The Newton Polynomial above can be expressed in a simplified form when are arranged consecutively with equal space. Introducing the notation for each and, the difference can be written as . So the Newton Polynomial above becomes:
is called the Newton Forward Divided Difference Formula.
If the nodes are reordered as, the Newton Polynomial becomes:
If are equally spaced with x= and for, then,
is called the Newton Backward Divided Difference Formula.
Read more about Newton Polynomial: Significance, Addition of New Points, Strengths and Weaknesses of Various Formulae, General Case, Main Idea, Taylor Polynomial, Application
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“I frame no hypotheses; for whatever is not deduced from the phenomena is to be called a hypothesis; and hypotheses, whether metaphysical or physical, whether of occult qualities or mechanical, have no place in experimental philosophy.”
—Isaac Newton (1642–1727)