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Topic: numerical gradient, Jacobian, Hessian: missing from Mathematica?
Replies: 3   Last Post: Apr 27, 2014 2:07 AM

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 Bob Hanlon Posts: 906 Registered: 10/29/11
Re: numerical gradient, Jacobian, Hessian: missing from Mathematica?
Posted: Apr 21, 2014 11:00 PM

\$Version

9.0 for Mac OS X x86 (64-bit) (January 24, 2013)

n=2;

vars=Table[x[m],{m,n}]

{x[1],x[2]}

funcs=Table[f[m]@@vars,{m,n}]

{f[1][x[1],x[2]],f[2][x[1],x[2]]}

{{Derivative[1, 0][f[1]][x[1], x[2]], Derivative[0, 1][f[1]][x[1], x[2]]},
{Derivative[1, 0][f[2]][x[1], x[2]], Derivative[0, 1][f[2]][x[1], x[2]]}}

Grad is new in v9; for some older versions use D[funcs, {vars}]

True

http://mathworld.wolfram.com/Jacobian.html

jacobian=Outer[D,funcs,vars]

{{Derivative[1, 0][f[1]][x[1], x[2]], Derivative[0, 1][f[1]][x[1], x[2]]},
{Derivative[1, 0][f[2]][x[1], x[2]], Derivative[0, 1][f[2]][x[1], x[2]]}}

Det[jacobian]

Derivative[0, 1][f[2]][x[1], x[2]]*Derivative[1, 0][f[1]][x[1], x[2]] -
Derivative[0, 1][f[1]][x[1], x[2]]*Derivative[1, 0][f[2]][x[1], x[2]]

http://mathworld.wolfram.com/Hessian.html

hessians=D[#,{vars,2}]&/@funcs

{{{Derivative[2, 0][f[1]][x[1], x[2]], Derivative[1, 1][f[1]][x[1], x[2]]},
{Derivative[1, 1][f[1]][x[1], x[2]],
Derivative[0, 2][f[1]][x[1], x[2]]}},
{{Derivative[2, 0][f[2]][x[1], x[2]],
Derivative[1, 1][f[2]][x[1], x[2]]},
{Derivative[1, 1][f[2]][x[1], x[2]],
Derivative[0, 2][f[2]][x[1], x[2]]}}}

Det/@hessians

{-Derivative[1, 1][f[1]][x[1], x[2]]^2 + Derivative[0, 2][f[1]][x[1], x[2]]*
Derivative[2, 0][f[1]][x[1],
x[2]], -Derivative[1, 1][f[2]][x[1], x[2]]^2 +
Derivative[0, 2][f[2]][x[1], x[2]]*Derivative[2, 0][f[2]][x[1], x[2]]}

Bob Hanlon

On Mon, Apr 21, 2014 at 5:11 AM, Alan <alan.isaac@gmail.com> wrote:

> Am I overlooking commands to produce numerical gradient, Jacobian, and
> Hessian? The closest I could find was ND (in the NumericalCalculus
> package), which seems to be only for univariate functions.
>
> Thanks,
> Alan Isaac
>
>

Date Subject Author
4/21/14 Bob Hanlon
4/25/14 Alan
4/27/14 Alexey