Intro
Kerala School
Terminology
π value
Madhava’s series
Sine table
Analysis
Comparison
Conclusion
On Sangamagr¯ama M¯adhava’s (c.1350 - c.1425) Algorithms for the Computation of Sine and Cosine Functions V.N. Krishnachandran, Reji C. Joy, Siji K.B. Vidya Academy of Science and Technology, Thrissur - 680501, Kerala.
V N Krishnachandran, Reji C Joy, Siji K B On Sangamagr¯ ama M¯ adhava’s (c.1350 - c.1425) Algorithms
Biblio
Intro
Kerala School
Terminology
π value
Madhava’s series
Sine table
Analysis
Outline 1 Introduction 2 Kerala School of Astronomy and Mathematics 3 Some terminology 4 Madhava’s value for π 5 Madhava’s series 6 Madhava’s sine table 7 Analysis of Madhava’s computational schemes 8 Comparison with modern algorithms 9 Conclusion 10 Bibliography V N Krishnachandran, Reji C Joy, Siji K B On Sangamagr¯ ama M¯ adhava’s (c.1350 - c.1425) Algorithms
Comparison
Conclusion
Biblio
Intro
Kerala School
Terminology
π value
Madhava’s series
Sine table
Analysis
Comparison
Introduction
Classical period in Indian mathematics: Aryabhata I (476 - 550 CE) : Author of Aryabhatiya. Varahamihira (505 - 587 CE) Brhamagupta (598 - 670 CE) Bhaskara I (c.600 - c.680 CE) Govindasvami (c.800 - c.860 CE) Aryabhata II (c.920 - c.1000 CE) Bhaskara II (1114 - 1185 CE) : Author of Lilavati.
V N Krishnachandran, Reji C Joy, Siji K B On Sangamagr¯ ama M¯ adhava’s (c.1350 - c.1425) Algorithms
Conclusion
Biblio
Intro
Kerala School
Terminology
π value
Madhava’s series
Sine table
Analysis
Comparison
Conclusion
Introduction
Development of mathematics in India did not end with Bhaskara II Mathematics continued to flourish in Kerala unknown to the rest of the world! C.M. Whish, an East India Company official, wrote about the Kerala School of Astronomy and Mathematics in 1832. But nobody noticed it.
V N Krishnachandran, Reji C Joy, Siji K B On Sangamagr¯ ama M¯ adhava’s (c.1350 - c.1425) Algorithms
Biblio
Intro
Kerala School
Terminology
π value
Madhava’s series
Sine table
Analysis
Comparison
Kerala School : Geographical area (Trikkandiyur)
Map showing Trikkandiyur and nearby places V N Krishnachandran, Reji C Joy, Siji K B On Sangamagr¯ ama M¯ adhava’s (c.1350 - c.1425) Algorithms
Conclusion
Biblio
Intro
Kerala School
Terminology
π value
Madhava’s series
Sine table
Analysis
Comparison
Conclusion
Kerala School : Period
Pre-Madhava period : Begins with Vararuci (4 th century CE) ending with Govinda Bhattathiri (1237 - 1295 CE) Madhava and his disciples (c.1350 - c.1650 CE) Later period (c.1650 - c.1850)
V N Krishnachandran, Reji C Joy, Siji K B On Sangamagr¯ ama M¯ adhava’s (c.1350 - c.1425) Algorithms
Biblio
Intro
Kerala School
Terminology
π value
Madhava’s series
Sine table
Analysis
Comparison
Conclusion
Kerala School : Founder
Sangamagrama Madhava (c.1350 - c.1425 CE) No personal details about M¯adhava has come to light. Sangamagr¯ama is surmised to be a reference to his place of residence. Some historians have identified Sangamagr¯ama as modern day Irinjalakuda in Thrissur District in Kerala. There are references and tributes to M¯adhava in the works of other authors about whom accurate details are available.
