1. Introduction Aryabhata Brahmagupta Bhaskara I Madhava References
On Finite Differences, Interpolation Methods
and Power Series Expansions in Indian
Mathematics
V. N. Krishnachandran
Vidya Academy of Science & Technology
Thrissur 680 501, Kerala
V. N. Krishnachandran Vidya Academy of Science & Technology Thrissur 680 501, Kerala
On Finite Differences, Interpolation Methods and Power Series Expansions in Indian Mathematics

2. Introduction Aryabhata Brahmagupta Bhaskara I Madhava References
Outline
1 Introduction
2 Aryabhata’s difference table
3 Brahmagupta’s interpolation formula
4 Bhaskara I’s approximation formula
5 Madhava’s power series expansions
6 References
V. N. Krishnachandran Vidya Academy of Science & Technology Thrissur 680 501, Kerala
On Finite Differences, Interpolation Methods and Power Series Expansions in Indian Mathematics

3. Introduction Aryabhata Brahmagupta Bhaskara I Madhava References
Introduction
Introduction
V. N. Krishnachandran Vidya Academy of Science & Technology Thrissur 680 501, Kerala
On Finite Differences, Interpolation Methods and Power Series Expansions in Indian Mathematics

4. Introduction Aryabhata Brahmagupta Bhaskara I Madhava References
Objectives
To present some of the greatest achievements of
pre-modern Indian mathematicians as
contributions to the development of numerical
analysis.
V. N. Krishnachandran Vidya Academy of Science & Technology Thrissur 680 501, Kerala
On Finite Differences, Interpolation Methods and Power Series Expansions in Indian Mathematics

5. Introduction Aryabhata Brahmagupta Bhaskara I Madhava References
Main themes
We present four themes:
1 Difference tables
2 Interpolation formulas
3 Rational polynomial approximations
4 Power series expansions
V. N. Krishnachandran Vidya Academy of Science & Technology Thrissur 680 501, Kerala
On Finite Differences, Interpolation Methods and Power Series Expansions in Indian Mathematics

6. Introduction Aryabhata Brahmagupta Bhaskara I Madhava References
Aryabhata’s difference table
Aryabhata’s difference table
V. N. Krishnachandran Vidya Academy of Science & Technology Thrissur 680 501, Kerala
On Finite Differences, Interpolation Methods and Power Series Expansions in Indian Mathematics

7. Introduction Aryabhata Brahmagupta Bhaskara I Madhava References
Aryabhata’s sine table
Aryabhata I’s (476 - 550 CE) celebrated work Aryabhatiyam
contains a sine table.
Aryabhata’s table was the first sine table ever constructed in
the history of mathematics.
The tables of Hipparchus (c.190 BC - c.120 BC), Menelaus
(c.70 - 140 CE) and Ptolemy (c.AD 90 - c.168) were all tables
of chords and not of half-chords.
V. N. Krishnachandran Vidya Academy of Science & Technology Thrissur 680 501, Kerala
On Finite Differences, Interpolation Methods and Power Series Expansions in Indian Mathematics

8. Introduction Aryabhata Brahmagupta Bhaskara I Madhava References
What Aryabhata tabulated
Aryabhata tabulated the values of jya (measured in minutes)
for arc equal to 225 minutes, 450 minutes, ... , 5400 minutes.
(Twenty-four values.)
V. N. Krishnachandran Vidya Academy of Science & Technology Thrissur 680 501, Kerala
On Finite Differences, Interpolation Methods and Power Series Expansions in Indian Mathematics

9. Introduction Aryabhata Brahmagupta Bhaskara I Madhava References
What others tabulated
Pre-Aryabhata astronomers tabulated
values of chords for various arcs.
V. N. Krishnachandran Vidya Academy of Science & Technology Thrissur 680 501, Kerala
On Finite Differences, Interpolation Methods and Power Series Expansions in Indian Mathematics

