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Mathematics · University

Numerical approximation and conditioning

Truncation error comes from approximation; round-off comes from finite-precision arithmetic.

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Prerequisites

Calculus and algebra.

Study notes

Error sources

Truncation error comes from approximation; round-off comes from finite-precision arithmetic.

Conditioning

A problem can amplify input perturbations even when the algorithm is carefully implemented.

Newton iteration

Use x_new=x−f(x)/f′(x) only when the derivative is nonzero and the iteration is meaningful.

Convergence claims

An asymptotic convergence order requires hypotheses and sufficiently close iterates, not just one successful step.

Worked examples

Perform one Newton step for x²−2=0 from x₀=1.

x₁=1−(1−2)/2=1.5.

1.5

Use one trapezoid over [0,2] to approximate the integral of x².

The interval width is 2 and the average endpoint height is (0+4)/2=2.

4

Check your understanding

  1. For f(x)=x² on [0,2], is the one-panel trapezoid estimate above or below the exact integral?

    1. Not a numerical value
    2. Above
    3. Always equal
    4. Below

  2. Explain catastrophic cancellation using subtraction of nearly equal approximate values.

  3. An approximation is 2.01 and the reference value is 2. Find absolute error.

  4. Which property describes sensitivity inherent in the problem rather than just the chosen algorithm?

    1. Source code indentation
    2. Conditioning
    3. Iteration count alone
    4. Font size

Answers and marking guidance

Answers and marking notes
  1. Above

    The exact value is 8/3; the chord lies above the convex graph.

  2. Leading digits cancel, leaving the small difference dependent on comparatively uncertain trailing digits. Relative error in the result can become large.

    Credit a loss of relative accuracy and the role of nearly equal inputs.

  3. 0.01

    Absolute error is abs(2.01−2).

  4. Conditioning

    Conditioning measures how outputs respond to small changes in data.

Further reading and review references

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