V N Krishnachandran, Reji C Joy, Siji K B On Sangamagr¯ ama M¯ adhava’s (c.1350 - c.1425) Algorithms
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Intro
Kerala School
Terminology
π value
Madhava’s series
Sine table
Analysis
Comparison
Conclusion
Kerala School : Prominent members
Paramesvara (c.1380 - c.1460), a pupil of Sangamagrama M¯adhava : Promulgator of Drigganita system of astronomical computations in Kerala. Da.modara, another prominent member of the Kerala school, was Paramesvara’s son and also his pupil. N¯ilakant.ha S¯omay¯aji (1444 - 1544), a pupil of Parameshvara. Author of Tantrasamgraha completed in 1501.
V N Krishnachandran, Reji C Joy, Siji K B On Sangamagr¯ ama M¯ adhava’s (c.1350 - c.1425) Algorithms
Biblio
Intro
Kerala School
Terminology
π value
Madhava’s series
Sine table
Analysis
Comparison
Conclusion
Kerala School : Other prominent members
Jy¯es.t.had¯eva (c.1500 - c.1600) Author of Yukt.ibh¯as.a. Yukt.ibh¯as.a is composed in Malayalam. It contains clear statements of the power series expansions of the sine and cosine functions and also their proofs. This treatise proves that the idea of proof is not alien to Indian mathematics.
V N Krishnachandran, Reji C Joy, Siji K B On Sangamagr¯ ama M¯ adhava’s (c.1350 - c.1425) Algorithms
Biblio
Intro
Kerala School
Terminology
π value
Madhava’s series
Sine table
Analysis
Comparison
Conclusion
Some terminology: C¯apa, jy¯a, koti-jy¯a, utkrama-jy¯a
Diagram showing c¯apa, jy¯a , etc.
V N Krishnachandran, Reji C Joy, Siji K B On Sangamagr¯ ama M¯ adhava’s (c.1350 - c.1425) Algorithms
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Intro
Kerala School
Terminology
π value
Madhava’s series
Sine table
Analysis
Comparison
Conclusion
Some terminology: C¯apa, jy¯a, koti-jy¯a, utkrama-jy¯a
jy¯a of arc AB = BM = R sin
s R
koti-jy¯a of arc AB = OM = R cos
s R
utkrama-jy¯a (or ´sara of arc AB) = MA = R 1 − cos V N Krishnachandran, Reji C Joy, Siji K B On Sangamagr¯ ama M¯ adhava’s (c.1350 - c.1425) Algorithms
s R
Biblio
Intro
Kerala School
Terminology
π value
Madhava’s series
Sine table
Analysis
Comparison
Conclusion
Some terminology: Katapayadi scheme The Kat.apay¯adi scheme is a method for representing numbers using letters of the Sanskrit alphabet. Table: Mapping of digits to letters in katapay¯adi scheme
1
2
3
4
5
6
7
8
9
0
ka
kha
ga
gha
na ˙
ca
cha
ja
jha
n˜a
t.a pa
t.ha pha
d.a ba
d.ha bha
n.a ma
ta
tha
da
dha
na
-
-
-
-
-
ya
ra
la
va
´sa
s.a
sa
ha
-
-
V N Krishnachandran, Reji C Joy, Siji K B On Sangamagr¯ ama M¯ adhava’s (c.1350 - c.1425) Algorithms
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Analysis
Comparison
Conclusion
Some terminology: Katapayadi scheme
Consonants have numerals assigned as per Table in previous slide. All stand-alone vowels like a and i are assigned to 0. In case of a conjunct, consonants attached to a non-vowel will be valueless. Numbers are written in increasing place values from left to right. The number 386 which denotes 3 × 100 + 8 × 10 + 6 in modern notations would be written as 683 in pre-modern Indian traditions.
V N Krishnachandran, Reji C Joy, Siji K B On Sangamagr¯ ama M¯ adhava’s (c.1350 - c.1425) Algorithms
Biblio
Intro
Kerala School
Terminology
π value
Madhava’s series
Sine table
Analysis
Comparison
Conclusion
Madhava’s value for π
Madhava derived the following series for the computation of π: 1 1 1 π = 1 − + − + ··· 4 3 5 7 The series is known as the Gregory (1638 1675) series. But it was known in Kerala more than two centuries before Gregory. Using this series and several correction terms M¯adhava computed the following value for π: π = 3.1415926535922.