10. Introduction Aryabhata Brahmagupta Bhaskara I Madhava References
Aryabhata’s table
The stanza specifying Aryabhata’s table is the tenth one (excluding
two preliminary stanzas) in the first section of Aryabhatiya titled
Dasagitikasutra.
V. N. Krishnachandran Vidya Academy of Science & Technology Thrissur 680 501, Kerala
On Finite Differences, Interpolation Methods and Power Series Expansions in Indian Mathematics

11. Introduction Aryabhata Brahmagupta Bhaskara I Madhava References
Aryabhata’s table in his notation
(Table values are encoded in a scheme invented by Aryabhata.)
V. N. Krishnachandran Vidya Academy of Science & Technology Thrissur 680 501, Kerala
On Finite Differences, Interpolation Methods and Power Series Expansions in Indian Mathematics

12. Introduction Aryabhata Brahmagupta Bhaskara I Madhava References
Aryabhata’s table in modern notation
225 224 222 219
215 210 205 199
191 183 174 164
154 143 131 119
106 93 79 65
51 37 22 7
(Read numbers row-wise.)
V. N. Krishnachandran Vidya Academy of Science & Technology Thrissur 680 501, Kerala
On Finite Differences, Interpolation Methods and Power Series Expansions in Indian Mathematics

13. Introduction Aryabhata Brahmagupta Bhaskara I Madhava References
Interpretation of Aryabhata’s table
Aryabhata’s table is not a table of the values of jyas.
Aryabhata’s table is a table of the first differences of the
values of jyas.
V. N. Krishnachandran Vidya Academy of Science & Technology Thrissur 680 501, Kerala
On Finite Differences, Interpolation Methods and Power Series Expansions in Indian Mathematics

14. Introduction Aryabhata Brahmagupta Bhaskara I Madhava References
Aryabahata’s table as a table of first differences
Angle (A) Value in A’bhata’s value Modern value
(in minutes) A’bhata’s table of jya (A) of jya (A)
225 225 225 224.8560
450 224 449 448.7490
675 222 671 670.7205
900 219 890 889.8199
1125 215 1105 1105.1089
1350 210 1315 1315.6656
1575 205 1520 1520.5885
1800 199 1719 1719.0000
.
. .
. .
. .
.
. . . .
Values in second column are differences of values in third column.
V. N. Krishnachandran Vidya Academy of Science & Technology Thrissur 680 501, Kerala
On Finite Differences, Interpolation Methods and Power Series Expansions in Indian Mathematics

15. Introduction Aryabhata Brahmagupta Bhaskara I Madhava References
Brahmagupata’s interpolation formula
Brahmagupata’s interpolation
formula
V. N. Krishnachandran Vidya Academy of Science & Technology Thrissur 680 501, Kerala
On Finite Differences, Interpolation Methods and Power Series Expansions in Indian Mathematics

16. Introduction Aryabhata Brahmagupta Bhaskara I Madhava References
Brahmagupta
Brahmagupta’s (598 - 668 CE) works contain Sanskrit verses
describing a second order interpolation formula.
The earliest such work is Dhyana-graha-adhikara, a treatise
completed in early seventh century CE.
Brahmagupta was the first to invent and use an interpolation
formula of the second order in the history of mathematics.
V. N. Krishnachandran Vidya Academy of Science & Technology Thrissur 680 501, Kerala
On Finite Differences, Interpolation Methods and Power Series Expansions in Indian Mathematics

17. Introduction Aryabhata Brahmagupta Bhaskara I Madhava References
Brahmagupta’s verse
(Earliest appearance: Dhyana-graha-adhikara, sloka 17)
V. N. Krishnachandran Vidya Academy of Science & Technology Thrissur 680 501, Kerala
On Finite Differences, Interpolation Methods and Power Series Expansions in Indian Mathematics

