Concept Architecture
Concept
Theoretically, the Secant Method is an iterative numerical algorithm for finding the roots of non-linear equations without requiring explicit evaluation of derivatives. It is founded on numerical analysis and approximates the derivative used in the Newton?Raphson Method by the slope of a secant line passing through two successive approximations. The method combines the simplicity of derivative-free algorithms with a convergence rate that is generally faster than the Bisection Method and Fixed-Point Iteration.
Mathematically, the Secant Method constructs successive approximations by replacing the derivative in Newton?Raphson's iteration with a finite difference approximation. Starting from two initial estimates, each new approximation is obtained from the intersection of the secant line with the x-axis. Under appropriate conditions, the method exhibits superlinear convergence, although convergence is not guaranteed unless the initial estimates are sufficiently close to the true root.
In practice, the Secant Method is widely applied in health economics to solve non-linear equations arising in optimisation, calibration of decision models, pharmacokinetic modelling and numerical estimation. It is particularly valuable when derivatives are unavailable, computationally expensive or difficult to evaluate analytically.
Purpose
Used to determine numerical roots of non-linear equations without derivative calculations, support optimisation and calibration procedures and enable efficient numerical computation in health economic modelling.
Mathematical Formulae
Primary Formula
Iteration Formula
x??? = x? ? f(x?)(x? ? x???) � [f(x?) ? f(x???)]
where:
- x???, x? = current approximations
- x??? = updated approximation
Supporting Formulae
Secant Slope
m = [f(x?) ? f(x???)] � (x? ? x???)
Stopping Criterion
|x??? ? x?| < �
or
|f(x???)| < �
where:
- � = numerical tolerance
Related Mathematical Methods
- Newton?Raphson Method
- Bisection Method
- Fixed-Point Iteration
- Root-finding algorithms
- Numerical optimisation
- Finite difference approximation
Example
Solve
f(x) = x� ? 2
using the Secant Method.
Initial estimates
x? = 1
x? = 2
First iteration
x?
= 2 ? (2� ? 2)(2 ? 1) � [(2� ? 2) ? (1� ? 2)]
= 2 ? (2 ? 1) � (2 ? (?1))
= 1.333333
Subsequent iterations converge rapidly to
�2
� 1.414214
The Secant Method therefore estimates the numerical root without evaluating the derivative.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| Formula | =B2-(C2*(B2-A2))/(C2-D2) | Perform one Secant Method iteration. |
| ABS | =ABS(B3-B2) | Calculate the difference between successive approximations. |
| IF | =IF(ABS(B3-B2)<1E-8,"Converged","Continue") | Evaluate the convergence criterion. |
| POWER | =POWER(A2,2) | Evaluate polynomial functions during iterations. |
| Solver | Refine solutions after Secant Method convergence. | Support calibration and optimisation in health economic models. |
VBA (Optional)
Automate Secant Method iterations, monitor convergence criteria, solve non-linear equations and generate numerical solutions for optimisation, calibration and simulation within health economic models.
Sources
- Burden RL, Faires JD. Numerical Analysis.
- Atkinson KE. An Introduction to Numerical Analysis.
- Stoer J, Bulirsch R. Introduction to Numerical Analysis.
- Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical Recipes: The Art of Scientific Computing.
- Ortega JM, Rheinboldt WC. Iterative Solution of Nonlinear Equations in Several Variables.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
Related Concepts (5)
Library
Publications
1
Numerical Analysis of Parameterized Nonlinear Equations — Werner C. Rheinboldt, 1st Edition ed., 1986 (John Wiley & Sons)
A specialist numerical-analysis reference on iterative solution, continuation and computation for parameterized nonlinear systems, linked here to the ontology's numerical-method and optimization branch.
BookView source →
Frequently Asked Questions (6)
What is the secant method?
An iterative root-finding method that approximates the derivative using two previous estimates to locate a function's root.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
How does the secant method locate a root?
The secant method locates a root by approximating the derivative from two previous estimates and using that approximation to step towards the root, iterating as it goes. Each new estimate comes from the line through the last two points. This use of two previous estimates to approximate the derivative and advance is how the secant method finds a root By forming a slope from the last two estimates instead of an exact derivative, the method finds a root without needing derivative information, which is useful when that information is hard to obtain.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
What does the secant method use to approximate the derivative?
The secant method uses two previous estimates to approximate the derivative, forming a slope from the pair of points rather than computing the derivative exactly. This approximate slope then guides the next step towards the root. Approximating the derivative from two previous estimates is the defining feature of the secant method By forming a slope from the last two estimates instead of an exact derivative, the method finds a root without needing derivative information, which is useful when that information is hard to obtain.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
Why does the secant method avoid computing the derivative directly?
The secant method avoids computing the derivative directly by approximating it from two previous estimates, which is useful when the exact derivative is unavailable or costly to evaluate. Using the two most recent points to form a slope replaces the need for an explicit derivative. This reliance on an approximate derivative is central to the secant method By forming a slope from the last two estimates instead of an exact derivative, the method finds a root without needing derivative information, which is useful when that information is hard to obtain.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
What kind of method is the secant method?
The secant method is an iterative root-finding method, refining its estimate step by step to locate a function's root while approximating the derivative from two previous estimates. Each iteration improves on the last using the latest pair of points. Being an iterative root-finding method that approximates the derivative is what the secant method is By forming a slope from the last two estimates instead of an exact derivative, the method finds a root without needing derivative information, which is useful when that information is hard to obtain.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
How does the secant method relate to the Newton-Raphson method?
The secant method approximates the derivative using two previous estimates to locate a root, while the Newton-Raphson method uses the actual derivative of the function to do so. The secant method replaces Newton-Raphson's exact derivative with an approximation from two points. The two are connected as iterative root-finding methods differing in whether the derivative is computed exactly or approximated By forming a slope from the last two estimates instead of an exact derivative, the method finds a root without needing derivative information, which is useful when that information is hard to obtain.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
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British health economist
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Verification date: 16 Jun 2026
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