V N Krishnachandran, Reji C Joy, Siji K B On Sangamagr¯ ama M¯ adhava’s (c.1350 - c.1425) Algorithms
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Kerala School
Terminology
π value
Madhava’s series
Sine table
Analysis
Comparison
Conclusion
Madhava’s series for sine: In Madhava’s own words
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 j¯iva, as collected together in the verse beginning with “vidv¯an” etc.
V N Krishnachandran, Reji C Joy, Siji K B On Sangamagr¯ ama M¯ adhava’s (c.1350 - c.1425) Algorithms
Biblio
Intro
Kerala School
Terminology
π value
Madhava’s series
Sine table
Analysis
Comparison
Conclusion
Biblio
Madhava’s series for sine: Rendering in modern notations Madhava’s series for sine function: j¯iva
= s h − s·
s2 (22 + 2)r 2 h s2 − s· 2 (2 + 2)r 2 h s2 − s· 2 (2 + 2)r 2 iii − ···
V N Krishnachandran, Reji C Joy, Siji K B On Sangamagr¯ ama M¯ adhava’s (c.1350 - c.1425) Algorithms
s2 (42 + 4)r 2 s2 s2 · 2 · (4 + 4)r 2 (62 + 6)r 2 ·
Intro
Kerala School
Terminology
π value
Madhava’s series
Sine table
Analysis
Comparison
Conclusion
Madhava’s series for sine: As power series for sine
Let θ be the angle subtended by the arc s at the center of the circle. s = r θ and j¯iva = rsinθ. Substitute these in the expression given by Madhava. We get θ3 θ5 θ7 + − + ··· 3! 5! 7! This is the infinite power series expansion of the sine function. sin θ = θ −
V N Krishnachandran, Reji C Joy, Siji K B On Sangamagr¯ ama M¯ adhava’s (c.1350 - c.1425) Algorithms
Biblio
Intro
Kerala School
Terminology
π value
Madhava’s series
Sine table
Analysis
Comparison
Conclusion
Biblio
Madhava’s series for sine: Reformulation for computation
M¯adhava considers one quarter of a circle. The length of the quarter-circle is taken as 5400 minutes (say C minutes). He computes the radius R of the circle: R = 2 × 5400/π = 3437.74677078493925 = 34370
V N Krishnachandran, Reji C Joy, Siji K B On Sangamagr¯ ama M¯ adhava’s (c.1350 - c.1425) Algorithms
4400
48000
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π value
Madhava’s series
Sine table
Analysis
Comparison
Conclusion
Biblio
Madhava’s series for sine: Reformulation for computation
The expression for j¯iva is now put in the following form: s5 s3 + − ··· R 2 (22 + 2) R 4 (22 + 2)(42 + 4) s 3 h R π 3 s 2 h R π 5 ii 2 2 =s− − − ··· C 3! C 5!
jiva = s −
V N Krishnachandran, Reji C Joy, Siji K B On Sangamagr¯ ama M¯ adhava’s (c.1350 - c.1425) Algorithms
Intro
Kerala School
Terminology
π value
Madhava’s series
Sine table
Analysis
Comparison
Conclusion
Biblio
Madhava’s series for sine: Pre-computaion of coefficients
No. 1
Expression R × (π/2)3 /3!
22200 3900 40000
Value
2
R × (π/2)5 /5!
2730 5700 47000
3
R × (π/2)7 /7!
160 0500 41000
4 5
R × (π/2)9 /9! R × (π/2)11 /11!