18. Introduction Aryabhata Brahmagupta Bhaskara I Madhava References
Translation of Brahmagupta’s verse
Multiply half the difference of the tabular differences crossed
over and to be crossed over by the residual arc and divide by
900 minutes (= h). By the result (so obtained) increase or
decrease half the sum of the same (two) differences, according
as this (semi-sum) is less than or greater than the difference
to be crossed over. We get the true functional differences to
be crossed over.
(Gupta, R.C.. “Second order interpolation in Indian mathematics
upto the fifteenth century”. Indian Journal of History of Science 4
(1 & 2): pp.86 - 98.)
V. N. Krishnachandran Vidya Academy of Science & Technology Thrissur 680 501, Kerala
On Finite Differences, Interpolation Methods and Power Series Expansions in Indian Mathematics

19. Introduction Aryabhata Brahmagupta Bhaskara I Madhava References
Brahamagupta’s verse : Interpretation (notations)
Consider a set of values of f (x) tabulated at equally spaced
values of x:
x x1 · · · xr xr +1 · · · xn
f (x) f1 · · · fr fr +1 · · · fn
Let Dj = fj − fj−1 .
Let it be required to find f (a) where xr < a < xr +1 .
Let t = a − xr and h = xj − xj−1 .
V. N. Krishnachandran Vidya Academy of Science & Technology Thrissur 680 501, Kerala
On Finite Differences, Interpolation Methods and Power Series Expansions in Indian Mathematics

20. Introduction Aryabhata Brahmagupta Bhaskara I Madhava References
Brahamagupta’s verse : Interpretation
True functional difference =
Dr + Dr +1 t |Dr − Dr +1 |
±
2 h 2
Dr + Dr +1
according as is less than or greater than Dr +1 .
2
True functional difference =
Dr + Dr +1 t Dr +1 − Dr
+
2 h 2
V. N. Krishnachandran Vidya Academy of Science & Technology Thrissur 680 501, Kerala
On Finite Differences, Interpolation Methods and Power Series Expansions in Indian Mathematics

21. Introduction Aryabhata Brahmagupta Bhaskara I Madhava References
Brahamagupta’s verse : Interpretation
The functional difference Dr +1 in the approximation formula
t
f (a) = f (xr ) + Dr +1
h
is replaced by this true functional difference.
The resulting approximation fromula is Brahmagupta’s
interpolation formula.
V. N. Krishnachandran Vidya Academy of Science & Technology Thrissur 680 501, Kerala
On Finite Differences, Interpolation Methods and Power Series Expansions in Indian Mathematics

22. Introduction Aryabhata Brahmagupta Bhaskara I Madhava References
Brahmagupta’s interpolation formula
Brahmagupta’s interpolation formula:
t Dr + Dr +1 t Dr +1 − Dr
f (a) = f (xr ) + +
h 2 h 2
This is the Stirlings interpolation formula truncated at the
second order.
V. N. Krishnachandran Vidya Academy of Science & Technology Thrissur 680 501, Kerala
On Finite Differences, Interpolation Methods and Power Series Expansions in Indian Mathematics

23. Introduction Aryabhata Brahmagupta Bhaskara I Madhava References
Bhaskara I’s approximation formula
Bhaskara I’s approximation
formula
V. N. Krishnachandran Vidya Academy of Science & Technology Thrissur 680 501, Kerala
On Finite Differences, Interpolation Methods and Power Series Expansions in Indian Mathematics

24. Introduction Aryabhata Brahmagupta Bhaskara I Madhava References
Bhaskara I
Bhaskara I (c.600 - c.680), a seventh century Indian
mathematician (not the author of Lilavati).
Mahabhaskariya, a treatise by Bhaskara I, contains a verse
describing a rational polynomial approximation to sin x.
V. N. Krishnachandran Vidya Academy of Science & Technology Thrissur 680 501, Kerala
On Finite Differences, Interpolation Methods and Power Series Expansions in Indian Mathematics

25. Introduction Aryabhata Brahmagupta Bhaskara I Madhava References
Bhaskara’s verse
(Mahabhaskariya, VII, 17 - 19)
V. N. Krishnachandran Vidya Academy of Science & Technology Thrissur 680 501, Kerala
On Finite Differences, Interpolation Methods and Power Series Expansions in Indian Mathematics

