Secant Method

A root-finding algorithm that uses a succession of roots of secant lines to better approximate a root of a function. The secant method can be thought of as a finite-difference approximation of Newton's method. It does not require the calculation of derivatives, but it requires two initial guesses and has a slower convergence rate than Newton's method.

Historical Context

The Secant method is one of the oldest known root-finding algorithms, predating Newton's method by over 3000 years. It was used in a rudimentary form by Babylonian mathematicians to approximate roots.

In modern numerical analysis, it serves as an excellent alternative to Newton's method when the analytical derivative of the function is unknown, too costly to compute, or cannot be evaluated cleanly. By keeping track of the last two evaluated points, it effectively constructs a secant line to approximate the slope that Newton's method would obtain via the true tangent line.

Though its convergence rate of $\approx 1.618$ is lower than Newton's quadratic $2$ , it only requires one new function evaluation per iteration (as opposed to evaluating both $f(x)$ and $f'(x)$ ), which often makes it faster in practice for complex functions.

Geometric Foundation: The Secant Line

Guess 1 (x₀)1.20

Guess 2 (x₁)1.80

Secant Intersection

By calculating the secant line between

x_0

and

x_1

, we can follow its slope down to the x-axis to find

x_2

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

f(x) = x^3 - 2x - 5

Geometric View

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

Convergence Plot

Each step reduces the error. On a logarithmic scale, convergence patterns form straight lines.

Target Function

Guess 1 (x₀)

Guess 2 (x₁)

Tolerance (ε)

Speed

○ Iter 0. Ready.

Iter	x_n-1	x_n	f(x_n-1)	f(x_n)	x_n+1	Error

Error Bound |x_n+1 - x_n|

Current State

No active state.

Algorithm

while |x_n+1 - x_n| > ε:if f(x_n) == f(x_n-1): breakx_n+1 = x_n - f(x_n) * (x_n - x_n-1) / (f(x_n) - f(x_n-1))x_n-1 = x_nx_n = x_n+1return x_n

Why it works

The Secant method is a root-finding algorithm that uses a succession of roots of secant lines to better approximate a root of a function $f$ . It can be thought of as a finite-difference approximation of Newton's method. Instead of requiring the derivative of the function, which may be difficult or impossible to compute, it uses two initial points to estimate the tangent line with a secant line.

Mathematically, we approximate the derivative using the finite difference from the last two points:

f'(x_n) \approx \frac{f(x_n) - f(x_{n-1})}{x_n - x_{n-1}}

Substituting this into Newton's method iteration formula gives the Secant iteration formula:

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

Where the Secant Method Fails

Like Newton's Method, the Secant Method can fail if the function behaves poorly or if the initial guesses are far from the root.

Division by Zero: If the function values at the two points are identical ( $f(x_n) = f(x_{n-1})$ ), the secant line is horizontal and never intersects the x-axis.
Runaway (Divergence): Like Newton's method, it can be thrown very far away if a secant line is nearly horizontal.
Convergence Rate: The secant method has a superlinear convergence rate (with order approximately $\phi \approx 1.618$ ), making it faster than bisection but slightly slower than the quadratic rate of Newton's method.

Newton's method, but without the derivative. Can you feel the difference?

The secant line approximates the tangent. One fewer evaluation per step — and yet it converges at the golden ratio. Here's why that trade-off matters.

Output

Output will appear here after running the code

Output

Output will appear here after running the code

Need something guaranteed to converge?Bisection Method →