May 23, 2012

The simplest way to solve "x=cos(x)"

The following is the simplest way to solve the equation "x=cos(x)" by using a scientific calculator:
  1. Bring your scientific calculator (you can use your iphone or a web calculator).
  2. Change the mode to "rad" (radian).
  3. Push "1" (one) key.
  4. Continue to push "cos" key until you feel bored.
The result should be an approximated value of the solution, x=0.73908513...


The procedure is exactly the following iteration:
x[n+1] = cos(x[n]),  n=0,1,2,...,  x[0]=1.
By a fixed-point theorem, you can prove that the sequence {x[n]} converges to the solution.

I used this demo when I taucht iteration methods in a "numerical computation" class. This is a nice example for motivating students to learn more abstract theory.

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