ON FINITE DIFFERENCES, INTERPOLATION METHODS AND POWER SERIES EXPANSIONS IN INDIAN MATHEMATICS V. N. Krishnachandrana a
Department of Computer Applications, Vidya Academy of Science & Technology, Thrissur – 680 501, Kerala
ABSTRACT This paper aims to bring to the attention of academicians, industry personnel, experts in modeling and simulation and young researchers and students, all of whom are specializing in mathematical modeling and scientific computations and all of who are coming from different parts of the world, to some of the greatest achievements of Indian astronomers and mathematicians in the area of numerical analysis. These are well known to historians of Indian mathematics. But the expositions of their findings are mainly restricted to literature on history of mathematics. Moreover, the focus of their studies is in the development of mathematical ideas and hence their presentations are couched in the associated specialised notations and terminology. Practicing numerical analysts and industrial personnel are unlikely to come across these writings. Even if they do they are likely to dismiss them as of none of their concern. In this paper we have discussed a few of these Indian accomplishments. We begin the paper with a discussion of Āryabhaṭa I's (476 – 550 CE) sine table which is the world's first instance of a difference table. Another world first, namely, the first use of a finite-difference interpolation formula, an achievement of Brahmagupta (598 – 668 CE), is then discussed. We then proceed to present a brief discussion on a non-polynomial interpolation formula for the sine function which is due to Bhaskara I (c. 600 – c. 680 CE) . One of the greatest advancements of Indian mathematics was the discovery of power series expansions of the sine, cosine and arctangent functions by Sangamagrama Madhava (c. 1350 – c. 1425 CE) in the fifteenth century CE, more than two centuries before these results were discovered in Europe. We have presented these series in Madhava's own words and also their renderings in modern notations. The paper concludes with a discussion on the suitability of using these ideas in the classroom as introductory and motivational material for teaching numerical analysis. Keywords: Finite difference, Interpolation method, Power series expansion, Āryabhaṭa I, Brahmagupta, Bhaskara I, Sangamagrama Madhava
INTRODUCTION There is a large body of literature dealing with the mathematical and astronomical achievements of India in the classical period. This period begins with Āryabhaṭa I who flourished during 476–550 CE. Āryabhaṭa I authored the great Indian classic Aryabhatiya which is considered to be the fountainhead of all subsequent investigations to mathematics and astronomy in India. Another great personality in the classical period of Indian mathematics was Brahmagupta (598 – 668 CE), the author the acclaimed Brahmashutasidhanta. Bhaskara I (c. 600 – c. 680 CE) was a mathematician-astronomer of the same period who is best known for his work Mahabhaskariya. (Bhaskara I is not to be confused with Bhaskara II, author of the much celebrated and studied Lilavati who lived during 1114 – 1185 CE.) Perhaps unknown to the rest of the world, a few centuries after Bhaskara II, a school of astronomy and mathematics was born, developed and flourished in a small area in the modern state of Kerala in India. The mathematical achievements of this school has only recently been studied, understood and commended upon by modern mathematical scholarship. The greatest single achievement of this school was the discovery of the power series expansions of the sine and cosine functions by Sangamagrama Madhava who is believed to have lived during 1350 1425. This paper has a very narrow focus. It discusses only a few results obtained by these great Indian mathematicians which have a bearing on the area of modern numerical analysis. ĀRYABHAṭA’S SINE TABLE : A TABLE OF DIFFERENCES Āryabhaṭa's sine table is a set of twenty-four numbers given in Āryabhaṭiya for the computation of the half-chords of a certain set of arcs of a circle. It is not a table in the modern sense of a mathematical table; that is, it is not a set of numbers arranged into rows and columns. These numbers are couched in the special numerical notation invented by Āryabhaṭa himself . These numbers appear as the tenth stanza (excluding the stanza containing the invocation and a stanza which is an explanation of Āryabhaṭa's numerical notation) in the first section of Āryabhatiya titled Daśagītikasūtra [1a]. What makes this table unique from a numerical analytic point of view is that it is not a set of values of the trigonometric sine function. It is a table of the first differences of the values of trigonometric sine function.
Fig 1. Diagram showing jya (or, jiva)
Āryabhaṭa‟s table was the first ever sine table to be constructed in the history of mathematics . The now lost tables of Hipparchus (c.190 BC – c.120 BC) and Menelaus (c.70–140 CE) and those of Ptolemy (c.AD 90 – c.168) were all tables of chords and not of halfchords.
