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Notes
of Numeric Calculus |
INTEGRATION
ALGORITHMS WITH VARIABLE STEP
For a large class of analiytical functions, assuming a fixed spaced step
can be quite inefficients.
Look at this simple example:
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Using the Cavalieri-Simpson formula we get an error of about 2,7E-9
with 257 nodes |
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Splitting the integration interval, we compute each integral with 65
nodes. So the final integral requires no more than 130 nodes with about the
same precision, increasing the efficency of more the 50%. |
The reason is that for x > 3 the function becames very regular
This simple observation is the basic of variable step integration: when
the function and its derivatives decrease quickly along the integration
interval, then we can increase the step without destroy the required precision
of computing. From this idea follow the integration algorithm with variable
step. It can be derived from Newton-Cotes formulas or from the Romberg
iterative process.