An Example of Interval Newton iteration

  Take

with

An interval extension of the Jacobi matrix for f is

and its value at is

The usual procedure (although not required in this special case) is to precondition the system

say, by the inverse of the midpoint matrix

to obtain

i.e.

The interval Gauss-Seidel method can then be used to compute sharper bounds on , beginning with . That is,

Thus, the first component of is

In the second step of the interval Gauss-Seidel method,

so, rounded out to four digits, is computed to be

This last inclusion proves that there exists a unique solution to F(x)=0 within , and hence, within . Furthermore, iteration of the procedure will result in bounds on the exact solution that become narrow quadratically.