Brahmagupta interpolation formula

The Brahmagupta interpolation formula is the second polynomial-order interpolation formula found by the Indian mathematician and astronomer Brahmagupta (598–668) at the beginning of the 7th century AD. A poetic description of this formula in Sanskrit is found in an additional part of Khandakhadyaki, the work completed by Brahmagupta in 665 ^[1] . The same verse is available in his earlier work, Dhyana-graha-adhikara, the exact date of creation of which has not been established. However, the internal interconnection of the works suggests that it was created earlier by the scientist's main work, , which was completed earlier in 628, so the creation of a second-order interpolation formula can be attributed to the first quarter of the 7th century ^[1] . Brahmagupta was the first to find and use the formula in finite differences of the second order in the history of mathematics ^[2] ^[3] .

The Brahmagupta formula coincides with Newton's second-order interpolation formula , which was found (rediscovered) after more than a thousand years.

Content

Task

As an astronomer, Brahmaguptta was interested in obtaining accurate sine values based on a small number of known tabulated values of this function. Thus, he was faced with the task of finding the quantity $f(x)$ ${\ displaystyle f (x)}$ , $x_{r}<x<x_{r+1}$ ${\ displaystyle x_ {r} <x <x_ {r + 1}}$ according to the values of the function in the table:

$x$ ${\ displaystyle x}$	$x_{1}$ ${\ displaystyle x_ {1}}$	$x_{2}$ ${\ displaystyle x_ {2}}$	...	$x_{r}$ ${\ displaystyle x_ {r}}$	$x_{r+1}$ ${\ displaystyle x_ {r + 1}}$	$x_{r+2}$ ${\ displaystyle x_ {r + 2}}$	...	$x_{n}$ ${\ displaystyle x_ {n}}$
$f(x_{r})$ ${\ displaystyle f (x_ {r})}$	$f_{1}$ ${\ displaystyle f_ {1}}$	$f_{2}$ ${\ displaystyle f_ {2}}$	...	$f_{r}$ ${\ displaystyle f_ {r}}$	$f_{r+1}$ ${\ displaystyle f_ {r + 1}}$	$f_{r+2}$ ${\ displaystyle f_ {r + 2}}$	...	$f_{n}$ ${\ displaystyle f_ {n}}$

Provided that the values of the function are calculated at points with a constant step $h$ ${\ displaystyle h}$ , ( $x_{r+1}-x_{r}=h$ ${\ displaystyle x_ {r + 1} -x_ {r} = h}$ for all $r$ ${\ displaystyle r}$ ), Ariabhata suggested using the first finite differences for the calculations (tabular):

D_{r}=f_{r+1}-f_{r}

{\ displaystyle D_ {r} = f_ {r + 1} -f_ {r}}

Mathematicians before Brahmagupta used the obvious linear interpolation formula

f(x)=f_{r}+tD_{r}

{\ displaystyle f (x) = f_ {r} + tD_ {r}}

,

Where $t=(x-x_{r})/h$ ${\ displaystyle t = (x-x_ {r}) / h}$ .

Brahmagupta replaced in this formula $D_{r}$ ${\ displaystyle D_ {r}}$ an arc function of finite differences, which allows one to obtain more exact values of the interpolated function in order.

Brahmagupta Computation Algorithm

In Brahmagupta terminology, the difference $D_{r-1}$ ${\ displaystyle D_ {r-1}}$ called the last segment (गत काण्ड), $D_{r}$ ${\ displaystyle D_ {r}}$ called the useful segment (भोग्य काण्ड). Cut length $x-x_{r}$ ${\ displaystyle x-x_ {r}}$ to the interpolation point in minutes is called the cut (विकल). A new expression that should replace $D_{r}$ ${\ displaystyle D_ {r}}$ is called a regular useful segment (स्फुट भोग्य काण्ड). The calculation of the correct useful segment is described in the couplet ^[4] ^[1] :

According to Bhuttopala's commentary (X century), verses are translated as follows ^[1] ^[5] : Multiply the stump by the half-difference of the useful and past segments and divide the result by 900. Add the result to the half-sum of the useful and past segments if this half-sum is less than the useful segment . If more, then subtract. You will get the correct useful difference ^[6] .

900 minutes (15 degrees) is the interval $h$ ${\ displaystyle h}$ between the arguments of the tabular sine values used by Brahmagupta.

Brahmagupta formula in modern notation

In modern notation, the Brahmagupta calculation algorithm is expressed by the formulas:

{\begin{aligned}f(x)&=f_{r}+t({\frac {D_{r}+D_{r-1}}{2}}+t{\frac {D_{r}-D_{r-1}}{2}})\\&=f_{r}+t{\frac {D_{r}+D_{r-1}}{2}}+t^{2}{\frac {D_{r}-D_{r-1}}{2}}.\end{aligned}}

{\ displaystyle {\ begin {aligned} f (x) & = f_ {r} + t ({\ frac {D_ {r} + D_ {r-1}} {2}} + t {\ frac {D_ { r} -D_ {r-1}} {2}}) \\ & = f_ {r} + t {\ frac {D_ {r} + D_ {r-1}} {2}} + t ^ {2 } {\ frac {D_ {r} -D_ {r-1}} {2}}. \ end {aligned}}}

This is Newton’s second-order interpolation formula ^[7] ^[8] .

