Madhava's correction term is a mathematical expression attributed to Madhava of Sangamagrama (c. 1340 – c. 1425), 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 (pi) than the partial sum approximation obtained by truncating the Madhava–Leibniz infinite series for . The Madhava–Leibniz infinite series for is
\pi | =1- | |
4 |
1 | + | |
3 |
1 | - | |
5 |
1 | |
7 |
+ …
Taking the partial sum of the first
n
\pi | |
4 |
≈ 1-
1 | + | |
3 |
1 | - | |
5 |
1 | |
7 |
+ … +(-1)n-1
1 | |
2n-1 |
Denoting the Madhava correction term by
F(n)
\pi | |
4 |
≈ 1-
1 | + | |
3 |
1 | - | |
5 |
1 | |
7 |
+ … +(-1)n-1
1 | |
2n-1 |
+(-1)nF(n)
Three different expressions have been attributed to Madhava as possible values of
F(n)
F | ||||
|
F | ||||
|
F | ||||
|
In the extant writings of the mathematicians of the Kerala school there are some indications regarding how the correction terms
F1(n)
F2(n)
F3(n)
The expressions for
F2(n)
F3(n)
F1(n)
F2(n)
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 (Chapter 2: Verses 271–274):[2] [3]
English translation of the verses:[2]
"To the diameter multiplied by 4 alternately add and subtract in order the diameter multiplied by 4 and divided separately by the odd numbers 3, 5, etc. That odd number at which this process ends, four times the diameter should be multiplied by the next even number, halved and [then] divided by one added to that [even] number squared. The result is to be added or subtracted according as the last term was subtracted or added. This gives the circumference more accurately than would be obtained by going on with that process."
In modern notations this can be stated as follows (where
d
Circumference
=4d-
4d | |
3 |
+
4d | |
5 |
- … \pm
4d | |
p |
\mp
4d\left(p+1\right)/2 | |
1+(p+1)2 |
If we set
p=2n-1
4dF2(n)
The same commentary also gives the correction term
F3(n)
English translation of the verses:[2]
"A subtler method, with another correction. [Retain] the first procedure involving division of four times the diameter by the odd numbers, 3, 5, etc. [But] then add or subtract it [four times the diameter] multiplied by one added to the next even number halved and squared, and divided by one added to four times the preceding multiplier [with this] multiplied by the even number halved."
In modern notations, this can be stated as follows:
Circumference =4d-
4d | |
3 |
+
4d | |
5 |
- … \pm
4d | |
p |
\mp
4dm | |
\left(1+4m\right)(p+1)/2 |
,
where the "multiplier" If we set
p=2n-1
4dF3(n)
Let
si=1-
1 | + | |
3 |
1 | - | |
5 |
1 | |
7 |
+ … +(-1)n-1
1 | |
2n-1 |
+(-1)nFi(n)
Then, writing
p=2n+1
\left| | \pi |
4 |
-si(n)\right|
\begin{align}&\begin{align} | 1 |
p3-p |
-
1 | |
(p+2)3-(p+2) |
&<\left|
\pi | |
4 |
-s1(n)\right|<
1 | |
p3-p |
,\\[10mu]
4 | |
p5+4p |
-
4 | |
(p+2)5+4(p+2) |
&<\left|
\pi | |
4 |
-s2(n)\right|<
4 | , \end{align}\\[20mu] &\begin{align} & | |
p5+4p |
36 | |
p7+7p5+28p3-36p |
-
36 | |