V N Krishnachandran, Reji C Joy, Siji K B On Sangamagr¯ ama M¯ adhava’s (c.1350 - c.1425) Algorithms
3300 06000 44000
In kat.apay¯adi system ni-rvi-ddh¯a-ngana-r¯e-ndra-rung sa-rv¯a-rtha´s¯i-la-sthi-ro ka-v¯i-´sani-ca-ya tu-nna-ba-la vi-dv¯an
Intro
Kerala School
Terminology
π value
Madhava’s series
Sine table
Analysis
Comparison
Conclusion
Biblio
Madhava’s series for sine: Reformulation for computation
Madhava’s polynomial approximation to sine function: jiva = s − (s/C )3 [(22200 3900 40000 ) − (s/C )2 [(2730 5700 47000 − (s/C )2 [(160 0500 41000 ) − (s/C )2 [(3300 06000 ) − (s/C )2 (44000 )]]]]
V N Krishnachandran, Reji C Joy, Siji K B On Sangamagr¯ ama M¯ adhava’s (c.1350 - c.1425) Algorithms
Intro
Kerala School
Terminology
π value
Madhava’s series
Sine table
Analysis
Comparison
Madhava’s series for cosine Madhava’s polynomial approximation to cosine function: ´sara = (s/C )2 [(42410 0900 00000 ) − (s/C )2 [(8720 03000 05000 ) − (s/C )2 [(0710 4300 24000 ) − (s/C )2 [(030 09000 37000 ) − (s/C )2 [(0500 12000 ) − (s/C )2 (06000 )]]]]]
V N Krishnachandran, Reji C Joy, Siji K B On Sangamagr¯ ama M¯ adhava’s (c.1350 - c.1425) Algorithms
Conclusion
Biblio
Intro
Kerala School
Terminology
π value
Madhava’s series
Sine table
Analysis
Comparison
Conclusion
Madhava’s sine table
(continued in next slide) V N Krishnachandran, Reji C Joy, Siji K B On Sangamagr¯ ama M¯ adhava’s (c.1350 - c.1425) Algorithms
Biblio
Intro
Kerala School
Terminology
π value
Madhava’s series
Madhava’s sine table
V N Krishnachandran, Reji C Joy, Siji K B On Sangamagr¯ ama M¯ adhava’s (c.1350 - c.1425) Algorithms
Sine table
Analysis
Comparison
Conclusion
Biblio
Intro
Kerala School
Terminology
π value
Madhava’s series
Sine table
Analysis
Comparison
Conclusion
Analysis: These are algorithms!
Madhava and his followers were developing algorithms for the computation of sine and cosine functions. The expansions are given as a step by step procedure for computations the function values. In Jy¯es.t.had¯eva’s Yuktibh¯as.a, the author has used the term kriya-krama which translates into procedure or an algorithm.
V N Krishnachandran, Reji C Joy, Siji K B On Sangamagr¯ ama M¯ adhava’s (c.1350 - c.1425) Algorithms
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Terminology
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Madhava’s series
Sine table
Analysis
Comparison
Conclusion
Analysis: Use of polynomial approximation
M¯adhava uses polynomial approximations. For sine function, an 11 th degree polynomial is used. For the cosine function, a 12 th degree polynomial is used. The orders of the polynomials were decided by the requirements of accuracy. The values computed by M¯adhava could also be obtained by other methods. But M¯adhava did seek and get a general method in the form of power series expansions.
V N Krishnachandran, Reji C Joy, Siji K B On Sangamagr¯ ama M¯ adhava’s (c.1350 - c.1425) Algorithms
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Conclusion
Analysis: Pre-computation of the coefficients
Madhava pre-computed the coefficients appearing in the polynomial approximations.
V N Krishnachandran, Reji C Joy, Siji K B On Sangamagr¯ ama M¯ adhava’s (c.1350 - c.1425) Algorithms
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Sine table
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Comparison
Conclusion
Analysis: Use of Horner’s scheme
M¯adhava had applied what is now known as the Horner’s scheme for the computation of polynomials. The scheme is now attributed to William George Horner (1786 1837) who was a British mathematician and schoolmaster.
V N Krishnachandran, Reji C Joy, Siji K B On Sangamagr¯ ama M¯ adhava’s (c.1350 - c.1425) Algorithms
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Sine table
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Conclusion
Analysis: Horner’s scheme
Given the polynomial p(x) =
n X
ai x i = a0 + a1 x + a2 x 2 + a3 x 3 + · · · + an x n ,
i=0
let it be required to evaluate p(x) at a specific value of x.
