Numerically computing partial derivatives

• To: mathgroup at smc.vnet.net
• Subject: [mg47940] Numerically computing partial derivatives
• From: Mark Coleman <mark at markscoleman.com>
• Date: Sun, 2 May 2004 04:50:34 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```Greetings,

I am working with a maximum likelihood problem in econometrics. With
some effort I can get Mathematica v5.0 to maximize the function. In order to
derive standard errors of the estimates, however, I need to calculate
the Hessian of the function at the optimal solution. This requires, of
course,  calculating the set of second derivatives of the function. Due
to the nature of the function, however, neither the built-in D or ND
operator seem to work (note: The function contains a term of the form
Log[Det[I-rho*W]], where I is the nxn Identity matrix, rho is a real,
and W is a non-symmetric (sparse) matrix of reals, or order n. In
practice, n > 1000 at times. As a result, symbolic differentiation is
not feasible. In addition, when I use ND [], I get nonsensical answers.

As a result, I'd like to use a simple numerical differentiation scheme
based upon  a Taylor-expansion, e.g., the simplest being df(x)/dx =
(f(x+h)-f(x)) / h. I thought that before I code or two or three-point
algorithm I would check on the list and see if anyone had already
written code for a numerical Hessian.

Thanks,

Mark

```

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