Abstract
When the values of a function are tabulated for some discrete values of the argument, the functional values corresponding to intermediary argumental values are obtained ordinarily by linear interpolation. For greater accuracy, higher order technique is necessary. It is known that the famous Indian mathematician Brahmagupta (seventh century ad) gave a rule for second-order interpolation.
Indian Journal of History of Science, Vol. 14, No. 1. (1979) pp. 66–72. Paper presented at the Annual Conference of the Indian Mathematical Society, Trivandrum, 1976.
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Notes
- 1.
Since, in the first quadrant, cotangent decreases and the greatest values of \(\theta \) and x are h and \(90^{\circ }-h\), respectively.
- 2.
Linear interpolation yields 48; 36.
Abbreviations
- a :
-
– the argument, circular arc measured in angular units
- \(a_0\), \(a_1\), \(a_2\) ...,:
-
– successive and equidistant values of a with \(a_0=0\).
- \(D_1\), \(D_2\), \(D_3\) ...:
-
– tabulated functional differences
\(D_1=f(a_1) - f(a_0)\),
\(D_2=f(a_2) - f(a_1)\), etc.
- \(D_p\), \(D_{p+1}\):
-
– tabulated functional difference just crossed over (bhukta-khaṇḍa) and the current tabulated functional difference (bhogya-khaṇḍa).
\(D_m = \dfrac{1}{2} \left( D_p+D_{p+1}\right) \).
- \(D_t\) :
-
– the true (sphuṭa) value of the current functional difference as given by Brahmagupta.
- f(a):
-
– functional value corresponding to the argumental value a. The function considered here is either the Indian Sine (\(= R \,sin \, a\)), or the Indian Versed Sine (\(R \,vers \, a\)). So that we have \(f(a_0) =0\).
- p :
-
– positive integer.
- h :
-
– equal (or common) arcual interval
\(h = a_1 - a_0 = a_2 - a_1 \) and so on.
\(n = \dfrac{\theta }{h}\).
\(x = p. h\), the arc crossed over.
- R :
-
– Sinus totus, radius of the circle of reference defining Sine and Versed Sine.
- T :
-
– mathematically exact value of the current functional difference, so that \(f(x+\theta ) = f(x) + \theta \, . \,\dfrac{T}{h}\).
\(T_1\), \(T_2\), etc., are successive approximations to T
\(T_\infty -\) the theoretically ultimate or limiting value
- \(\Delta \) :
-
– first-order forward finite difference operator
\(\Delta f(a) = f(a +h) - f(a)\)
\(\Delta f(x) = D_{p+1}\).
- \(\nabla \) :
-
– first-order backward finite difference operator
\(\nabla f(a) = f(a) - f (a - h)\)
\(\nabla f(x) = D_p\).
- \(\theta \) :
-
– residual arc such that \(f(x+\theta )\) is required to be found out or interpolated, \(\theta \) being positive and less than h.
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Ramasubramanian, K. (2019). Munīśvara’s Modification of Brahmagupta’s Rule for Second-Order Interpolation. In: Ramasubramanian, K. (eds) Gaṇitānanda. Springer, Singapore. https://doi.org/10.1007/978-981-13-1229-8_28
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