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Cancellation
Subtracting nearly equal numbers can destroy significant digits. Rewrite formulas before evaluating them numerically.
Stable algebra
Choose algebraic forms that avoid subtractive cancellation, overflow, underflow, and unnecessary loss of significance.
Residual warning
A small residual is useful, but it may hide a large forward error when the condition number is large.
Backward error
A small backward error means the computed answer is exact for a nearby problem. This is often the most reliable stability interpretation.
Absolute vs relative error
Absolute error measures size in the original units; relative error measures error compared with the scale of the exact value.
Floating-point model
Floating-point arithmetic replaces real arithmetic by operations in a finite set. Separate the mathematical problem from its floating-point realization.
Conditioning
Conditioning belongs to the problem. Stability belongs to the algorithm. A stable algorithm cannot fix an ill-conditioned problem.
Unit roundoff
The unit roundoff \(u\) controls the local model \(\operatorname{fl}(x\circ y)=(x\circ y)(1+\delta)\), \(|\delta|\le u\), when no overflow/underflow occurs.