## The Secant Method

Instead of using the derivative of a functionto find the next iterate in the search for the root ofas in the Newton-Raphson method, we may to use the last two iterates,to find an estimate for

By definition,

We putto obtain

We then use this estimate into obtain

The Newton Raphson method needs an initial guess for the root,The secant method needs two initial guesses,and

Example: Solvestarting withand

0 | 1 | 2 | 3 | 4 | |

1.000000 | 1.500000 | 1.400000 | 1.413793 | 1.414216 | |

0.414214 | 0.085786 | 0.014214 | 0.000442 | 0.000002 |

For this example, the secant method requires one more iteration than Newton's method to approximatewith the same accuracy The method is slightly slower than the Newton Raphson method, but it does not require the evaluation of a derivative. It does need two initial points but these do not have to straddle the root.