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

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November 23

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Relationship between a function's fixed points and roots?

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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)[reply]

For a given function, no. The fixed points of a function   are exactly the zeroes of  , and the zeroes of a function   are exactly the fixed points of  . 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)[reply]
Ah, well of course. Thanks! Earl of Arundel (talk) 18:16, 23 November 2021 (UTC)[reply]
Here is a connection that is more a curiosity than anything deep. Given a function   defined on the reals and a real number  , consider the following three propositions:
    (A)      is a fixed point of  ;
    (B)      is a zero of  ;
    (C)     .
If any two among these three propositions hold, so too does the third.  --Lambiam 22:36, 23 November 2021 (UTC)[reply]
Interesting! Which trivially implies that any given polynomial function   lacking any sort of constant term must also therefore have at least one fixed point at  . It is a rather simple relationship as you say still pretty elegant... Earl of Arundel (talk) 00:22, 24 November 2021 (UTC)[reply]

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)[reply]

Let   be any function such that  . Given function  , define   by   Then a zero of   is a fixed point of   (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   The larger generality can sometimes be used to achieve convergence in fixed-point iteration where the choice   would diverge. See also Cobweb plot.  --Lambiam 10:02, 24 November 2021 (UTC)[reply]
Neat! It's such a nice result. Does this theorem have a name? Here the function   (blue) is used to construct a synthetic fixed point with   (orance) precisely at the real root of   (green), which in this case just happens to be 1/4. Earl of Arundel (talk) 00:30, 25 November 2021 (UTC)[reply]