f(root) approx 0

Yes No↓

Return root

We start with a function and an initial guess or interval. The root finding method iteratively searches for a point where the function value is zero, adjusting guesses until it finds the root.

Execution Sample

SciPy

from scipy.optimize import root, brentq

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

sol = root(f, 1.5)
root_brentq = brentq(f, 1, 2)

This code finds the root of f(x) = x^3 - x - 2 using two methods: root with initial guess 1.5, and brentq on interval [1, 2].

Execution Table

Step	Method	Input	Function eval	Action	Result
1	root	Initial guess x=1.5	f(1.5)=1.875	Check if close to 0	Not close, iterate
2	root	Next guess x=1.7	f(1.7)=3.213	Check if close to 0	Not close, iterate
3	root	Next guess x=1.4	f(1.4)=0.744	Check if close to 0	Not close, iterate
4	root	Next guess x=1.521	f(1.521)=0.045	Check if close to 0	Close, iterate
5	root	Next guess x=1.5214	f(1.5214)=0.0001	Check if close to 0	Very close, stop
6	root	Return root	f(1.5214)=~0	Root found	Root approx 1.5214
1	brentq	Interval [1, 2]	f(1)=-2, f(2)=4	Check sign change	Sign change detected
2	brentq	Midpoint 1.5	f(1.5)=1.875	Check sign	Root in [1,1.5]
3	brentq	Midpoint 1.25	f(1.25)=-0.797	Check sign	Root in [1.25,1.5]
4	brentq	Midpoint 1.375	f(1.375)=-0.123	Check sign	Root in [1.375,1.5]
5	brentq	Midpoint 1.4375	f(1.4375)=0.388	Check sign	Root in [1.375,1.4375]
6	brentq	Midpoint 1.40625	f(1.40625)=0.132	Check sign	Root in [1.375,1.40625]
7	brentq	Midpoint 1.390625	f(1.390625)=0.004	Check sign	Root in [1.375,1.390625]
8	brentq	Midpoint 1.3828125	f(1.3828125)=-0.059	Check sign	Root in [1.3828125,1.390625]
9	brentq	Midpoint 1.38671875	f(1.38671875)=-0.027	Check sign	Root in [1.38671875,1.390625]
10	brentq	Midpoint 1.388671875	f(1.388671875)=-0.011	Check sign	Root in [1.388671875,1.390625]
11	brentq	Midpoint 1.3896484375	f(1.3896484375)=-0.003	Check sign	Root in [1.3896484375,1.390625]
12	brentq	Midpoint 1.39013671875	f(1.39013671875)=0.0005	Check sign	Root in [1.3896484375,1.39013671875]
13	brentq	Midpoint 1.389892578125	f(1.389892578125)=-0.001	Check sign	Root in [1.389892578125,1.39013671875]
14	brentq	Midpoint 1.3900146484375	f(1.3900146484375)=-0.0003	Check sign	Root in [1.3900146484375,1.39013671875]
15	brentq	Midpoint 1.39007568359375	f(1.39007568359375)=0.0001	Check sign	Root in [1.3900146484375,1.39007568359375]
16	brentq	Midpoint 1.390045166015625	f(1.390045166015625)=-0.0001	Check sign	Root in [1.390045166015625,1.39007568359375]
17	brentq	Midpoint 1.3900604248046875	f(1.3900604248046875]=0.0	Check sign	Root found
18	brentq	Return root	f(1.3900604248046875)=0	Root found	Root approx 1.39006

💡 root stops when function value is close to zero; brentq stops when interval is narrowed to root

Variable Tracker

Variable	Start	After 1	After 2	After 3	After 4	After 5	Final
root_guess	1.5	1.7	1.4	1.521	1.5214	1.5214	1.5214
brentq_interval_low	1	1	1.25	1.375	1.375	1.3896484375	1.390045166015625
brentq_interval_high	2	1.5	1.5	1.5	1.4375	1.39013671875	1.39007568359375
function_value	f(1.5)=1.875	f(1.7)=3.213	f(1.4)=0.744	f(1.521)=0.045	f(1.5214)=0.0001	f(1.5214)=~0	f(1.5214)=~0

Key Moments - 3 Insights

Why does the root method try different guesses instead of just returning the first one?

Because the first guess usually does not make the function zero. The method tries new guesses to get closer to the root, as shown in execution_table rows 1-5.

Why does brentq need an interval where the function changes sign?

Brentq relies on the function values at interval ends having opposite signs to guarantee a root inside. This is shown in execution_table row 6 where f(1) and f(2) have opposite signs.

How do we know when the root finding stops?

The process stops when the function value at the guess is very close to zero or the interval is very small, as shown in execution_table rows 5 and 17-18.

Visual Quiz - 3 Questions

Look at the execution_table for the root method. What is the function value at the final root guess?

A Approximately 0

B Approximately 1.5

C Approximately 3.2

D Approximately -2

💡 Hint

Check the 'Function eval' column at step 6 for root method.

At which step does brentq first detect a sign change in the function values?

A Step 1

B Step 6

C Step 2

D Step 10

💡 Hint

Look at the 'Action' column for brentq method in execution_table row 6.

If the initial interval for brentq was [2, 3] instead of [1, 2], what would happen?

A The method would find the root faster

B The method would fail because no sign change in [2,3]

C The method would return 2 immediately

D The method would ignore the interval and use initial guess

💡 Hint

Brentq requires a sign change in the interval to work, see execution_table row 6.

Concept Snapshot

Root finding finds x where f(x)=0.
root() uses initial guess and iterative solver.
brentq() needs interval with sign change.
Both stop when f(x) approx 0.
Use root for general cases, brentq for bracketed roots.

Full Transcript

This lesson shows how root finding works using scipy's root and brentq methods. We start with a function and either an initial guess or an interval. The root method tries guesses and checks function values until it finds a root where f(x) is near zero. Brentq requires an interval where the function changes sign, then narrows the interval by checking midpoints until it finds the root. The execution table traces each step, showing guesses, function values, and decisions. Key moments clarify why guesses change, why sign changes matter, and when the process stops. The visual quiz tests understanding of function values at steps, sign change detection, and interval requirements. The snapshot summarizes usage and behavior of root and brentq methods.