Āryabhaṭa‟s sine table, both in his own notations and in modern notations, is given in Table 1. Fig. 1 indicates the terms used in the table. Table 1 : Āryabhaṭa‟s sine table Value in Āryabhaṭa’s table
Modern value In his Āryabhaṭa’s Sl. Arc (A) (in In his of jya (A) numerical value of No arcminutes) numerical In Arabic notation jya (A) (3438 × sin notation numerals (A)) (in ISO 15919 (in Devanagari) transliteration) 1 225 makhi 225 225′ 224.8560 मखि 2
BRAHMAGUPTA’S INTERPOLATION FORMULA Brahmagupata in early seventh century CE developed a second order interpolation formula. The Sanskrit couplet describing the formula can be found in Khandakadyaka a work of
Brahmagupta completed in 665 CE. The same couplet appears in Dhyana-graha-adhikara an earlier work of Brahmagupta but of uncertain date. However internal evidences suggest that Dhyana-graha-adhikara could be dated earlier than Brahmasphuta-siddhanta a work of Brahmagupta composed in 628 CE. Hence the invention of the second order interpolation formula by Brahmagupta should be placed near the beginning of the second quarter of the seventh century CE. Brahmagupta was the first to invent and use a finite difference interpolation formula in the history of mathematics [3,4]. Brahmagupta‟s interpolation formula is equivalent to modern-day second order NewtonStirling interpolation formula. Brahmagupta’s description of the scheme Given a set of tabulated values of a function f(x), let it be required to compute the value of f(x) at a given value of x, say, x=a. Let the tabulated values be as in the table below and let x r < a < xr+1. X x1 x2 … xr xr+1 xr+2 … xn f(x) f1 f2 … fr fr+1 fr+2 … fn Assuming that that the successively tabulated values of x are equally spaced with a common spacing of h, Āryabhaṭa had considered the table of first differences of the table of values of the function f(x). Writing Di = fi+1 – fi the following table can be formed: x x2 … f(x) D1 …
xr xr+1 … Dr Dr+1 …
Mathematicians prior to Brahmagupta used a simple liner interpolation formula. The linear interpolation formula to compute f(a) is, where t denotes a-x, f(a) = f(x) + (t/h) ( fr+1 – fr ) For the computation of f(a), Brhamagupta replaces fr+1 – fr with another expression which gives more accurate values and which amounts to using a second order interpolation formula. In Brahmagupta‟s terminology the difference Dr is the gatakhanda, meaning past difference or the difference that was crossed over, the difference fr+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 t. The new expression which replaces fr+1 – fr is called sphutabhogyakhanda. The description of shutabhogyakhanda is contained in a Sanskrit couplet (Dhyana-Graha-Upadesa-Adhyaya, 17; Khandaka Khadyaka, IX, 8)  which has been translated as follows: 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 shutabhogyakhanda 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 h.) Brahmagupta‟s method of computation of shutabhogyakhanda can be formulated in our notations, as follows: shutabhogyakhanda = (Dr + Dr+1)/2 ± (t/h) |Dr – Dr+1|/2 The „ + „ or „ – „ sign is to be taken according as (1/2)( Dr + Dr+1 ) is less than or greater than Dr+1, or equivalently, according as Dr < Dr+1 or Dr > Dr+1. Brahmagupta‟s expression can be put in the following form: shutabhogyakhanda = (Dr + Dr+1)/2 +(t/h) (Dr+1 – Dr )/2 This correction factor yields the following approximate value for f(a) : f(a) = f(xr) + (t/h) × sphuta-bhogya-khanda = f(xr) + (t/h) (Dr + Dr+1)/2 + (t2/h2) ) (Dr+1 – Dr )/2 This is the Stirling‟s interpolation formula truncated at the second order. It is not known how Brahmagupta arrived at his interpolation formula. It is also interesting to note that Brahmagupta has given a separate formula for the case where the values of the independent variable are not equally spaced. BHASKARA I’S SINE APPROXIMATION FORMULA In the seventh century there were several attempts were made to obtain approximation of functions. A rational approximation to the sine function was discovered by Bhaskara I. This formula is given in his treatise titled Mahabhaskariya. It is not known how Bhaskara I arrived at his approximation formula. The formula is elegant and simple and enables one to compute reasonably accurate values of trigonometric sines without using any geometry whatsoever. The approximation formula The formula is given in verses 17 – 19, Chapter VII, Mahabhaskariya of Bhaskara I. A translation of the verses is given below : (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. (The reference “Rsine-differences 225” is an allusion to Āryabhaṭa‟s sine table.) In modern mathematical notations, for an angle x in degrees, this formula gives
