On finite differences, interpolation methods and power series expansions in indian mathematics

Introduction Aryabhata Brahmagupta Bhaskara I Madhava References

On Finite Differences, Interpolation Methodsand Power Series Expansions in Indian

Mathematics

V. N. KrishnachandranVidya 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


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




Introduction

Introduction




Objectives

To present some of the greatest achievements ofpre-modern Indian mathematicians ascontributions to the development of numericalanalysis.




Main themes

We present four themes:

1 Difference tables

2 Interpolation formulas

3 Rational polynomial approximations

4 Power series expansions




Aryabhata’s difference table

Aryabhata’s difference table




Aryabhata’s sine table

Aryabhata I’s (476 - 550 CE) celebrated work Aryabhatiyamcontains a sine table.

Aryabhata’s table was the first sine table ever constructed inthe 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 tablesof chords and not of half-chords.




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.)




What others tabulated

Pre-Aryabhata astronomers tabulatedvalues of chords for various arcs.




Aryabhata’s table

The stanza specifying Aryabhata’s table is the tenth one (excludingtwo preliminary stanzas) in the first section of Aryabhatiya titledDasagitikasutra.




Aryabhata’s table in his notation

(Table values are encoded in a scheme invented by Aryabhata.)




Aryabhata’s table in modern notation

225 224 222 219215 210 205 199191 183 174 164154 143 131 119106 93 79 65

51 37 22 7

(Read numbers row-wise.)




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 thevalues of jyas.




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)

225 225 225 224.8560450 224 449 448.7490675 222 671 670.7205900 219 890 889.8199

1125 215 1105 1105.10891350 210 1315 1315.66561575 205 1520 1520.58851800 199 1719 1719.0000

......

......

Values in second column are differences of values in third column.




Brahmagupata’s interpolation formula

Brahmagupata’s interpolationformula




Brahmagupta

Brahmagupta’s (598 - 668 CE) works contain Sanskrit versesdescribing a second order interpolation formula.

The earliest such work is Dhyana-graha-adhikara, a treatisecompleted in early seventh century CE.

Brahmagupta was the first to invent and use an interpolationformula of the second order in the history of mathematics.




Brahmagupta’s verse

(Earliest appearance: Dhyana-graha-adhikara, sloka 17)




Translation of Brahmagupta’s verse

Multiply half the difference of the tabular differences crossedover and to be crossed over by the residual arc and divide by900 minutes (= h). By the result (so obtained) increase ordecrease half the sum of the same (two) differences, accordingas this (semi-sum) is less than or greater than the differenceto be crossed over. We get the true functional differences tobe crossed over.

(Gupta, R.C.. “Second order interpolation in Indian mathematicsupto the fifteenth century”. Indian Journal of History of Science 4(1 & 2): pp.86 - 98.)




Brahamagupta’s verse : Interpretation (notations)

Consider a set of values of f (x) tabulated at equally spacedvalues of x :

x x1 · · · xr xr+1 · · · xnf (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.




Brahamagupta’s verse : Interpretation

True functional difference =

Dr + Dr+1

2± t

h

|Dr − Dr+1|2

according asDr + Dr+1

2is less than or greater than Dr+1.

True functional difference =

Dr + Dr+1

2+

t

h

Dr+1 − Dr

2




Brahamagupta’s verse : Interpretation

The functional difference Dr+1 in the approximation formula

f (a) = f (xr ) +t

hDr+1

is replaced by this true functional difference.

The resulting approximation fromula is Brahmagupta’sinterpolation formula.




Brahmagupta’s interpolation formula

Brahmagupta’s interpolation formula:

f (a) = f (xr ) +t

h

[Dr + Dr+1

2+

t

h

Dr+1 − Dr

2

]This is the Stirlings interpolation formula truncated at thesecond order.




Bhaskara I’s approximation formula

Bhaskara I’s approximationformula




Bhaskara I

Bhaskara I (c.600 - c.680), a seventh century Indianmathematician (not the author of Lilavati).

Mahabhaskariya, a treatise by Bhaskara I, contains a versedescribing a rational polynomial approximation to sin x .




Bhaskara’s verse

(Mahabhaskariya, VII, 17 - 19)




Bhaskara’s verse: Translation

(Now) I briefly state the rule (for finding the bhujaphala and thekotiphala, etc.) without making use of the Rsine-differences 225,etc. Subtract the degrees of a bhuja (or koti) from the degrees ofa half circle (that is, 180 degrees). Then multiply the remainder bythe degrees of the bhuja or koti and put down the result at twoplaces. At one place subtract the result from 40500. By one-fourthof the remainder (thus obtained), divide the result at the otherplace as multiplied by the anthyaphala (that is, the epicyclicradius). Thus is obtained the entire bahuphala (or, kotiphala) forthe sun, moon or the star-planets. So also are obtained the directand inverse Rsines.

