Secant Method
Newton-Raphson without the derivative — superlinear convergence using two starting points.
Try Secant — No Sign-in NeededWhat is the Secant Method?
The Secant method is a root-finding algorithm that achieves near-Newton-Raphson speed without requiring the derivative of f(x). Instead of computing f′(xₙ) analytically, it approximates the derivative using the slope of the line (the secant line) connecting the two most recent iterates. This finite-difference approximation replaces the tangent line of Newton-Raphson with a chord, making the method applicable to functions whose derivatives are unavailable, expensive to compute, or simply inconvenient to express.
The method was known to ancient Babylonian mathematicians as a tool for approximating division, and was later formalized in the numerical analysis literature. It is sometimes called the chord method or the two-point method.
The Secant method's convergence order is approximately 1.618 — the golden ratio φ. This places it between Newton-Raphson (order 2) and Bisection (order 1). In practice, it typically needs 20–40% more iterations than Newton-Raphson for the same accuracy, but saves the cost and complexity of deriving or coding the derivative. For many real-world problems, this is an excellent trade-off.
The Formula
The derivative f′(xₙ) is replaced by the finite-difference approximation [f(xₙ) − f(xₙ₋₁)] / [xₙ − xₙ₋₁], the slope of the secant line through the last two points.
Step-by-Step Algorithm
- 1
Choose two starting points x₀ and x₁ (they need not bracket the root).
- 2
Evaluate f(x₀) and f(x₁).
- 3
Compute x₂ = x₁ − f(x₁) · (x₁ − x₀) / (f(x₁) − f(x₀)).
- 4
Check convergence: if |f(x₂)| or |x₂ − x₁| is below tolerance, return x₂.
- 5
Set x₀ ← x₁, x₁ ← x₂ and repeat from step 3.
Convergence & Behaviour
The Secant method's convergence order of φ ≈ 1.618 means the error satisfies eₙ₊₁ ≈ C · eₙ^φ. This is slower than the quadratic convergence of Newton-Raphson but much faster than the linear convergence of Bisection. Like Newton-Raphson, the method can fail if the secant slope becomes nearly zero (i.e., f(xₙ) ≈ f(xₙ₋₁)) — a division-by-zero situation. It can also diverge if both starting points are far from the root. Unlike Bisection, there is no bracket guarantee, so it is not unconditionally convergent.
Use the Secant method when f′(x) cannot be computed analytically, when symbolic differentiation is impractical (e.g., the function comes from a simulation or experimental data), or when you want faster-than-bisection convergence without the overhead of automatic differentiation.
Avoid it when f(xₙ) ≈ f(xₙ₋₁) for likely starting points (near-flat function — division will blow up), or when you need a guaranteed result (use Bisection first to bracket, then switch to Secant).
Worked Example
Find the root of f(x) = x³ − 2x − 5 using x₀ = 2, x₁ = 3.
- 1.f(2) = 8 − 4 − 5 = −1; f(3) = 27 − 6 − 5 = 16
- 2.x₂ = 3 − 16 · (3 − 2)/(16 − (−1)) = 3 − 16/17 ≈ 2.0588
- 3.f(2.0588) ≈ −0.3794
- 4.x₃ = 2.0588 − (−0.3794)(2.0588 − 3)/(−0.3794 − 16) ≈ 2.0813
- 5.Continuing: x₄ ≈ 2.0946, x₅ ≈ 2.0945515 (converged)
Result: Root at x ≈ 2.09455 in 5 iterations. No derivative was needed.
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