Brahmagupta'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 description of sphuta-bhogyakhanda is contained in the following Sanskrit couplet (Dhyana-Graha-Upadesa-Adhyaya, 17; Khandaka Khadyaka, IX, 8):[1] [clarification needed (text needed)] This has been translated using Bhattolpala's (10th century CE) commentary as follows:[1][4] This formula was originally stated for the computation of the values of the sine function for which the common interval in the underlying base table was 900 minutes or 15 degrees.
Brahmagupta's expression can be put in the following form: This correction factor yields the following approximate value for f(a): This is Stirling's interpolation formula truncated at the second-order differences.
[1] Brahmagupta has given a separate formula for the case where the values of the independent variable are not equally spaced.