12.2 Determining an Inverse Function Accurately by Iterating the Linear Approximation

How?

Given a pair of numbers, (x₀, f(x₀)) the linear approximation, fLx₀ to f defined at x₀, allows us to compute fLx₀(x) as an approximation to f(x).

If we know the inverse function to f, we can compute f ^-1(fLx₀(x)) and that gives us a new pair of numbers, (f ^-1(fLx₀(x)), fLx₀(x)) that we can call (x₁, f(x₁)) and repeat (or iterate) this operation to produce x₂, then x₃, ... , until it converges.

This in the old days was so horribly boring a procedure that it could not be inflicted on students. Now it is duck soup for a spreadsheet, and can be set up and computed in a matter of minutes for all the inverse functions we encounter: which are the roots (x^1/j), the (natural) logarithm, arcsin and arctan.

What has to be done?

The linear approximation fLx_j defined at x_j evaluated at x is given by

fLx_j(x) = f(x_j) + f '(x_j) (x - x_j)

setting this up, setting

x_j+1 = f ^-1 (fLx_j(x))

and iterating is all that need be done.

Exercises:

12.1 Set up a general root finding spreadsheet so that you can input x and j and it will spit out the j-th root of x by using this method where the machine only computes integer powers. (Clues on how to do this are in the next section.)

12.2 Set up a spreadsheet to find ln x using the ability of the machine to compute exp x.

12.3 Do the same for the inverse functions to the sine and tangent. These are generally written as something like arcsine and arctangent or asin or atan or something in between.

12.4 Can this method fail? If so how?