# Questions tagged [computable-analysis]

computability and complexity in real or complex analysis

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### Decide whether a polynomial has a root

Let $A$ be a ring such that all elements of $A$ are complex computable numbers. I'm interested in knowing whether the decision problem that asks, given $P\in A[X]$, if $P$ has a root in $A$ is ...
611 views

### What is the "continuity" as a term in computable analysis?

Background I once implemented a datatype representing arbitrary real numbers in Haskell. It labels every real numbers by having a Cauchy sequence converging to it. That will let $\mathbb{R}$ be in the ...
44 views

### Semidecidable properties of computable reals

By computable real I mean $x\in\mathbb{R}$ such that there is some computable total function $p_x$ that takes a natural number $n$ and returns a dyadic rational $r$ such that $|x-r|<2^{-n}$. I ...
168 views

### What is the computational cost of automatic differentiation compared to symbolic and numerical differentiation?

Automatic differentiation is a set of techniques to numerically evaluate the derivative of a function. Quoting from Wikipedia (emphasis mine): These classical methods run into problems: symbolic ...
173 views

### What is the relationship between two definitions of Turing-computability of a partial function?

When one wants to know that whether a partial function $f \colon \Sigma^{*} \supsetneq \mathrm{dom}(f) \rightarrow \Sigma^{*}$ is Turing-computable, there are two methods that I think they are both ...
547 views

### Decidable properties of computable reals

Is "Rice's theorem for the computable reals" -- that is, no nontrivial property of the number represented by a given computable real is decidable -- true? Does this correspond in some direct way to ...
110 views

### What does Sigma notation mean, in the context of computability of functions?

I was reading a paper on the computability of AIXI  and came across the notion of $\Sigma^0_n$-computability for real-valued functions in section 2.3. I'd like to read about this in more detail. ...
215 views

### Is Newton's Method to compute the zeros of a function an algorithm?

Looking for Newton's method in Wikipedia, I read the following: In numerical analysis, Newton's method (also known as the Newton-Raphson method), named after Isaac Newton and Joseph Raphson, is ...
A student recently asked me to check an NP-hardness proof for them. They performed a reduction along the lines of: I reduce this problem $P'$ that is known to be NP-complete to my problem $P$ (with ...