Brahmagupta's interpolation formula
Brahmagupata's interpolation formula is a second-order polynomial interpolation formula developed by the Indian mathematician and astronomer Brahmagupta (598–668 CE) in the early 7th century CE. The Sanskrit couplet describing the formula can be found in the supplementary part of Khandakadyaka a work of Brahmagupta completed in 665 CE. The same couplet appears in Dhyana-graha-adhikara an earlier work of Brahmagupta but of uncertain date. However internal evidences suggest that Dhyana-graha-adhikara could be dated earlier than Brahmasphuta-siddhanta a work of Brahmagupta composed in 628 CE. "Hence the invention of the second-order interpolation formula by Brahmagupta should be placed near the beginning of the second quarter of the 7th century CE, if not earlier." Brahmagupta was the first to invent and use an interpolation formula using second-order differences in the history of mathematics.
Brahmagupa's interpolation formula is equivalent to modern-day second-order Newton–Stirling interpolation formula.
Given a set of tabulated values of a function f(x) in the table below, let it be required to compute the value of f(a), xr < a < xr+1.
Assuming that the successively tabulated values of x are equally spaced with a common spacing of h, Aryabhata had considered the table of first differences of the table of values of a function. Writing
the following table can be formed:
Mathematicians prior to Brahmagupta used a simple linear interpolation formula. The linear interpolation formula to compute f(a) is
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