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November 23

Relationship between a function's fixed points and roots?

Is there any connection between the zeros of a function and its fixed points? It almost seems as if there must be. (However to be quite honest I can't even articulate why I would think such a thing in the first place!) Earl of Arundel (talk) 17:06, 23 November 2021 (UTC)

For a given function, no. The fixed points of a function $f$ are exactly the zeroes of $x\mapsto x-f(x)$, and the zeroes of a function $g$ are exactly the fixed points of $x\mapsto x-g(x)$. So you can turn any "find a zero" problem into a "find a fixed point" problem and vice versa. —Kusma (talk) 17:43, 23 November 2021 (UTC)

Ah, well of course. Thanks! Earl of Arundel (talk) 18:16, 23 November 2021 (UTC)

Here is a connection that is more a curiosity than anything deep. Given a function $f$ defined on the reals and a real number $x$, consider the following three propositions:

    (A)    $x$ is a fixed point of $f$;

    (B)    $x$ is a zero of $f$;

    (C)    $x=0$.

If any two among these three propositions hold, so too does the third.  --Lambiam 22:36, 23 November 2021 (UTC)

Interesting! Which trivially implies that any given polynomial function $f(x)$ lacking any sort of constant term must also therefore have at least one fixed point at $f(0)=0$. It is a rather simple relationship as you say still pretty elegant... Earl of Arundel (talk) 00:22, 24 November 2021 (UTC)

Searching for a root that way is called fixed point iteration fwiw. 2601:648:8202:350:0:0:0:69F6 (talk) 08:13, 24 November 2021 (UTC)

Let $u$ be any function such that $u(0)=0$. Given function $g$, define $f$ by $f(x)=x+u(g(x)).$ Then a zero of $g$ is a fixed point of $f$ (but the converse is not necessarily true). This is a more general version of the schema given above by Kusma, which corresponds to the choice $u(x)=-x.$ The larger generality can sometimes be used to achieve convergence in fixed-point iteration where the choice $u(x)=-x$ would diverge. See also Cobweb plot.  --Lambiam 10:02, 24 November 2021 (UTC)

Neat! It's such a nice result. Does this theorem have a name? Here the function $u(x)=2x^{2}-x$ (blue) is used to construct a synthetic fixed point with $f(x)=x+u(g(x))$ (orance) precisely at the real root of $g(x)=12x^{2}+x-1$ (green), which in this case just happens to be 1/4. Earl of Arundel (talk) 00:30, 25 November 2021 (UTC)