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Introduction

Aryabhata

Brahmagupta

Bhaskara I

Madhava

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

References

Introduction

Aryabhata

Brahmagupta

Bhaskara I

Madhava

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

References

Introduction

Aryabhata

Brahmagupta

Bhaskara I

Madhava

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

References

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

Introduction

Aryabhata

Brahmagupta

Bhaskara I

Madhava

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

References

Introduction

Aryabhata

Brahmagupta

Bhaskara I

Madhava

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

References

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

Introduction

Aryabhata

Brahmagupta

Bhaskara I

Madhava

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

References

Introduction

Aryabhata

Brahmagupta

Bhaskara I

Madhava

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

References

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

Introduction

Aryabhata

Brahmagupta

Bhaskara I

Madhava

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

References

Introduction

Aryabhata

Brahmagupta

Bhaskara I

Madhava

Aryabhata’s table in modern notation

225 215 191 154 106 51

224 210 183 143 93 37

222 205 174 131 79 22

219 199 164 119 65 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

References

Introduction

Aryabhata

Brahmagupta

Bhaskara I

Madhava

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

References

Introduction

Aryabhata

Brahmagupta

Bhaskara I

Madhava

References

Aryabahata’s table as a table of first differences Angle (A) (in minutes) 225 450 675 900 1125 1350 1575 1800 .. .

Value in A’bhata’s table 225 224 222 219 215 210 205 199 .. .

A’bhata’s value of jya (A) 225 449 671 890 1105 1315 1520 1719 .. .

Modern value of jya (A) 224.8560 448.7490 670.7205 889.8199 1105.1089 1315.6656 1520.5885 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

Introduction

Aryabhata

Brahmagupta

Bhaskara I

Madhava

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

References

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

Introduction

Aryabhata

Brahmagupta

Bhaskara I

Madhava

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

References

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

Introduction

Aryabhata

Brahmagupta

Bhaskara I

Madhava

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

References

Introduction

Aryabhata

Brahmagupta

Bhaskara I

Madhava

Brahamagupta’s verse : Interpretation

True functional difference = Dr + Dr +1 t |Dr − Dr +1 | ± 2 h 2 Dr + Dr +1 is less than or greater than Dr +1 . 2 True functional difference = according as

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

References

Introduction

Aryabhata

Brahmagupta

Bhaskara I

Madhava

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

References

Introduction

Aryabhata

Brahmagupta

Bhaskara I

Madhava

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

References

Introduction

Aryabhata

Brahmagupta

Bhaskara I

Madhava

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

References

Introduction

Aryabhata

Brahmagupta

Bhaskara I

Madhava

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

References

Introduction

Aryabhata

Brahmagupta

Bhaskara I

Madhava

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

References

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

Introduction

Aryabhata

Brahmagupta

Bhaskara I

Madhava

Bhaskara I’s approximation formula

Let x be an angle measured in degrees. sin x =

4x(180 − 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

References

Introduction

Aryabhata

Brahmagupta

Bhaskara I

Madhava

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

References

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

Introduction

Aryabhata

Brahmagupta

Bhaskara I

Madhava

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

References

Introduction

Aryabhata

Brahmagupta

Bhaskara I

Madhava

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

References

Introduction

Aryabhata

Brahmagupta

Bhaskara I

Madhava

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

References

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

Introduction

Aryabhata

Brahmagupta

Bhaskara I

Madhava

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 (22 + 2)r 2 h s2 − s· 2 (2 + 2)r 2 h s2 − s· 2 (2 + 2)r 2

h − s·

s2 (42 + 4)r 2 iii s2 s2 · 2 · − · · · (4 + 4)r 2 (62 + 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

References

Introduction

Aryabhata

Brahmagupta

Bhaskara I

Madhava

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

References

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: iii  s 3 h R π
3  s  2 h R π
5  s 2 h R π
7 2 2 2 − − −· · · jiva = s− C 3! C 5! C 7!


3 5 11 R π2 R π2 R π2 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

Introduction

Aryabhata

Brahmagupta

Bhaskara I

Madhava

References

Madhava’s power series for sine : Computational scheme jiva = s−  s 3 h (22200 3900 40000 )− C  s 2 h (2730 5700 47000 )− C  s 2 h (160 0500 41000 )− C  s 2 h (3300 06000 )− C  s 2 iiii (44000 ) − 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

Introduction

Aryabhata

Brahmagupta

Bhaskara I

Madhava

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

References

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

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

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

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

Introduction

Aryabhata

Brahmagupta

Bhaskara I

Madhava

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

References

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

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

Introduction

Aryabhata

Brahmagupta

Bhaskara I

Madhava

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

References