Which of the following is a characteristic of the secant method?

Question

BlackTom AI · Accepted Answer

The secant method approximates the derivative by using the slope of a chord through two recent iterates, so the derivative is not required in closed form.

1) Relevant knowledge points and concepts
- Newton’s method update: x_{n+1} = x_n − f(x_n)/f'(x_n).
- The secant method replaces the unknown exact derivative f'(x_n) with a finite-difference approximation using two previous points: (f(x_n) − f(x_{n−1}))/(x_n − x_{n−1}).
- The secant update becomes x_{n+1} = x_n − f(x_n) * (x_n − x_{n−1}) / (f(x_n) − f(x_{n−1})).
- The method requires two initial guesses and has superlinear (≈1.618) convergence under appropriate conditions.

2) Step-by-step reasoning and derivation
- Start from Newton: x_{n+1} = x_n − f(x_n)/f'(x_n).
- Approximate f'(x_n) by the chord slope between x_{n} and x_{n−1}: f'(x_n) ≈ [f(x_n) − f(x_{n−1})]/[x_n − x_{n−1}].
- Substitute into Newton’s step:
  x_{n+1} = x_n − f(x_n) / ( [f(x_n) − f(x_{n−1})]/[x_n − x_{n−1}] )
  = x_n − f(x_n) * (x_n − x_{n−1}) / (f(x_n) − f(x_{n−1})).
- This shows explicitly that the method uses a chord (secant line) slope for the derivative approximation.

3) Intermediate steps shown above (algebraic substitution and simplification).

4) Why the correct answer is correct
- Option b states “Derivative is approximated using a chord,” which matches the finite-difference substitution and the algebraic secant update. The secant method is precisely Newton’s method with the derivative replaced by the chord slope.

5) Why the other options are incorrect
- a. “Exact derivative of the function is required” — false: secant avoids needing the exact derivative (that is Newton’s requirement).
- c. “One initial guess and a perturbation is required” — false: secant requires two initial approximations x_0 and x_1 (not a single guess plus perturbation).
- d. “Midpoint is calculated to determine the root estimate” — false: computing a midpoint is characteristic of the bisection method, not the secant method.

Conclusion: The correct characteristic is b. Derivative is approximated using a chord, which matches the provided answer.

Which of the following is a characteristic of the secant method?单项选择题

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