# Tagged Questions

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### Comparing the order of convergence $\mathcal{O}( h^2 |\log(h)|)$

I don't have any intuition in judging how fast a term of the order $\mathcal{O}( h^2 |\log(h)|)$ is decreasing as $h \to 0$, so i tried comparing it with terms of the form $\mathcal{O}( h^\alpha )$ ...
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### Numerical analysis problem

Given: $f(x)= x^2\sin x+2x-3$ on the interval $[0,2]$. Question: Show that the function has exactly one root in $(0,2)$ My work: I plugged $a=0$, $b=2$ into the function. So $f(0)=-3$ and ...
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### A sequence of intervals- Trying to find a fixed point -

This might be a trivial question but I couldn't come up with a clever trick,theorems or whatnot. Suppose $I_0=\left[\frac{1}{h_0},\frac{1}{l_0}\right]$ where $h_0=1$ and $l_0=\frac{1}{2}$. Given ...
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### Prove that a function is in “Big-O”

Prove that $f(h)=h^2+5h^{17}$ is in $O(h^2)$. I don't understand this problem. Big O notation continues to befuddle me. I think that what I need to show is that there exists a constant $C$ such that ...
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### Proving that $\exp(-x^2)$ has a unique fixed point on the interval $[0,1]$

Consider the function $g(x)=e^{-x^2}$. Prove that g has a unique fixed point on the interval [0,1]. So, our teacher did not go over this section, but assigned it for homework and I have no idea ...
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### “Big-O” notation with a Taylor Series Expansion

Use a Taylor's expansion to rid the expression $1-\cos x$ of subtractive cancellation for $x$ small. Use a $\mathcal{O}(x^5)$ approximate. I understand Taylor series and I know that the expansion of ...
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### Compute $\lim\limits_{x \to 1}[f(x)]$ and $\lim\limits_{x \to -1}[f(x)]$ for $f(x)=\frac{1}{1-x}-\frac{1}{1+x}$

Is it possible to rewrite expression $\frac{1}{1-x}-\frac{1}{1+x}$ in order to be able to find its values near $x=1$ and $x=-1$ more precisely? This is a question in a numerical methods course. Is the ...
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### When $e^x$ ~ $e^{-2x}$ ? - Numerical analysis

For what $x$, $e^x$ ~ $e^{-2x}$ ? And how one can change this expression to avoid significant digits loss? I am able to think only about $x =0$, but then both are equal and you lose nothing.
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### $a + b = a$ in machine precision [closed]

I have the following statement: "If $a + b = a$, then $b = 0$" may not true with the floating point operations. Actually, if $|y| ‎< (\varepsilon / B) |x|$, then $fl(x+y) = x$, where ...
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### Proving something is a spline of degree $n$

I'm trying to understand this theorem: In the proof they only explain why $S_{(n)}(x)$ is identically $0$ outside the interval $(0,(n+1)h)$. But it is not clear to me that this is a spline of ...
Given the equation $\displaystyle{\int_{-x}^x\exp({-t^2})dt}=-\ln(x)$: a. Simplify the integral using Gauss method with 3 points. b. Solve given equation by Newton Raphson iterative ...