[net.math] Accelerating fixed point solutions

bas@calmasd.UUCP (Bharath 'Al' Sethuraman) (05/25/85)

One way of solving the fixed point equation z = f(z), where z is an 
n-dim vector, and f is a (vector) function from Rn to Rn, is by iteration.
That is, one starts with a guess z0, and recursively calculates
                  z1 = f(z0)
                  z2 = f(z1)
                  .........
                  zn = f(zn-1),
hoping that the sequence z0,z1,z2,... converges.  I need some suggestions
on how to accelerate this procedure. (Techniques that might, for example,
compute an improved zn based on a certain number of previous z-sub-i(s).)
I primarily need acceleration techniques for 3 or less dimensions.  Any 
comments on speed,complexity, etc. would also be appreciated.

Thanks!