Derivation

Whatever generates a linearly converging iteration sequence, can we deduce information about the limit of the sequence from already calculated values?

For a linear method, , or, for sufficiently large ,

> lambda=(p[n+1]-pe)/(p[n]-pe);

> lambda=(p[n+2]-pe)/(p[n+1]-pe);

with the exact solution, or

> eq:=(p[n+1]-pe)/(p[n]-pe)-(p[n+2]-pe)/(p[n+1]-pe)=0;

which can be solved for the true solution, :

> pexact:=solve(eq,pe);

which is more commonly written as .

This formula allows us to calculate a more accurate approximation to the real solution using only already calculated values in the series. This can be very useful when each iteration is costly in terms of computer resources.