Application
As can be seen from the definition of the divided differences new data points can be added to the data set to create a new interpolation polynomial without recalculating the old coefficients. And when a data point changes we usually do not have to recalculate all coefficients. Furthermore if the xi are distributed equidistantly the calculation of the divided differences becomes significantly easier. Therefore the Newton form of the interpolation polynomial is usually preferred over the Lagrange form for practical purposes, although, in fact (and contrary to widespread claims), Lagrange, too, allows calculation of the next higher degree interpolation without re-doing previous calculations—and is considerably easier to evaluate.
Read more about this topic: Newton Polynomial
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