UNIT 1 • FINDING ROOTS OF EQUATIONS

The Secant Method
Step-by-Step Friendly Guide

The Secant Method is an open root-finding method that approximates derivatives using two recent function evaluations — making it faster than Bisection with no need for analytical derivatives. It achieves super-linear convergence (order ≈ 1.618, the golden ratio).

HOW THE SECANT METHOD WORKSSECANT LINE APPROXIMATION

x_{n+1} = x_n - f(x_n) \cdot \frac{x_n - x_{n-1}}{f(x_n) - f(x_{n-1})}

1. Two Initial GuessesChoose

x_0

and

x_1

— no bracketing required (unlike Bisection).

2. Draw a Secant LineApproximate the derivative by a secant through

(x_{n-1}, f(x_{n-1}))

and

(x_n, f(x_n))

3. Find x-InterceptWhere the secant crosses the x-axis becomes the new estimate

x_{n+1}

4. Repeat Until ConvergenceSlide the window forward: old

x_1

becomes new

x_0

, new estimate becomes

x_1

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM TO SOLVE

Find the root of $f(x) = x^3 - 2x - 5 = 0$ using the Secant Method with $x_0 = 2$ , $x_1 = 3$ , tolerance = 0.0001.

STEP 01 • EVALUATE INITIAL POINTS

Calculate $f(x_0)$ and $f(x_1)$ to set up the first secant line:

f(x_0) = f(2) = 2^3 - 2(2) - 5 = 8 - 4 - 5 = -1

f(x_1) = f(3) = 3^3 - 2(3) - 5 = 27 - 6 - 5 = 16

STEP 02 • APPLY THE SECANT FORMULA

Plug into the secant formula to find the first new estimate $x_2$ :

x_2 = x_1 - f(x_1) \cdot \frac{x_1 - x_0}{f(x_1) - f(x_0)}

x_2 = 3 - 16 \cdot \frac{3 - 2}{16 - (-1)} = 3 - 16 \cdot \frac{1}{17} \approx 2.0588

STEP 03 • SLIDE THE WINDOW FORWARD

Update the pair: the old $x_1 = 3$ becomes the new $x_0$ , and $x_2 \approx 2.0588$ becomes the new $x_1$ . Repeat until $|f(x_{n+1})| < \epsilon$ .

New pair:

x_0 = 3,\; x_1 = 2.0588

→ Continue iterating

STEP 04 • ITERATION SUMMARY TABLE

k	x₀	x₁	x₂ (New)	f(x₀)	f(x₁)	Ea (%)
1	2.000000	3.000000	2.058824	-1.000000	16.000000	2.857%
2	3.000000	2.058824	2.094568	16.000000	-0.351...	1.705%
3	2.058824	2.094568	2.094552	-0.351...	-0.002...	0.001%

FINAL CONCLUSION

Root Converged: x ≈ 2.094552

f(2.094552) = (2.094552)^3 - 2(2.094552) - 5 \approx 0

The Secant Method converged in just 3 iterations, compared to ~14 for Bisection on the same problem. This is the power of the $\phi$ -order (≈1.618) convergence rate — each step dramatically improves accuracy! Note: unlike Newton-Raphson, no derivative $f'(x)$ was required.

LABORATORY

Interactive Live Calculator & Visualizer

ROOT FINDER LABORATORY

Secant Method Interactive Engine

PreviousFalse Position Method

Unit 1•3 of 34

NextIteration Method

∑

f(x)∫dxπ

NETZ Engine

Evaluating numerical algorithm

UNIT 1 • FINDING ROOTS OF EQUATIONS

The Secant Method
Step-by-Step Friendly Guide

HOW THE SECANT METHOD WORKSSECANT LINE APPROXIMATION

x_{n+1} = x_n - f(x_n) \cdot \frac{x_n - x_{n-1}}{f(x_n) - f(x_{n-1})}

1. Two Initial GuessesChoose

x_0

and

x_1

— no bracketing required (unlike Bisection).

2. Draw a Secant LineApproximate the derivative by a secant through

(x_{n-1}, f(x_{n-1}))

and

(x_n, f(x_n))

3. Find x-InterceptWhere the secant crosses the x-axis becomes the new estimate

x_{n+1}

4. Repeat Until ConvergenceSlide the window forward: old

x_1

becomes new

x_0

, new estimate becomes

x_1

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM TO SOLVE

Find the root of $f(x) = x^3 - 2x - 5 = 0$ using the Secant Method with $x_0 = 2$ , $x_1 = 3$ , tolerance = 0.0001.

STEP 01 • EVALUATE INITIAL POINTS

Calculate $f(x_0)$ and $f(x_1)$ to set up the first secant line:

f(x_0) = f(2) = 2^3 - 2(2) - 5 = 8 - 4 - 5 = -1

f(x_1) = f(3) = 3^3 - 2(3) - 5 = 27 - 6 - 5 = 16

STEP 02 • APPLY THE SECANT FORMULA

Plug into the secant formula to find the first new estimate $x_2$ :

x_2 = x_1 - f(x_1) \cdot \frac{x_1 - x_0}{f(x_1) - f(x_0)}

x_2 = 3 - 16 \cdot \frac{3 - 2}{16 - (-1)} = 3 - 16 \cdot \frac{1}{17} \approx 2.0588

STEP 03 • SLIDE THE WINDOW FORWARD

Update the pair: the old $x_1 = 3$ becomes the new $x_0$ , and $x_2 \approx 2.0588$ becomes the new $x_1$ . Repeat until $|f(x_{n+1})| < \epsilon$ .

New pair:

x_0 = 3,\; x_1 = 2.0588

→ Continue iterating

STEP 04 • ITERATION SUMMARY TABLE

k	x₀	x₁	x₂ (New)	f(x₀)	f(x₁)	Ea (%)
1	2.000000	3.000000	2.058824	-1.000000	16.000000	2.857%
2	3.000000	2.058824	2.094568	16.000000	-0.351...	1.705%
3	2.058824	2.094568	2.094552	-0.351...	-0.002...	0.001%

FINAL CONCLUSION

Root Converged: x ≈ 2.094552

f(2.094552) = (2.094552)^3 - 2(2.094552) - 5 \approx 0

LABORATORY

Interactive Live Calculator & Visualizer

ROOT FINDER LABORATORY

Secant Method Interactive Engine

PreviousFalse Position Method

Unit 1•3 of 34

NextIteration Method

The Secant Method Step-by-Step Friendly Guide

Step-by-Step Worked Example

Root Converged: x ≈ 2.094552

Interactive Live Calculator & Visualizer

Secant Method Interactive Engine

NETZ Engine

The Secant Method Step-by-Step Friendly Guide

Step-by-Step Worked Example

Root Converged: x ≈ 2.094552

Interactive Live Calculator & Visualizer

Secant Method Interactive Engine

The Secant Method
Step-by-Step Friendly Guide

The Secant Method
Step-by-Step Friendly Guide