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Mathematics · lesson 3 of 3

Numerical approximation and conditioning

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

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Learn the ideas

Prerequisites: Calculus and algebra.

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

Example 1

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

x₁=1−(1−2)/2=1.5.

1.5

Example 2

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

8 questions

Decimals or fractions are accepted for numeric answers. Proofs and written responses use sample answers and require human marking. This is practice, not a protected examination.

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

Show answer and explanation

1.5

x₁=1−(1−2)/2=1.5.

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

Show answer and explanation

4

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

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

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Above

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

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

Sample answer and marking guidance

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.

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

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0.01

Absolute error is abs(2.01−2).

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

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Conditioning

Conditioning measures how outputs respond to small changes in data.

7 Why can Newton's method fail when starting far from a root?

Sample answer and marking guidance

The tangent approximation can lead to a poor next point, a near-zero derivative, divergence or a cycle. Local convergence guarantees do not cover arbitrary starts.

Credit a concrete failure mechanism and the local nature of common guarantees.

8 A method has error about C h². By what factor should the error decrease when h is halved in its asymptotic regime?

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4

Replacing h by h/2 multiplies h² by 1/4.

Further reading and review references

These are reference links, not copied exercises, institutional endorsement or proof of independent review.