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Results tagged with fixed-point-theorems
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user 62673
A fixed-point theorem is a result saying that a function $F$ will have at least one fixed point (a point $x$ for which $F(x) = x$), under some conditions on $F$ that can be stated in general terms.
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On a matrix algorithm involving rank-one projections
Let $\{v_i\}_{i=1}^N$ be a set of $n$-dimensional real vectors spanning $\mathbb{R}^n$. Let $p\in [0,1]$ be a rational number and consider the iteration
\begin{equation}
X_{k+1}=\frac{1}{N}\sum_{i=1}^ …
- 2,712
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votes
0
answers
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On the existence of fixed points of a matrix iteration
Let $A\in\mathbb{R}^{n\times n}$ be a Hurwitz stable matrix (i.e. all eigenvalues of $A$ have negative real part). Let $\succeq$ denote the standard partial order in the cone of positive semidefinite …
- 2,712
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Fixed-point iteration depending on a parameter
Let $f\colon X\times \mathbb{R}\to X, (x,\varepsilon)\mapsto y$, with $X$ open, be a continuous function in both arguments. Consider the following fixed-point iteration
\begin{align}
x_{k+1} = f(x_k,\ …
- 2,712