13.7.3 Hessian matrix
Recall, the Hessian of a function F of n variables
x1,…,xn is the matrix of second order derivatives:
| | ⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣ | | ⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦ |
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The hessian command
computes the Hessian of a function.
- hessian takes two arguments:
- expr, an expression involving several variables.
- vars, a list of the variable names.
- hessian(expr,vars) returns the
Hessian of the expression.
Examples
Find the Hessian matrix of F(x,y,z)=2x2y−xz3.
| hessian(2*x^2*y-x*z^3,[x,y,z]) |
|
| |
| ⎡
⎢
⎢
⎣ | | 4 y | 4 x | −3 z2 |
| 2· 2 x | 0 | 0 |
| −3 z2 | 0 | −2· 3 x z |
| ⎤
⎥
⎥
⎦ |
|
| | | | | | | | | | |
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To get the Hessian matrix at the critical points:
| solve(derive(2*x^2*y-x*z^3,[x,y,z]),[x,y,z]) |
Output (the critical points):
Input to evaluate the Hessian at these points:
| subst([[4*y,4*x,-3*z^2],[2*2*x,0,0],[-3*z^2,0,6*x*z]],[x,y,z],[0,y,0]) |