On Sangamagrama Madhava's (c.1350 - c.1425) Algorithms for the Computation of Sine and Cosine...

Intro Kerala School Terminology π value Madhava’s series Sine table Analysis Comparison Conclusion Biblio

On Sangamagrama Madhava’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. Vidya Academy of Science and Technology, Thrissur - 680501, Kerala.

On Sangamagrama Madhava’s (c.1350 - c.1425) Algorithms for the Computation of Sine and Cosine Functions


International Conference onComputational Engineering Practices and Techniques

MES College of Engineering, Kuttippuram, Kerala.25 - 26 November 2010




We pay obeisance to the computational genius ofthe great Kerala mathematicians and astronomers

of yesteryears. This paper is a tribute to theirunparalleled achievements.

V.N. Krishnachandran, Reji C. Joy, Siji K.B.




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




Introduction

Introduction




Introduction

Classical period in Indian Mathematics begins withAryabhata I.




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.




Introduction

Classical period in Indian Mathematics ends withBhaskara II.




Introduction

Development of mathematics in India did not endwith Bhaskara II !




Introduction

Mathematics continued to flourish in Keralaunknown to the rest of the world!

C.M. Whish, an East India Company official, wrote about this in1832. But nobody noticed it.




Kerala School of Astronomy and Mathematics

Kerala School of Astronomyand Mathematics




Kerala School : Geographical area (Kerala)

Map of Kerala




Kerala School : Geographical area (Trikkandiyur)

Map showing Trikkandiyur and nearby places




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)




Kerala School : Founder

Sangamagrama Madhava(c.1350 - c.1425 CE)

No personal details about Madhava has come to light.

Sangamagrama is surmised to be a reference to his place ofresidence.

Some historians have identified Sangamagrama as modern dayIrinjalakuda in Thrissur District in Kerala.

There are references and tributes to Madhava in the works ofother authors about whom accurate details are available.




Kerala School : Iringatappilli temple

Iringatappilli temple near IrinjalakkudaThe granite slabs in the picture were said to have been used by

Madhava for temple rituals and astronomical studies.




Keral School : Prominent members

Paramesvara (c.1380 - c.1460), a pupil of SangamagramaMadhava : Promulgator of Drigganita system of astronomicalcomputations in Kerala.

Da.modara, another prominent member of the Kerala school,was Paramesvara’s son and also his pupil.

Nilakant.ha Somayaji (1444 - 1544), a pupil of Parameshvara.Author of Tantrasamgraha completed in 1501.




Kerala School : Tantrasangraha

Image of the cover page of Tantrasangraha being published bySpringer.




Kerala School : Other prominent members

Jyes.t.hadeva (c.1500 - c.1600). Author of Yukt.ibhas.a.




Kerala School : Yukt. ibhas.a

Image of the cover page of Yukt.ibhas.a published by Springer.




Kerala School : Yukt. ibhas.a

Composed in Malayalam.

It contains clear statements of the power series expansions ofthe sine and cosine functions and also their proofs.

This treatise proves that the idea of proof is not alien toIndian mathematics.




Kerala School : Other members

Citrabhanu (c.1550)

Sankara Variar (c.1500 - c.1560, author of Kriya-kramakari )

Acyuta Pis.arat.i (c.1550 - 1621)

Putumana Somayaji (author of Karan. apadhat. i)

Sankara Varman (1774 - 1839, author of Sadratnamala)




Kerala School : Pre-Madhava figures




Kerala School : Madhava and his disciples

Diagram showing teacher-pupil relationships




Kerala School: Later figures




Some terminology

Some terminology




Some terminology: Capa, jya, koti-jya, utkrama-jya

Diagram showing capa, jya , etc.




Some terminology: Capa, jya, koti-jya, utkrama-jya

jya of arc AB = BM = R sin(sR

)koti-jya of arc AB = OM = R cos

(sR

)utkrama-jya (or sara of arc AB) = MA = R

(1 − cos

(sR

))V.N. Krishnachandran,, Reji C. Joy,, Siji K.B. Vidya Academy of Science and Technology, Thrissur - 680501, Kerala.



Some terminology: Katapayadi scheme

The Kat.apayadi scheme is a method for representing numbersusing letters of the Sanskrit alphabet.





Table: Mapping of digits to letters in katapayadi scheme

1 2 3 4 5 6 7 8 9 0

ka kha ga gha na ca cha ja jha na

t.a t.ha d. a d. ha n.a ta tha da dha na

pa pha ba bha ma - - - - -

ya ra la va sa s.a sa ha - -





Consonants have numerals assigned as per Table in previousslide.

All stand-alone vowels like a and i are assigned to 0.

In case of a conjunct, consonants attached to a non-vowel willbe valueless.

Numbers are written in increasing place values from left toright. The number 386 which denotes 3 × 100 + 8 × 10 + 6 inmodern notations would be written as 683 in pre-modernIndian traditions.




