Conditioning, Stability and Floating-Point Error · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Sensitive problems and honest algorithms

Mathematics · Calculus & Analysis · ages 23-24
Name ______________________   Date ____________

  1. A stable algorithm on an ill-conditioned problem can still give a poor answer.

    Circle one:   True   False

  2. Which subtraction risks catastrophic cancellation?

    • 1.0001 - 1.0000 with 4-digit arithmetic
    • 2.5 - 1.5
    • 100 + 200
    • 0.1 + 0.2

  3. A problem is ill-conditioned but the algorithm used is stable. A student says the answer can still be poor. Is that right?

    Circle one:   True   False

  4. A matrix has norm 4 and its inverse has norm 0.5. What is its condition number?

    Answer: ______________

  5. For f(x) = sqrt(x), what is the relative condition number |x f'(x)/f(x)| at x = 4?

    • 2
    • 4
    • 1
    • 0.5

  6. A backward stable algorithm runs on a well-conditioned problem. A student expects an accurate answer. Is that right?

    Circle one:   True   False

  7. A computation returns 0.0001 where the true value is 0.0002. What is the relative error?

    Answer: ______________

  8. Adding a tiny number to a huge float changes nothing. Why?

    • The processor skips small additions
    • The huge value leaves no digits for the tiny part
    • The tiny number is always exactly zero

  9. Pat doubles working precision but keeps the cancelling formulas. Damage barely moves. What should she learn?

    • Precision alone cannot restore destroyed information; rework the algebra
    • She should halve precision instead
    • Cancellation only ever affects addition

  10. Which task is an ill-conditioned problem rather than a stable technique?

    • Adding numbers in increasing order
    • Solving a Hilbert system
    • Applying an orthogonal transformation
    • Backward substitution on a triangular system
LightMySky · lightmysky.comW1-mt_q8IoCElXIV-s1

Answer key

For grown-ups. Fold this page away before handing over the rest.

Sensitive problems and honest algorithms W1-mt_q8IoCElXIV-s1

  1. True · Stability controls the method, not the problem sensitivity.
  2. 1.0001 - 1.0000 with 4-digit arithmetic · The leading digits cancel and the result lives on rounding garbage, so 1.0001 - 1.0000 with 4-digit arithmetic is the dangerous one.
  3. True · Stability controls the algorithm's own errors, not the problem's sensitivity, so bad answers can survive.
  4. 2 · Multiply the two norms: 4 times 0.5 is 2.
  5. 0.5 · The relative condition |x f'/f| at x = 4 is 4 times (1/4) over 2 = 0.5.
  6. True · Backward stability plus a calm problem means the computed answer sits near the true one.
  7. 0.5 · The error 0.0001 against the true 0.0002 is a relative error of 0.5.
  8. The huge value leaves no digits for the tiny part · Fifteen digits cannot span the gap, so the tiny part rounds away.
  9. Precision alone cannot restore destroyed information; rework the algebra · Destroyed digits stay destroyed; only new algebra helps.
  10. Solving a Hilbert system · Hilbert matrices have astronomically large condition numbers, so solving a Hilbert system is the ill-conditioned problem.
Worksheet · LightMySky