Brahmagupta's interpolation formula

In Brahmagupta's terminology the difference {{math|D_r}} is the gatakhanda, meaning past difference or the difference that was crossed over, the difference {{math|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 {{math|a − x_r}}. The new expression which replaces {{math|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):

File:Brahmagupas Interpolation Formula In Devanagari.jpg{{clarify|date=January 2016|reason=This is an image of text and should be replaced by the text itself.|post-text=(text needed)}}

This has been translated using Bhattolpala's (10th century CE) commentary as follows:{{cite book|last=Raju |first=C K|title=Cultural foundations of mathematics: the nature of mathematical proof and the transmission of the calculus from India to Europe in the 16th c. CE|year=2007|publisher=Pearson Education India|isbn=9788131708712|pages=138–140}}

: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 {{math|h}}.

Brahmagupta's method computation of shutabhogyakhanda can be formulated in modern notation as follows:

:sphuta-bhogyakhanda $\backslash displaystyle\; =\; \backslash frac\{D\_r\; +\; D\_\{r-1\}\}\{2\}\; \backslash pm\; t\backslash frac$

The ± sign is to be taken according to whether {{math|{{sfrac|1|2}}(D_r + D_r+1)}} is less than or greater than {{math|D_r+1}}, or equivalently, according to whether {{math|D_r < D_r+1}} or {{math|D_r > D_r+1}}. Brahmagupta's expression can be put in the following form:

:sphuta-bhogyakhanda $\backslash displaystyle\; =\; \backslash frac\{D\_r\; +\; D\_\{r-1\}\}\{2\}\; +\; t\backslash frac\{D\_r-D\_\{r-1\}\}\{2\}.$

This correction factor yields the following approximate value for {{math|f(a)}}:

: $$

\begin{align}

f(a) & = f_r + t\times\text{sphuta-bhogyakhanda}\\

& = f_r + t \frac{D_r + D_{r-1}}{2} + t^2\frac{D_r - D_{r-1}}{2}.

\end{align}

This is Stirling's interpolation formula truncated at the second-order differences.{{cite book|last=Milne-Thomson|first=Louis Melville|title=The Calculus of Finite Differences|year=2000|publisher=AMS Chelsea Publishing|isbn=9780821821077|pages=67–68}}{{cite book|last=Hildebrand|first=Francis Begnaud|title=Introduction to numerical analysis|year=1987|publisher=Courier Dover Publications|isbn=9780486653631|pages=[https://archive.org/details/introductiontonu00hild_0/page/138 138–139]|url-access=registration|url=https://archive.org/details/introductiontonu00hild_0/page/138}} It is not known how Brahmagupta arrived at his interpolation formula. Brahmagupta has given a separate formula for the case where the values of the independent variable are not equally spaced.

• Brahmagupta's identity
• Brahmagupta matrix
• Brahmagupta–Fibonacci identity

{{reflist}}

Category:Interpolation

Category:Indian mathematics

Category:History of mathematics