Helmut Knaust
Department of Mathematical Sciences
University of Texas at El Paso
The CORDIC algorithm does not use Calculus based methods such as polynomial or rational function approximation.
The CORDIC (= COordinate Rotation DIgital Computer) algorithm was developed by Jack E. Volder in 1959.
His objective was to build a real-time navigational computer for use on aircrafts, so he was primarily interested in computing trigonometric functions.
Subsequently, the CORDIC scheme was extended by J. Walther in 1971 to other transcendental functions.
Hand-held calculators do not convert numbers to base 2. They use a binary-coded decimal (BCD) system instead.
Calculators can only perform four operations inexpensively:
The CORDIC Algorithm is a unified computational scheme to perform
To compute and for we let
We also define
The scheme becomes
The Cordic Representation Theorem
Suppose is a decreasing sequence of positive real numbers satisfying
for , and suppose r is a real number such that
If , and , for , where
then
Proof: By induction.
The previous Theorem and some computations tell us that it is basically possible to write the angle as a combination of the angles , more precisely, we can choose so that
Theorem. If and , then and .
Proof (by induction, sketch):
, ,
, ,
, ,
Hyperbolic Functions
(repeated), ,
, ,
(repeated), ,
Multiplication and Division
, ,
, ,