What is the order of the polynomial that is applied for the Simpson's 3/8 rule?

Question

BlackTom AI · Accepted Answer

Simpson's 3/8 rule uses a cubic polynomial (degree 3) as its interpolant.

1) Relevant knowledge points and concepts
- Newton/Lagrange interpolation: an interpolating polynomial that passes through n+1 distinct points has degree at most n.
- Newton–Cotes rules: numerical integration formulas obtained by integrating an interpolating polynomial over the interval.
- Simpson's 3/8 is a closed Newton–Cotes rule that uses four equally spaced points, so the interpolant is degree 3.

2) Complete reasoning, step by step
- Simpson's 3/8 rule takes four nodes x0, x1, x2, x3 (three subintervals, equally spaced) and constructs the unique polynomial P3(x) of degree ≤3 that satisfies P3(xi) = f(xi) for i = 0..3.
- Integrate P3(x) over [x0, x3] to approximate ∫_{x0}^{x3} f(x) dx. Because P3 is cubic, the rule is exact for all polynomials up to degree 3.

3) Calculation and intermediate steps (formula)
- Let h = (x3 − x0)/3. The 3/8 rule result is
  ∫_{x0}^{x3} f(x) dx ≈ (3h/8) [ f(x0) + 3 f(x1) + 3 f(x2) + f(x3) ].
- This formula comes from integrating the Lagrange basis polynomials of degree 3; the presence of four function values confirms a cubic interpolant.

4) Why the correct answer is correct
- Because four points determine a polynomial of degree at most 3, Simpson's 3/8 rule is based on a cubic interpolating polynomial and is exact for polynomials of degree ≤3. The error term involves the fourth derivative, showing degree-3 exactness.

5) Why the other options are incorrect
- a. 2: Degree 2 corresponds to Simpson's 1/3 rule (uses three points), not the 3/8 rule.
- b. 1: Degree 1 corresponds to the trapezoidal rule (uses two points).
- d. 4: Degree 4 would require five points (closed Newton–Cotes with five nodes), not the four used in 3/8.

6) Conclusion
- Therefore the polynomial order for Simpson's 3/8 rule is 3, matching the provided answer c. 3.

What is the order of the polynomial that is applied for the Simpson's 3/8 rule?单项选择题

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