HP 15c - Manual (Page 135)

Section 4: Using Matrix Operations 135
Labels used: E and 8.
Registers used: no additional registers.
Matrices used: A (from previous program) and E (eigenvalues).
To use the combined eigenvalue, eigenvalue storage, and eigenvector programs for an A
matrix up to 3×3:
1. Execute the eigenvalue program as described earlier.
2. Press E to store the eigenvalues.
3. Enter again the elements of the original matrix into A.
4. Recall the desired eigenvalue from matrix E using l E.
5. Execute the eigenvector program as described above.
6. Repeat steps 4 and 5 for each eigenvalue.
Optimization
Optimization describes a class of problems in which the object is to find the minimum or
maximum value of a specified function. Often, the interest is focused on the behavior of the
function in a particular region.
The following program uses the method of steepest descent to determine local minimums or
maximums for a real-valued function of two or more variables. This method is an iterative
procedure that uses the gradient of the function to determine successive sample points. Four
input parameters control the sampling plan.
For the function
f (x) = f (x
1
,x
2
, … ,x
n
)
the gradient of f,

f, is defined by












 
 
 
 
n
2
1
) (
x f
x f
x f
f

x
The critical points of f(x) are the solutions to

f (x) = 0. A critical point may be a local
minimum, a local maximum, or a point that is neither.
The gradient of f(x) evaluated at a point x gives the direction of steepest ascent—that is, the
way in which x should be changed in order to cause the most rapid increase in f(x). The
negative gradient gives the direction of steepest descent. The direction vector is