26. Introduction Aryabhata Brahmagupta Bhaskara I Madhava References
Bhaskara’s verse: Translation
(Now) I briefly state the rule (for finding the bhujaphala and the
kotiphala, etc.) without making use of the Rsine-differences 225,
etc. Subtract the degrees of a bhuja (or koti) from the degrees of
a half circle (that is, 180 degrees). Then multiply the remainder by
the degrees of the bhuja or koti and put down the result at two
places. At one place subtract the result from 40500. By one-fourth
of the remainder (thus obtained), divide the result at the other
place as multiplied by the anthyaphala (that is, the epicyclic
radius). Thus is obtained the entire bahuphala (or, kotiphala) for
the sun, moon or the star-planets. So also are obtained the direct
and inverse Rsines.
(R.C. Gupta (1967). Bhaskara I’ approximation to sine. Indian
Journal of HIstory of Science 2 (2)
V. N. Krishnachandran Vidya Academy of Science & Technology Thrissur 680 501, Kerala
On Finite Differences, Interpolation Methods and Power Series Expansions in Indian Mathematics

27. Introduction Aryabhata Brahmagupta Bhaskara I Madhava References
Bhaskara I’s approximation formula
Let x be an angle measured in degrees.
4x(180 − x)
sin x =
40500 − x(180 − x)
V. N. Krishnachandran Vidya Academy of Science & Technology Thrissur 680 501, Kerala
On Finite Differences, Interpolation Methods and Power Series Expansions in Indian Mathematics

28. Introduction Aryabhata Brahmagupta Bhaskara I Madhava References
Bhaskara I’s approximation formula
This is a rational polynomial approximation to sin x when
angle x is expressed in degrees.
It is not known how Bhaskara arrived at this formula.
V. N. Krishnachandran Vidya Academy of Science & Technology Thrissur 680 501, Kerala
On Finite Differences, Interpolation Methods and Power Series Expansions in Indian Mathematics

29. Introduction Aryabhata Brahmagupta Bhaskara I Madhava References
Accuracy of Bhaskara’s approximation formula
The maximum absolute error in using the formula is around 0.0016.
V. N. Krishnachandran Vidya Academy of Science & Technology Thrissur 680 501, Kerala
On Finite Differences, Interpolation Methods and Power Series Expansions in Indian Mathematics

30. Introduction Aryabhata Brahmagupta Bhaskara I Madhava References
Madhava’s power series expansions
Madhava’s power series
expansions
V. N. Krishnachandran Vidya Academy of Science & Technology Thrissur 680 501, Kerala
On Finite Differences, Interpolation Methods and Power Series Expansions in Indian Mathematics

31. Introduction Aryabhata Brahmagupta Bhaskara I Madhava References
Sangamagrama Madhava
Madhava flourished during c.1350 - c.1425.
Madhava founded the so called Kerala School of Astronomy
and Mathematics.
Only a few minor works of Madhava have survived.
There are copious references and tributes to Madhava in the
works of his followers.
V. N. Krishnachandran Vidya Academy of Science & Technology Thrissur 680 501, Kerala
On Finite Differences, Interpolation Methods and Power Series Expansions in Indian Mathematics

32. Introduction Aryabhata Brahmagupta Bhaskara I Madhava References
Madhava’s power series for sine in Madhava’s words
V. N. Krishnachandran Vidya Academy of Science & Technology Thrissur 680 501, Kerala
On Finite Differences, Interpolation Methods and Power Series Expansions in Indian Mathematics

33. Introduction Aryabhata Brahmagupta Bhaskara I Madhava References
Madhava’s power series for sine in English
Multiply the arc by the square of itself (multiplication being
repeated any number of times) and divide the result by the
product of the squares of even numbers increased by that
number and the square of the radius (the multiplication being
repeated the same number of times). The arc and the results
obtained from above are placed one above the other and are
subtracted systematically one from its above. These together
give jiva collected here as found in the expression beginning
with vidwan etc.
(A.K. Bag (1975). Madhava’s sine and cosine series. Indian
Journal of History of Science 11 (1): pp.54-57.)
V. N. Krishnachandran Vidya Academy of Science & Technology Thrissur 680 501, Kerala
On Finite Differences, Interpolation Methods and Power Series Expansions in Indian Mathematics

