Madhava's correction term

Madhava's correction term is a mathematical expression attributed to Madhava of Sangamagrama, the founder of the Kerala school of astronomy and mathematics, that can be used to give a better approximation to the value of the mathematical constant than the partial sum approximation obtained by truncating the Madhava–Leibniz infinite series for. The Madhava–Leibniz infinite series for is
Taking the partial sum of the first terms we have the following approximation to :
Denoting the Madhava correction term by, we have the following better approximation to :
Three different expressions have been attributed to Madhava as possible values of, namely,
In the extant writings of the mathematicians of the Kerala school there are some indications regarding how the correction terms and have been obtained, but there are no indications on how the expression has been obtained. This has led to a lot of speculative work on how the formulas might have been derived.

Correction terms as given in Kerala texts

The expressions for and are given explicitly in the Yuktibhasha, a major treatise on mathematics and astronomy authored by the Indian astronomer Jyesthadeva of the Kerala school of mathematics around 1530, but that for appears there only as a step in the argument leading to the derivation of.
The Yuktidipika–Laghuvivrthi commentary of Tantrasangraha, a treatise written by Nilakantha Somayaji an astronomer/mathematician belonging to the Kerala school of astronomy and mathematics and completed in 1501, presents the second correction term in the following verses :
English translation of the verses:
In modern notations this can be stated as follows :
If we set, the last term in the right hand side of the above equation reduces to.
The same commentary also gives the correction term in the following verses :
English translation of the verses:
In modern notations, this can be stated as follows:
where the "multiplier" If we set, the last term in the right hand side of the above equation reduces to.

Accuracy of the correction terms

Let
Then, writing, the errors have the following bounds:

Numerical values of the errors in the computation of

The errors in using these approximations in computing the value of are
The following table gives the values of these errors for a few selected values of.


11
21
51
101
151

Continued fraction expressions for the correction terms

It has been noted that the correction terms are the first three convergents of the following continued fraction expressions:
The function that renders the equation
exact can be expressed in the following form:
The first three convergents of this infinite continued fraction are precisely the correction terms of Madhava. Also, this function has the following property:

Speculative derivation by Hayashi ''et al.''

In a paper published in 1990, a group of three Japanese researchers proposed an ingenious method by which Madhava might have obtained the three correction terms. Their proposal was based on two assumptions: Madhava used as the value of and he used the Euclidean algorithm for division.
Writing
and taking compute the values express them as a fraction with 1 as numerator, and finally ignore the fractional parts in the denominator to obtain approximations:
This suggests the following first approximation to which is the correction term talked about earlier.
The fractions that were ignored can then be expressed with 1 as numerator, with the fractional parts in the denominators ignored to obtain the next approximation. Two such steps are:
This yields the next two approximations to exactly the same as the correction terms
and
attributed to Madhava.