An Introduction To Numerical Analysis (4): The Secant Method

 The secant method of numerical approximation basically is a further refinement of the Newton- Raphson method. We begin by taking the secant line joining   ( x_o ,  f (x_o) ) and  ( x₁ , f (x₁)) for the curve shown below.  

Then we can write by the point-slope formula familiar to anyone who's taken Algebra II:

y= [f(x₁) -  f (x_o)]  (x -  x_o) / [ x₁ -  x_o ]     +   f (x_o)

And the root of this linear function is the value of x such that y = 0, or:

x =   x₁ -  f(x₁) { x₁ -  x_o / f(x₁)  -    f(x_o)}

This new value x is then used as  x₂  to repeat the process, i.e. using x₁  and x₂ instead of  x_o  and x₁.   The iteration process is then continued on, e.g. for x₃ , x₄  etc. until sufficiently high precision is achieved.   This yields the general formula:

x _n   =   x _n-1   -  f(x _n-1) { x _n-1   -  x _n-2 / f(x _n-1)  - f(x _n-2) }

Suggested Problems:

Use the secant method to find an approximation to:

x ³  - x  - 1   =  0

Compare it with what you obtained using Newton's method