Finite Romberg table from nested composite trapezoid levels

Input a real function $f$ on a nondegenerate closed interval $[a,b]$ and an integer $L\ge 0$. The output is the finite triangular table $(R_{\mathrm{\ell },j}{)}_{0\le j\le \mathrm{\ell }\le L}$ and its top entry $R_{L,L}$.

1. For each $\mathrm{\ell }=0,\dots ,L$, set $N_{\mathrm{\ell }}:=2^{\mathrm{\ell }}\in \mathbb{N}$ using natural exponentiation of Exponentiation of natural numbers, $m^n$, and its agreement with the integer power in $\mathbb{R}$. Since $2^0=1$ and $2^{\mathrm{\ell }+1}=2^{\mathrm{\ell }}\cdot 2$, induction gives $N_{\mathrm{\ell }}\ge 1$, as required by Composite trapezoidal rule. Set $h_{\mathrm{\ell }}:=(b-a)\mathrm{/}{N}_{\mathrm{\ell }}$ and $R_{\mathrm{\ell },0}:=T_{N_{\mathrm{\ell }}}(f)$ using its composite trapezoidal value. Natural powers in real scalar expressions are understood through their canonical real images.
2. For $j=1,\dots ,L$, and then for each $\mathrm{\ell }=j,\dots ,L$, set $R_{\mathrm{\ell },j}:=R_{\mathrm{\ell },j-1}+\frac{R_{\mathrm{\ell },j-1}-R_{\mathrm{\ell }-1,j-1}}{4^j-1}\mathrm{.}$

For $j\ge 1$, claim 3 of Monotonicity of $x\mapsto x^n$ and of $n\mapsto a^n$ gives $4^j\ge 4>1$, using the natural-to-real power identity of Exponentiation of natural numbers, $m^n$, and its agreement with the integer power in $\mathbb{R}$; thus every displayed denominator $4^j-1$ is the positive real $4^j-1$ and is invertible by The reals form a field. This definition returns only the stated finite table; it makes no accuracy claim for nonsmooth inputs or finite-precision arithmetic.