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$(a)$ Derive adaptive quadrature formula to evalute $\int_a^bf(x)dx$.
$(b)$ Given, $I=\int_0^{\frac{\pi}{4}}cos^2xdx$ compute $S(0,\frac{\pi}{4}),S(0,\frac{\pi}{8})$ & $S(\frac{\pi}{8},\frac{\pi}{4})$. Also verify the error estimate $$\frac{1}{15}\Biggl|S(a,b)-S\left(a,\frac{a+b}{2}\right)-S\left(\frac{a+b}{2},b\right)\Biggr|<\epsilon$$ for this problem.

For $(a)$ I think this may be work$($Theorem $2)$. But What to do with $(b)?$ I haven't mentioned that kind of error bound in my book Numerical Analysis $10$th Edition by Richard L. Burden. How to get that. Ant help will be appreciated.
Thanks in advanced.
emonHR
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Dr.Antidode
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  • Is "adaptive quadrature formula" meant in the sense of this question on stackoverflow? Then it should really be $S([a,b],ϵ)$ and part b) is trivial, as each term has error $ϵ$, so by triangle inequality the error of the whole expression is less than $3ϵ/15=ϵ/5$. – Lutz Lehmann Nov 18 '19 at 22:34

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