(R.C. Gupta (1967). Bhaskara I’ approximation to sine. IndianJournal of HIstory of Science 2 (2)





Let x be an angle measured in degrees.

sin x =4x(180− x)

40500− x(180− x)





This is a rational polynomial approximation to sin x whenangle x is expressed in degrees.

It is not known how Bhaskara arrived at this formula.




Accuracy of Bhaskara’s approximation formula

The maximum absolute error in using the formula is around 0.0016.




Madhava’s power series expansions

Madhava’s power seriesexpansions




Sangamagrama Madhava

Madhava flourished during c.1350 - c.1425.

Madhava founded the so called Kerala School of Astronomyand Mathematics.

Only a few minor works of Madhava have survived.

There are copious references and tributes to Madhava in theworks of his followers.




Madhava’s power series for sine in Madhava’s words




Madhava’s power series for sine in English

Multiply the arc by the square of itself (multiplication beingrepeated any number of times) and divide the result by theproduct of the squares of even numbers increased by thatnumber and the square of the radius (the multiplication beingrepeated the same number of times). The arc and the resultsobtained from above are placed one above the other and aresubtracted systematically one from its above. These togethergive jiva collected here as found in the expression beginningwith vidwan etc.

(A.K. Bag (1975). Madhava’s sine and cosine series. IndianJournal of History of Science 11 (1): pp.54-57.)




Madhava’s power series for sine in modern notations

Let θ be the angle subtended at the center of a circle of radius rby an arc of length s. Then jiva ( = jya) of s is r sin θ.

jiva = s

−[s · s2

(22 + 2)r 2

−[s · s2

(22 + 2)r 2· s2

(42 + 4)r 2

−[s · s2

(22 + 2)r 2· s2

(42 + 4)r 2· s2

(62 + 6)r 2− · · ·

]]]




Madhava’s power series for sine : reformulation forcomputations

Chose a circle the length of a quarter of which is C = 5400minutes.

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.




Madhava’s power series for sine : reformulation forcomputations

For an arc s of a circle of radius R:

jiva = s−( s

C

)3 [R(π2

)3

3!−( s

C

)2 [R(π2

)5

5!−( s

C

)2 [R(π2

)7

7!−· · ·

]]]

The five coefficientsR(π2

)3

3!,

R(π2

)5

5!, ... ,

R(π2

)11

11!were

pre-computed to the desired degree of accuracy.




Madhava’s power series for sine : Computational scheme

jiva = s−( s

C

)3 [(2220′ 39′′ 40′′′)−( s

C

)2 [(273′ 57′′ 47′′′)−( s

C

)2 [(16′ 05′′ 41′′′)−( s

C

)2 [(33′′ 06′′′)−( s

C

)2(44′′′)−

]]]]




Madhava’s sine table




Madhava’s sine table

The table is a set of numbers encoded in the katapayadischeme.

The table contains the values of jya (or, jiva) for arcs equal to225 minutes, ... , 5400 minutes (twenty-four values).

The values are correct up to seven decimal places.

Madhava computed these values using the power seriesexpansion of the sine function.




Madhava’s method vs. modern algorithm

Madhava formulated his result on the power series expansion as acomputational algorithm. This algorithm anticipates many ideasused in the modern algorithm for computation of sine function.

Details in next slide ...




Madhava’s method vs. modern algorithm

The first point is that Madhava’s method was indeed analgorithm!

Madhava used an eleventh degree polynomial to computesine. Madhava used Taylor series approximation. Modernalgorithms use minmax polynomial of the same degree.

Madhava pre-computed the coefficients to the desiredaccuracy. Modern algorithms also do the same.

Madhava essentially used Horner’s method for the efficientcomputation of polynomials. Modern algorithms also use thesame method.




Madhava’s power series for cosine and arctangent functions

Madhava had developed similar results for the computation of thecosine function and also the arctangent function. See referencesfor details.




References

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 InterpolationFrom Ancient Astronomy to Modern Signal and ImageProcessing”. Proceedings of the IEEE 90 (3): 319 - 342.

Gupta, R.C.. “Second order interpolation in Indianmathematics upto the fifteenth century”. Indian Journal ofHistory of Science 4 (1 & 2): 86 - 98.

R.C. Gupta (1967). “Bhaskara I’ approximation to sine”.Indian Journal of HIstory of Science 2 (2)




References (continued)

Bag, A.K. (1976). “Madhava’s sine and cosine series”. IndianJournal of History of Science (Indian National Academy ofScience) 11 (1): 54 - 57.

C.K. Raju (2007). Cultural foundations of mathematics: Thenature of mathematical proof and the transmission of calculusfrom 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.




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