Secant method

Definition

The secant method is an iterative technique for finding the roots of a function by approximating the derivative. It uses two initial approximations to generate a sequence converging to the root.

5 Must Know Facts For Your Next Test

1. The secant method does not require the calculation of the derivative, unlike Newton's Method.
2. It uses the formula $x_{n+1} = x_n - f(x_n) \frac{x_n - x_{n-1}}{f(x_n) - f(x_{n-1})}$ to find successive approximations of the root.
3. Convergence of the secant method is generally slower than that of Newton's Method but faster than simple iteration methods.
4. The method can fail if $f(x_n)$ is equal to $f(x_{n-1})$, causing division by zero.
5. Choosing initial guesses that are close to the actual root improves convergence speed and accuracy.

Review Questions

• What are the main differences between the secant method and Newton's Method?
• Write down and explain the formula used in the secant method for updating successive approximations.
• Why might choosing initial guesses close to the actual root be important in using the secant method?

Related terms

Newton's Method: An iterative technique for finding roots of a function using tangent lines, requiring computation of derivatives. The update formula is $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$.

Root-Finding Algorithm: Any algorithm used to find solutions (roots) where a function equals zero.

Derivative: A measure of how a function changes as its input changes; it represents an instantaneous rate of change.