# Tagged Questions

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### Finding the proper g(x) for fixed point iteration on $2\sin{\pi x} + x = 0$

After spending over an hour trying to get this problem I realize my trig is weak. I found: Fixed point iteration .Numerical method. The selected solution is informative, but lacking detail to really ...
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### Fixed point method where the derivative is one - does it converge

I'm trying to see if the iterative method $x_n=g(x_{n-1})$ where $g(x)=2\sqrt{x-1}$ will converge to $2$, if I take $x_0$ that is sufficiently close to $2$. Indeed notice that $g(2)=2$. and we have a ...
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I saw some sentences and proofs related to fixed-point iteration in my numerical method textbook: $$e_{k+1} = g'(\theta) e_{k}$$ if $|g'(x^\ast)|<1$, there is a constant such that $|g'(\theta)|\le ... 0answers 53 views ### Aitkens Extrapolation The Aitken's extrapolation can be written as $$X^n = X_{n-2} + \dfrac{(X_{n-1}- X_{n-2})^2}{(X_{n-1}- X_{n-2})-(X_n- X_{n-1})}$$ Verify it? And$X^n$can be viewed being defined recursively by ... 0answers 24 views ### Heuristics for sequence convergence Having a finite sequence of double precision floating point numbers (obtained using the fixed point iteration of a function), is there any algorithm which can be used to determine that this sequence ... 0answers 26 views ### Show that$x_{k+1}:=g^{-1}(g(x_k)-f(x_k))$converges to a root of$f$If$f:[-1,1]\to\mathbb R$continously differentiable and$g:[-1,1]\to[-2,2]$continuously differentiable and bijective such that,$|f'(x)-g'(x)|\le 1/2 \inf\limits_{y\in[-1,1]}g'(y)$... 1answer 65 views ### Root of a function? (Proof with Banach theorem) Given is a function$f$:$\left [ -1,1 \right ]\rightarrow \mathbb{R}$, which is continuous and differentiable. The function$g$:$\left [ -1,1 \right ]\rightarrow \left [ -2,2 \right ]$is a ... 1answer 88 views ### Fixed-point iteration, Convergence of a sequence? Given is the function$f(x)=x^{3}+x-1$on$\mathbb{R}$. Use the Fixed-point iteration for$x\in \left [ 0.5 , 1 \right ]$to show that the sequence$\left \{ x \right \}_{n}$converges to the ... 1answer 45 views ### Show an iterative fixed point method does not converge I was asked the following question: let$g(x)=\frac{30}{1+x}$. Notice that$g(5)=5$. Is there an$\epsilon >0$such that the series$\{x_k\}_{k=0}^{\infty}$defined by$x_{k+1}=g(x_k)$and ... 1answer 37 views ### Trouble with fixed point iteration. Find the fixed point(s) of$g(x) = (1/2)x^2 + (1/2)x$. Does the fixed point iteration(s) converge(s) to the the fixed point(s) if you start with a close enough approximation? Then choose$x_0 ...
Find the fixed point(s) of $g(x) = x^2 + 3x - 3$. Does the fixed point iteration(s) converge(s) to the fixed points if you start with a close enough first approximation? I set $g(x) = x$ and got ...