# Questions tagged [na.numerical-analysis]

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### Complexity of Finding the Eigendecomposition of a Matrix

My question is simple: What is the worst-case running time of the best known algorithm for computing an eigendecomposition of an $n \times n$ matrix? Does eigendecomposition reduce to matrix ...
1k views

### Computational complexity of pi

Let $L = \{ n : \text{the }n^{th}\text{ binary digit of }\pi\text{ is }1 \}$ (where $n$ is thought of as encoded in binary). Then what can we say about the computational complexity of $L$? It's ...
2k views

### How are real numbers specified in computation?

This may be a basic question, but I've been reading and trying to understand papers on such subjects as Nash equilibrium computation and linear degeneracy testing and have been unsure of how real ...
13k views

### Universal Approximation Theorem — Neural Networks

I posted this earlier on MSE, but it was suggested that here may be a better place to ask. Universal approximation theorem states that "the standard multilayer feed-forward network with a single ...
2k views

### How to compute powers of square matrices?

Suppose we are given a matrix $A \in \mathbb R^{N\times N}$, and let $m \in \mathbb N_0$. How fast can we compute the power $A^m$ of that matrix? The next best thing in comparison to computing $m$-...
416 views

### Motivation for volume estimation

What are some concrete and compelling applications for estimating the volume of convex polyhedra of the sort considered in the more recent papers on random walk methods? These papers on volume ...
523 views

### Numerical stability of Simplex method

The simplex algorithm is often treated either within real arithmetic, or in the discrete world with exact computations. However, it seems to be implemented most often with floating-point arithmetic. ...
What algorithm can we use to find all integer roots of a polynomial $f(x)$ with integer coefficients? I observe that Sage can find the roots within a few seconds even when all coefficients of $f(x)$ ...