Questions about the error in quadrature formulas

Question

everyone, I am new to these subjects and I would like to clarify a subject that is little covered in my textbook.

It tells me that fixed an n, a number of sub-intervals to use to calculate the integral using the composite trapezoidal or Simpson rule, we can calculate the error for $E_n$ and $E_{2n}$ that is relative to the same integral considering the n sub-intervals and 2n sub-intervals. In this way we can obtain a more accurate estimate of the error, it says. Could someone explain me better what is meant? And especially if I do $E_n/E_{2n}$, what information do I get?

Answer 1

If you apply the trapezoidal rule with $N=n, 2n, 4n$ segments, then you get a value

$$ I_N = I_* + c_2 N^{-2} + c_4 N^{-4} + ... $$ The estimation of the terms of this expansion from two or more numerical evaluations for different $N$ is called Richardson extrapolation or Romberg integration (successively eliminating the terms for $n^{-2k}$).

Now if you compute the error from two integrations for $N=n,2n$ by eliminating the exact value $I_*$, you get $$ I_n-I_{2n}=c_2n^{-2}(1-2^{-2})+c_4n^{-4}(1-2^{-4})+ $$ so that you get the error estimate $$ E_n=\frac43(I_n-I_{2n}) = c_2n^{-2} + \frac54 c_4n^{-4}+... $$

If you now compute $$ \frac{E_n}{E_{2n}} =\frac{c_2 + \frac54 c_4n^{-2}+...}{c_22^{-2} + \frac54 c_42^{-4}n^{-2}+...} =4+\frac{\frac{15}{4}c_4n^{-2}+...}{c_2+\frac5{16}c_4n^{-2}+...} $$ you can see from the numerical value of this fraction that if it is far away from $4$ that with some high probability there is some error in the implementation.

If you compute a series of these fractions, you should find some middle region of values for $n$ where the quotient is close to $4$. For large values of $n$ one has to add the floating point noise of size $\mu\cdot n$ to the error, it will dominate the error for $n>1/\sqrt[3]\mu\sim 10^5$, where $\mu=2^{-52}\sim 10^{-16}$ is the machine constant of the floating point data type. Thus for large $n$ the error will increase again in a vaguely linear way and the quotient will fluctuate randomly.

If such a middle region does not exist, then the coefficient $c_4$ or a higher one is very large against $c_2$, the function is then either rapidly growing or highly oscillatory and you will need more adapted methods.

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See also similar observations in a BDF ODE solver, a fourth order Taylor ODE solver, classical RK4, and one-sided and symmetric and further improved difference quotients.