Wikipedia:Reference desk/Archives/Mathematics/2021 November 23

= 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)