Unit circle/Circle length/Bisection/Program/Exercise

We consider the unit circle, i.e.

${}{\left\{{\begin{pmatrix}x\\y\end{pmatrix}}\mid x^{2}+y^{2}=1\right\}}\subseteq \mathbb {R} ^{2}\,.$

We put ${}P_{0}={\begin{pmatrix}1\\0\end{pmatrix}}$ and ${}P_{1}={\begin{pmatrix}0\\1\end{pmatrix}}$, and we define recursively the sequence ${}P_{n}$ (in the plane) by

${}Q_{n}={\frac {1}{2}}{\left(P_{0}+P_{n-1}\right)}\,$

(that is, ${}Q_{n}$ is the bisection point of the line segment between ${}P_{0}$ and ${}P_{n-1}$), and ${}P_{n}$ is the intersection point of the half-line through ${}{\begin{pmatrix}0\\0\end{pmatrix}}$ and ${}Q_{n}$ and the circle. We consider the lengths ${}d_{n}=d(P_{0},P_{n})$ as an approximation for the length of the circle arc between ${}P_{0}$ and ${}P_{n}$, and therefore

${}x_{n}=2^{n}d_{n}\,$

is an approximation for the length of the half circle arc (that is, ${}\pi$). Since in the computation of the points ${}P_{n}$ and the lengths ${}d_{n}$, square roots occure (due to the Pythagorean theorem), we can approximate these by rational numbers only with certain errors.

Write a computer-program (in pseudocode), which computes and prints a sequence ${}y_{n}$ of approximations (${}n\geq 1$) for ${}x_{n}$. In the computation of ${}y_{n}$, all square roots, which are used in the computation of ${}x_{n}$, shall be computed with ${}n$ steps of Heron's method, starting with the initial value ${}1$. Also, the program shall use better and better approximations for the auxiliary points, the computation of ${}y_{n}$ requires that better and better approximations for ${}P_{2},\ldots ,P_{n}$ are determined.

• The computer has as many memory units as needed, which can contain rational numbers.
• The natural numbers are provided in a data base (they do not have to be computed).
• It can write the content of a memory unit into another memory unit.
• It can do the arithmetic operations (addition, subtraction, multiplication, division by a number ${}\neq 0$) on rational numbers and write the result in another memory unit.
• It can compare the content of memory units and can, depending on the outcome, switch to a certain program line.
• It can print contents of memory units and it can print given texts.

The program shall run to infinity and write down the approximations ${}y_{1},y_{2},y_{3},...$.

Create a solution