Consider f : ^{2 2}

Value the Jacobian matrix of f at x = (1, 1) three different ways:

a. analytically,

b. with a central approximation,

c. with a forward approximation.

In items (b) and (c), use h = .00001.

The Jacobian of f is given by

[s1]

We can solve this problem by valuing each of the four derivatives—analytically or using finite differences—individually. Solutions are:

a. The analytic solution is

[s2]

b. The central approximation matches the analytic solution to the number of decimal places indicated.

[s3]

c. The forward approximation is not as good:

[s4]