Brahmagupta's interpolation formula

Brahmagupta's interpolation formula is equivalent to modern-day second-order Newton–Stirling interpolation formula.

Mathematicians prior to Brahmagupta used a simple linear interpolation formula. The linear interpolation formula to compute f(a) is

${\displaystyle f(a)=f_r+t{D}_r}$ where ${\displaystyle t=\frac{a-x_r}{h}}$.

For the computation of f(a), Brahmagupta replaces D_r with another expression which gives more accurate values and which amounts to using a second-order interpolation formula.

Brahmagupta's description of the scheme

In Brahmagupta's terminology the difference D_r is the gatakhanda, meaning past difference or the difference that was crossed over, the difference D_r+1 is the bhogyakhanda which is the difference yet to come. Vikala is the amount in minutes by which the interval has been covered at the point where we want to interpolate. In the present notations it is a − x_r. The new expression which replaces f_r+1 − f_r is called sphuta-bhogyakhanda. The description of sphuta-bhogyakhanda is contained in the following Sanskrit couplet (Dhyana-Graha-Upadesa-Adhyaya, 17; Khandaka Khadyaka, IX, 8):

This has been translated using Bhattolpala's (10th century CE) commentary as follows:

Multiply the vikala by the half the difference of the gatakhanda and the bhogyakhanda and divide the product by 900. Add the result to half the sum of the gatakhanda and the bhogyakhanda if their half-sum is less than the bhogyakhanda, subtract if greater. (The result in each case is sphuta-bhogyakhanda the correct tabular difference.)

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. So the reference to 900 is in fact a reference to the common interval h.