V N Krishnachandran, Reji C Joy, Siji K B On Sangamagr¯ ama M¯ adhava’s (c.1350 - c.1425) Algorithms
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Conclusion
Analysis: Horner’s scheme To compute p(x), the polynomial is expressed in the form p(x) = a0 + x(a1 + x(a2 + · · · + x(an−1 + an x) · · · )). Then apply the following algorithm for computing p(x): bn = an bn−1 = an−1 + bn x ··· b1 = a1 + b2 x b0 = a0 + b1 x. b0 is the required value of the polynomial. V N Krishnachandran, Reji C Joy, Siji K B On Sangamagr¯ ama M¯ adhava’s (c.1350 - c.1425) Algorithms
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Analysis: Use of Horner’s scheme
M¯adhava had actually implemented Horner’s scheme in his algorithms. The method was known to Isaac Newton in 1669, the Chinese mathematician Qin Jiushao in the 13th century, and even earlier to the Persian Muslim mathematicians. M¯adhava’s was the first conscious and deliberate application of the scheme in a computational algorithm with the intention of reducing the complexity of numerical procedures.
V N Krishnachandran, Reji C Joy, Siji K B On Sangamagr¯ ama M¯ adhava’s (c.1350 - c.1425) Algorithms
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Conclusion
Analysis: Simultaneous computation of sine and cosine In many modern implementations of routines for the calculations of the sine and cosine functions, there would be one routine for the simultaneous computation of sine and cosine. Whenever sine or cosine is required, the other would also be required. So a common algorithm which returns both values simultaneously would be more time efficient and economical. It would appear that M¯adhava had anticipated such a scenario. This is evidenced by the description of one common procedure for the evaluation of sine and cosine functions in Yuktibh¯as.a.
V N Krishnachandran, Reji C Joy, Siji K B On Sangamagr¯ ama M¯ adhava’s (c.1350 - c.1425) Algorithms
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Comparison with programmes in Open64 Compiler
Compare, for example, with the programme included in the Open64 Compiler developed by Computer Architecture and Parallel Systems Laboratory in University of Delaware. These programs are not using the polynomials used by M¯adhava. They are using the minimax polynomial computed using the Remez algorithm to improve the accuracy of computations.
V N Krishnachandran, Reji C Joy, Siji K B On Sangamagr¯ ama M¯ adhava’s (c.1350 - c.1425) Algorithms
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Comparison: Coefficients for computation of sine 00135 /* coefficients for polynomial approximation of sin on +/- pi/4 */ 00136 00137 static const du S[] = 00138 00139 D(0x3ff00000, 0x00000000), 00140 D(0xbfc55555, 0x55555548), 00141 D(0x3f811111, 0x1110f7d0), 00142 D(0xbf2a01a0, 0x19bfdf03), 00143 D(0x3ec71de3, 0x567d4896), 00144 D(0xbe5ae5e5, 0xa9291691), 00145 D(0x3de5d8fd, 0x1fcf0ec1), 00146 ; V N Krishnachandran, Reji C Joy, Siji K B On Sangamagr¯ ama M¯ adhava’s (c.1350 - c.1425) Algorithms
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Comparison: Coefficients for computation of cosine 00148 /* coefficients for polynomial approximation of cos on +/- pi/4 */ 00149 00150 static const du C[] = 00151 00152 D(0x3ff00000, 0x00000000), 00153 D(0xbfdfffff, 0xffffff96), 00154 D(0x3fa55555, 0x5554f0ab), 00155 D(0xbf56c16c, 0x1640aaca), 00156 D(0x3efa019f, 0x81cb6a1d), 00157 D(0xbe927df4, 0x609cb202), 00158 D(0x3e21b8b9, 0x947ab5c8), 00159 ; V N Krishnachandran, Reji C Joy, Siji K B On Sangamagr¯ ama M¯ adhava’s (c.1350 - c.1425) Algorithms
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Comparison: Polynomial approximations
00329 00330 00331 00332 00333 00334 00335 00336 00337
xsq = x*x; cospoly = (((((C[6].d*xsq + C[5].d)*xsq + C[4].d)*xsq + C[3].d)*xsq + C[2].d)*xsq + C[1].d)*xsq + C[0].d; sinpoly = (((((S[6].d*xsq + S[5].d)*xsq + S[4].d)*xsq + S[3].d)*xsq + S[2].d)*xsq + S[1].d)*(xsq*x) + x;
V N Krishnachandran, Reji C Joy, Siji K B On Sangamagr¯ ama M¯ adhava’s (c.1350 - c.1425) Algorithms
Conclusion
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Conclusion
Madhava was a great computationalist. Madhava pioneered several new ideas and techniques for the efficient computaion of functions. In particular, Madhava implemented the following concepts to devise a scheme for the computation of sine and cosine functions: Use of an approximating polynomial. Pre-computation of coefficients. Use of Horner’s scheme for the evaluation of polynomials. Simultaneous computation of sine and cosine functions.