34. Introduction Aryabhata Brahmagupta Bhaskara I Madhava References
Madhava’s power series for sine in modern notations
Let θ be the angle subtended at the center of a circle of radius r
by an arc of length s. Then jiva ( = jya) of s is r sin θ.
jiva = s
s2
− s·
(22 + 2)r 2
s2 s2
− s· 2 ·
(2 + 2)r 2 (42 + 4)r 2
s2 s2 s2
− s· 2 · 2 · 2 − ···
(2 + 2)r 2 (4 + 4)r 2 (6 + 6)r 2
V. N. Krishnachandran Vidya Academy of Science & Technology Thrissur 680 501, Kerala
On Finite Differences, Interpolation Methods and Power Series Expansions in Indian Mathematics

35. Introduction Aryabhata Brahmagupta Bhaskara I Madhava References
Madhava’s power series for sine : reformulation for
computations
Chose a circle the length of a quarter of which is C = 5400
minutes.
Let R be the radius of such a circle.
Choose Madhava’s value for π: π = 3.1415926536.
The radius R can be computed as follows:
R = 2 × 5400/π
= 3437 minutes, 44 seconds, 48 sixtieths of a second.
V. N. Krishnachandran Vidya Academy of Science & Technology Thrissur 680 501, Kerala
On Finite Differences, Interpolation Methods and Power Series Expansions in Indian Mathematics

36. Introduction Aryabhata Brahmagupta Bhaskara I Madhava References
Madhava’s power series for sine : reformulation for
computations
For an arc s of a circle of radius R:
π 3 π 5 π 7
s 3 R 2 s 2 R 2 s 2 R 2
jiva = s− − − −· · ·
C 3! C 5! C 7!
3 5 11
R π2 R π2 R π
2
The five coefficients , , ... , were
3! 5! 11!
pre-computed to the desired degree of accuracy.
V. N. Krishnachandran Vidya Academy of Science & Technology Thrissur 680 501, Kerala
On Finite Differences, Interpolation Methods and Power Series Expansions in Indian Mathematics

37. Introduction Aryabhata Brahmagupta Bhaskara I Madhava References
Madhava’s power series for sine : Computational scheme
jiva = s−
s 3
(2220 39 40 )−
C
s 2
(273 57 47 )−
C
s 2
(16 05 41 )−
C
s 2
(33 06 )−
C
s 2
(44 ) −
C
V. N. Krishnachandran Vidya Academy of Science & Technology Thrissur 680 501, Kerala
On Finite Differences, Interpolation Methods and Power Series Expansions in Indian Mathematics

38. Introduction Aryabhata Brahmagupta Bhaskara I Madhava References
Madhava’s sine table
V. N. Krishnachandran Vidya Academy of Science & Technology Thrissur 680 501, Kerala
On Finite Differences, Interpolation Methods and Power Series Expansions in Indian Mathematics

39. Introduction Aryabhata Brahmagupta Bhaskara I Madhava References
Madhava’s sine table
The table is a set of numbers encoded in the katapayadi
scheme.
The table contains the values of jya (or, jiva) for arcs equal to
225 minutes, ... , 5400 minutes (twenty-four values).
The values are correct up to seven decimal places.
Madhava computed these values using the power series
expansion of the sine function.
V. N. Krishnachandran Vidya Academy of Science & Technology Thrissur 680 501, Kerala
On Finite Differences, Interpolation Methods and Power Series Expansions in Indian Mathematics

