Calculate the cosine of a sequence

Question

I have to calculate the following:

float2 y = CONSTANT;
for (int i = 0; i < totalN; i++)
   h[i] = cos(y*i);

totalN is a large number, so I would like to make this in a more efficient way. Is there any way to improve this? I suspect there is, because, after all, we know what's the result of cos(n), for n=1..N, so maybe there's some theorem that allows me to compute this in a faster way. I would really appreciate any hint.

Thanks in advance,

Federico

Answer 1

Using one of the most beautiful formulas of mathematics, Euler's formula
exp(i*x) = cos(x) + i*sin(x),

substituting x := n * phi:

cos(n*phi) = Re( exp(i*n*phi) )
sin(n*phi) = Im( exp(i*n*phi) )

exp(i*n*phi) = exp(i*phi) ^ n

Power ^n is n repeated multiplications. Therefore you can calculate cos(n*phi) and simultaneously sin(n*phi) by repeated complex multiplication by exp(i*phi) starting with (1+i*0).

Code examples:

Python:

from math import *

DEG2RAD = pi/180.0 # conversion factor degrees --> radians
phi = 10*DEG2RAD # constant e.g. 10 degrees

c = cos(phi)+1j*sin(phi) # = exp(1j*phi)
h=1+0j
for i in range(1,10):
  h = h*c
  print "%d %8.3f"%(i,h.real)

or C:

#include <stdio.h>
#include <math.h>

// numer of values to calculate:
#define N 10

// conversion factor degrees --> radians:
#define DEG2RAD (3.14159265/180.0)

// e.g. constant is 10 degrees:
#define PHI (10*DEG2RAD)

typedef struct
{
  double re,im;
} complex_t;


int main(int argc, char **argv)
{
  complex_t c;
  complex_t h[N];
  int index;

  c.re=cos(PHI);
  c.im=sin(PHI);

  h[0].re=1.0;   
  h[0].im=0.0;
  for(index=1; index<N; index++)
  {
    // complex multiplication h[index] = h[index-1] * c;
    h[index].re=h[index-1].re*c.re - h[index-1].im*c.im; 
    h[index].im=h[index-1].re*c.im + h[index-1].im*c.re; 
    printf("%d: %8.3f\n",index,h[index].re);
  }
}