Heron's method/Program/Natural register/Exercise

Write a computer-program (pseudocode) for the computation of rational approximations for the square root of a rational number using Heron's method.

• The computer has as many memory units as needed, which can contain natural numbers.
• It can compare the content of memory units and can, depending on the outcome, switch to a certain program line.
• It can add the content of two memory units and write the result into another memory unit.
• It can multiply the content of two memory units and write the result into another memory unit.
• It can print contents of memory units and it can print given texts.
• There is a stop command.

The initial configuration is

$(a,b,c,d,e,0,0,\ldots )$

with ${}b,c,e\neq 0$. Here, ${}a/b$ is the number from which we want to compute the square root, ${}x_{0}=c$ is the initial value and ${}d/e$ is the wished-for accuracy. The program shall compute and print the Heron-sequence ${}x_{0},x_{1},x_{2},\ldots$ (the numerators and denominators are printed successively) and it shall stop when the member ${}x_{n}$ printed last fulfills the property

${}\vert {x_{n}^{2}-{\frac {a}{b}}}\vert \leq {\frac {d}{e}}\,.$

Create a solution