← Main book chapter 1
Contents / index
Browser back
← Previous chapter
Next chapter →
Laboratory
Memory
Cards
Sort
Match
Visuals
Coding
Quiz
Memory Cards
Memory Cards — Chapter 1
Each card contains a compact summary of what must be remembered from the chapter.
Shuffle on page
Reveal all
Hide all
Go to game cards
Stable algebra
Choose algebraic forms that avoid subtractive cancellation, overflow, underflow, and unnecessary loss of significance.
Absolute vs relative error
Absolute error measures size in the original units; relative error measures error compared with the scale of the exact value.
Cancellation
Subtracting nearly equal numbers can destroy significant digits. Rewrite formulas before evaluating them numerically.
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.
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.
Conditioning
Conditioning belongs to the problem. Stability belongs to the algorithm. A stable algorithm cannot fix an ill-conditioned problem.
Floating-point model
Floating-point arithmetic replaces real arithmetic by operations in a finite set. Separate the mathematical problem from its floating-point realization.