ImportMaster at 16:41, 3 May 2022

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'''Brahmagupta's interpolation formula''' is a second-order polynomial [[interpolation formula]] developed by the [[India]]n [[mathematician]] and [[astronomer]] [[Brahmagupta]] (598–668 [[Common era|CE]]) in the early 7th century [[Common Era|CE]]. The [[Sanskrit]] couplet describing the formula can be found in the supplementary part of ''Khandakadyaka'' a work of [[Brahmagupta]] completed in 665 CE.<ref name="Gupta">{{cite journal|last=Gupta|first=R. C. |title=Second-order interpolation in Indian mathematics upto the fifteenth century|journal=Indian Journal of History of Science|volume=4|issue=1 & 2|pages=86–98}}</ref> The same couplet appears in Brahmagupta's earlier ''Dhyana-graha-adhikara'', which was probably written "near the beginning of the second quarter of the 7th century CE, if not earlier."<ref name="Gupta"/> Brahmagupta was the one of the first to describe and use an [[interpolation formula]] using second-order [[finite difference|differences]].<ref>{{cite book|last=Van Brummelen|first=Glen|authorlink=Glen Van Brummelen|title=The mathematics of the heavens and the earth: the early history of trigonometry|year=2009|publisher=Princeton University Press|isbn=9780691129730|pages=329}} (p.111)</ref><ref>{{cite journal|last=Meijering|first=Erik|title=A Chronology of Interpolation From Ancient Astronomy to Modern Signal and Image Processing|journal=Proceedings of the IEEE|date=March 2002|volume=90|issue=3|pages=319–321|doi=10.1109/5.993400|url=http://infoscience.epfl.ch/record/63085}}</ref>

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

{| class="wikitable"
|-
|  {{math|''x''}}  ||  {{math|''x''2}}  ||  ...   ||   {{math|''x''''r''}}  ||   {{math|''x''''r''+1}}  ||  ...   ||  {{math|''x''''n''}} 
|-
|  Differences ||  {{math|''D''1}} ||  ... ||  {{math|''D''''r''}} ||  {{math|''D''''r''+1}} || ... ||   {{math|''D''''n''}}
|}

Mathematicians prior to Brahmagupta used a simple [[linear interpolation]] formula. The linear interpolation formula to compute {{math|''f''(''a'')}} is

: <math>f(a)=f_r+ t D_r</math> where <math>t=\frac{a-x_r}{h}</math>.

For the computation of {{math|''f''(''a'')}}, Brahmagupta replaces {{math|''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 {{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''):<ref name="Gupta"/>

[[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:<ref name="Gupta"/><ref name="Raju">{{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}}</ref>

: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''}}.

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

:''sphuta-bhogyakhanda'' <math>\displaystyle = \frac{D_r + D_{r-1}}{2} \pm t\frac{|D_r - D_{r-1}|}{2}.</math>

The [[plus or minus|±]] 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'' <math> \displaystyle = \frac{D_r + D_{r-1}}{2} + t\frac{D_r-D_{r-1}}{2}.</math>

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

: <math>
\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}
</math>

This is Stirling's [[interpolation formula]] truncated at the second-order differences.<ref>{{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}}</ref><ref>{{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}}</ref> It is not known how Brahmagupta arrived at his interpolation formula.<ref name="Gupta"/> Brahmagupta has given a separate formula for the case where the values of the independent variable are not equally spaced.

==See also==
* [[Brahmagupta's identity]]
* [[Brahmagupta matrix]]
* [[Brahmagupta–Fibonacci identity]]

==References==
{{reflist}}

[[Category:Interpolation]]
[[Category:Indian mathematics]]
[[Category:History of mathematics]]

Brahmagupta's interpolation formula - Revision history

ImportMaster at 16:41, 3 May 2022