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Re: numerical gradient, Jacobian, Hessian: missing from Mathematica?

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  • Subject: [mg132618] Re: numerical gradient, Jacobian, Hessian: missing from Mathematica?
  • From: Bob Hanlon <hanlonr357 at gmail.com>
  • Date: Mon, 21 Apr 2014 22:59:52 -0400 (EDT)
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$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]]}

http://reference.wolfram.com/mathematica/ref/Grad.html

Grad[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]]}}

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

Grad[funcs, vars]==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 at 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
>
>


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