Madhava’s value for π






Madhava derived the following series for the computation of π:

Madhava series for π

π

4= 1 − 1

3+

1

5− 1

7+ · · ·





The series is known as the Gregory series.

Its discovery has been attributed to Gottfried Wilhelm Leibniz(1646 - 1716) and James Gregory (1638 1675).

The series was known in Kerala more than two centuriesbefore its European discovers were born.





Using this series and several correction terms Madhava computedthe following value for π:

π = 3.1415926535922.




Madhava’s series

Madhava’s series




Madhava’s series for sine: In Madhava’s own words

Multiply the arc by the square of the arc, and take theresult of repeating that (any number of times). Divide(each of the above numerators) by the squares of thesuccessive even numbers increased by that number andmultiplied by the square of the radius. Place the arc andthe successive results so obtained one below the other,and subtract each from the one above. These togethergive the jiva, as collected together in the verse beginningwith “vidvan” etc.




Madhava’s series for sine: Rendering in modern notations

The following numerators are formed first:

s · s2, s · s2 · s2, s · s2 · s2 · s2, ·

These are then divided by quantities specified in the verse.

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,

· · ·





Place the arc and the successive results so obtained one belowthe other, and subtract each from the one above to get theexpression for jiva.





Madhava’s series for sine function

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 series for sine: As power series for sine

Let θ be the angle subtended by the arc s at the center of thecircle. Then s = rθ and jiva = rsinθ. Substituting these in the lastexpression and simplifying we get

sin θ = θ − θ3

3!+θ5

5!− θ7

7!+ · · ·

which is the infinite power series expansion of the sine function.




Madhava’s series for sine: Reformulation for computation

Madhava considers one quarter of a circle.

The length of the quarter-circle is taken as 5400 minutes (sayC minutes).

He computes the radius R of the circle:

R = 2 × 5400/π

= 3437.74677078493925

= 3437′ 44′′ 48′′′





The expression for jiva is now put in the following form:

jiva = s − s3

R2(22 + 2)+

s5

R4(22 + 2)(42 + 4)− · · ·

= s −( s

C

)3 [R(π2

)3

3!−( s

C

)2 [R(π2

)5

5!− · · ·

]]




Madhava’s series for sine: Pre-computaion of coefficients

No. Expression Value In kat.apayadi system

1 R × (π/2)3/3! 2220′ 39′′ 40′′′ ni-rvi-ddha-nga-na-re-ndra-rung

2 R × (π/2)5/5! 273′ 57′′ 47′′′ sa-rva-rtha-si-la-sthi-ro

3 R × (π/2)7/7! 16′ 05′′ 41′′′ ka-vi-sa-ni-ca-ya

4 R × (π/2)9/9! 33′′ 06′′′ tu-nna-ba-la

5 R × (π/2)11/11! 44′′′ vi-dvan





Madhava’s polynomial approximation to sinefunction:

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 series for cosine

Madhava’s polynomial approximation to cosinefunction:

sara = (s/C )2[(4241′09′′00′′′)

− (s/C )2[(872′03′′′05′′′)

− (s/C )2[(071′43′′24′′′)

− (s/C )2[(03′09′′′37′′′)

− (s/C )2[(05′′12′′′)

− (s/C )2(06′′′)]]]]]




Madhava’s sine table






(continued in next slide)




Analysis of Madhava’s computational schemes

Analysis of Madhava’scomputational schemes




Analysis: These are algorithms!

Madhava and his followers were developing algorithms for thecomputation of sine and cosine functions.

This algorithmic aspect is evident in the way the seriesexpansions were formulated. It is given as a step by stepprocedure for computations the function values.

In Jyes.t.hadeva’s Yuktibhas.a, the author has used the termkriya-krama which translates into procedure or an algorithm.




Analysis: Use of polynomial approximation

Madhava uses polynomial approximations.

For sine function, an 11 th degree polynomial is used. For thecosine function, a 12 th degree polynomial is used.

The orders of the polynomials were decided by therequirements of accuracy.

The values computed by Madhava could also be obtained byother methods.

But Madhava did seek and get a general method in the formof power series expansions.




Analysis: Pre-computation of the coefficients

Madhava pre-computed the coefficients appearing in thepolynomial approximations.




Analysis: Use of Horner’s scheme

Madhava had applied what is now known as the Horner’sscheme for the computation of polynomials.

The scheme is now attributed to William George Horner (17861837) who was a British mathematician and schoolmaster.




Analysis: Horner’s scheme

Given the polynomial

p(x) =n∑

i=0

aixi = a0 + a1x + a2x2 + a3x3 + · · · + anxn,

let it be required to evaluate p(x) at a specific value of x .




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 + anx) · · · )).

Then apply the following algorithm for computing p(x):

bn = an

bn−1 = an−1 + bnx

· · ·b1 = a1 + b2x

b0 = a0 + b1x .

b0 is the required value of the polynomial.




