Measure how much the mathematical answer can change

Try

Suppose a measurement a=10.01 and a reference b=10.00 are subtracted. The answer is 0.01. If each input is uncertain by 0.005, the difference can change by as much as 0.01, the entire answer. Even exact arithmetic cannot remove this uncertainty. The problem itself is sensitive because it asks for a small difference between larger quantities. This is conditioning, a property of the mathematical task and its perturbation model.

Predict

Predict whether exact subtraction guarantees an accurate real-world difference. It does not: exact subtraction of uncertain inputs preserves their uncertainty. Conversely, a sensitive problem can have a stable implementation whose output is the exact answer for slightly perturbed data. Keep sensitive problem and unstable algorithm separate; otherwise a solver may be blamed for information the inputs never contained. Both absolute and relative errors are useful, but they answer different questions.

Experiment

Compare subtraction tasks with componentwise relative input changes at most 0.001.

Inputs	Difference	Relative amplification factor
10 and 2	8	12/8=1.5
10 and 9.9	0.1	19.9/0.1=199

Choose opposite signs for the input perturbations and calculate the output change. These are worst-case bounds: giving both inputs the same relative change can produce a much smaller effect. Record your perturbations alongside the answers.

Understand

For y=f(x), local absolute sensitivity is |f′(x)|. If x and f(x) are nonzero, relative sensitivity is κ(x)=|xf′(x)/f(x)|. Divide δy≈f′(x)δx by output magnitude and compare with relative input change. This is a first-order quantity, not an exact finite-perturbation guarantee. For multiple inputs, choose a normwise or componentwise model explicitly. Subtraction with componentwise relative errors bounded by ε has absolute error at most ε(|a|+|b|). The resulting relative factor is the one in the table. The perturbation model is part of the theorem, not an optional label.

Apply

Near a zero output, relative error may be undefined or huge while absolute error remains small. For f(x)=x−1, absolute sensitivity is one, but |x/(x−1)| diverges. For a simple root r, adding η to the function changes the root to first order by −η/f′(r). Multiple roots can be more sensitive: perturbing x²=0 to x²=η splits its root for positive η and removes real roots for negative η. For linear systems, singular values quantify directional sensitivity; κ₂(A)=σmax/σmin for invertible A. A zero smallest singular value means no full-space inverse exists.

A change of units can alter an absolute condition number without changing the physical problem. Relative quantities often remove that scale, but only where denominators are meaningful. When comparing solvers across different variables, specify whether errors are measured in meters, seconds, normalized coordinates or a weighted norm. A single unscaled Euclidean norm can hide a large error in a small-magnitude component.

Continue

Before comparing algorithms, state what inputs may change, how changes are measured and which output accuracy matters. Conditioning limits what accurate algorithms can promise. Later lessons combine condition estimates with residuals and backward errors. A small residual alone is not an accuracy certificate for a sensitive problem, and extra arithmetic digits cannot repair uncertainty already present in measurements. In an implementation report, distinguish assumed uncertainty from measured numerical error; neither can be inferred solely from the number of printed decimal places.

Review calculus and linear algebra when a prerequisite needs practice. Continue to Ask which nearby problem a computation solves.