Concept Architecture
Concept
Theoretically, Conditioning is a mathematical property that describes the sensitivity of the solution of a problem to small perturbations in its input data. It is a fundamental concept in numerical analysis because it distinguishes errors that arise from the mathematical problem itself from those introduced by the numerical algorithm used to solve it. A well-conditioned problem exhibits only small changes in its solution when the input changes slightly, whereas an ill-conditioned problem may produce large changes in the solution from very small input perturbations.
Mathematically, Conditioning is quantified using a condition number, which measures the relative amplification of input errors into output errors. The condition number depends only on the mathematical problem and not on the numerical method employed. Large condition numbers indicate ill-conditioned problems that are highly sensitive to measurement error, rounding error or uncertainty in the input data.
In practice, Conditioning is important in health economics because regression models, optimisation procedures, Markov models, simulation algorithms and matrix computations frequently involve numerical problems whose stability depends on conditioning. Assessing condition numbers enables analysts to identify unstable models, evaluate numerical reliability and interpret computational results appropriately.
Purpose
Used to quantify the sensitivity of mathematical problems to input uncertainty, evaluate numerical reliability, identify ill-conditioned problems and support robust implementation of health economic models.
Mathematical Formulae
Primary Formula
For a differentiable function,
?(x) = |(x � f(x)) ? f?(x)|
where:
- ?(x) = condition number
- f(x) = function value
- f?(x) = first derivative
Supporting Formulae
Matrix Condition Number
?(A) = ?A? ? ?A???
Relative Error Relationship
Relative Output Error � ? ? Relative Input Error
Related Mathematical Methods
- Condition number
- Numerical stability
- Matrix inversion
- Floating-point arithmetic
- Error propagation
- Sensitivity analysis
Example
Suppose a matrix has a condition number of
?(A) = 500
If the relative error in the input data is
0.001 (0.1%)
the approximate relative error in the computed solution is
500 ? 0.001
= 0.5
or 50%.
The problem is therefore ill-conditioned, indicating that even very small input errors may produce large errors in the computed solution.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| MMULT | =MMULT(A2:C4,D2:F4) | Perform matrix multiplication used in condition number calculations. |
| MINVERSE | =MINVERSE(A2:C4) | Calculate the matrix inverse. |
| MDETERM | =MDETERM(A2:C4) | Assess matrix properties associated with conditioning. |
| ABS | =ABS((B2-A2)/A2) | Calculate relative input or output error. |
| Solver | Evaluate numerical sensitivity in optimisation models. | Assess the effects of conditioning on health economic computations. |
VBA (Optional)
Automate estimation of matrix condition numbers, identify ill-conditioned numerical problems and generate diagnostics assessing computational reliability within health economic models.
Sources
- Higham NJ. Accuracy and Stability of Numerical Algorithms.
- Trefethen LN, Bau D. Numerical Linear Algebra.
- Golub GH, Van Loan CF. Matrix Computations.
- Burden RL, Faires JD. 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 (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 conditioning?
The sensitivity of a mathematical problem to small changes in its input data or numerical calculations.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
What does conditioning describe?
Conditioning describes the sensitivity of a mathematical problem to small changes in its input data or numerical calculations. A well-conditioned problem responds to small input changes with small changes in its solution, while a poorly conditioned one can amplify them greatly. Conditioning is therefore a property of the problem itself, indicating how much the solution can be disturbed by slight perturbations to the inputs Because it is a property of the problem rather than the method, poor conditioning limits the accuracy attainable by any algorithm, and recognising it helps separate unavoidable sensitivity from method-induced error.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
Why does conditioning matter in numerical computation?
Conditioning matters in numerical computation because it captures how sensitive a problem is to small changes in its input data or calculations, and finite-precision arithmetic inevitably introduces such small perturbations. When a problem is poorly conditioned, these tiny disturbances can produce large errors in the result regardless of the method used. Understanding conditioning therefore helps distinguish errors inherent to the problem from those introduced by an algorithm Because it is a property of the problem rather than the method, poor conditioning limits the accuracy attainable by any algorithm, and recognising it helps separate unavoidable sensitivity from method-induced error.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
Is conditioning a property of the problem or of the method?
Conditioning is a property of the mathematical problem, since it describes the sensitivity of that problem to small changes in its input data or numerical calculations, independent of how the problem is solved. A poorly conditioned problem is difficult to solve accurately by any method, because the sensitivity lies in the problem itself. This distinguishes conditioning from properties that depend on the particular algorithm chosen Because it is a property of the problem rather than the method, poor conditioning limits the accuracy attainable by any algorithm, and recognising it helps separate unavoidable sensitivity from method-induced error.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
What does good conditioning imply about a problem?
Good conditioning implies that small changes in a problem's input data or numerical calculations produce only small changes in its solution, since conditioning measures exactly that sensitivity. A well-conditioned problem can therefore be solved accurately even when its inputs carry slight uncertainty or rounding. This resilience to small perturbations is what makes good conditioning desirable in numerical computation Because it is a property of the problem rather than the method, poor conditioning limits the accuracy attainable by any algorithm, and recognising it helps separate unavoidable sensitivity from method-induced error.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
How does conditioning relate to numerical stability?
Conditioning is the sensitivity of a mathematical problem to small changes in its input, while numerical stability is the property of a numerical method that limits the growth of rounding and computational errors during calculations. Conditioning concerns the problem and numerical stability concerns the method. The two are connected in that an accurate result generally requires both a well-conditioned problem and a numerically stable method to solve it Because it is a property of the problem rather than the method, poor conditioning limits the accuracy attainable by any algorithm, and recognising it helps separate unavoidable sensitivity from method-induced error.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
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Verified by Dr Darrin Baines
British health economist
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Verification date: 23 Mar 2026
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