Concept Architecture
Concept
Theoretically, a Numerical Method is a mathematical procedure that approximates the solution to problems for which an exact analytical solution is unavailable, impractical or computationally inefficient. Numerical methods form the foundation of scientific computing and computational mathematics by replacing exact symbolic solutions with systematically computed approximations whose accuracy can be quantified. In health economics, numerical methods underpin decision modelling, optimisation, simulation, differential equation solving, parameter estimation and probabilistic analysis.
Mathematically, numerical methods generate a sequence of approximations that converge towards the true solution of a mathematical problem. Depending on the problem, they employ iterative algorithms, discretisation schemes, interpolation, numerical integration or optimisation procedures. The quality of a numerical method is evaluated through convergence, stability, consistency and computational efficiency, while approximation errors are characterised using recognised error measures.
In practice, numerical methods are implemented within decision-analytic models, Markov models, microsimulation, survival extrapolation, Bayesian estimation and optimisation procedures used throughout health economic evaluation. Modern software packages and spreadsheet models employ numerical algorithms to solve equations, estimate parameters, evaluate integrals and simulate complex healthcare systems where analytical solutions are not feasible.
Purpose
Used to obtain accurate approximate solutions to mathematical problems that cannot be solved analytically, enabling computation, simulation, optimisation and statistical estimation in health economic models.
Mathematical Formulae
Primary Formula
x??? = F(x?)
where successive approximations converge towards the desired solution.
Supporting Formulae
Absolute error:
E = |x? ? x|
Relative error:
E? = |x? ? x| / |x|
Convergence criterion:
|x??? ? x?| < �
Root-finding residual:
|f(x?)| < �
Related Mathematical Methods
- Newton?Raphson Method
- Bisection Method
- Euler Method
- Runge?Kutta Methods
- Finite Difference Method
- Finite Element Method
- Numerical Integration
- Optimisation Algorithms
Example
A health economic model requires the solution of a nonlinear equation describing equilibrium treatment uptake.
The iterative algorithm produces:
x? = 0.620
x? = 0.648
x? = 0.651
x? = 0.6512
With a convergence tolerance of � = 0.0001:
|0.6512 ? 0.6510| = 0.0002
The algorithm continues until the convergence criterion is satisfied, after which the estimated value is used throughout the economic model.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| Solver | Minimise or maximise an objective function | Numerical optimisation of model parameters |
| Goal Seek | Set model output to a target value | Solve nonlinear equations within decision models |
| ABS | =ABS(B3-B2) | Assess convergence between successive iterations |
| SUMPRODUCT | =SUMPRODUCT(A2:A100,B2:B100) | Numerical calculations in simulation and optimisation |
| IF | =IF(ABS(B3-B2)<0.0001,"Converged","Continue") | Apply iterative stopping criteria |
VBA (Optional)
A VBA procedure can automate iterative numerical algorithms until predefined convergence criteria are satisfied and return the computed solution to the health economic model.
Sources
- Burden RL, Faires JD. Numerical Analysis.
- Quarteroni A, Sacco R, Saleri F. Numerical Mathematics.
- Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical Recipes.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
- Heath MT. Scientific Computing: An Introductory Survey.
Related Concepts (8)
Library
Publications
1
Parameter Estimation and Uncertainty: A Report of the ISPOR-SMDM Modeling Good Research Practices Task Force-6 — Briggs, Weinstein, Fenwick, Karnon, Sculpher & Paltiel, Task Force Report 6 ed., 2012 (Value in Health / Medical Decision Making)
Best-practice guidance on parameter estimation and the characterisation of uncertainty in decision models, covering probabilistic sensitivity analysis, distributional choices, and correlation between parameters.
Journal ArticleView source →
Frequently Asked Questions (6)
What is a numerical method?
A computational technique for obtaining an approximate solution to a mathematical problem when an exact analytical solution is unavailable or impractical.
Source: Press et al. 2007
Why does a numerical method give an approximate rather than exact answer?
A numerical method is used when a problem has no formula that yields an exact answer, or when such a formula is too hard to evaluate, so the solution is approached by computation instead. Because it works by successive approximation or by sampling rather than by solving exactly, its result is close to the true value but not identical, carrying a small error that can usually be made smaller with more effort. The approximation is deliberate, the price of solving problems that resist exact treatment. Press and colleagues (2007) describe such methods.
Source: Press et al. 2007
Why are numerical methods used?
Numerical methods are used because many mathematical problems arising in modelling have no closed-form analytical solution, or one too complex to obtain, so an approximate computational solution is the practical alternative. They allow integrals, differential equations, and optimisations that cannot be solved exactly to be evaluated to a chosen accuracy. As models grow complex, analytical solutions become unavailable, and numerical methods provide the means to obtain results, making them indispensable tools wherever exact solutions are out of reach but approximate answers suffice.
Source: Briggs, Claxton & Sculpher 2006
What are examples of numerical methods?
Examples of numerical methods include numerical integration, such as quadrature or Monte Carlo integration, for evaluating integrals; root-finding methods, such as the Newton-Raphson method, for solving equations; iterative optimisation methods for finding maxima or minima; and methods for solving differential equations. They also include matrix computations and simulation. Each provides an approximate solution to a class of problems that may lack a tractable analytical form, and the choice depends on the problem type, the accuracy required, and the computational cost.
Source: Press et al. 2007
How is the accuracy of a numerical method assessed?
The accuracy of a numerical method is assessed by how close its approximate solution is to the true value, characterised by error that depends on the method and its settings, such as the step size or the number of iterations or samples. Reducing the step size or increasing the samples generally improves accuracy at greater computational cost. Convergence is checked by seeing whether the result stabilises as the method is refined. Understanding the error behaviour allows the settings to be chosen so the approximation meets the required precision.
Source: Press et al. 2007
What are the limitations of numerical methods?
Numerical methods give approximate rather than exact solutions, carrying error that must be controlled, and improving accuracy generally increases computational cost, so there is a trade-off between precision and effort. Some methods can be unstable or converge slowly for certain problems, and results can be sensitive to settings such as step size or starting values. Rounding in finite-precision arithmetic adds further error. These limitations mean numerical methods are applied with attention to their error, stability, and cost, and their results checked for convergence and plausibility.
Source: Press et al. 2007
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 28 Oct 2025
Content version: 1.0.0
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