Concept Architecture
Concept
Theoretically, Truncation Error is the numerical error that arises when an exact mathematical process is approximated by a finite representation or finite sequence of operations. It is a fundamental concept in numerical analysis and scientific computing, occurring whenever infinite series, continuous functions or exact differential equations are replaced by finite approximations. Unlike rounding error, which results from finite numerical precision, truncation error results from deliberately omitting higher-order terms or simplifying mathematical procedures.
Mathematically, Truncation Error is defined as the difference between the exact mathematical solution and the approximate solution produced by a numerical method. Its magnitude depends on the approximation technique, step size and order of the numerical algorithm. In many numerical methods, truncation error decreases according to a known order of convergence as the step size becomes smaller.
In practice, Truncation Error is important in health economics because numerical integration, differential equation models, Markov models, optimisation algorithms and simulation techniques all rely on finite approximations. Understanding truncation error enables analysts to select appropriate numerical methods and step sizes while ensuring that approximation errors remain negligible relative to decision-making thresholds.
Purpose
Used to quantify approximation error arising from finite numerical methods, evaluate algorithm accuracy, assess convergence and support reliable implementation of health economic and scientific computing models.
Mathematical Formulae
Primary Formula
Truncation Error = Exact Solution ? Approximate Solution
Supporting Formulae
Taylor Series Remainder
R?(x) = f?�???(?)(x ? a)�?? � (n + 1)!
where:
- ? lies between a and x
Order of Truncation Error
Error = O(h?)
where:
- h = step size
- p = order of the numerical method
Related Mathematical Methods
- Taylor series
- Euler method
- Runge?Kutta methods
- Finite difference methods
- Numerical integration
- Error propagation
Example
The exponential function is approximated using the first two terms of its Taylor series:
e? � 1 + x
For x = 0.1:
Exact value
e?�? = 1.105170918
Approximation
1 + 0.1 = 1.100000000
Truncation Error
= 1.105170918 ? 1.100000000
= 0.005170918
The approximation therefore introduces a Truncation Error of approximately 0.00517 because higher-order Taylor series terms have been omitted.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| EXP | =EXP(A2) | Calculate the exact exponential value. |
| Formula | =1+A2 | Calculate a first-order Taylor approximation. |
| ABS | =ABS(EXP(A2)-(1+A2)) | Calculate the Truncation Error. |
| POWER | =POWER(A2,2) | Calculate higher-order approximation terms. |
| IF | =IF(ABS(B2-C2)<0.001,"Acceptable","Reduce Step Size") | Assess whether numerical approximation error is acceptable. |
VBA (Optional)
Automate estimation of truncation error in numerical integration, differential equation solvers and health economic simulation models while evaluating convergence and selecting appropriate numerical step sizes.
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.
- Higham NJ. Accuracy and Stability of Numerical Algorithms.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
Related Concepts (6)
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 truncation error?
The error introduced by approximating a mathematical process using a finite number of terms or steps.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
What causes truncation error?
Truncation error is caused by approximating a mathematical process using a finite number of terms or steps, when the exact process would require infinitely many. Cutting the process short at a finite point leaves out the remainder, and that omission is the truncation error. This approximation by a finite number of terms or steps is what produces truncation error Because it stems from cutting a process short rather than from storing numbers, truncation error would remain even with exact arithmetic, and it typically shrinks as more terms or steps are kept.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
What is truncation error the result of?
Truncation error is the result of representing a mathematical process with a finite number of terms or steps rather than carrying it to completion. The part left out by stopping early is the source of the error. Being the error from this finite approximation of a process is what defines truncation error Because it stems from cutting a process short rather than from storing numbers, truncation error would remain even with exact arithmetic, and it typically shrinks as more terms or steps are kept.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
How can truncation error be reduced?
Truncation error can generally be reduced by using more terms or steps in the approximation, since it arises from approximating a mathematical process with a finite number of them. Including more of the process leaves less out and shrinks the error. This dependence on how many terms or steps are used is characteristic of truncation error Because it stems from cutting a process short rather than from storing numbers, truncation error would remain even with exact arithmetic, and it typically shrinks as more terms or steps are kept.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
Does truncation error come from rounding?
Truncation error does not come from rounding; it arises from approximating a mathematical process using a finite number of terms or steps, independent of how numbers are stored. It would occur even with exact arithmetic, because the process itself is cut short. This origin in finite approximation rather than rounding distinguishes truncation error Because it stems from cutting a process short rather than from storing numbers, truncation error would remain even with exact arithmetic, and it typically shrinks as more terms or steps are kept.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
How does truncation error relate to rounding error?
Truncation error is the error from approximating a mathematical process using a finite number of terms or steps, while rounding error is the difference introduced when a value is rounded to fit a limited number of digits. Truncation error comes from cutting a process short, whereas rounding error comes from representing numbers. The two are connected as distinct sources of numerical error in a computation Because it stems from cutting a process short rather than from storing numbers, truncation error would remain even with exact arithmetic, and it typically shrinks as more terms or steps are kept.
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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