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When using the left Riemann sum algorithm, one does not know a priori what step size or number of intervals to choose so that the
When using the left Riemann sum algorithm, one does not know a priori what step size or
number of intervals to choose so that the result is accurate to within a desired error tolerance the
maximum error that can be tolerated This choice depends on the given function to integrate.
For example, if one integrates the constant function, an algorithm using produces the
exact result, ie no error.
a Modify the rmlsum file to include a whileloop so that the program code terminates when
an accuracy based on relative approximate error is achieved. In each iteration of the whileloop, double the number of the intervals ie the step size is reduced by one half in the
numerical integration algorithm. The functions inputoutput structure should be:
function I N rmlsumNfunabmaxtol
where, I is the approximate result of the integration and N is the number of intervals that
would yield a result to satisfy the error tolerance criterion. Note that this output value of N is
not necessarily the minimum number of intervals to achieve the error tolerance.
Apply this numerical solver to the following integral using maxtole
One potential difficulty of the above approach is that it might take many iterations to achieve the
error tolerance. In general, one would like to set a maximum number of iterations.
b Modify the above numerical solver so that the whileloop checks both the error tolerance
and the maximum number of iterations. In other words, the loop terminates when either the
error is sufficiently small, or the maximum number of iterations is exceeded. The function
structure should be:
function I N rmlsumNfunabmaxtol,maxitr
Test your program code on the same integral from part a using maxtole and
maxitr Note that because the number of intervals is doubled in each iteration,
iterations would result in
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