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Useful Computational Methods

### The Secant Method for Square-roots

The Secant Method, like the Regula Falsi method, starts with two initial values x1 and x2 near the root, except that they do not have to enclose the root. The next value is given by the same formula as in Regula Falsi:
xn+1 = (xnxn-1 + Q)/(xn + xn-1)    ----- Equation (1).
The next interval is [xn, xn+1], regardless of whether this interval encloses the root. The sequence of values {xn} converges to √Q. So the difference between Regula Falsi and the Secant Method is not only that the initial interval in the latter does not have to enclose the root, but the decision step in the latter consists in just discarding xn-1 and using [xn, xn+1] as the next interval. The advantge of this is that the successive values, xn, approach the root faster.

The speed of convergence is greater than linear, but less than quadratic. So the method is faster than the Bisection method and the Regula Falsi for finding square-roots.

Using the Secant Method to find the roots of functions
The Secant Method has wider applicability than just finding approximations to square-roots. Consider a continuous and continuously differentiable function f. All that this means for our purpose is that the graph of such a function can be approximated by a straight line segment over very short intervals. The Secant Method assumes more than this. It assumes the graph to be a straight line over the interval chosen initially and in those obtained thereafter. Thus it takes any two numbers x0 and x1 close to the root. Under the assumption that the graph is a straight line, it looks for the point (x2,0) where the straight line passing through the two points (x0,f(x0)) and (x1,f(x1)) crosses the x-axis. In reality, this straight line is a secant of the graph, and hence the name Secant Method. Now, the slope m of the straight line is calculated in two ways and equated thus:

The speed of convergence of this algorithm is approximately 1.62. That is, the error of xn+1 is a constant times the k-th power of the error of xn, where k = (1 + √5)/2 @ 1.618. A method for which k = 1 has linear convergence, whereas one with k = 2 is said to have quadratic convergence. The Secant Method is super-linear in the speed of convergence. That is, 1 < k < 2.

Since the Secant Method is super-linear in speed of convergence, why use the Bisection Method or the Regula Falsi Method. Each method has advantages. The latter two methods bracket the root so that convergence is guaranteed for all continuous functions near "simple" roots. This is not always the case with the Secant method. For instance, when a new xn falls outside the domain of the function, we can't get xn+1. An example is in finding the root of lnx. For finding square-roots, the Secant Method is superior.

Here are some animations of the Secant Method.

If we take the function f(x) = x2 - Q, Equation (2) gives Equation (1).

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