How can I check if the relative error of two successive diagonal elements is smaller than e.g. $10^{-3}$?
$\left\vert \frac{T_{1,2}-T_{1,3}}{T_{1,3}}\right\vert<0.001$
for a Romberg Tableau of this form
$\begin{array}{cccccc} T_{1,1}\\ &\backslash\\ T_{2,2}&-&T_{1,2}\\ &\backslash&&\backslash\\ T_{3,3}&-&T_{2,3}&-&T_{1,3}\\ &\backslash&&\backslash&&\backslash\\ \end{array}$
I am using for $h=\frac{b-a}{N_i}$ with $a$, $b$ as integral limits and $N_i=2^i$, $i=1,...$ as Romberg sequence.
I am using the trapezoidal sum to compute
$T_{i,1}=T(h_i)=\frac{h_i}{2}\left(f(a)+f(b)+\sum_{j=1}^{N_i-1}f(a+j\cdot h_i)\right)$
and all other elements are computed with the following formula:
$T_{(j,j+k)}(f)=T_{(j+1,j+k)}(f)+\frac{T_{(j+1,k+1)}(f)-T_{(j,j+k-1)}(f)}{\left(\frac{h_j}{h_{j+k}}\right)^2-1}$
Thank you in advance.