Numerical methodsLesson 4 of 4 in the course Root finding

Secant Method: Iteration, Convergence, and Implementation

5 flashcards · 5 quiz questions · 5 match pairs · 5 fill-in sentences · English

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Summary

The secant method approximates a root of a function by repeatedly finding where a line through two points on the function crosses the x-axis. It is useful when a root is needed and calculating the function's derivative is inconvenient.

A root of ff is a value rr for which f(r)=0f(r)=0. Begin with two estimates, x0x_0 and x1x_1, and evaluate the function at both. The points (x0,f(x0))(x_0,f(x_0)) and (x1,f(x1))(x_1,f(x_1)) determine a straight line, called a secant line. Its intersection with the x-axis is the next estimate of the root.

All flashcards

What is a secant line in root finding?
The straight line through two points (xa,f(xa))(x_a,f(x_a)) and (xb,f(xb))(x_b,f(x_b)) on a function's graph; its x-intercept supplies a new root estimate.
What is the secant method's goal?
To approximate a root rr satisfying f(r)=0f(r)=0 using successive secant-line x-intercepts.
What is the secant recurrence?
xn=xn−1−f(xn−1)(xn−1−xn−2)/[f(xn−1)−f(xn−2)]x_n=x_{n-1}-f(x_{n-1})(x_{n-1}-x_{n-2})/[f(x_{n-1})-f(x_{n-2})]. It computes an estimate from the previous two estimates and their function values.
Why does the secant method need two initial estimates?
Two function points are needed to determine the first secant line and its x-intercept.
What makes a secant iteration's formula undefined?
A zero denominator, f(xn−1)−f(xn−2)=0f(x_{n-1})-f(x_{n-2})=0, makes the recurrence undefined.

Quiz questions

  1. How does the secant method avoid explicitly evaluating a derivative?Answer: It approximates the derivative with a finite-difference slope from two recent estimates.
  2. Near which kind of root does the secant method have its stated local convergence behavior?Answer: A simple root of a sufficiently smooth function, with starting values sufficiently close.
  3. What distinguishes false position from the secant method?Answer: False position selects endpoints that preserve opposite function signs.
  4. What does a small change between successive secant estimates establish by itself?Answer: It is a practical stopping signal, but it does not prove the residual is small.
  5. Why is the correction form of the secant update generally preferred in approximate arithmetic?Answer: It expresses the update as the current estimate minus a correction proportional to the function value.

Based on the Wikipedia article Secant method, licensed CC BY-SA 4.0. Adapted into a summary, flashcards and quizzes.

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