Newton Gauss (1670) Interpolation Formula Discovered By Govindaswami

gacchad-yata-gunantharavapuryathaishya-disvasanaa cchedaabhyaasa-

samuha-kaarmukakrti-praapthath tribhisthaadithah

vedaihi sadbhir avaaptam antyagunaje rasyo:

kramad antyabhe ganthavaahata-varthamaana-gunajaaccha paatham ekaadibhi:

antyad utkramatah kramena vishamai: sankhyaviseshai:

khsipedbhankthvaptam, yadi maurvikavidhir ayam makhyah kramad vartate sodhyam

vyutkramathaa stathakrthaphlam…..

  Mathematicaly this formula is summarised as follows: F(x+nh)=f(x)+nf(x)+½n(n-1)(f(x)-f(x-h) Multiply the difference of the last and the current sine differences by the square of the elemental arc and further mutiply by three. Now divide the result so obtained by four in the first rasi, or by six in the second rasi. The final result thus obtained should be added to the portion of the current sine difference (got by linear proportion). In the last rasi, multiply the linearly promotional part of the current sine differences by the remaining part of the elemental arc and divide by the elemental arc. Now, divide the result by the odd numbers according to the current sine difference, when counted from the end in the reverse order. Add the final result thus obtained to the portion of the current sine difference. These are the rules for computing true sine differences for sines. In the case of versed sines, apply the rules in the reverse order and the above corrections are to be subtracted from the respective differences.  ]]>