The Secant method is just a variation on the Newton method. You may recall that Newton's
method was derived from use of the Taylor series expansion, beginning with an equation in the
All iterative solution methods must begin with some guess x0 for the value of x that solves the
equation. The derivation of the solution method begins with an application of a Taylor Series
expansion of the function about the point x0.
We continue by using this expanded equation to find the x such that f(x)=0. To accomplish this
we make the assumption that x0 is close enough to the final solution value of x that (x-x0)2, (x-x0)3,
and higher powers of this difference are small enough to ignore. This leads to the equation:
At this point we need to be honest, realize that x in the above equation is not the true solution of
our original equation, and replace it with x1 to designate the next approximation to the solution.
Solving this equation for x1 we get:
We now proceed with the most recent approximation as the next guess to improve the solution iterating through application of the following equation until changes in successive approximations to the solution are small enough.
All the Secant method does is to approximate the derivative term in the Newton method with results from previous iterations.
If you look at this formula carefully you will realize that you need not one, but two initial guesses
to get this started. Make the first guess just as you would for a Newton iteration, and if you have
no better information, make the second guess close to the first (e.g. x1=.999 x0)
The Secant Method does not generalize well to multiple equations, but the basic idea of creating numerical derivatives can be carried through in such cases, requiring extra evaluations of equations near the latest guess at the solution values.