# how to check convergence of a vector in practice?

Let's say I have a final solution vector that I know, denoted by $$v_{\mathrm{final}}$$ and access to a series of vectors from different time steps $$\left\{v_{0}, v_{1},\ldots, v_{\mathrm{final} - 1}\right\}$$.

How would you check in practice whether the series of vectors converge to $$v_{\mathrm{final}}$$. For now I am computing the sum of the absolute difference between consecutive vectors and checking whether it goes to $$0$$.

My question is: how can I check that it indeed goes to $$0$$ ? fit a linear regression through the difference and check whether the slope is negative ?. And at the same time check that $$\left\vert v_{\mathrm{final} - 1} - v_{\mathrm{final}}\right\vert < \epsilon\ ?$$.

Thanks :)

• Your question is not clear to me. Do you have an infinite sequence of vectors? Or only finitely many? Aug 16, 2020 at 15:44
• Only a finite set of observations.
– Tom
Aug 16, 2020 at 15:45

A common technique used in root-finding is to track not a sequence of points, but a sequence of intervals. Start with two endpoints $$a_0, b_0$$ for which the function values $$f(a_0)$$ and $$f(b_0)$$ have opposite signs. Then you calculate some estimate $$c_0\in (a_0, b_0)$$ of the root and calculate $$f(c_0)$$. If it has the same sign as $$f(a_0)$$, then you set $$a_1 = c_0, b_1 = b_0$$, otherwise you set $$a_1 = a_0, b_1 = c_0$$ and repeat. As you can always choose $$c_0 = \frac {a_0+b_0}2$$, any well-designed technique can be expected to cut the interval in at least half with each step, so you are guaranteed convergence (though convergence by bisection is slow, so we prefer other methods).