Applications of function HESS are easier to visualize in the RPN mode.
Consider as an example the function φ(X,Y,Z) = X
function HESS to function φ in the following example. The screen shots show
the 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., ∂φ/∂x
0, 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
follows in RPN mode:
'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
['X=1','Y=0], respectively. The Hessian matrix is at level 1 at this point.
J @@@H@@@ @@s1@@ SUBST ‚ï
The resulting matrix A has a
φ/∂X∂Y = 0. The discriminant, for this critical point
2., and a
s1(-1,0) is ∆ = (∂
<0, point s1 represents a relative maximum.
Next, we substitute the second point, s2, into H:
J @@@H@@@ @@s2@@ SUBST ‚ï
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
+ XY + XZ, we'll apply
+5, proceed as
= -6., a