# Numerical approximation and conditioning

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

Course: [Differential equations and numerical methods](https://studybrix.com/en/courses/differential-equations/index.md)
Level: university
Prerequisites: Calculus and algebra.

## 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.

## Practice questions

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

Answer: 1.5

Explanation: x₁=1−(1−2)/2=1.5.

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

Answer: 4

Explanation: 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?

A. Not a numerical value
B. Above
C. Always equal
D. Below

Answer: Above

Explanation: 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 / marking guidance: Leading digits cancel, leaving the small difference dependent on comparatively uncertain trailing digits. Relative error in the result can become large.

Explanation: 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.

Answer: 0.01

Explanation: Absolute error is abs(2.01−2).

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

A. Source code indentation
B. Conditioning
C. Iteration count alone
D. Font size

Answer: Conditioning

Explanation: 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 / 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.

Explanation: 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?

Answer: 4

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

## Further reading

- https://ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/
- https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/

Original AI-assisted educational draft; not independently teacher-reviewed. References support further study, not copied exercises, affiliation or curriculum accreditation.