Tamura-Kanada Circuit Method
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Algorithm
The following algorithm can be used to calculate $\pi$ (pi):
- $(1): \quad$ Set $A = X = 1$, set $B = \dfrac 1 {\sqrt 2}$, and set $C = \dfrac 1 4$
- $(2): \quad$ Set $Y = A$.
- $(3): \quad$ Set $A = \dfrac {A + B} 2$
- $(4): \quad$ Set $B = \sqrt {B Y}$.
- $(5): \quad$ Set $C = C - X \paren {A - Y}^2$
- $(6): \quad$ Set $X = 2 X$
- $(7): \quad$ Output $\dfrac {\paren {A + B}^2} {4 C}$ as an approximation to $\pi$.
- $(8): \quad$ For a better approximation to $\pi$, set $Y$ equal to the output, return to step $(2)$ and continue.
Example
Starting with $A = X = 1$, $B = \dfrac 1 {\sqrt 2}$, $C = \dfrac 1 4$, the successive values of $\dfrac {\paren {A + B}^2} {4 C}$ on the first $3$ loops are:
\(\text {(1)}: \quad\) | \(\ds \) | \(\) | \(\ds 2 \cdotp 91421 \, 35\) | |||||||||||
\(\text {(2)}: \quad\) | \(\ds \) | \(\) | \(\ds 3 \cdotp 14057 \, 97\) | |||||||||||
\(\text {(3)}: \quad\) | \(\ds \) | \(\) | \(\ds 3 \cdotp 14159 \, 28\) |
and it is seen that the value for $\pi$ is already correct to $6$ decimal places.
Proof
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Source of Name
This entry was named for Yoshiaki Tamura and Yasumasa Kanada.
Historical Note
This algorithm was used by Yasumasa Kanada, Sayaka Yoshino and Yoshiaki Tamura to calculate $\pi$ (pi) to $16 \, 777 \, 216$ digits in $1983$.
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41972 \ldots$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41971 \ldots$