Secant Method

What happens if we abandon the requirement to bracket the root? Geometrically we show the situation in Fig 8.3. The points A and B are the original ones bracketing the root, and the point C is derived form the chord AB as before. We have shown, in D the next iteration using the reguli falsi noting that the other end of the ``bracket'', namely A isn't contributing much to the convergence. However the most recent two points, namely B and C are fairly close to the required root, and if this pair is used instead of the bracketing pair AC in the next step, we move to E which looks as though it is a very significant improvement. This is the secant method; we still use a chord on the curve to estimate the next approximation to the root, but now the chord is drawn between the last two points on the curve, rather than the most recent bracketing pair.

This leads to the following algorithm

• let p_n = a_n - f (a)$${\frac{{a_{n+1}-a_n}}{{f(a_{n+1}) - f(a_n)}}}$$;
• if p_n is a good enough approximation to the root, stop;
• if continuing, let a_n be a_n+1 and let a_n+1 = p_n.

Note that, just as before, we need two points to start the process. There is no requirement that successive approximations bracket the root, but obtaining such a pair initially is at least evidence that we are near a root!

Although we have derived this method in a fairly ad hoc way, it turns out that the secant method is one of the best methods to use for ``difficult' roots providing we can abandon the guarantee of the bisection method. We discuss this further after describing one more method, which is essentially the secant method taken to the limit in the sense of the calculus.

Figure 8.3: The Secant Method: extrapolate using the last two points found on the curve.

Figure 8.4: Newtons Method: the tangent helps find where the curve cuts the axis.