Concept Architecture
Concept
Theoretically, Polynomial Approximation is a mathematical technique that represents a function by a polynomial in order to approximate its behaviour over a specified interval. It is founded on approximation theory and numerical analysis and exploits the fact that polynomials are computationally efficient, differentiable and integrable. Polynomial approximation provides the basis for many numerical algorithms used in scientific computing, optimisation and health economic modelling.
Mathematically, Polynomial Approximation constructs a polynomial of finite degree that minimises approximation error according to a specified criterion, such as interpolation or least squares. Common approaches include Taylor polynomials, Chebyshev approximations and least squares polynomial fitting. The approximation error generally decreases as the polynomial degree increases, although excessively high-degree polynomials may exhibit oscillatory behaviour.
In practice, Polynomial Approximation is used in health economics to approximate complex cost functions, survival curves, utility functions and pharmacoeconomic models. It supports numerical integration, optimisation, calibration and simulation where exact analytical solutions are unavailable or computationally expensive.
Purpose
Used to approximate complex mathematical functions, improve computational efficiency, support numerical methods and enable estimation, optimisation and simulation in health economic modelling.
Mathematical Formulae
Primary Formula
Polynomial Approximation
P?(x) = a? + a?x + a?x� + ? + a?x�
where:
- P?(x) = polynomial approximation of degree n
- a?, a?, ?, a? = polynomial coefficients
Supporting Formulae
Approximation Error
E(x) = f(x) ? P?(x)
Taylor Polynomial
P?(x) = �???� [f???(a) � k!] (x ? a)?
Least Squares Criterion
RSS = �(y? ? P?(x?))�
Related Mathematical Methods
- Taylor series
- Least squares approximation
- Chebyshev approximation
- Interpolation
- Numerical integration
- Numerical optimisation
Example
Suppose a quadratic polynomial is used to approximate healthcare expenditure as a function of patient age:
P?(x) = 500 + 12x + 0.15x�
For a patient aged 60 years:
P?(60)
= 500 + (12 ? 60) + (0.15 ? 60�)
= 500 + 720 + 540
= �1,760
The quadratic polynomial therefore estimates healthcare expenditure of �1,760 for a patient aged 60 years.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| LINEST | =LINEST(B2:B101,A2:C101,TRUE,TRUE) | Estimate polynomial regression coefficients. |
| POWER | =POWER(A2,2) | Generate polynomial terms. |
| TREND | =TREND(B2:B101,A2:C101,A102:C102) | Predict values using the fitted polynomial. |
| SUMXMY2 | =SUMXMY2(B2:B101,C2:C101) | Calculate the residual sum of squares. |
| RSQ | =RSQ(B2:B101,C2:C101) | Assess goodness of fit of the polynomial approximation. |
VBA (Optional)
Automate polynomial curve fitting, estimation of polynomial coefficients, approximation of complex health economic functions and generation of prediction and calibration reports.
Sources
- Cheney EW, Kincaid D. Numerical Mathematics and Computing.
- 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.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
Related Concepts (4)
Library
Publications
1
Accuracy and Stability of Numerical Algorithms — Nicholas J. Higham, 2nd Edition ed., 2002 (Society for Industrial and Applied Mathematics (SIAM))
The standard reference on floating-point error, conditioning, stability and the numerical reliability of matrix, polynomial and linear-algebra algorithms.
BookView source →
Frequently Asked Questions (6)
What is polynomial approximation?
The representation of a function or dataset using a polynomial that provides a sufficiently accurate estimate over a specified interval.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
What does polynomial approximation represent a function with?
Polynomial approximation represents a function or dataset with a polynomial that provides a sufficiently accurate estimate over a specified interval. The polynomial stands in for the original, matching it closely enough across the interval of interest. Using a polynomial as this stand-in is what polynomial approximation does Because polynomials are easy to evaluate and manipulate, a polynomial that stays close enough over the interval of interest makes a convenient stand-in for a more awkward function.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
How close must a polynomial approximation be?
A polynomial approximation must be sufficiently accurate over the specified interval, meaning the polynomial estimate stays close enough to the function or dataset across that range for the purpose at hand. It need not match exactly, only well enough over the interval. This criterion of sufficient accuracy over a specified interval defines a polynomial approximation Because polynomials are easy to evaluate and manipulate, a polynomial that stays close enough over the interval of interest makes a convenient stand-in for a more awkward function.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
Over what range does polynomial approximation apply?
Polynomial approximation applies over a specified interval, the range across which the polynomial is required to provide a sufficiently accurate estimate of the function or dataset. Accuracy is judged over this interval rather than everywhere. Being tied to a specified interval is part of what polynomial approximation means Because polynomials are easy to evaluate and manipulate, a polynomial that stays close enough over the interval of interest makes a convenient stand-in for a more awkward function.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
Why use a polynomial in polynomial approximation?
A polynomial is used in polynomial approximation because polynomials are simple to evaluate and manipulate while still able to represent a function or dataset to sufficient accuracy over a specified interval. This makes the polynomial a convenient replacement for a more awkward function. The tractability of polynomials is why they are chosen for the approximation Because polynomials are easy to evaluate and manipulate, a polynomial that stays close enough over the interval of interest makes a convenient stand-in for a more awkward function.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
How does polynomial approximation relate to polynomial interpolation?
Polynomial approximation represents a function or dataset with a polynomial that is sufficiently accurate over a specified interval, while polynomial interpolation constructs a polynomial that passes exactly through a specified set of data points. Approximation seeks closeness across an interval, whereas interpolation demands exact agreement at the chosen points. The two are connected as polynomial methods differing in whether they fit closely or exactly Because polynomials are easy to evaluate and manipulate, a polynomial that stays close enough over the interval of interest makes a convenient stand-in for a more awkward function.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 26 May 2026
Content version: 1.0.0
Canonical Identity
- Term code
- CS-NA-AP-002
Stable URI · Machine-readable · Resolvable · CC BY 4.0