Accuracy of the formula The formula is applicable for values of x° in the range from 0 to 180. The formula is remarkably accurate in this range. It can be shown that the maximum absolute error in using the formula is around 0.0016. From a plot of the percentage value of the absolute error, it is clear that the maximum percentage error is less than 1.8. The approximation formula thus gives sufficiently accurate values of sines for all practical purposes. However it was not sufficient for the more accurate computational requirements of astronomy. The search for more accurate formulas by Indian astronomers eventually led to the discovery the power series expansions of sin x and cos x by Madhava of Sangamagrama (c. 1350 – c. 1425), the founder of the Kerala school of astronomy and mathematics. MADHAVA’S SERIES No surviving works of Madhava contain explicit statements regarding the expressions which are now referred to as Madhava series. However, in the writing of later members of the Kerala school of astronomy and mathematics like Nilakantha Somayaji and Jyeshthadeva one can find unambiguous attributions of these series to Madhava. It is also in the works of these later astronomers and mathematicians one can trace the Indian proofs of these series expansions [7,8]. These present-day counterparts of the infinite series expressions discovered by Madhava are given in Table 2. Table 2 : Madhava‟s series
sin x = x – x3 / 3! + x5 / 5! – x7 / 7! + … Power series for sine
cos x = 1 – x2 / 2! + x4 / 4! – x6 / 6! + Power series … for cosine
tan−1x = x – x3 / 3 + x5 / 5 – x7 / 7 + …
Power series for arctangent
π/4 = 1–1/3+1/5–1/7+…
Series for π
Western discoverers of the series and approximate dates of discovery Isaac Newton (1670) and Wilhelm Leibnitz (1676) Isaac Newton (1670) and Wilhelm Leibnitz (1676) James Gregory (1671) and Wilhelm Leibnitz (1676) James Gregory (1671) and Wilhelm Leibnitz (1676)
Due to lack of space, in this paper we have discussed only the power series expansion of the sine function. (For more details on other series, one may consult [9,10,11,12]).
Madhava series in Madhava’s own words Madhava‟s sine series is stated in verses 2.440 and 2.441 in Yukti-dipika commentary (Tantrasamgraha-vyakhya) by Sankara Variar. A translation of the verses follows : Multiply the arc by the square of the arc, and take the result of repeating that (any number of times). Divide (each of the above numerators) by the squares of the successive even numbers increased by that number and multiplied by the square of the radius. Place the arc and the successive results so obtained one below the other, and subtract each from the one above. These together give the jiva, as collected together in the verse beginning with “vidvan” etc. Rendering in modern notations Let r denote the radius of the circle and s the arc-length. Then the above passage translates into the following expression for jiva (or, jya).
Transformation to current notation Let θ be the angle subtended by the arc s at the centre of the circle. Then s = rθ and jiva = r sin θ. Substituting these in the last expression and simplifying we get
which is the infinite power series expansion of the sine function. Madhava’s reformulation for numerical computation The last line in the verse ′as collected together in the verse beginning with “vidvan” etc.′ is a reference to a reformulation of the series introduced by Madhava himself to make it convenient for easy computations for specified values of the arc and the radius. For such a reformulation, Madhava considers a circle one quarter of which measures 5400 minutes (say C minutes) and develops a scheme for the easy computations of the jiva′s of the various arcs of such a circle. Let R be the radius of a circle one quarter of which measures C. Madhava had already computed the value of π using his series formula for π. Using this value of π, namely 3.1415926535922, the radius R is computed as follows: R = 2 × 5400 / π = 3437.74677078493925 = 3437 arcminutes 44 arcseconds 48 sixtieths of an arcsecond = 3437′ 44′′ 48′′′ With this value for R,Madhava pre-computed the coefficients in the expression for jiva.The expressions vidvan, etc are these coefficients expressed in the Katapayadi scheme.
Substituting the numerical equivalents of these expressions, we get the following schem for the computation of jiva. Jiva = s – (s / C)3 [ (2220′ 39′′ 40′′′) – (s / C)2 [ (273′ 57′′ 47′′′) – (s / C)2 [ (16′ 5′′ 41′′′) – (s / C)2[ (33′′ 6′′′) – (s / C)2 (44′′′ ) ] ] ] ] This gives an approximation of jiva by its Taylor polynomial of the 11th order. It involves one division, six multiplications and five subtractions only. Madhava used this series to actually construct a table of values of the sine function. These values are correct to seven decimal places.
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