Proof

It is not known how Brahmagupta received this formula ^[1] . Nowadays, such formulas are proved using the expansion of functions $f(x+kh),k=1,2,...$ ${\ displaystyle f (x + kh), k = 1,2, ...}$ in the right grow equality in a taylor series at a point $x$ ${\ displaystyle x}$ . However, one can prove the formula by elementary methods: after replacement $t=(x-x_{r})/h$ ${\ displaystyle t = (x-x_ {r}) / h}$ Brahmagupta formula defines a parabola passing through three points $(x_{r-1},f_{r-1}),(x_{r},f_{r}),(x_{r+1},f_{r+1})$ ${\ displaystyle (x_ {r-1}, f_ {r-1}), (x_ {r}, f_ {r}), (x_ {r + 1}, f_ {r + 1})}$ . To derive this formula, it suffices to find the coefficients of this parabola by solving the system of three linear equations defined by these points.

Formula Accuracy

Computer calculation shows that having a table of 7 sine values in nodes with a step of 15 degrees, Brahmagupta could calculate this function with a maximum error of not more than 0.0012 and an average error of not more than 0,00042.

Notes

↑ ¹ ² ³ ⁴ ⁵ Gupta, RC Second-order interpolation in Indian mathematics upto the fifteenth century (Eng.) // Indian Journal of History of Science: journal. - Vol. 4 , no. 1 & 2 . - P. 86-98 .
↑ Van Brummelen, Glen. The mathematics of the heavens and the earth: the early history of trigonometry. - Princeton University Press, 2009 .-- P. 329. - ISBN 9780691129730 . (p.111)
↑ Meijering, Erik. A Chronology of Interpolation From Ancient Astronomy to Modern Signal and Image Processing (Eng.) // Proceedings of the IEEE : journal. - 2002 .-- March ( vol. 90 , no. 3 ). - P. 319—342 . - DOI : 10.1109 / 5.993400 .
↑ Dhyana-Graha-Upadesa-Adhyaya, 17; Khandaka Khadyaka, IX, 8
↑ Raju, C K. Cultural foundations of mathematics: the nature of mathematical proof and the transmission of the calculus from India to Europe in the 16th c. CE - Pearson Education India, 2007. - P. 138-140. - ISBN 9788131708712 .
↑ The final part of the algorithm is due to the fact that mathematicians before Brahmagupta and for a long time after it did not use the concept of a negative number. Therefore, it was not the difference that was actually calculated, but the modulus of the difference ${\frac {|D_{r-1}-D_{r}|}{2}}$ ${\ displaystyle {\ frac {| D_ {r-1} -D_ {r} |} {2}}}$ , and then this non-negative number was added or subtracted, depending on the sign of the difference determined by inequality.
↑ Milne-Thomson, Louis Melville. The Calculus of Finite Differences. - AMS Chelsea Publishing, 2000. - P. 67–68. - ISBN 9780821821077 .
↑ Hildebrand, Francis Begnaud. Introduction to numerical analysis. - Courier Dover Publications, 1987. - P. 138–139. - ISBN 9780486653631 .

[Gupta-1] ¹ ² ³ ⁴ ⁵ Gupta, RC Second-order interpolation in Indian mathematics upto the fifteenth century (Eng.) // Indian Journal of History of Science: journal. - Vol. 4 , no. 1 & 2 . - P. 86-98 .

[2] Van Brummelen, Glen. The mathematics of the heavens and the earth: the early history of trigonometry. - Princeton University Press, 2009 .-- P. 329. - ISBN 9780691129730 . (p.111)

[3] Meijering, Erik. A Chronology of Interpolation From Ancient Astronomy to Modern Signal and Image Processing (Eng.) // Proceedings of the IEEE : journal. - 2002 .-- March ( vol. 90 , no. 3 ). - P. 319—342 . - DOI : 10.1109 / 5.993400 .

[4] Dhyana-Graha-Upadesa-Adhyaya, 17; Khandaka Khadyaka, IX, 8

[Raju-5] Raju, C K. Cultural foundations of mathematics: the nature of mathematical proof and the transmission of the calculus from India to Europe in the 16th c. CE - Pearson Education India, 2007. - P. 138-140. - ISBN 9788131708712 .

[6] The final part of the algorithm is due to the fact that mathematicians before Brahmagupta and for a long time after it did not use the concept of a negative number. Therefore, it was not the difference that was actually calculated, but the modulus of the difference ${\frac {|D_{r-1}-D_{r}|}{2}}$ ${\ displaystyle {\ frac {| D_ {r-1} -D_ {r} |} {2}}}$ , and then this non-negative number was added or subtracted, depending on the sign of the difference determined by inequality.

[7] Milne-Thomson, Louis Melville. The Calculus of Finite Differences. - AMS Chelsea Publishing, 2000. - P. 67–68. - ISBN 9780821821077 .

[8] Hildebrand, Francis Begnaud. Introduction to numerical analysis. - Courier Dover Publications, 1987. - P. 138–139. - ISBN 9780486653631 .