(p+2)7+7(p+2)5+28(p+2)3-36(p+2) |
… \\[10mu] &\phantom{
4 | |
p5+4p |
-
4 | |
(p+2)5+4(p+2) |
The errors in using these approximations in computing the value of are
E(n)=\pi-4\left(1-
1 | + | |
3 |
1 | - | |
5 |
1 | |
7 |
+ … +(-1)n-1
1 | |
2n-1 |
\right)
Ei(n)=E(n)-4 x (-1)nFi(n)
The following table gives the values of these errors for a few selected values of
n
n | E(n) | E1(n) | E2(n) | E3(n) | |
---|---|---|---|---|---|
11 | -9.07 x 10-2 | 1.86 x 10-4 | -1.51 x 10-6 | 2.69 x 10-8 | |
21 | -4.76 x 10-2 | 2.69 x 10-5 | -6.07 x 10-8 | 3.06 x 10-10 | |
51 | -1.96 x 10-2 | 1.88 x 10-6 | -7.24 x 10-10 | 6.24 x 10-13 | |
101 | -9.90 x 10-3 | 2.43 x 10-7 | -2.38 x 10-11 | 5.33 x 10-15 | |
151 | -6.62 x 10-3 | 7.26 x 10-8 | -3.18 x 10-12 | ≈ 1 x 10-16 |
It has been noted that the correction terms
F1(n),F2(n),F3(n)
\cfrac{1}{4n+\cfrac{1}{n+\cfrac{1}{n+ … }}}
\cfrac{1}{4n+\cfrac{12}{n+\cfrac{22}{4n+\cfrac{32}{n+\cfrac{ … }{ … +\cfrac{r2}{n[4-3(r\bmod2)]+ … }}}}}}=\cfrac{1}{4n+\cfrac{22}{4n+\cfrac{42}{4n+\cfrac{62}{4n+\cfrac{82}{4n+ … }}}}}
The function
f(n)
\pi | |
4 |
=1-
1 | + | |
3 |
1 | |
5 |
- … \pm
1 | |
n |
\mpf(n+1)
exact can be expressed in the following form:[3]
f(n)=
1 | |
2 |
x \cfrac{1}{n+\cfrac{12}{n+\cfrac{22}{n+\cfrac{32}{n+ … }}}}
The first three convergents of this infinite continued fraction are precisely the correction terms of Madhava. Also, this function
f(n)
f(2n)=\cfrac{1}{4n+\cfrac{22}{4n+\cfrac{42}{4n+\cfrac{62}{4n+\cfrac{82}{4n+ … }}}}}
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
355/113
Writing
S(n)=\left|1-
1 | + | |
3 |
1 | - | |
5 |
1 | |
7 |
+ … +
(-1)n-1 | |
2n-1 |
-
\pi | |
4 |
\right|
and taking
\pi=355/113,
S(n),
\begin{alignat}{3} S(1)&=
97 | |
452 |
&&=
1 | |||||
|
&& ≈
1 | |
4 |
,\\[6mu] S(2)&=
161 | |
1356 |
&&=
1 | |||||
|
&& ≈
1 | |
8 |
,\\[6mu] S(3)&=
551 | |
6780 |
&&=
1 | |||||
|
&& ≈
1 | |
12 |
,\\[6mu] S(4)&=
2923 | |
47460 |
&&=
1 | |||||
|
&& ≈
1 | |
16 |
,\\[6mu] S(5)&=
21153 | |
427140 |
&&=
1 | |||||
|
&& ≈
1 | |
20 |
. \end{alignat}
This suggests the following first approximation to
S(n)
F1(n)
S(n) ≈
1 | |
4n |
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:
\begin{alignat}{5}
64 | |
97 |
&=
1 | |||||
|
&& ≈
1 | |
1 |
, &
33 | |
64 |
&=
1 | |||||
|
&& ≈
1 | |
1 |
,\\[6mu]
68 | |
161 |
&=
1 | |||||
|
&& ≈
1 | |
2 |
, &
25 | |
68 |
&=
1 | |||||
|
&& ≈
1 | |
2 |
,\\[6mu]
168 | |
551 |
&=
1 | |||||
|
&& ≈
1 | |
3 |
, &
47 | |
168 |
&=
1 | |||||
|
&& ≈
1 | |
3 |
,\\[6mu]
692 | |
2923 |
&=
1 | |||||
|
&& ≈
1 | |
4 |
, &
155 | |
692 |
&=
1 | |||||
|
&& ≈
1 | |
4 |
,\\[6mu]
4080 | |
21153 |
&=
1 | |||||
|
&& ≈
1 | |
5 |
, &
753 | |
4080 |
&=
1 | |||||
|
&& ≈
1 | |
5 |
. \end{alignat}
This yields the next two approximations to
S(n),
F2(n),
S(n) ≈
1 | |
4n+\dfrac{1 |
{n}}=
n | |
4n2+1 |
,
and
F3(n),
S(n) ≈ \dfrac{1}{4n+\dfrac{1}{n+\dfrac{1}{n}}}=
n2+1 | |
n(4n2+5) |
,
attributed to Madhava.