3-1. CORDIC method([1])

The CORDIC, it is one way to compute the value of the elementary functions, the calculation is performed while rotating the (or coordinates) vector plane.

$θ$ as the angle to be obtained finally, shows a schematic diagram of a method of determining the $sinθ$ and $cosθ$ in the CORDIC method in Figure 1. As shown in Figure 1, CORDIC method by repeating the process angle $z$ is to approach the angle $θ$, $cosθ$ as the coordinates $x$, is calculated as $sinθ$ the coordinates $y$. Also from Figure 1, there is a relationship, such as the following angle the $z$.

To approaches the angle $θ$, (1.1) shows that it will add to the angle $Z$ angle $A$. At this time, the angle between each of the relationship, such as the following.

As shown in (1.10), the angle $z$ is closer to $θ$ eventually. To determine $sinθ, cosθ$ in this way. The following will describe the main principle of the CORDIC method.

Shows the how the angle $z$ whether the angle $θ$ closer to the CORDIC method in Figure 2. I to R the length of the line segment connecting the origin ($x_i$, $y_i$) coordinates first. I and z the angle at that time. Coordinates ($x_i$, $y_i$) and angle z_i+1 is expressed by the following equation.

Consider the update value $α$ to approximate to the angle $θ$ the angle $z$ here.

When defined by the equation (1.6), $sinα_i$ and $cosα_i$ can be expressed as follows.

By using (1.7), (1.8) equation, it can be expressed as follows: coordinate transformation $(x_(i+1),y_(i+1))$ to the point from $(x_i,y_i)$ point.

Similarly,

Then, how $sinθ$ and $cosθ$ or go is calculated as an example, or try to follow the process.

As shown in Figure 3，It is assumed that the x-axis on the initial position of the vector will rotate，Initial coordinates point is $(x_0 ,y_0 (=0))$．Assuming that $z_n＝θ$,

In determining the initial value of $x_0$ from (1.13) and (1.11), by obtaining the $x_n$ and $y_n$ coordinates after the rotation of $n$ times， $sinθ$ and $cosθ$ of the angle $θ$ given can be determined as follows.

Attention must be paid to, It is assumed that the angle $z_n$ approaches the angle $θ$ by repeating n times the rotation, Assuming that the error $θ$，

As (1.16), there is an error occurs even after repeated $n$ times.