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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.

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  • $\begingroup$ What are the trapezoidal and what the Simpson sums? Are these the diagonals? Is the difference that you want to estimate between trapezoidal sums? $\endgroup$ – LutzL Jan 26 at 17:15
  • $\begingroup$ I think/hope I added all the missing information. $\endgroup$ – baxbear Jan 27 at 2:12

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