iterationschemelispracketsicp

# Using fixed point to show square root

In going through the exercises of SICP, it defines a fixed-point as a function that satisfies the equation F(x)=x. And iterating to find where the function stops changing, for example `F(F(F(x)))`.

The thing I don't understand is how a square root of, say, 9 has anything to do with that.

For example, if I have `F(x) = sqrt(9)`, obviously x=3. Yet, how does that relate to doing:

``````F(F(F(x))) --> sqrt(sqrt(sqrt(9)))
``````

Which I believe just converges to zero:

``````>>> math.sqrt(math.sqrt(math.sqrt(math.sqrt(math.sqrt(math.sqrt(9))))))
1.0349277670798647
``````

Since F(x) = sqrt(x) when x=1. In other words, how does finding the square root of a constant have anything to do with finding fixed points of functions?

Solution

• When calculating the square-root of a number, say `a`, you essentially have an equation of the form `x^2 - a = 0`. That is, to find the square-root of `a`, you have to find an `x` such that `x^2 = a` or `x^2 - a = 0` -- call the latter equation as (1). The form given in (1) is an equation which is of the form `g(x) = 0`, where `g(x) := x^2 - a`.

To use the fixed-point method for calculating the roots of this equation, you have to make some subtle modifications to the existing equation and bring it to the form `f(x) = x`. One way to do this is to rewrite (1) as x = a/x -- call it (2). Now in (2), you have obtained the form required for solving an equation by the fixed-point method: f(x) is a/x.

Observe that this method requires both sides of the equation to have an 'x' term; an equation of the form `sqrt(a) = x` doesn't meet the specification and hence can't be solved (iteratively) using the fixed-point method.

The thing I don't understand is how a square root of, say, 9 has anything to do with that.

For example, if I have F(x) = sqrt(9), obviously x=3. Yet, how does that relate to doing: F(F(F(x))) --> sqrt(sqrt(sqrt(9)))

These are standard methods for numerical calculation of roots of non-linear equations, quite a complex topic on its own and one which is usually covered in Engineering courses. So don't worry if you don't get the "hang of it", the authors probably felt it was a good example of iterative problem solving.