Concept Architecture
Concept
Theoretically, Fixed-Point Iteration is a numerical root-finding method that solves equations by repeatedly applying a transformation until successive approximations converge to a fixed point. It is founded on fixed-point theory and numerical analysis and is based on the principle that a solution to an equation can often be expressed as a value satisfying (x = g(x)). The method provides a simple iterative framework for solving non-linear equations when direct analytical solutions are unavailable.
Mathematically, Fixed-Point Iteration begins with an initial estimate and repeatedly evaluates a transformation function to generate increasingly accurate approximations. Convergence depends on the mathematical properties of the transformation, particularly that the magnitude of its derivative remains less than one in the neighbourhood of the fixed point. Under this condition, the sequence converges to the desired solution.
In practice, Fixed-Point Iteration is widely applied in health economics for solving non-linear equations arising in pharmacokinetic models, calibration of decision models, Markov equilibrium calculations and optimisation procedures. Although generally slower than Newton?Raphson methods, it offers computational simplicity and forms the basis of many iterative numerical algorithms.
Purpose
Used to obtain numerical solutions to non-linear equations, support iterative model calibration, solve equilibrium problems and enable computational optimisation in health economic modelling.
Mathematical Formulae
Primary Formula
Iteration Formula
x??? = g(x?)
where:
- x? = current approximation
- x??? = updated approximation
- g(x) = transformation function
Supporting Formulae
Fixed-Point Condition
x = g(x)
Convergence Criterion
|g?(x)| < 1
Stopping Criterion
|x??? ? x?| < �
where:
- � = numerical tolerance
Related Mathematical Methods
- Bisection Method
- Newton?Raphson Method
- Secant Method
- Root-finding algorithms
- Numerical optimisation
- Contraction Mapping Theorem
Example
Solve
x = cos(x)
using Fixed-Point Iteration.
Initial estimate
x? = 1
Iteration 1
x? = cos(1)
� 0.5403
Iteration 2
x? = cos(0.5403)
� 0.8576
Further iterations gradually converge to
x � 0.739085
which satisfies
x = cos(x)
within the desired numerical tolerance.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| COS | =COS(A2) | Perform one Fixed-Point Iteration step. |
| ABS | =ABS(B2-A2) | Calculate the difference between successive approximations. |
| IF | =IF(ABS(B2-A2)<1E-6,"Converged","Continue") | Evaluate the convergence criterion. |
| ROW | =ROW()-1 | Track iteration number. |
| Solver | Refine solutions after iterative convergence. | Support calibration and optimisation in health economic models. |
VBA (Optional)
Automate Fixed-Point Iteration, monitor convergence criteria, terminate iterations at predefined tolerances and solve non-linear equations arising in health economic simulation and calibration models.
Sources
- Burden RL, Faires JD. Numerical Analysis.
- Atkinson KE. An Introduction to Numerical Analysis.
- Stoer J, Bulirsch R. Introduction to Numerical Analysis.
- Ortega JM, Rheinboldt WC. Iterative Solution of Nonlinear Equations in Several Variables.
- Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical Recipes: The Art of Scientific Computing.
- 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 fixed-point iteration?
An iterative numerical method that repeatedly applies a function until successive estimates converge to a stable solution.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
How does fixed-point iteration find a solution?
Fixed-point iteration finds a solution by repeatedly applying a function to its own output until successive estimates converge to a stable value. Starting from an initial guess, each application produces a new estimate, and the process continues until the estimates settle. This repeated application of a function until the estimates stabilise is how fixed-point iteration reaches a solution Provided the function is chosen so that repetition draws estimates inward, the method converges on a value it leaves unchanged, which is the fixed point the iteration is named for.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
What does fixed-point iteration repeatedly apply?
Fixed-point iteration repeatedly applies a function, feeding each estimate back into the same function to produce the next, until successive estimates converge to a stable solution. The function is chosen so that its repeated application draws the estimates towards the answer. This repeated application of a single function is the defining action of fixed-point iteration Provided the function is chosen so that repetition draws estimates inward, the method converges on a value it leaves unchanged, which is the fixed point the iteration is named for.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
When does fixed-point iteration stop?
Fixed-point iteration stops when successive estimates converge to a stable solution, that is, when repeated application of the function no longer changes the estimate appreciably. At that point the value is a fixed point of the function, unchanged by further application. Reaching this stable, converged estimate is the stopping condition for fixed-point iteration Provided the function is chosen so that repetition draws estimates inward, the method converges on a value it leaves unchanged, which is the fixed point the iteration is named for.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
What is a fixed point in fixed-point iteration?
A fixed point in fixed-point iteration is a value left unchanged when the function is applied to it, which is why the estimates settle there once they converge. Because fixed-point iteration repeatedly applies a function until successive estimates converge to a stable solution, that stable solution is a fixed point. Converging on such a point is the goal of the method Provided the function is chosen so that repetition draws estimates inward, the method converges on a value it leaves unchanged, which is the fixed point the iteration is named for.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
How does fixed-point iteration relate to an iterative method?
Fixed-point iteration is an iterative numerical method that repeatedly applies a function until successive estimates converge to a stable solution, while an iterative method more generally is a computational approach that repeatedly applies the same calculation until a specified accuracy or convergence is achieved. Fixed-point iteration is therefore a specific iterative method whose repeated calculation is the application of a function. The two are connected as the general approach and a particular instance of it Provided the function is chosen so that repetition draws estimates inward, the method converges on a value it leaves unchanged, which is the fixed point the iteration is named for.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
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British health economist
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