Concept Architecture
Concept
Theoretically, Polynomial Interpolation is a numerical method that constructs a polynomial passing exactly through a given set of distinct data points. It is founded on approximation theory, algebra and numerical analysis and provides a continuous representation of discrete observations. For n + 1 data points with distinct independent-variable values, there exists a unique interpolating polynomial of degree no greater than n.
Mathematically, Polynomial Interpolation may be expressed in Lagrange, Newton divided-difference or monomial form. The interpolating polynomial satisfies P(x?) = y? at every observed point. Its approximation error depends on the degree of the polynomial, the placement of interpolation nodes and the derivatives of the underlying function. High-degree interpolation with equally spaced nodes may produce substantial oscillation, known as Runge?s phenomenon.
In practice, Polynomial Interpolation is used in health economics to estimate intermediate costs, utilities, survival probabilities and model parameters from discrete observations. It supports calibration, numerical integration, simulation and approximation of non-linear relationships, although spline methods are often preferred when many data points are present because they provide greater numerical stability.
Purpose
Used to construct a continuous polynomial through observed data points, estimate intermediate values, approximate non-linear functions and support numerical modelling in health economic analysis.
Mathematical Formulae
Primary Formula
Lagrange Interpolating Polynomial
P?(x) = �???� y?L?(x)
where:
L?(x) = ????,???� (x ? x?) � (x? ? x?)
and:
- x? = interpolation nodes
- y? = observed values
- L?(x) = Lagrange basis polynomial
Supporting Formulae
Interpolation Condition
P?(x?) = y?
Newton Interpolating Polynomial
P?(x) = f[x?] + f[x?,x?](x ? x?) + ? + f[x?,?,x?]????�??(x ? x?)
Interpolation Error
f(x) ? P?(x) = [f?�???(?) � (n + 1)!]????�(x ? x?)
where ? lies within the interval containing the interpolation nodes and x.
Related Mathematical Methods
- Lagrange interpolation
- Newton divided differences
- Linear interpolation
- Cubic spline
- Polynomial approximation
- Chebyshev nodes
Example
Suppose annual healthcare costs are observed at three ages:
- Age 40: �1,200
- Age 50: �1,800
- Age 60: �3,000
A quadratic interpolating polynomial passing through these three points is:
P?(x) = 3x� ? 258x + 6,720
At age 55:
P?(55)
= 3(55�) ? 258(55) + 6,720
= 1,200?
Using the polynomial correctly:
= 9,075 ? 14,190 + 6,720
= �1,605
The interpolating polynomial therefore estimates healthcare costs at age 55 from the exact curve defined by the three observed points.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| LINEST | =LINEST(B2:B4,A2:C4,TRUE,TRUE) | Estimate polynomial coefficients from observed data. |
| POWER | =POWER(A2,2) | Generate quadratic interpolation terms. |
| SUMPRODUCT | =SUMPRODUCT($F$2:$F$4,G2:I2) | Evaluate an interpolating polynomial from coefficients and powers. |
| PRODUCT | =PRODUCT((E2-$A$2:$A$4)/(A2-$A$2:$A$4)) | Construct Lagrange basis components with appropriate exclusions. |
| IF | =IF(AND(E2>=MIN(A2:A4),E2<=MAX(A2:A4)),"Interpolation","Extrapolation") | Determine whether the estimated value lies within the observed range. |
VBA (Optional)
Automate construction and evaluation of Lagrange or Newton interpolating polynomials, calculate divided-difference tables and estimate intermediate health economic parameters from discrete observations.
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.
- Cheney EW, Kincaid D. Numerical Mathematics and Computing.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
Related Concepts (5)
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 interpolation?
The construction of a polynomial that passes exactly through a specified set of data points to estimate intermediate values.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
What does polynomial interpolation construct?
Polynomial interpolation constructs a polynomial that passes exactly through a specified set of data points, then uses it to estimate intermediate values. The polynomial is fitted so that it agrees with the data at every given point. Building this exact-fitting polynomial to read off values between the points is what polynomial interpolation does By fixing a polynomial to agree exactly at the chosen points, the method reads off values in between, though a single high-degree polynomial through many points can oscillate between them.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
How does the polynomial in polynomial interpolation fit the data points?
The polynomial in polynomial interpolation fits the data points exactly, passing through each specified point rather than merely near it. This exact agreement at the given points is the defining requirement. Constructing a polynomial that matches every data point precisely is central to polynomial interpolation By fixing a polynomial to agree exactly at the chosen points, the method reads off values in between, though a single high-degree polynomial through many points can oscillate between them.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
What does polynomial interpolation estimate?
Polynomial interpolation estimates intermediate values, using a polynomial that passes exactly through the specified data points to give values between them. Once the polynomial is fixed by the points, it can be evaluated anywhere in the range. Estimating these in-between values is the purpose of polynomial interpolation By fixing a polynomial to agree exactly at the chosen points, the method reads off values in between, though a single high-degree polynomial through many points can oscillate between them.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
What determines the polynomial in polynomial interpolation?
The specified set of data points determines the polynomial in polynomial interpolation, since it is constructed to pass exactly through them. The points fix the polynomial that agrees with the data and is then used to estimate intermediate values. This determination by the given data points defines how polynomial interpolation builds its polynomial By fixing a polynomial to agree exactly at the chosen points, the method reads off values in between, though a single high-degree polynomial through many points can oscillate between them.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
How does polynomial interpolation relate to a cubic spline?
Polynomial interpolation constructs a single polynomial passing exactly through a set of data points, while a cubic spline fits smooth cubic pieces between the points and keeps the function and its derivatives continuous. A cubic spline interpolates piecewise with low-degree polynomials rather than one global polynomial. The two are connected as interpolation methods differing in whether they use one polynomial or many joined pieces By fixing a polynomial to agree exactly at the chosen points, the method reads off values in between, though a single high-degree polynomial through many points can oscillate between them.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
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British health economist
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Verification date: 26 May 2026
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