Applications of function HESS are easier to visualize in the RPN mode.
Consider as an example the function (X,Y,Z) = X
function HESS to function
RPN stack before and after applying function HESS.
When applied to a function of two variables, the gradient in level 2, when
made equal to zero, represents the equations for critical points, i.e.,
while the matrix in level 3 represent second derivatives. Thus, the results from
the HESS function can be used to analyze extrema in functions of two variables.
For example, for the function f(X,Y) = X
'X^3-3*X-Y^2+5' ` ['X','Y'] `
's1' K 's2' K
The variables s1 and s2, at this point, contain the vectors ['X=-1','Y=0] and
J @@@H@@@ @@s1@@ SUBST ‚ï
The resulting matrix A has a
-2., and a
= ( f/ x ) ( f/ y )-[ f/ x y]
<0, point s1 represents a relative maximum.
Next, we substitute the second point, s2, into H:
J @@@H@@@ @@s2@@ SUBST ‚ï
in the following example. The screen shots show the
/ X Y = 0. The discriminant, for this critical point
+ XY + XZ, we'll apply
+5, proceed as follows in RPN
Enter function and variables
Apply function HESS
Find critical points
Store critical points
Store Hessian matrix
Substitute s1 into H
= (-6.)(-2.) = 12.0 > 0. Since
Substitute s2 into H
= -6., a