Analysis: Use of Horner’s scheme

Madhava had actually implemented Horner’s scheme in hisalgorithms.

The method was known to Isaac Newton in 1669, the Chinesemathematician Qin Jiushao in the 13th century, and evenearlier to the Persian Muslim mathematicians.

Madhava’s was the first conscious and deliberate applicationof the scheme in a computational algorithm with the intentionof reducing the complexity of numerical procedures.




Analysis: Simultaneous computation of sine and cosine

In many modern implementations of routines for thecalculations of the sine and cosine functions, there would beone routine for the simultaneous computation of sine andcosine.

Whenever sine or cosine is required, the other would also berequired. So a common algorithm which returns both valuessimultaneously would be more time efficient and economical.

It would appear that Madhava had anticipated such ascenario. This is evidenced by the description of one commonprocedure for the evaluation of sine and cosine functions inYuktibhas.a.




Comparison with modern algorithms

Comparison with modernalgorithms




Comparison: Programmes in Open64 Compiler

Compare, for example, with the programme included in theOpen64 Compiler developed by Computer Architecture andParallel Systems Laboratory in University of Delaware.

These programs are not using the polynomials used byMadhava.

They are using the minimax polynomial computed using theRemez algorithm to improve the accuracy of computations.




Comparison: Coefficients for computation of sine

The program segment in the next slide specifies the pre-computedcoefficients in the polynomial approximation for the sine function.The values are given in the IEEE:754 floating point format.




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 ;




Comparison: Coefficients for computation of cosine

The program segment in the next slide specifies the pre-computedcoefficients in the polynomial approximation for the cosinefunction. The values are also given in the IEEE:754 floating pointformat.




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 ;




Comparison: Polynomial approximations

The program segment in the next slide describes the computationsof the polynomial approximations for the sine and cosine functionssimultaneously.




Comparison: Polynomial approximations

00329 xsq = x*x;

00330

00331 cospoly = (((((C[6].d*xsq + C[5].d)*xsq +

00332 C[4].d)*xsq + C[3].d)*xsq +

00333 C[2].d)*xsq + C[1].d)*xsq + C[0].d;

00334

00335 sinpoly = (((((S[6].d*xsq + S[5].d)*xsq +

00336 S[4].d)*xsq + S[3].d)*xsq +

00337 S[2].d)*xsq + S[1].d)*(xsq*x) + x;




Conclusion

Conclusion




Conclusion

It is true that this programme having more than 500 lines of codehas made use of several other ideas as well. But the criticalcomponents continues to be the following which are essentially theideas enshrined in Madhava’s computational scheme developedmore than six centuries ago.

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.




Bibliography

Bibliography




Bibliography

1 G. G. Joseph, A Passage to Infinity: Medieval IndianMathematics from Kerala and Its Impact. New Delhi: SagePublications Pvt. Ltd, 2009.

2 C. M. Whish, On the hindu quadrature of the circle and theinfinite series of the proportion of the circumference to thediameter exhibited in the four sastras, the tantra sahgraham,yucti bhasha, carana padhati and sadratnamala, Transactionsof 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 Indianmathematics. New Delhi: New Age International Publishers,1992.




Bibliography

4 V. J. Katz, The mathematics of Egypt, Mesopotemia, China,India and Islam: A source book. Princeton: PrincetonUniversity Press, 2007, ch. Chapter 4 : Mathematics in IndiaIV. Kerala School (pp. 480 - 495).

5 K. Plofker, Mathematics in India. Princeton, NJ: PrincetonUniversity 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, IndianJournal of History of Science, vol. 33 (1), Mar. 1998.




Bibliography

8 K. V. Sarma and S. Hariharan, Yuktibhasa of jyesthadeva : abook of rationales in indian mathematics and astronomy - ananalytical appraisal, Indian Journal of History of Science, vol.26 (2), pp. 185 207, 1991.

9 A. V. Raman, The katapayadi formula and the modernhashing 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 byLeibniz, Gregory, and Nilakantha in Sherlock Holmes inBabylon and other tales of mathematical history, R. W.Marlow Anderson, Victor Katz, Ed. The MathematicalAssociation of America, 2004.




Bibliography

11 C. K. Raju, Cultural foundations of mathematics : The natureof mathematical proof and the transmission of the calculusfrom India to Europe in the 16th c. CE. Delhi: Centre forStudies in Civilizations, 2007.

12 F. Cajori, A history of mathematics, 5th ed. ChelseaPublishing Series, 1999.

13 Jyes.t. hadeva, Gan. ita-yukti-bhas. a, K. V. Sarma, Ed. NewDelhi: 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/ksin.c

16 k cos.c. [Online]. Available: http://www.netlib.org/fdlibm/kcos.c




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On Sangamagrama Madhava's (c.1350 - c.1425) Algorithms for the Computation of Sine and Cosine...

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