Calculators - CORDIC Method

Date: 05/16/99 at 03:15:37
From: Warren
Subject: How can a computer or calculator calculate?

How can a computer or calculator calculate square roots, cube roots, 
Xth root, x^y, logarithm, sine, cosine, ...?

Date: 05/16/99 at 06:44:01
From: Doctor Jerry
Subject: Re: How can a computer or calculator calculate?

Hi Warren,

Calculators use what is called the CORDIC method.

TI and HP use several stored constants and calculate the values of 
several sequences of numbers, using only addition and multiplication.  
They continue the calculation until sufficient accuracy is obtained.

The algorithm looks like this: I'll use x_k to mean x sub k, and 
x_{k+1} to mean x sub k+1, and so on.

Let

x_{k+1} = x_k - d_k*y_k*2^(-k)
y_{k+1} = y_k + d_k*x_k*2^(-k)
z_{k+1} = z_k - d_k*s_k

The numbers d_k are equal to the sign of z_k (if z_k >= 0, d_k = 1; if 
z_k < 0, then d_k = -1). Also, s_k = arctan(2^(-k)). The numbers s_k 
are permanently stored in the calculator, maybe up to k = 50 or so.

Starting values for the calculation are calculated. If z_0 = t is 
given, where t is a given angle (in radians), then y_0 = 0 and 
x_0 = cos(s_0)*cos(s_1)*...*cos(s_{47}).

As k increases, x_k approaches cos(t) and y_k approaches sin(t).

Similar calculations give other trig functions.

- Doctor Jerry, The Math Forum
  http://mathforum.org/dr.math/