V N Krishnachandran, Reji C Joy, Siji K B On Sangamagr¯ ama M¯ adhava’s (c.1350 - c.1425) Algorithms
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Bibliography 1 G. G. Joseph, A Passage to Infinity: Medieval Indian Mathematics from Kerala and Its Impact. New Delhi: Sage Publications Pvt. Ltd, 2009. 2 C. M. Whish, On the hindu quadrature of the circle and the infinite series of the proportion of the circumference to the diameter exhibited in the four sastras, the tantra sahgraham, yucti bhasha, carana padhati and sadratnamala, Transactions of the Royal Asiatic Society of Great Britain and Ireland (Royal Asiatic Society of Great Britain and Ireland, vol. 3 (3), pp. 509 523, 1834. 3 I. S. B. Murthy, A modern introduction to ancient Indian mathematics. New Delhi: New Age International Publishers, 1992. V N Krishnachandran, Reji C Joy, Siji K B On Sangamagr¯ ama M¯ adhava’s (c.1350 - c.1425) Algorithms
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Bibliography 4 V. J. Katz, The mathematics of Egypt, Mesopotemia, China, India and Islam: A source book. Princeton: Princeton University Press, 2007, ch. Chapter 4 : Mathematics in India IV. Kerala School (pp. 480 - 495). 5 K. Plofker, Mathematics in India. Princeton, NJ: Princeton University Press, 2010. 6 Madhava of sangamagramma. [Online]. Available: http://wwwgap. dcs.st-and.ac.uk/ history/Projects/Pearce/Chapters/Ch9 3.html 7 K. V. Sarma and V. S. Narasimhan, Tantrasamgraha, Indian Journal of History of Science, vol. 33 (1), Mar. 1998.
V N Krishnachandran, Reji C Joy, Siji K B On Sangamagr¯ ama M¯ adhava’s (c.1350 - c.1425) Algorithms
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Bibliography 8 K. V. Sarma and S. Hariharan, Yuktibhasa of jyesthadeva : a book of rationales in indian mathematics and astronomy - an analytical appraisal, Indian Journal of History of Science, vol. 26 (2), pp. 185 207, 1991. 9 A. V. Raman, The katapayadi formula and the modern hashing technique, Annals of the History of Computing, vol. 19 (4), pp. 4952, 1997. 10 R. Roy, Article The discovery of the series formula for π by Leibniz, Gregory, and Nilakantha in Sherlock Holmes in Babylon and other tales of mathematical history, R. W. Marlow Anderson, Victor Katz, Ed. The Mathematical Association of America, 2004. V N Krishnachandran, Reji C Joy, Siji K B On Sangamagr¯ ama M¯ adhava’s (c.1350 - c.1425) Algorithms
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Bibliography 11 C. K. Raju, Cultural foundations of mathematics : The nature of mathematical proof and the transmission of the calculus from India to Europe in the 16th c. CE. Delhi: Centre for Studies in Civilizations, 2007. 12 F. Cajori, A history of mathematics, 5th ed. Chelsea Publishing Series, 1999. 13 Jyeshthadeva, Ganita-yukti-bhasha, K. V. Sarma, Ed. New Delhi: Hindustan Book Agency, 2008. 14 osprey/libm/mips/sincos.c. [Online]. Available: http:// www.open64.net/ doc/d6/d08/ sincos 8c-source.html 15 k sin.c. [Online]. Available: http://www.netlib.org/fdlibm/k sin.c 16 k cos.c. [Online]. Available: http://www.netlib.org/fdlibm/k cos.c V N Krishnachandran, Reji C Joy, Siji K B On Sangamagr¯ ama M¯ adhava’s (c.1350 - c.1425) Algorithms
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Thanks
V N Krishnachandran, Reji C Joy, Siji K B On Sangamagr¯ ama M¯ adhava’s (c.1350 - c.1425) Algorithms
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Comparison
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