# Questions tagged [numerical-calculus]

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85 questions
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### How do you compute relative error when the exact solution is unknown?

I'd have a rather complex system of non linear ODEs and with a lot of help I've written an algorithm that solves them. I'd now like to compute the relative error, but I do not have a known solution to ...
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### Floating-point rounding error in numerical differentiation formula

In Numerical Analysis by Timothy Sauer (Pearson, 2nd Edition) it says that $\tilde{f'}(x+h) = f(x+h) + \epsilon_{\text{mach}}$, where $\tilde{f'}(x)$ is the floating-point representation of the given ...
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### If the initial points for secant iteration method are sufficiently close to the root, the iteration converges to the root

Well I wish to prove that in case I may choose $x_0,x_1$ close enough to the root $a$ of $f(x)$, then the secant method $x_{n+1} = x_n -\frac{x_n -x_{n-1}}{f(x_n)-f(x_{n-1})}f(x_n)$ converges to the ...
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### limit of the ratio of two divergent integrals

I have to calculate two ratios of two pairs of proper integrals, for a range of parameters. I am stuck when it comes to calculating these ratios when the parameters are such that the integrands start ...
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### How I can to prove the succession $x_n$ converge to 1? [duplicate]

Let $0<x_0<1$ if $x_{n+1} = sin(x_n)$ show that $\lim_{n\to\infty} \frac{x_n}{\sqrt{3}/n} = 1$