40. Introduction Aryabhata Brahmagupta Bhaskara I Madhava References
Madhava’s method vs. modern algorithm
Madhava formulated his result on the power series expansion as a
computational algorithm. This algorithm anticipates many ideas
used in the modern algorithm for computation of sine function.
Details in next slide ...
V. N. Krishnachandran Vidya Academy of Science & Technology Thrissur 680 501, Kerala
On Finite Differences, Interpolation Methods and Power Series Expansions in Indian Mathematics

41. Introduction Aryabhata Brahmagupta Bhaskara I Madhava References
Madhava’s method vs. modern algorithm
The first point is that Madhava’s method was indeed an
algorithm!
Madhava used an eleventh degree polynomial to compute
sine. Madhava used Taylor series approximation. Modern
algorithms use minmax polynomial of the same degree.
Madhava pre-computed the coefficients to the desired
accuracy. Modern algorithms also do the same.
Madhava essentially used Horner’s method for the efficient
computation of polynomials. Modern algorithms also use the
same method.
V. N. Krishnachandran Vidya Academy of Science & Technology Thrissur 680 501, Kerala
On Finite Differences, Interpolation Methods and Power Series Expansions in Indian Mathematics

42. Introduction Aryabhata Brahmagupta Bhaskara I Madhava References
Madhava’s power series for cosine and arctangent functions
Madhava had developed similar results for the computation of the
cosine function and also the arctangent function. See references
for details.
V. N. Krishnachandran Vidya Academy of Science & Technology Thrissur 680 501, Kerala
On Finite Differences, Interpolation Methods and Power Series Expansions in Indian Mathematics

43. Introduction Aryabhata Brahmagupta Bhaskara I Madhava References
References
References
V. N. Krishnachandran Vidya Academy of Science & Technology Thrissur 680 501, Kerala
On Finite Differences, Interpolation Methods and Power Series Expansions in Indian Mathematics

44. Introduction Aryabhata Brahmagupta Bhaskara I Madhava References
References
Walter Eugene Clark (1930). The Aryabhatiya of Aryabhata:
An ancient Indian work on mathematics and astronomy.
Chicago: The University of Chicago Press (p.19).
Meijering, Erik (March 2002). “A Chronology of Interpolation
From Ancient Astronomy to Modern Signal and Image
Processing”. Proceedings of the IEEE 90 (3): 319 - 342.
Gupta, R.C.. “Second order interpolation in Indian
mathematics upto the fifteenth century”. Indian Journal of
History of Science 4 (1 & 2): 86 - 98.
R.C. Gupta (1967). “Bhaskara I’ approximation to sine”.
Indian Journal of HIstory of Science 2 (2)
V. N. Krishnachandran Vidya Academy of Science & Technology Thrissur 680 501, Kerala
On Finite Differences, Interpolation Methods and Power Series Expansions in Indian Mathematics

45. Introduction Aryabhata Brahmagupta Bhaskara I Madhava References
References (continued)
Bag, A.K. (1976). “Madhava’s sine and cosine series”. Indian
Journal of History of Science (Indian National Academy of
Science) 11 (1): 54 - 57.
C.K. Raju (2007). Cultural foundations of mathematics: The
nature of mathematical proof and the transmission of calculus
from India to Europe in the 16 thc. CE. History of Philosophy,
Science and Culture in Indian Civilization. X Part 4. Delhi:
Centre for Studies in Civilizations. pp. 114 - 123.
Kim Plofker (2009). Mathematics in India. Princeton:
Princeton University Press. pp. 217 - 254.
Joseph, George Gheverghese (2009). A Passage to Infinity :
Medieval Indian Mathematics from Kerala and Its Impact.
Delhi: Sage Publications (Inda) Pvt. Ltd.
V. N. Krishnachandran Vidya Academy of Science & Technology Thrissur 680 501, Kerala
On Finite Differences, Interpolation Methods and Power Series Expansions in Indian Mathematics

46. Introduction Aryabhata Brahmagupta Bhaskara I Madhava References
Thanks
Thanks ...
V. N. Krishnachandran Vidya Academy of Science & Technology Thrissur 680 501, Kerala
On Finite Differences, Interpolation Methods and Power Series Expansions in Indian Mathematics